Index: sage/algebras/free_algebra.py =================================================================== --- sage/algebras/free_algebra.py (revision 4007) +++ sage/algebras/free_algebra.py (revision 1860) @@ -13,26 +13,4 @@ sage: G.base_ring() Free Algebra on 3 generators (x, y, z) over Integer Ring - -TESTS: - sage: F = FreeAlgebra(GF(5),3,'x') - sage: F == loads(dumps(F)) - True - - sage: F. = FreeAlgebra(GF(5),3) - sage: F == loads(dumps(F)) - True - - sage: F = FreeAlgebra(GF(5),3, ['xx', 'zba', 'Y']) - sage: F == loads(dumps(F)) - True - - sage: F = FreeAlgebra(GF(5),3, 'abc') - sage: F == loads(dumps(F)) - True - - sage: F = FreeAlgebra(FreeAlgebra(ZZ,1,'a'), 2, 'x') - sage: F == loads(dumps(F)) - True - """ @@ -220,5 +198,5 @@ if P is self: return x - if not (P is self.base_ring()): + if not (P == self.base_ring()): return FreeAlgebraElement(self, x) # ok, not a free algebra element (or should not be viewed as one). @@ -226,5 +204,5 @@ R = self.base_ring() # coercion from free monoid - if isinstance(x, FreeMonoidElement) and x.parent() is F: + if isinstance(x, FreeMonoidElement) and x.parent() == F: return FreeAlgebraElement(self,{x:R(1)}) # coercion via base ring @@ -290,5 +268,5 @@ # monoid - if R is self.__monoid: + if R == self.__monoid: return self(x) Index: sage/algebras/free_algebra_element.py =================================================================== --- sage/algebras/free_algebra_element.py (revision 4007) +++ sage/algebras/free_algebra_element.py (revision 1859) @@ -3,9 +3,4 @@ AUTHOR: David Kohel, 2005-09 - -TESTS: - sage: R. = FreeAlgebra(QQ,2) - sage: x == loads(dumps(x)) - True """ @@ -49,5 +44,5 @@ self.__monomial_coefficients = { x:R(1) } elif True: - self.__monomial_coefficients = dict([ (A.monoid()(e1),R(e2)) for e1,e2 in x.items()]) + self.__monomial_coefficients = x else: raise TypeError, "Argument x (= %s) is of the wrong type."%x Index: sage/algebras/free_algebra_quotient.py =================================================================== --- sage/algebras/free_algebra_quotient.py (revision 4149) +++ sage/algebras/free_algebra_quotient.py (revision 1840) @@ -1,19 +1,4 @@ """ Free algebra quotients - -TESTS: - sage: n = 2 - sage: A = FreeAlgebra(QQ,n,'x') - sage: F = A.monoid() - sage: i, j = F.gens() - sage: mons = [ F(1), i, j, i*j ] - sage: r = len(mons) - sage: M = MatrixSpace(QQ,r) - sage: mats = [M([0,1,0,0, -1,0,0,0, 0,0,0,-1, 0,0,1,0]), M([0,0,1,0, 0,0,0,1, -1,0,0,0, 0,-1,0,0]) ] - sage: H2. = A.quotient(mons,mats) - sage: H2 == loads(dumps(H2)) - True - sage: i == loads(dumps(i)) - True """ @@ -92,6 +77,6 @@ raise TypeError, "Argument A must be an algebra." R = A.base_ring() -# if not R.is_field(): # TODO: why? -# raise TypeError, "Base ring of argument A must be a field." + if not R.is_field(): + raise TypeError, "Base ring of argument A must be a field." n = A.ngens() assert n == len(mats) @@ -103,13 +88,4 @@ self.__monomial_basis = mons # elements of free monoid ParentWithGens.__init__(self, R, names, normalize=False) - - def __eq__(self,right): - return type(self) == type(right) and \ - self.ngens() == right.ngens() and \ - self.rank() == right.rank() and \ - self.module() == right.module() and \ - self.matrix_action() == right.matrix_action() and \ - self.monomial_basis() == right.monomial_basis() - def __call__(self, x): @@ -163,10 +139,4 @@ return self.__dim - def matrix_action(self): - return self.__matrix_action - - def monomial_basis(self): - return self.__monomial_basis - def rank(self): """ Index: sage/algebras/free_algebra_quotient_element.py =================================================================== --- sage/algebras/free_algebra_quotient_element.py (revision 4149) +++ sage/algebras/free_algebra_quotient_element.py (revision 1855) @@ -46,11 +46,6 @@ if isinstance(x, FreeAlgebraQuotientElement) and x.parent() is A: return x - AlgebraElement.__init__(self, A) Q = self.parent() - - if isinstance(x, FreeAlgebraQuotientElement) and x.parent() == Q: - self.__vector = Q.module()(x.vector()) - return if isinstance(x, (int, long, Integer)): self.__vector = Q.module().gen(0) * x @@ -66,5 +61,4 @@ F = A.monoid() B = A.monomial_basis() - if isinstance(x, (int, long, Integer)): self.__vector = x*M.gen(0) @@ -95,4 +89,6 @@ self.__vector = x.ambient_algebra_element().vector() else: + print "x =", x + print "type(x) =", type(x) raise TypeError, "Argument x (= %s) is of the wrong type."%x @@ -108,22 +104,7 @@ else: return x - - def _latex_(self): - Q = self.parent() - M = Q.monoid() - with localvars(M, Q.variable_names()): - cffs = list(self.__vector) - mons = Q.monomial_basis() - x = repr_lincomb(mons, cffs, True).replace("*1 "," ") - if x[len(x)-2:] == "*1": - return x[:len(x)-2] - else: - return x def vector(self): return self.__vector - - def __cmp__(self, right): - return cmp(self.vector(),right.vector()) def __neg__(self): @@ -160,12 +141,18 @@ if c != 0: z.__vector += monomial_product(A,c*u,B[i]) return z - - def _rmul_(self, c): - return self.parent([c*a for a in self.__vector]) - - def _lmul_(self, c): - return self.parent([a*c for a in self.__vector]) + + def __pow__(self, n): + if not isinstance(n, (int, long, Integer)): + raise TypeError, "Argument n (= %s) must be an integer."%n + if n < 0: + raise IndexError, "Argument n (= %s) must be positive."%n + elif n == 0: + return self.parent()(1) + elif n == 1: + return self + elif n == 2: + return self * self + k = n//2 + return self**k * self**(n-k) - - Index: sage/algebras/quaternion_algebra.py =================================================================== --- sage/algebras/quaternion_algebra.py (revision 4069) +++ sage/algebras/quaternion_algebra.py (revision 3729) @@ -3,13 +3,4 @@ AUTHOR: David Kohel, 2005-09 - -TESTS: - sage: A = QuaternionAlgebra(QQ, -1,-1, names=list('ijk')) - sage: A == loads(dumps(A)) - True - sage: i, j, k = A.gens() - sage: i == loads(dumps(i)) - True - """ @@ -120,5 +111,5 @@ _cache = {} -def QuaternionAlgebra(K, a, b, names=['i','j','k'], denom=1): +def QuaternionAlgebra(K, a, b, names, denom=1): """ Return the quaternion algebra over $K$ generated by $i$, $j$, and $k$ @@ -150,16 +141,4 @@ raise ValueError, "Base ring K (= %s) must be a field."%K - if K(2) == 0: - raise ValueError, "Base ring K (= %s) must be a field of characteristic different from 2."%K - - try: - a = K(a) - except TypeError: - raise ValueError, "Arguments a = %s and b = %s must coerce into K (= %s)."%(a,b,K) - try: - b = K(b) - except TypeError: - raise ValueError, "Arguments a = %s and b = %s must coerce into K (= %s)."%(a,b,K) - if a == 0 or b == 0: raise ValueError, "Arguments a = %s and b = %s must be nonzero."%(a,b) @@ -180,7 +159,7 @@ if K is RationalField(): prms = ramified_primes(a,b) - H = QuaternionAlgebra_generic(K, [2,0,0,0], prms) + H = QuaternionAlgebra_generic(K, prms) else: - H = QuaternionAlgebra_generic(K, [2,0,0,0]) + H = QuaternionAlgebra_generic(K) A = FreeAlgebra(K,3, names=names) F = A.monoid() @@ -243,5 +222,5 @@ vki = V([ -t13, t3, 0, t1 ]) - vik vjk = V([ -t23, 0, t3, t2 ]) - vkj - H = QuaternionAlgebra_generic(K, [2,t1,t2,t3]) + H = QuaternionAlgebra_generic(K) A = FreeAlgebra(K,3, names=names) F = A.monoid() @@ -266,5 +245,5 @@ return QuaternionAlgebraWithInnerProduct(K,norms,traces,names=names) -def QuaternionAlgebraWithDiscriminants(D1, D2, T, names=['i','j','k'], M=2): +def QuaternionAlgebraWithDiscriminants(D1, D2, T, names, M=2): r""" Return the quaternion algebra over the rationals generated by $i$, @@ -327,9 +306,8 @@ mj = MQ([0,0,1,0, 0,0,0,1, -n2,0,t2,0, 0,-n2,0,t2]) # N.B. mk = mi*mj - t3 = t1*t2-t12 - mk = MQ([0,0,0,1, 0,0,-n1,t1, -n2*t1,n2,t3,0, -n1*n2,0,0,t3]) + mk = MQ([0,0,0,1, 0,0,-n1,t1, -n2*t1,n2,t1*t2-t12,0, -n1*n2,0,0,t1*t2-t12]) mats = [ mi, mj, mk ] prms = ramified_primes_from_discs(D1,D2,T) - H = QuaternionAlgebra_generic(QQ, [2,t1,t2,t3], prms) + H = QuaternionAlgebra_generic(QQ, prms) A = FreeAlgebra(QQ,3, names=names) F = A.monoid() @@ -340,11 +318,10 @@ class QuaternionAlgebra_generic(FreeAlgebraQuotient): - def __init__(self, K, basis_traces = None, ramified_primes=None): - """ - """ + def __init__(self, K, ramified_primes=None): + """ + """ + self.__vector_space = VectorSpace(K,4) if ramified_primes != None: - self._ramified_primes = ramified_primes - if basis_traces != None: - self._basis_traces = basis_traces + self.__ramified_primes = ramified_primes def __call__(self, x): @@ -380,22 +357,21 @@ def gram_matrix(self): - """ - The Gram matrix of the inner product determined by the norm. - """ - try: - M = self.__vector_space.inner_product_matrix() - except: - K = self.base_ring() - M = MatrixSpace(K,4)(0) - B = self.basis() - for i in range(4): - x = B[i] - M[i,i] = 2*(x.reduced_norm()) - for j in range(i+1,4): - y = B[j] - c = (x * y.conjugate()).reduced_trace() - M[i,j] = c - M[j,i] = c - self.__vector_space = VectorSpace(K,4,inner_product_matrix = M) + V = self.__vector_space + if not V._FreeModule_generic__inner_product_is_dot_product: + return V.inner_product_matrix() + K = self.base_ring() + M = MatrixSpace(K,4)(0) + B = self.basis() + for i in range(4): + x = B[i] + M[i,i] = 2*(x.reduced_norm()) + for j in range(i+1,4): + y = B[j] + c = (x * y.conjugate()).reduced_trace() + M[i,j] = c + M[j,i] = c + # TODO: Make it so one can correctly set these things. + V._FreeModule_generic__inner_product_is_dot_product = False + V._FreeModule_generic__inner_product_matrix = M return M @@ -412,7 +388,10 @@ return False + def _set_ramified_primes(self, prms): + self.__ramified_primes = prms + def ramified_primes(self): try: - return self._ramified_primes + return self.__ramified_primes except: raise AttributeError, "Ramified primes have not been computed." @@ -423,10 +402,10 @@ def vector_space(self): + V = self.__vector_space try: - V = self.__vector_space + _ = V._FreeModule_generic__inner_product_matrix except: - M = self.gram_matrix() # induces construction of V - V = self.__vector_space - return V + V._FreeModule_generic__inner_product_matrix = self.gram_matrix() + return self.__vector_space Index: sage/all.py =================================================================== --- sage/all.py (revision 4317) +++ sage/all.py (revision 3696) @@ -71,5 +71,5 @@ from sage.interfaces.all import * from sage.functions.all import * -from sage.calculus.all import * +#from sage.calculus.all import * from sage.server.all import * from sage.dsage.all import * @@ -114,4 +114,5 @@ # very useful 2-letter shortcuts CC = ComplexField() +I = CC.gen(0) QQ = RationalField() RR = RealField() # default real field @@ -135,5 +136,5 @@ oo = infinity -#x = PolynomialRing(QQ,'x').gen() +x = PolynomialRing(QQ,'x').gen() # grab signal handling back from PARI or other C libraries @@ -232,22 +233,4 @@ from sage.interfaces.quit import expect_quitall expect_quitall(verbose=verbose) - - - # The following code close all open file descriptors, - # so that on shared file systems the delete_tmpfiles - # command below works. - # AUTHOR: - # * Kate Minola (2007-05-03) - import resource # Resource usage information. - maxfd = resource.getrlimit(resource.RLIMIT_NOFILE)[1] - if maxfd != resource.RLIM_INFINITY: - # Iterate through and close all file descriptors. - for fd in range(0, maxfd): - try: - os.close(fd) - except OSError: # ERROR, fd wasn't open to begin with (ignored) - pass - - # Now delete the temp files from sage.misc.misc import delete_tmpfiles delete_tmpfiles() Index: sage/calculus/all.py =================================================================== --- sage/calculus/all.py (revision 4308) +++ sage/calculus/all.py (revision 2327) @@ -1,78 +1,7 @@ -from equations import SymbolicEquation, forget, assume, assumptions, solve - -from calculus import (SymbolicExpressionRing, - is_SymbolicExpressionRing, - is_SymbolicExpression, - CallableSymbolicExpressionRing, - is_CallableSymbolicExpressionRing, - is_CallableSymbolicExpression, - SR, - sin, cos, sec, tan, log, erf, sqrt, asin, acos, atan, - tanh, sinh, cosh, coth, sech, csch, ln, - ceil, floor, - polylog, - abs_symbolic, exp, - is_SymbolicExpression, - is_SymbolicExpressionRing) +from calculus import (SER, + var, + sin, cos, sec, + x, y, z, w, t) -from functional import (diff, derivative, - laplace, inverse_laplace, - expand, - integrate, limit, lim, - taylor, simplify) - -from var import (var, function) - -from predefined import (a, - b, - c, - d, - f, - g, - h, - j, - k, - l, - m, - n, - o, - p, - q, - r, - s, - t, - u, - v, - w, - x, - y, - z, - A, - B, - C, - D, - E, - F, - G, - H, - J, - K, - L, - M, - N, - P, - Q, - R, - S, - T, - U, - V, - W, - X, - Y, - Z) - -def symbolic_expression(x): - return SR(x) - +from functional import diff Index: sage/calculus/calculus.py =================================================================== --- sage/calculus/calculus.py (revision 4313) +++ sage/calculus/calculus.py (revision 2630) @@ -1,200 +1,9 @@ -r""" -Symbolic Computation. - -AUTHORS: - Bobby Moretti and William Stein: 2006--2007 - -The \sage calculus module is loosely based on the \sage Enhahcement Proposal -found at: http://www.sagemath.org:9001/CalculusSEP. - -EXAMPLES: - - The basic units of the calculus package are symbolic expressions - which are elements of the symbolic expression ring (SR). There are - many subclasses of SymbolicExpression. The most basic of these is - the formal indeterminate class, SymbolicVariable. To create a - SymbolicVariable object in \sage, use the var() method, whose - argument is the text of that variable. Note that \sage is - intelligent about {\LaTeX}ing variable names. - - sage: x1 = var('x1'); x1 - x1 - sage: latex(x1) - x_{1} - sage: theta = var('theta'); theta - theta - sage: latex(theta) - \theta - - \sage predefines upper and lowercase letters as global - indeterminates. Thus the following works: - sage: x^2 - x^2 - sage: type(x) - - - More complicated expressions in SAGE can be built up using - ordinary arithmetic. The following are valid, and follow the rules - of Python arithmetic: (The '=' operator represents assignment, and - not equality) - sage: f = x + y + z/(2*sin(y*z/55)) - sage: g = f^f; g - (z/(2*sin(y*z/55)) + y + x)^(z/(2*sin(y*z/55)) + y + x) - - Differentiation and integration are available, but behind the - scenes through maxima: - - sage: f = sin(x)/cos(2*y) - sage: f.derivative(y) - 2*sin(x)*sin(2*y)/(cos(2*y)^2) - sage: g = f.integral(x); g - -cos(x)/(cos(2*y)) - - Note that these methods require an explicit variable name. If none - is given, \sage will try to find one for you. - sage: f = sin(x); f.derivative() - cos(x) - - However when this is ambiguous, \sage will raise an exception: - sage: f = sin(x+y); f.derivative() - Traceback (most recent call last): - ... - ValueError: must supply an explicit variable for an expression containing more than one variable - - Substitution works similarly. We can substitute with a python dict: - sage: f = sin(x*y - z) - sage: f({x: t, y: z}) - sin(t*z - z) - - Also we can substitute with keywords: - sage: f = sin(x*y - z) - sage: f(x = t, y = z) - sin(t*z - z) - - If there is no ambiguity of variable names, we don't have to specify them: - sage: f = sin(x) - sage: f(y) - sin(y) - sage: f(pi) - 0 - - However if there is ambiguity, we must explicitly state what - variables we're substituting for: - - sage: f = sin(2*pi*x/y) - sage: f(4) - sin(8*pi/y) - - We can also make CallableSymbolicExpressions, which is a SymbolicExpression - that are functions of variables in a fixed order. Each - SymbolicExpression has a function() method used to create a - CallableSymbolicExpression. - - sage: u = log((2-x)/(y+5)) - sage: f = u.function(x, y); f - (x, y) |--> log((2 - x)/(y + 5)) - - There is an easier way of creating a CallableSymbolicExpression, which - relies on the \sage preparser. - - sage: f(x,y) = log(x)*cos(y); f - (x, y) |--> log(x)*cos(y) - - Then we have fixed an order of variables and there is no ambiguity - substituting or evaluating: - - sage: f(x,y) = log((2-x)/(y+5)) - sage: f(7,t) - log(-5/(t + 5)) - - Some further examples: - - sage: f = 5*sin(x) - sage: f - 5*sin(x) - sage: f(2) - 5*sin(2) - sage: f(pi) - 0 - sage: float(f(pi)) # random low order bits - 6.1232339957367663e-16 - -COERCION EXAMPLES: - -We coerce various symbolic expressions into the complex numbers: - - sage: CC(I) - 1.00000000000000*I - sage: CC(2*I) - 2.00000000000000*I - sage: ComplexField(200)(2*I) - 2.0000000000000000000000000000000000000000000000000000000000*I - sage: ComplexField(200)(sin(I)) - 1.1752011936438014568823818505956008151557179813340958702296*I - sage: f = sin(I) + cos(I/2); f - I*sinh(1) + cosh(1/2) - sage: CC(f) - 1.12762596520638 + 1.17520119364380*I - sage: ComplexField(200)(f) - 1.1276259652063807852262251614026720125478471180986674836290 + 1.1752011936438014568823818505956008151557179813340958702296*I - sage: ComplexField(100)(f) - 1.1276259652063807852262251614 + 1.1752011936438014568823818506*I - -We illustrate construction of an inverse sum where each denominator -has a new variable name: - sage: f = sum(1/var('n%s'%i)^i for i in range(10)) - sage: print f - 1 1 1 1 1 1 1 1 1 - --- + --- + --- + --- + --- + --- + --- + --- + -- + 1 - 9 8 7 6 5 4 3 2 n1 - n9 n8 n7 n6 n5 n4 n3 n2 - -Note that after calling var, the variables are immediately available for use: - sage: (n1 + n2)^5 - (n2 + n1)^5 - -We can, of course, substitute: - sage: print f(n9=9,n7=n6) - 1 1 1 1 1 1 1 1 387420490 - --- + --- + --- + --- + --- + --- + --- + -- + --------- - 8 6 7 5 4 3 2 n1 387420489 - n8 n6 n6 n5 n4 n3 n2 - -TESTS: -We test pickling: - sage: f = -sqrt(pi)*(x^3 + sin(x/cos(y))) - sage: bool(loads(dumps(f)) == f) - True - -Substitution: - sage: f = x - sage: f(x=5) - 5 - -The symbolic Calculus package uses its own copy of Maxima for -simplification, etc., which is separate from the default system-wide -version: - sage: maxima.eval('[x,y]: [1,2]') - '[1,2]' - sage: maxima.eval('expand((x+y)^3)') - '27' - -If the copy of maxima used by the symbolic calculus package were -the same as the default one, then the following would return 27, -which would be very confusing indeed! - sage: expand((x+y)^3) - y^3 + 3*x*y^2 + 3*x^2*y + x^3 -""" - -import weakref +"""nodoctest""" from sage.rings.all import (CommutativeRing, RealField, is_Polynomial, - is_MPolynomial, is_MPolynomialRing, is_RealNumber, is_ComplexNumber, RR, - Integer, Rational, CC, - QuadDoubleElement, - PolynomialRing, ComplexField) - -from sage.structure.element import RingElement, is_Element + Integer, Rational, CC) +#import sage.rings.rational +from sage.structure.element import RingElement from sage.structure.parent_base import ParentWithBase @@ -204,69 +13,16 @@ from sage.interfaces.maxima import MaximaElement, Maxima - -# The calculus package uses its own copy of maxima, which is -# separate from the default system-wide version. -maxima = Maxima() +from sage.interfaces.all import maxima from sage.misc.sage_eval import sage_eval -from sage.calculus.equations import SymbolicEquation -from sage.rings.real_mpfr import RealNumber -from sage.rings.complex_number import ComplexNumber -from sage.rings.real_double import RealDoubleElement -from sage.rings.complex_double import ComplexDoubleElement -from sage.rings.real_mpfi import RealIntervalFieldElement -from sage.rings.infinity import InfinityElement - -from sage.libs.pari.gen import pari, PariError, gen as PariGen - -from sage.rings.complex_double import ComplexDoubleElement - -import sage.functions.constants - -import math - -is_simplified = False - -infixops = {operator.add: '+', - operator.sub: '-', - operator.mul: '*', - operator.div: '/', - operator.pow: '^'} - - -def is_SymbolicExpression(x): - """ - EXAMPLES: - sage: is_SymbolicExpression(sin(x)) - True - sage: is_SymbolicExpression(2/3) - False - sage: is_SymbolicExpression(sqrt(2)) - True - """ - return isinstance(x, SymbolicExpression) - -def is_SymbolicExpressionRing(x): - """ - EXAMPLES: - sage: is_SymbolicExpressionRing(QQ) - False - sage: is_SymbolicExpressionRing(SR) - True - """ - return isinstance(x, SymbolicExpressionRing_class) - - +from sage.functions.constants import Constant +import sage.functions.constants as c + +# There will only ever be one instance of this class class SymbolicExpressionRing_class(CommutativeRing): - """ - The ring of all formal symbolic expressions. - - EXAMPLES: - sage: SR - Symbolic Ring - sage: type(SR) - - """ + ''' + A Ring of formal expressions. + ''' def __init__(self): self._default_precision = 53 # default precision bits @@ -274,32 +30,8 @@ def __call__(self, x): - """ - Coerce x into the symbolic expression ring SR. - - EXAMPLES: - sage: a = SR(-3/4); a - -3/4 - sage: type(a) - - sage: a.parent() - Symbolic Ring - - If a is already in the symblic expression ring, coercing returns - a itself (not a copy): - sage: SR(a) is a - True - - A Python complex number: - sage: SR(complex(2,-3)) - 2.00000000000000 - 3.00000000000000*I - """ - if is_Element(x) and x.parent() is self: - return x - elif hasattr(x, '_symbolic_'): - return x._symbolic_(self) return self._coerce_impl(x) def _coerce_impl(self, x): - if isinstance(x, CallableSymbolicExpression): + if isinstance(x, CallableFunction): return x._expr elif isinstance(x, SymbolicExpression): @@ -307,34 +39,16 @@ elif isinstance(x, MaximaElement): return symbolic_expression_from_maxima_element(x) - elif is_Polynomial(x) or is_MPolynomial(x): + elif is_Polynomial(x): return SymbolicPolynomial(x) - elif isinstance(x, (RealNumber, - RealDoubleElement, - RealIntervalFieldElement, - float, - sage.functions.constants.Constant, - Integer, - int, - Rational, - PariGen, - ComplexNumber, - ComplexDoubleElement, - QuadDoubleElement, - InfinityElement - )): - return SymbolicConstant(x) - elif isinstance(x, complex): - return evaled_symbolic_expression_from_maxima_string('%s+%%i*%s'%(x.real,x.imag)) + elif isinstance(x, Integer): + return Constant_object(x) else: - raise TypeError, "cannot coerce type '%s' into a SymbolicExpression."%type(x) - - def _repr_(self): - return 'Symbolic Ring' - - def _latex_(self): - return 'SymbolicExpressionRing' - - def var(self, x): - return var(x) + return Symbolic_object(x) + + def _repr_(self): + return "Ring of Symbolic Expressions" + + def _latex_(self): + return "SymbolicExpressionRing" def characteristic(self): @@ -347,409 +61,88 @@ return True - def is_exact(self): - return False - -# Define the unique symbolic expression ring. -SR = SymbolicExpressionRing_class() - -# The factory function that returns the unique SR. -def SymbolicExpressionRing(): - """ - Return the symbolic expression ring. - - EXAMPLES: - sage: SymbolicExpressionRing() - Symbolic Ring - sage: SymbolicExpressionRing() is SR - True - """ - return SR +# ... and here it is: +SymbolicExpressionRing = SymbolicExpressionRing_class() +SER = SymbolicExpressionRing +# conversions is the dict of the form system:command class SymbolicExpression(RingElement): """ - A Symbolic Expression. - - EXAMPLES: - Some types of SymbolicExpressions: - - sage: a = SR(2+2); a - 4 - sage: type(a) - - + A Symbolic Expression. """ - def __init__(self): - RingElement.__init__(self, SR) - if is_simplified: - self._simp = self - - def __hash__(self): - return hash(self._repr_(simplify=False)) - - def __nonzero__(self): - try: - return self.__nonzero + def __init__(self, conversions={}): + RingElement.__init__(self, SymbolicExpressionRing) + + def _maxima_(self, maxima=maxima): + try: + return self.__maxima except AttributeError: - ans = not bool(self == SR.zero_element()) - self.__nonzero = ans - return ans - - def __str__(self): - """ - Printing an object explicitly gives ASCII art: - - EXAMPLES: - sage: f = y^2/(y+1)^3 + x/(x-1)^3 - sage: f - y^2/(y + 1)^3 + x/(x - 1)^3 - sage: print f - 2 - y x - -------- + -------- - 3 3 - (y + 1) (x - 1) - - """ - return self.display2d(onscreen=False) - - def show(self): - from sage.misc.functional import _do_show - return _do_show(self) - - def display2d(self, onscreen=True): - """ - Display self using ASCII art. - - INPUT: - onscreen -- string (optional, default True) If True, - displays; if False, returns string. - - EXAMPLES: - We display a fraction: - sage: f = (x^3+y)/(x+3*y^2+1); f - (y + x^3)/(3*y^2 + x + 1) - sage: print f - 3 - y + x - ------------ - 2 - 3 y + x + 1 - - Use onscreen=False to get the 2d string: - sage: f.display2d(onscreen=False) - '\t\t\t\t\t 3\r\n\t\t\t\t y + x\r\n \t\t\t ------------\r\n\t\t\t\t 2\r\n\t\t\t\t 3 y + x + 1' - - ASCII art is really helps for the following integral: - sage: f = integral(sin(x^2)); f - sqrt(pi)*((sqrt(2)*I + sqrt(2))*erf((sqrt(2)*I + sqrt(2))*x/2) + (sqrt(2)*I - sqrt(2))*erf((sqrt(2)*I - sqrt(2))*x/2))/8 - sage: print f - (sqrt(2) I + sqrt(2)) x - sqrt( pi) ((sqrt(2) I + sqrt(2)) erf(------------------------) - 2 - (sqrt(2) I - sqrt(2)) x - + (sqrt(2) I - sqrt(2)) erf(------------------------))/8 - 2 - """ - if not self.is_simplified(): - self = self.simplify() - s = self._maxima_().display2d(onscreen=False) - s = s.replace('%pi',' pi').replace('%i',' I').replace('%e', ' e') - if onscreen: - print s - else: - return s - - def is_simplified(self): - return hasattr(self, '_simp') and self._simp is self - - def _declare_simplified(self): - self._simp = self - - def hash(self): - return hash(self._repr_(simplify=False)) - - def plot(self, *args, **kwds): - from sage.plot.plot import plot - - # see if the user passed a variable in. - if kwds.has_key('param'): - param = kwds['param'] - else: - param = None - for i in range(len(args)): - if isinstance(args[i], SymbolicVariable): - param = args[i] - args = args[:i] + args[i+1:] - break - - F = self.simplify() - if isinstance(F, Symbolic_object): - if hasattr(F._obj, '__call__'): - f = lambda x: F._obj(x) - else: - y = float(F._obj) - f = lambda x: y - - elif param is None: - if isinstance(self, CallableSymbolicExpression): - A = self.arguments() - if len(A) == 0: - raise ValueError, "function has no input arguments" - else: - param = A[0] - f = lambda x: self(x) - else: - A = self.variables() - if len(A) == 0: - y = float(self) - f = lambda x: y - else: - param = A[0] - f = self.function(param) - else: - f = self.function(param) - return plot(f, *args, **kwds) - - def __lt__(self, right): - return SymbolicEquation(self, SR(right), operator.lt) - - def __le__(self, right): - return SymbolicEquation(self, SR(right), operator.le) - - def __eq__(self, right): - return SymbolicEquation(self, SR(right), operator.eq) - - def __ne__(self, right): - return SymbolicEquation(self, SR(right), operator.ne) - - def __ge__(self, right): - return SymbolicEquation(self, SR(right), operator.ge) - - def __gt__(self, right): - return SymbolicEquation(self, SR(right), operator.gt) + m = maxima(self._maxima_init_()) + self.__maxima = m + return m + + def _neg_(self): + return SymbolicArithmetic([self], operator.neg) def __cmp__(self, right): - """ - Compares self and right. - - This is by definition the comparison of the underlying Maxima - objects. - - EXAMPLES: - These two are equal: - sage: cmp(e+e, e*2) - 0 - """ return cmp(maxima(self), maxima(right)) - - def _neg_(self): - """ - Return the formal negative of self. - - EXAMPLES: - sage: -a - -a - sage: -(x+y) - -y - x - """ - return SymbolicArithmetic([self], operator.neg) - - ################################################################## - # Coercions to interfaces - ################################################################## - # The maxima one is special: - def _maxima_(self, session=None): - if session is None: - return RingElement._maxima_(self, maxima) - else: - return RingElement._maxima_(self, session) - - def _maxima_init_(self): - return self._repr_(simplify=False) - - - # The others all go through _sys_init_, which is defined below, - # and does all interfaces in a unified manner. - - def _axiom_init_(self): - return self._sys_init_('axiom') - - def _gp_init_(self): - return self._sys_init_('pari') # yes, gp goes through pari - - def _maple_init_(self): - return self._sys_init_('maple') - - def _magma_init_(self): - return '"%s"'%self.str() - - def _kash_init_(self): - return self._sys_init_('kash') - - def _macaulay2_init_(self): - return self._sys_init_('macaulay2') - - def _mathematica_init_(self): - return self._sys_init_('mathematica') - - def _octave_init_(self): - return self._sys_init_('octave') - - def _pari_init_(self): - return self._sys_init_('pari') - - def _sys_init_(self, system): - return repr(self) - - ################################################################## - # These systems have no symbolic or numerical capabilities at all, - # really, so we always just coerce to a string - ################################################################## - def _gap_init_(self): - """ - Conversion of symbolic object to GAP always results in a GAP string. - - EXAMPLES: - sage: gap(e+pi^2 + x^3) - "x^3 + pi^2 + e" - """ - return '"%s"'%repr(self) - - def _singular_init_(self): - """ - Conversion of symbolic object to Singular always results in a Singular string. - - EXAMPLES: - sage: singular(e+pi^2 + x^3) - x^3 + pi^2 + e - """ - return '"%s"'%repr(self) - - ################################################################## - # Non-canonical coercions to compiled built-in rings and fields - ################################################################## - def __int__(self): - """ - EXAMPLES: - sage: int(sin(2)*100) - 90 - """ - try: - return int(repr(self)) - except (ValueError, TypeError): - return int(float(self)) - - def __long__(self): - """ - EXAMPLES: - sage: long(sin(2)*100) - 90L - """ - return long(int(self)) - - - def _mpfr_(self, field): - raise TypeError - - def _complex_mpfr_field_(self, field): - raise TypeError - - def _complex_double_(self, C): - raise TypeError - - def _real_double_(self, R): - raise TypeError - - def _real_rqdf_(self, R): - raise TypeError - - def _rational_(self): - return Rational(repr(self)) - - def __abs__(self): - return abs_symbolic(self) - - def _integer_(self): - """ - EXAMPLES: - - """ - return Integer(repr(self)) - + def _add_(self, right): - """ - EXAMPLES: - sage: x + y - y + x - sage: x._add_(y) - y + x - """ - return SymbolicArithmetic([self, right], operator.add) + # if we are adding a negation, instead subtract the operand of negation + #if isinstance(right, SymbolicArithmetic): + # if right._operator is operator.neg: + # return SymbolicArithmetic([self, right._operands[0]], operator.sub) + #elif isinstance(right, Symbolic_object) and right < 0: + # return SymbolicArithmetic([self, SER(abs(right._obj))], operator.sub) + #else: + return SymbolicArithmetic([self, right], operator.add) def _sub_(self, right): - """ - EXAMPLES: - sage: x - y - x - y - """ return SymbolicArithmetic([self, right], operator.sub) def _mul_(self, right): - """ - EXAMPLES: - sage: x * y - x*y - """ + # do some simplification... pull out negatives from the operands and put it + # in front of this multiplication + #if isinstance(self, SymbolicArithmetic) and isinstance(right, SymbolicArithmetic): + # if self._operator is operator.neg and right._operator is operator.neg: + # s_unneg = self._operands[0] + # r_unneg = right._operands[0] + # return SymbolicArithmetic([s_unneg, r_unneg], operator.mul) + #if isinstance(self, SymbolicArithmetic): + # if self._operator is operator.neg and (not isinstance(right, SymbolicArithmetic) \ + # or not (right._operator is operator.neg)): + # s_unneg = self._operands[0] + # return -SymbolicArithmetic([s_unneg, right], operator.mul) + #if isinstance(right, SymbolicArithmetic): + # if not isinstance(self, SymbolicArithmetic) or (not (self._operator is operator.neg)) \ + # and right._operator is operator.neg: + # r_unneg = right._operands[0] + # return -SymbolicArithmetic([self, r_unneg], operator.mul) + return SymbolicArithmetic([self, right], operator.mul) def _div_(self, right): - """ - EXAMPLES: - sage: x / y - x/y - """ + # do some simplification... pull out negatives from the operands and put it + # in front of this division + #if isinstance(self, SymbolicArithmetic) and isinstance(right, SymbolicArithmetic): + # if self._operator is operator.neg and right._operator is operator.neg: + # s_unneg = self._operands[0] + # r_unneg = right._operands[0] + # return SymbolicArithmetic([s_unneg, r_unneg], operator.div) + #elif isinstance(self, SymbolicArithmetic): + # if self._operator is operator.neg and (not isinstance(right, SymbolicArithmetic) \ + # or not (right._operator is operator.neg)): + # s_unneg = self._operands[0] + # return -SymbolicArithmetic([s_unneg, right], operator.div) + #elif isinstance(right, SymbolicArithmetic): + # if not isinstance(self, SymbolicArithmetic) or (not (self._operator is operator.neg)) \ + # and right._operator is operator.neg: + # r_unneg = right._operands[0] + # return -SymbolicArithmetic([self, r_unneg], operator.div) + return SymbolicArithmetic([self, right], operator.div) def __pow__(self, right): - """ - EXAMPLES: - sage: x^(n+1) - x^(n + 1) - """ right = self.parent()(right) return SymbolicArithmetic([self, right], operator.pow) - - def variables(self, vars=tuple([])): - """ - Return sorted list of variables that occur in the simplified - form of self. - - OUTPUT: - a Python set - - EXAMPLES: - sage: f = x^(n+1) + sin(pi/19); f - x^(n + 1) + sin(pi/19) - sage: f.variables() - (n, x) - - sage: a = e^x - sage: a.variables() - (x,) - """ - return vars - - def _first_variable(self): - try: - return self.__first_variable - except AttributeError: - pass - v = self.variables() - if len(v) == 0: - ans = var('x') - else: - ans = v[0] - self.__first_variable = ans - return ans def _has_op(self, operator): @@ -793,166 +186,11 @@ - def __call__(self, dict=None, **kwds): - return self.substitute(dict, **kwds) - - def power_series(self, base_ring): - """ - Return algebraic power series associated to this symbolic - expression, which must be a polynomial in one variable, with - coefficients coercible to the base ring. - - The power series is truncated one more than the degree. - - EXAMPLES: - sage: var('theta') - theta - sage: f = theta^3 + (1/3)*theta - 17/3 - sage: g = f.power_series(QQ); g - -17/3 + 1/3*theta + theta^3 + O(theta^4) - sage: g^3 - -4913/27 + 289/9*theta - 17/9*theta^2 + 2602/27*theta^3 + O(theta^4) - sage: g.parent() - Power Series Ring in theta over Rational Field - """ - v = self.variables() - if len(v) != 1: - raise ValueError, "self must be a polynomial in one variable but it is in the variables %s"%tuple([v]) - f = self.polynomial(base_ring) - from sage.rings.all import PowerSeriesRing - R = PowerSeriesRing(base_ring, names=f.parent().variable_names()) - return R(f, f.degree()+1) - - def polynomial(self, base_ring): - r""" - Return self as an algebraic polynomial over the given base ring, if - possible. - - The point of this function is that it converts purely symbolic - polynomials into optimized algebraic polynomials over a given - base ring. - - WARNING: This is different from \code{self.poly(x)} which is used - to rewrite self as a polynomial in x. - - INPUT: - base_ring -- a ring - - EXAMPLES: - sage: f = x^2 -2/3*x + 1 - sage: f.polynomial(QQ) - x^2 - 2/3*x + 1 - sage: f.polynomial(GF(19)) - x^2 + 12*x + 1 - - sage: f = x^2*e + x + pi/e - sage: f.polynomial(RDF) - 2.71828182846*x^2 + 1.0*x + 1.15572734979 - sage: g = f.polynomial(RR); g - 2.71828182845905*x^2 + 1.00000000000000*x + 1.15572734979092 - sage: g.parent() - Univariate Polynomial Ring in x over Real Field with 53 bits of precision - sage: f.polynomial(RealField(100)) - 2.7182818284590452353602874714*x^2 + 1.0000000000000000000000000000*x + 1.1557273497909217179100931833 - sage: f.polynomial(CDF) - 2.71828182846*x^2 + 1.0*x + 1.15572734979 - sage: f.polynomial(CC) - 2.71828182845905*x^2 + 1.00000000000000*x + 1.15572734979092 - - We coerce a multivariate polynomial with complex symbolic coefficients: - sage: f = pi^3*x - y^2*e - I; f - -1*e*y^2 + pi^3*x - I - sage: f.polynomial(CDF) - -1.0*I + (-2.71828182846)*y^2 + 31.0062766803*x - sage: f.polynomial(CC) - -1.00000000000000*I + (-2.71828182845905)*y^2 + 31.0062766802998*x - sage: f.polynomial(ComplexField(70)) - -1.0000000000000000000*I + (-2.7182818284590452354)*y^2 + 31.006276680299820175*x - - Another polynomial: - sage: f = sum((e*I)^n*x^n for n in range(5)); f - e^4*x^4 - e^3*I*x^3 - e^2*x^2 + e*I*x + 1 - sage: f.polynomial(CDF) - 54.5981500331*x^4 + (-20.0855369232*I)*x^3 + (-7.38905609893)*x^2 + 2.71828182846*I*x + 1.0 - sage: f.polynomial(CC) - 54.5981500331442*x^4 + (-20.0855369231877*I)*x^3 + (-7.38905609893065)*x^2 + 2.71828182845905*I*x + 1.00000000000000 - - A multivariate polynomial over a finite field: - sage: f = (3*x^5 - 5*y^5)^7; f - (3*x^5 - 5*y^5)^7 - sage: g = f.polynomial(GF(7)); g - 2*y^35 + 3*x^35 - sage: parent(g) - Polynomial Ring in x, y over Finite Field of size 7 - """ - vars = self.variables() - if len(vars) == 0: - vars = ['x'] - R = PolynomialRing(base_ring, names=vars) - G = R.gens() - V = R.variable_names() - return self.substitute_over_ring( - dict([(var(V[i]),G[i]) for i in range(len(G))]), ring=R) - - def _polynomial_(self, R): - """ - Coerce this symbolic expression to a polynomial in R. - - EXAMPLES: - sage: R = QQ[x,y,z] - sage: R(x^2 + y) - y + x^2 - sage: R = QQ[w] - sage: R(w^3 + w + 1) - w^3 + w + 1 - sage: R = GF(7)[z] - sage: R(z^3 + 10*z) - z^3 + 3*z - - NOTE: If the base ring of the polynomial ring is the symbolic - ring, then a constant polynomial is always returned. - sage: R = SR[x] - sage: a = R(sqrt(2) + x^3 + y) - sage: a - y + x^3 + sqrt(2) - sage: type(a) - - sage: a.degree() - 0 - - We coerce to a double precision complex polynomial ring: - sage: f = e*x^3 + pi*y^3 + sqrt(2) + I; f - pi*y^3 + e*x^3 + I + sqrt(2) - sage: R = CDF[x,y] - sage: R(f) - 1.41421356237 + 1.0*I + 3.14159265359*y^3 + 2.71828182846*x^3 - - We coerce to a higher-precision polynomial ring - sage: R = ComplexField(100)[x,y] - sage: R(f) - 1.4142135623730950066967437806 + 1.0000000000000000000000000000*I + 3.1415926535897932384626433833*y^3 + 2.7182818284590452353602874714*x^3 - - """ - vars = self.variables() - B = R.base_ring() - if B == SR: - if is_MPolynomialRing(R): - return R({tuple([0]*R.ngens()):self}) - else: - return R([self]) - G = R.gens() - sub = [] - for v in vars: - r = repr(v) - for g in G: - if repr(g) == r: - sub.append((v,g)) - if len(sub) == 0: - return R(B(self)) - return self.substitute_over_ring(dict(sub), ring=R) + def __call__(self, **kwds): + return self.substitute(**kwds) def function(self, *args): """ - Return a CallableSymbolicExpression, fixing a variable order - to be the order of args. + Return a CallableFunction, fixing a variable order to be the order of + args. EXAMPLES: @@ -960,37 +198,11 @@ sage: g = u.function(x,y) sage: g(x,y) - x*cos(y) + sin(x) + sin(x) + x*cos(y) sage: g(t,z) - t*cos(z) + sin(t) - sage: g(x^2, x^y) - x^2*cos(x^y) + sin(x^2) - - sage: f = (x^2 + sin(a*w)).function(a,x,w); f - (a, x, w) |--> x^2 + sin(a*w) - sage: f(1,2,3) - sin(3) + 4 - - Using the function method we can obtain the above function f, - but viewed as a function of different variables: - sage: h = f.function(w,a); h - (w, a) |--> x^2 + sin(a*w) - - This notation also works: - sage: h(w,a) = f - sage: h - (w, a) |--> x^2 + sin(a*w) - - You can even make a symbolic expression $f$ into a function by - writing \code{f(x,y) = f}: - sage: f = x^n + y^n; f - y^n + x^n - sage: f(x,y) = f - sage: f - (x, y) |--> y^n + x^n - sage: f(2,3) - 3^n + 2^n - """ - R = CallableSymbolicExpressionRing(args) - return R(self) + sin(t) + t*cos(z) + sage: g(x^2, log(y)) + sin(x^2) + x^2*cos(log(y)) + """ + return CallableFunction(self, args) @@ -1000,18 +212,4 @@ def derivative(self, *args): """ - Returns the derivative of itself. If self has exactly one variable, then - it differentiates with respect to that variable. If there is more than one - variable in the expression, then you must explicitly supply a variable. - If you supply a variable $x$ followed by a number $n$, then it will - differentiate with respect to $n$ times with respect to $n$. - - You may supply more than one variable. Each variable may optionally be - followed by a positive integer. Then SAGE will differentiate with - respect to the first variable $n$ times, where $n$ is the number - immediately following the variable in the parameter list. If the - variable is not followed by an integer, then SAGE will differentiate - once. Then SAGE will differentiate by the second variables, and if that - is followed by a number $m$, it will differentiate $m$ times, and so on. - EXAMPLES: sage: h = sin(x)/cos(x) @@ -1024,48 +222,14 @@ sage: diff(u,x,y) sin(x)*sin(y) - cos(x)*cos(y) - sage: f = ((x^2+1)/(x^2-1))^(1/4) - sage: g = diff(f, x); g # this is a complex expression - x/(2*(x^2 - 1)^(1/4)*(x^2 + 1)^(3/4)) - (x*(x^2 + 1)^(1/4)/(2*(x^2 - 1)^(5/4))) - sage: g.simplify_rational() - -x/((x^2 - 1)^(5/4)*(x^2 + 1)^(3/4)) - - sage: f = y^(sin(x)) - sage: diff(f, x) - cos(x)*y^sin(x)*log(y) - - sage: g(x) = sqrt(5-2*x) - sage: g_3 = diff(g, x, 3); g_3(2) - -3 - - sage: f = x*e^(-x) - sage: diff(f, 100) - x*e^(-x) - 100*e^(-x) - - sage: g = 1/(sqrt((x^2-1)*(x+5)^6)) - sage: diff(g, x) - -3/((x + 5)^3*sqrt(x^2 - 1)*abs(x + 5)) - (x/((x^2 - 1)^(3/2)*abs(x + 5)^3)) """ # check each time s = "" - # see if we can implicitly supply a variable name - try: - a = args[0] - except IndexError: - # if there were NO arguments, try assuming - a = 1 - if a is None or isinstance(a, (int, long, Integer)): - vars = self.variables() - if len(vars) == 1: - s = "%s, %s" % (vars[0], a) - else: - raise ValueError, "must supply an explicit variable for an " +\ - "expression containing more than one variable" for i in range(len(args)): if isinstance(args[i], SymbolicVariable): - s = s + '%s, ' %repr(args[i]) + s = s + '%s, ' %str(args[i]) # check to see if this is followed by an integer try: if isinstance(args[i+1], (int, long, Integer)): - s = s + '%s, ' %repr(args[i+1]) + s = s + '%s, ' %str(args[i+1]) else: s = s + '1, ' @@ -1089,655 +253,19 @@ f = self.parent()(t) return f - - differentiate = derivative - diff = derivative ################################################################### - # Taylor series - ################################################################### - def taylor(self, v, a, n): - """ - Expands self in a truncated Taylor or Laurent series in the - variable v around the point a, containing terms through $(x - a)^n$. - - INPUT: - v -- variable - a -- number - n -- integer - + # integral + ################################################################### + def integral(self, v): + """ EXAMPLES: - sage: taylor(a*log(z), z, 2, 3) - log(2)*a + a*(z - 2)/2 - (a*(z - 2)^2/8) + a*(z - 2)^3/24 - sage: taylor(sqrt (sin(x) + a*x + 1), x, 0, 3) - 1 + (a + 1)*x/2 - ((a^2 + 2*a + 1)*x^2/8) + (3*a^3 + 9*a^2 + 9*a - 1)*x^3/48 - sage: taylor (sqrt (x + 1), x, 0, 5) - 1 + x/2 - (x^2/8) + x^3/16 - (5*x^4/128) + 7*x^5/256 - sage: taylor (1/log (x + 1), x, 0, 3) - 1/x + 1/2 - (x/12) + x^2/24 - (19*x^3/720) - sage: taylor (cos(x) - sec(x), x, 0, 5) - -x^2 - (x^4/6) - sage: taylor ((cos(x) - sec(x))^3, x, 0, 9) - -x^6 - (x^8/2) - sage: taylor (1/(cos(x) - sec(x))^3, x, 0, 5) - -1/x^6 + 1/(2*x^4) + 11/(120*x^2) - 347/15120 - (6767*x^2/604800) - (15377*x^4/7983360) - """ - v = var(v) - l = self._maxima_().taylor(v, SR(a), Integer(n)) - return self.parent()(l) - - ################################################################### - # limits - ################################################################### - def limit(self, dir=None, **argv): - r""" - Return the limit as the variable v approaches a from the - given direction. - - \begin{verbatim} - expr.limit(x = a) - expr.limit(x = a, dir='above') - \end{verbatim} - - INPUT: - dir -- (default: None); dir may have the value `plus' (or 'above') - for a limit from above, `minus' (or 'below') for a limit from - below, or may be omitted (implying a two-sided - limit is to be computed). - **argv -- 1 named parameter - - NOTE: Output it may also use `und' (undefined), `ind' - (indefinite but bounded), and `infinity' (complex infinity). - - EXAMPLES: - sage: f = (1+1/x)^x - sage: f.limit(x = oo) - e - sage: f.limit(x = 5) - 7776/3125 - sage: f.limit(x = 1.2) - 2.069615754672029 - sage: f.limit(x = I) - e^(I*log(1 - I)) - sage: f(1.2) - 2.069615754672029 - sage: f(I) - (1 - I)^I - sage: CDF(f(I)) - 2.06287223508 + 0.74500706218*I - sage: CDF(f.limit(x = I)) - 2.06287223508 + 0.74500706218*I - - More examples: - sage: limit(x*log(x), x = 0, dir='above') - 0 - sage: lim((x+1)^(1/x),x = 0) - e - sage: lim(e^x/x, x = oo) - +Infinity - sage: lim(e^x/x, x = -oo) - 0 - sage: lim(-e^x/x, x = oo) - -Infinity - sage: lim((cos(x))/(x^2), x = 0) - +Infinity - sage: lim(sqrt(x^2+1) - x, x = oo) - 0 - sage: lim(x^2/(sec(x)-1), x=0) - 2 - sage: lim(cos(x)/(cos(x)-1), x=0) - -Infinity - sage: lim(x*sin(1/x), x=0) - 0 - - Traceback (most recent call last): - ... - TypeError: Computation failed since Maxima requested additional constraints (use assume): - Is x positive or negative? - - sage: f = log(log(x))/log(x) - sage: forget(); assume(x<-2); lim(f, x=0) - limit(log(log(x))/log(x), x=0) - - The following means "indefinite but bounded": - sage: lim(sin(1/x), x = 0) - ind - """ - if len(argv) != 1: - raise ValueError, "call the limit function like this, e.g. limit(expr, x=2)." - else: - k = argv.keys()[0] - v = var(k, create=False) - a = argv[k] - if dir is None: - l = self._maxima_().limit(v, a) - elif dir == 'plus' or dir == 'above': - l = self._maxima_().limit(v, a, 'plus') - elif dir == 'minus' or dir == 'below': - l = self._maxima_().limit(v, a, 'minus') - else: - raise ValueError, "dir must be one of 'plus' or 'minus'" - return self.parent()(l) - - ################################################################### - # Laplace transform - ################################################################### - def laplace(self, t, s): - r""" - Attempts to compute and return the Laplace transform of self - with respect to the variable t and transform parameter s. If - Laplace cannot find a solution, a formal function is returned. - - The function that is returned maybe be viewed as a function of s. - - DEFINITION: - The Laplace transform of a function $f(t)$, defined for all - real numbers $t \geq 0$, is the function $F(s)$ defined by - $$ - F(s) = \int_{0}^{\infty} e^{-st} f(t) dt. - $$ - - EXAMPLES: - We compute a few Laplace transforms: - sage: sin(x).laplace(x, s) - 1/(s^2 + 1) - sage: (z + exp(x)).laplace(x, s) - z/s + 1/(s - 1) - sage: var('t0') - t0 - sage: log(t/t0).laplace(t, s) - (-log(t0) - log(s) - euler_gamma)/s - - We do a formal calculation: - sage: f = function('f', x) - sage: g = f.diff(x); g - diff(f(x), x, 1) - sage: g.laplace(x, s) - s*laplace(f(x), x, s) - f(0) - - EXAMPLE: A BATTLE BETWEEN the X-women and the Y-men (by David Joyner): - Solve - $$ - x' = -16y, x(0)=270, y' = -x + 1, y(0) = 90. - $$ - This models a fight between two sides, the "X-women" - and the "Y-men", where the X-women have 270 initially and - the Y-men have 90, but the Y-men are better at fighting, - because of the higher factor of "-16" vs "-1", and also get - an occasional reinforcement, because of the "+1" term. - - sage: var('t') - t - sage: t = var('t') - sage: x = function('x', t) - sage: y = function('y', t) - sage: de1 = x.diff(t) + 16*y - sage: de2 = y.diff(t) + x - 1 - sage: de1.laplace(t, s) - 16*laplace(y(t), t, s) + s*laplace(x(t), t, s) - x(0) - sage: de2.laplace(t, s) - s*laplace(y(t), t, s) + laplace(x(t), t, s) - (1/s) - y(0) - - Next we form the augmented matrix of the above system: - sage: A = matrix([[s, 16, 270],[1, s, 90+1/s]]) - sage: E = A.echelon_form() - sage: xt = E[0,2].inverse_laplace(s,t) - sage: yt = E[1,2].inverse_laplace(s,t) - sage: print xt - 4 t - 4 t - 91 e 629 e - - -------- + ----------- + 1 - 2 2 - sage: print yt - 4 t - 4 t - 91 e 629 e - -------- + ----------- - 8 8 - sage: p1 = plot(xt,0,1/2,rgbcolor=(1,0,0)) - sage: p2 = plot(yt,0,1/2,rgbcolor=(0,1,0)) - sage: (p1+p2).save() - """ - return self.parent()(self._maxima_().laplace(var(t), var(s))) - - def inverse_laplace(self, t, s): - r""" - Attempts to compute the inverse Laplace transform of self with - respect to the variable t and transform parameter s. If - Laplace cannot find a solution, a formal function is returned. - - The function that is returned maybe be viewed as a function of s. - - - DEFINITION: - The inverse Laplace transform of a function $F(s)$, - is the function $f(t)$ defined by - $$ - F(s) = \frac{1}{2\pi i} \int_{\gamma-i\infty}^{\gamma + i\infty} e^{st} F(s) dt, - $$ - where $\gamma$ is chosen so that the contour path of - integration is in the region of convergence of $F(s)$. - - EXAMPLES: - sage: f = (1/(w^2+10)).inverse_laplace(w, m); f - sin(sqrt(10)*m)/sqrt(10) - sage: laplace(f, m, w) - 1/(w^2 + 10) - """ - return self.parent()(self._maxima_().ilt(var(t), var(s))) - - - ################################################################### - # a default variable, for convenience. - ################################################################### - def default_variable(self): - vars = self.variables() - if len(vars) < 1: - return var('x') - else: - return vars[0] - - ################################################################### - # integration - ################################################################### - def integral(self, v=None, a=None, b=None): - """ - Returns the indefinite integral with respect to the variable - $v$, ignoring the constant of integration. Or, if endpoints - $a$ and $b$ are specified, returns the definite integral over - the interval $[a, b]$. - - If \code{self} has only one variable, then it returns the - integral with respect to that variable. - - INPUT: - v -- (optional) a variable or variable name - a -- (optional) lower endpoint of definite integral - b -- (optional) upper endpoint of definite integral - - - EXAMPLES: - sage: h = sin(x)/(cos(x))^2 + sage: h = sin(x)/cos(x) sage: h.integral(x) - 1/cos(x) - - sage: f = x^2/(x+1)^3 - sage: f.integral() - log(x + 1) + (4*x + 3)/(2*x^2 + 4*x + 2) - - sage: f = x*cos(x^2) - sage: f.integral(x, 0, sqrt(pi)) - 0 - sage: f.integral(a=-pi, b=pi) - 0 - - sage: f(x) = sin(x) - sage: f.integral(x, 0, pi/2) - 1 - - Constraints are sometimes needed: - sage: integral(x^n,x) - Traceback (most recent call last): - ... - TypeError: Computation failed since Maxima requested additional constraints (use assume): - Is n+1 zero or nonzero? - sage: assume(n > 0) - sage: integral(x^n,x) - x^(n + 1)/(n + 1) - sage: forget() - - NOTE: Above, putting assume(n == -1) does not yield the right behavior. - Directly in maxima, doing - - The examples in the Maxima documentation: - sage: integral(sin(x)^3) - cos(x)^3/3 - cos(x) - sage: integral(x/sqrt(b^2-x^2)) - x*log(2*sqrt(b^2 - x^2) + 2*b) - sage: integral(x/sqrt(b^2-x^2), x) - -sqrt(b^2 - x^2) - sage: integral(cos(x)^2 * exp(x), x, 0, pi) - 3*e^pi/5 - 3/5 - sage: integral(x^2 * exp(-x^2), x, -oo, oo) - sqrt(pi)/2 - - We integrate the same function in both Mathematica and SAGE (via Maxima): - sage: f = sin(x^2) + y^z - sage: g = mathematica(f) # optional -- requires mathematica - sage: print g # optional - z 2 - y + Sin[x ] - sage: print g.Integrate(x) # optional - z Pi 2 - x y + Sqrt[--] FresnelS[Sqrt[--] x] - 2 Pi - sage: print f.integral(x) - z (sqrt(2) I + sqrt(2)) x - x y + sqrt( pi) ((sqrt(2) I + sqrt(2)) erf(------------------------) - 2 - (sqrt(2) I - sqrt(2)) x - + (sqrt(2) I - sqrt(2)) erf(------------------------))/8 - 2 - - We integrate the above function in maple now: - sage: g = maple(f); g # optional -- requires maple - sin(x^2)+y^z - sage: g.integrate(x) # optional -- requires maple - 1/2*2^(1/2)*Pi^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*x)+y^z*x - - We next integrate a function with no closed form integral. Notice that - the answer comes back as an expression that contains an integral itself. - sage: A = integral(1/ ((x-4) * (x^3+2*x+1)), x); A - log(x - 4)/73 - (integrate((x^2 + 4*x + 18)/(x^3 + 2*x + 1), x)/73) - sage: print A - / 2 - [ x + 4 x + 18 - I ------------- dx - ] 3 - log(x - 4) / x + 2 x + 1 - ---------- - ------------------ - 73 73 - - ALIASES: - integral() and integrate() are the same. - - """ - - if v is None: - v = self.default_variable() - + """ if not isinstance(v, SymbolicVariable): - v = var(repr(v)) - #raise TypeError, 'must integrate with respect to a variable' - if (a is None and (not b is None)) or (b is None and (not a is None)): - raise TypeError, 'only one endpoint given' - if a is None: - return self.parent()(self._maxima_().integrate(v)) - else: - return self.parent()(self._maxima_().integrate(v, a, b)) - - integrate = integral - - def nintegral(self, x, a, b, - desired_relative_error='1e-8', - maximum_num_subintervals=200): - r""" - Return a numerical approximation to the integral of self from - a to b. - - INPUT: - x -- variable to integrate with respect to - a -- lower endpoint of integration - b -- upper endpoint of integration - desired_relative_error -- (default: '1e-8') the desired - relative error - maximum_num_subintervals -- (default: 200) maxima number - of subintervals - - OUTPUT: - -- float: approximation to the integral - -- float: estimated absolute error of the approximation - -- the number of integrand evaluations - -- an error code: - 0 -- no problems were encountered - 1 -- too many subintervals were done - 2 -- excessive roundoff error - 3 -- extremely bad integrand behavior - 4 -- failed to converge - 5 -- integral is probably divergent or slowly convergent - 6 -- the input is invalid - - ALIAS: - nintegrate is the same as nintegral - - REMARK: - There is also a function \code{numerical_integral} that implements - numerical integration using the GSL C library. It is potentially - much faster and applies to arbitrary user defined functions. - - EXAMPLES: - sage: f(x) = exp(-sqrt(x)) - sage: f.nintegral(x, 0, 1) - (0.52848223531423055, 4.1633141378838452e-11, 231, 0) - - We can also use the \code{numerical_integral} function, which calls - the GSL C library. - sage: numerical_integral(f, 0, 1) # random low-order bits - (0.52848223225314706, 6.8392846084921134e-07) - """ - v = self._maxima_().quad_qags(var(x), - a, b, desired_relative_error, - maximum_num_subintervals) - return float(v[0]), float(v[1]), Integer(v[2]), Integer(v[3]) - - nintegrate = nintegral - - - ################################################################### - # Manipulating epxressions - ################################################################### - def coeff(self, x, n=1): - """ - Returns the coefficient of $x^n$ in self. - - INPUT: - x -- variable, function, expression, etc. - n -- integer, default 1. - - Sometimes it may be necessary to expand or factor first, since - this is not done automatically. - - EXAMPLES: - sage: f = (a*sqrt(2))*x^2 + sin(y)*x^(1/2) + z^z - sage: f.coeff(sin(y)) - sqrt(x) - sage: f.coeff(x^2) - sqrt(2)*a - sage: f.coeff(x^(1/2)) - sin(y) - sage: f.coeff(1) - 0 - sage: f.coeff(x, 0) - z^z - """ - return self.parent()(self._maxima_().coeff(x, n)) - - def coeffs(self, x=None): - """ - Coefficients of self as a polynomial in x. - - INPUT: - x -- optional variable - OUTPUT: - list of pairs [expr, n], where expr is a symbolic expression and n is a power. - - EXAMPLES: - sage: p = x^3 - (x-3)*(x^2+x) + 1 - sage: p.coeffs() - [[1, 0], [3, 1], [2, 2]] - sage: p = expand((x-a*sqrt(2))^2 + x + 1); p - x^2 - 2*sqrt(2)*a*x + x + 2*a^2 + 1 - sage: p.coeffs(a) - [[x^2 + x + 1, 0], [-2*sqrt(2)*x, 1], [2, 2]] - sage: p.coeffs(x) - [[2*a^2 + 1, 0], [1 - 2*sqrt(2)*a, 1], [1, 2]] - - A polynomial with wacky exponents: - sage: p = (17/3*a)*x^(3/2) + x*y + 1/x + x^x - sage: p.coeffs(x) - [[1, -1], [x^x, 0], [y, 1], [17*a/3, 3/2]] - """ - f = self._maxima_() - P = f.parent() - P._eval_line('load(coeflist)') - if x is None: - x = self.default_variable() - x = var(x) - G = f.coeffs(x) - S = symbolic_expression_from_maxima_string(repr(G)) - return S[1:] - - def poly(self, x=None): - r""" - Express self as a polynomial in x. Is self is not a polynomial - in x, then some coefficients may be functions of x. - - WARNING: This is different from \code{self.polynomial()} which - returns a SAGE polynomial over a given base ring. - - EXAMPLES: - sage: p = expand((x-a*sqrt(2))^2 + x + 1); p - x^2 - 2*sqrt(2)*a*x + x + 2*a^2 + 1 - sage: p.poly(a) - (x^2 + x + 1)*1 + -2*sqrt(2)*x*a + 2*a^2 - sage: bool(expand(p.poly(a)) == p) - True - sage: p.poly(x) - (2*a^2 + 1)*1 + (1 - 2*sqrt(2)*a)*x + 1*x^2 - """ - f = self._maxima_() - P = f.parent() - P._eval_line('load(coeflist)') - if x is None: - x = self.default_variable() - x = var(x) - G = f.coeffs(x) - ans = None - for i in range(1, len(G)): - Z = G[i] - coeff = SR(Z[0]) - n = SR(Z[1]) - if repr(coeff) != '0': - if repr(n) == '0': - xpow = SR(1) - elif repr(n) == '1': - xpow = x - else: - xpow = x**n - if ans is None: - ans = coeff*xpow - else: - ans += coeff*xpow - ans._declare_simplified() - return ans - - def combine(self): - """ - Simplifies self by combining all terms with the same - denominator into a single term. - - EXAMPLES: - sage: f = x*(x-1)/(x^2 - 7) + y^2/(x^2-7) + 1/(x+1) + b/a + c/a - sage: print f - 2 - y (x - 1) x 1 c b - ------ + --------- + ----- + - + - - 2 2 x + 1 a a - x - 7 x - 7 - sage: print f.combine() - 2 - y + (x - 1) x 1 c + b - -------------- + ----- + ----- - 2 x + 1 a - x - 7 - """ - return self.parent()(self._maxima_().combine()) - - def numerator(self): - """ - EXAMPLES: - sage: f = x*(x-a)/((x^2 - y)*(x-a)) - sage: print f - x - ------ - 2 - x - y - sage: f.numerator() - x - sage: f.denominator() - x^2 - y - """ - return self.parent()(self._maxima_().num()) - - def denominator(self): - """ - EXAMPLES: - sage: f = (sqrt(x) + sqrt(y) + sqrt(z))/(x^10 - y^10 - sqrt(var('theta'))) - sage: print f - sqrt(z) + sqrt(y) + sqrt(x) - --------------------------- - 10 10 - - y + x - sqrt(theta) - sage: f.denominator() - -y^10 + x^10 - sqrt(theta) - """ - return self.parent()(self._maxima_().denom()) - - ################################################################### - # solve - ################################################################### - def roots(self, x=None): - r""" - Returns the roots of self, with multiplicities. - - INPUT: - x -- variable to view the function in terms of - (use default variable if not given) - OUTPUT: - list of pairs (root, multiplicity) - - If there are infinitely many roots, e.g., a function - like sin(x), only one is returned. - - EXAMPLES: - A simple example: - sage: ((x^2-1)^2).roots() - [(-1, 2), (1, 2)] - - A complicated example. - sage: f = expand((x^2 - 1)^3*(x^2 + 1)*(x-a)); f - x^9 - a*x^8 - 2*x^7 + 2*a*x^6 + 2*x^3 - 2*a*x^2 - x + a - - The default variable is a, since it is the first in alphabetical order: - sage: f.roots() - [(x, 1)] - - As a polynomial in a, x is indeed a root: - sage: f.poly(a) - (x^9 - 2*x^7 + 2*x^3 - x)*1 + (-x^8 + 2*x^6 - 2*x^2 + 1)*a - sage: f(a=x) - 0 - - The roots in terms of x are what we expect: - sage: f.roots(x) - [(a, 1), (-1*I, 1), (I, 1), (1, 3), (-1, 3)] - - Only one root of $\sin(x) = 0$ is given: - sage: f = sin(x) - sage: f.roots(x) - [(0, 1)] - - We derive the roots of a general quadratic polynomial: - sage: (a*x^2 + b*x + c).roots(x) - [((-sqrt(b^2 - 4*a*c) - b)/(2*a), 1), ((sqrt(b^2 - 4*a*c) - b)/(2*a), 1)] - """ - if x is None: - x = self.default_variable() - S, mul = self.solve(x, multiplicities=True) - return [(S[i].rhs(), mul[i]) for i in range(len(mul))] - - def solve(self, x, multiplicities=False): - r""" - Solve the equation \code{self == 0} for the variable x. - - INPUT: - x -- variable to solve for - multiplicities -- bool (default: False); if True, return corresponding multiplicities. - - EXAMPLES: - sage: (z^5 - 1).solve(z) - [z == e^(2*I*pi/5), z == e^(4*I*pi/5), z == e^(-(4*I*pi/5)), z == e^(-(2*I*pi/5)), z == 1] - """ - x = var(x) - return (self == 0).solve(x, multiplicities=multiplicities) + raise TypeError, "second argument to diff must be a variable" + return self.parent()(self._maxima_().integrate(v)) + ################################################################### @@ -1745,20 +273,13 @@ ################################################################### def simplify(self): - try: - return self._simp - except AttributeError: - S = evaled_symbolic_expression_from_maxima_string(self._maxima_init_()) - S._simp = S - self._simp = S - return S + return self.parent()(self._maxima_()) def simplify_trig(self): r""" - First expands using trig_expand, then employs the identities - $\sin(x)^2 + \cos(x)^2 = 1$ and $\cosh(x)^2 - \sin(x)^2 = 1$ - to simplify expressions containing tan, sec, etc., to sin, - cos, sinh, cosh. - - ALIAS: trig_simplify and simplify_trig are the same + Employs the identities $\sin(x)^2 + \cos(x)^2 = 1$ and + $\cosh(x)^2 - \sin(x)^2 = 1$ to simplify expressions + containing tan, sec, etc., to sin, cos, sinh, cosh. + + trig_simplify() and simplify_trig() are the same method. EXAMPLES: @@ -1769,121 +290,9 @@ sage: f.simplify_trig() 1 - """ - # much better to expand first, since it often doesn't work - # right otherwise! - return self.parent()(self._maxima_().trigexpand().trigsimp()) + + """ + return self.parent()(self._maxima_().trigsimp()) trig_simplify = simplify_trig - - def simplify_rational(self): - """ - Simplify by expanding repeatedly rational expressions. - - ALIAS: rational_simplify and simplify_rational are the same - - EXAMPLES: - sage: f = sin(x/(x^2 + x)) - sage: f - sin(x/(x^2 + x)) - sage: f.simplify_rational() - sin(1/(x + 1)) - - sage: f = ((x - 1)^(3/2) - (x + 1)*sqrt(x - 1))/sqrt((x - 1)*(x + 1)) - sage: print f - 3/2 - (x - 1) - sqrt(x - 1) (x + 1) - -------------------------------- - sqrt((x - 1) (x + 1)) - sage: print f.simplify_rational() - 2 sqrt(x - 1) - - ------------- - 2 - sqrt(x - 1) - """ - return self.parent()(self._maxima_().fullratsimp()) - - rational_simplify = simplify_rational - - # TODO: come up with a way to intelligently wrap Maxima's way of - # fine-tuning all simplificationsrational - - def simplify_radical(self): - r""" - Wraps the Maxima radcan() command. From the Maxima documentation: - - Simplifies this expression, which can contain logs, exponentials, - and radicals, by converting it into a form which is canonical over a - large class of expressions and a given ordering of variables; that - is, all functionally equivalent forms are mapped into a unique form. - For a somewhat larger class of expressions, produces a regular form. - Two equivalent expressions in this class do not necessarily have the - same appearance, but their difference can be simplified by radcan to - zero. - - For some expressions radcan is quite time consuming. This is the - cost of exploring certain relationships among the components of the - expression for simplifications based on factoring and partial - fraction expansions of exponents. - - ALIAS: radical_simplify, simplify_log, log_simplify, exp_simplify, - simplify_exp are all the same - - EXAMPLES: - - sage: f = log(x*y) - sage: f.simplify_radical() - log(y) + log(x) - - sage: f = (log(x+x^2)-log(x))^a/log(1+x)^(a/2) - sage: f.simplify_radical() - log(x + 1)^(a/2) - - sage: f = (e^x-1)/(1+e^(x/2)) - sage: f.simplify_exp() - e^(x/2) - 1 - """ - return self.parent()(self._maxima_().radcan()) - - radical_simplify = simplify_log = log_simplify = simplify_radical - simplify_exp = exp_simplify = simplify_radical - - ################################################################### - # factor - ################################################################### - def factor(self, dontfactor=[]): - """ - Factors self, containing any number of variables or functions, - into factors irreducible over the integers. - - INPUT: - self -- a symbolic expression - dontfactor -- list (default: []), a list of variables with - respect to which factoring is not to occur. - Factoring also will not take place with - respect to any variables which are less - important (using the variable ordering - assumed for CRE form) than those on the - `dontfactor' list. - - EXAMPLES: - sage: (x^3-y^3).factor() - (-(y - x))*(y^2 + x*y + x^2) - sage: factor(-8*y - 4*x + z^2*(2*y + x)) - (2*y + x)*(z - 2)*(z + 2) - sage: f = -1 - 2*x - x^2 + y^2 + 2*x*y^2 + x^2*y^2 - sage: F = factor(f/(36*(1 + 2*y + y^2)), dontfactor=[x]) - sage: print F - 2 - (x + 2 x + 1) (y - 1) - ---------------------- - 36 (y + 1) - """ - if len(dontfactor) > 0: - m = self._maxima_() - name = m.name() - cmd = 'block([dontfactor:%s],factor(%s))'%(dontfactor, name) - return evaled_symbolic_expression_from_maxima_string(cmd) - else: - return self.parent()(self._maxima_().factor()) ################################################################### @@ -1893,10 +302,5 @@ """ """ - try: - return self.__expand - except AttributeError: - e = self.parent()(self._maxima_().expand()) - self.__expand = e - return e + return self.parent()(self._maxima_().expand()) def expand_trig(self): @@ -1909,5 +313,5 @@ cos(2*x)*cos(y) - sin(2*x)*sin(y) - ALIAS: trig_expand and expand_trig are the same + ALIAS: trig_expand """ return self.parent()(self._maxima_().trigexpand()) @@ -1916,304 +320,196 @@ + ################################################################### # substitute ################################################################### - def substitute(self, in_dict=None, **kwds): + def substitute(self, dict=None, **kwds): """ Takes the symbolic variables given as dict keys or as keywords and replaces them with the symbolic expressions given as dict values or as - keyword values. Also run when you call a SymbolicExpression. - - INPUT: - in_dict -- (optional) dictionary of inputs - **kwds -- named parameters - + keyword values. Also run when you call a SymbolicExpression. + EXAMPLES: - sage: x,y,t = var('x,y,t') - sage: u = 3*y - sage: u.substitute(y=t) - 3*t - - sage: u = x^3 - 3*y + 4*t + sage: u = (x^3 - 3*y + 4*t) sage: u.substitute(x=y, y=t) - y^3 + t + y^3 - 3*t + 4*t sage: f = sin(x)^2 + 32*x^(y/2) - sage: f(x=2, y = 10) - sin(2)^2 + 1024 + sage: f.substitute(x=2, y = 10) + sin(2)^2 + 32*2^(10/2) sage: f(x=pi, y=t) - 32*pi^(t/2) - - sage: f = x - sage: f.variables() - (x,) - - sage: f = 2*x - sage: f.variables() - (x,) - - sage: f = 2*x^2 - sin(x) - sage: f.variables() - (x,) - sage: f(pi) - 2*pi^2 - sage: f(x=pi) - 2*pi^2 + sin(pi)^2 + 32*pi^(t/2) + + sage: f(x=pi, y=t).simplify() + AUTHORS: -- Bobby Moretti: Initial version """ - X = self.simplify() - kwds = self.__parse_in_dict(in_dict, kwds) - kwds = self.__varify_kwds(kwds) - return X._recursive_sub(kwds) - - def subs(self, *args, **kwds): - return self.substitute(*args, **kwds) + if dict is not None: + for k, v in dict.iteritems(): + kwds[str(k)] = v + # find the keys from the keywords + return self._recursive_sub(kwds) def _recursive_sub(self, kwds): - raise NotImplementedError, "implement _recursive_sub for type '%s'!"%(type(self)) - - def _recursive_sub_over_ring(self, kwds, ring): - raise NotImplementedError, "implement _recursive_sub_over_ring for type '%s'!"%(type(self)) - - def __parse_in_dict(self, in_dict, kwds): - if in_dict is None: - return kwds - - if not isinstance(in_dict, dict): - if len(kwds) > 0: - raise ValueError, "you must not both give the variable and specify it explicitly when doing a substitution." - in_dict = SR(in_dict) - vars = self.variables() - #if len(vars) > 1: - # raise ValueError, "you must specify the variable when doing a substitution, e.g., f(x=5)" - if len(vars) == 0: - return {} + # if we have operands, call ourself on each operand + try: + ops = self._operands + except AttributeError: + pass + if isinstance(self, SymbolicVariable): + s = str(self) + if s in kwds: + return kwds[s] else: - return {vars[0]: in_dict} - - # merge dictionaries - kwds.update(in_dict) - return kwds - - def __varify_kwds(self, kwds): - return dict([(var(k),w) for k,w in kwds.iteritems()]) - - def substitute_over_ring(self, in_dict=None, ring=None, **kwds): - X = self.simplify() - kwds = self.__parse_in_dict(in_dict, kwds) - kwds = self.__varify_kwds(kwds) - if ring is None: - return X._recursive_sub(kwds) + return self + elif isinstance(self, SymbolicConstant): + return self + elif isinstance(self, SymbolicArithmetic): + new_ops = [op._recursive_sub(kwds) for op in ops] + return SymbolicArithmetic([new_ops[0], new_ops[1]], self._operator) + elif isinstance(self, SymbolicComposition): + return SymbolicComposition(ops[0], ops[1]._recursive_sub(kwds)) + +class PrimitiveFunction(SymbolicExpression): + def __init__(self): + pass + + def __call__(self, x): + # if we're calling with a symbolic expression, do function composition + if isinstance(x, SymbolicExpression) and not isinstance(x, Constant): + return SymbolicComposition(self, x) + + # if x is a polynomial object, first turn it into a function and + # compose self with x + elif is_Polynomial(x): + return Function_composition(self, SymbolicPolynomial(x)) + + # if we can't figure out what x is, return a symbolic version of x + elif isinstance(x, (Integer, Rational, + int, long, float, complex)): + return self(Constant_object(x)) + else: - return X._recursive_sub_over_ring(kwds, ring) - - - ################################################################### - # Real and imaginary parts - ################################################################### - def real(self): - """ - Return the real part of self. - - EXAMPLES: - sage: a = log(3+4*I) - sage: print a - log(4 I + 3) - sage: print a.real() - log(5) - sage: print a.imag() - 4 - atan(-) - 3 - - Now make a and b symbolic and compute the general real part: - sage: restore('a,b') - sage: f = log(a + b*I) - sage: f.real() - log(b^2 + a^2)/2 - """ - return self.parent()(self._maxima_().real()) - - def imag(self): - """ - Return the imaginary part of self. - - EXAMPLES: - sage: sqrt(-2).imag() - sqrt(2) - - We simplify Ln(Exp(z)) to z for -Pi %s" % (self._varlist[0], self._expr._repr_()) + else: + vars = ", ".join(map(str, self._varlist)) + return "(%s) |--> %s" % (vars, self._expr._repr_()) + + def _latex_(self): + if len(self._varlist) == 1: + return "%s \\ {\mapsto}\\ %s" % (self._varlist[0], + self._expr._latex_()) + else: + vars = ", ".join(map(latex, self._varlist)) + # the weird TeX is to workaround an apparent JsMath bug + return "\left(%s \\right)\\ {\\mapsto}\\ %s" % (vars, self._expr._latex_()) + + def derivative(self, dt): + return CallableFunction(self._expr.derivative(dt), self._varlist) + + def integral(self, dx): + return CallableFunction(self._expr.integral(dx), self._varlist) + + def substitute(self, dict=None, **kwds): + return CallableFunction(self._expr.substitute(dict, kwds), self._varlist) + + def simplify(self): + return CallableFunction(self._expr.simplify(), self._varlist) + + def trig_simplify(self): + return CallableFunction(self._expr.trig_simplify(), self._varlist) + + #TODO: Arithmetic, expand, trig_expand, etc... class Symbolic_object(SymbolicExpression): - r""" - A class representing a symbolic expression in terms of a SageObject (not - necessarily a 'constant'). - """ + r''' + A class representing a symbolic expression in terms of a SageObject. + ''' + def __init__(self, obj): SymbolicExpression.__init__(self) self._obj = obj - def __hash__(self): - return hash(self._obj) - - #def derivative(self, *args): - # TODO: remove - # return self.parent().zero_element() - def obj(self): - """ - EXAMPLES: - """ return self._obj - def __float__(self): - """ - EXAMPLES: - """ - return float(self._obj) - - def __complex__(self): - """ - EXAMPLES: - """ - return complex(self._obj) - - def _mpfr_(self, field): - """ - EXAMPLES: - """ - return field(self._obj) - - def _complex_mpfr_field_(self, C): - """ - EXAMPLES: - """ - return C(self._obj) - - def _complex_double_(self, C): - """ - EXAMPLES: - """ - return C(self._obj) - - def _real_double_(self, R): - """ - EXAMPLES: - """ - return R(self._obj) - - def _real_rqdf_(self, R): - """ - EXAMPLES: - """ - return R(self._obj) - - def _repr_(self, simplify=True): - """ - EXAMPLES: - """ - return repr(self._obj) - - def _latex_(self): - """ - EXAMPLES: - """ + def _repr_(self): + return str(self._obj) + + def _latex_(self): return latex(self._obj) + + def __pow__(self, right): + if isinstance(right, Symbolic_object): + try: + return Symbolic_object(self._obj + right._obj) + except TypeError: + pass + return SymbolicExpression.__pow__(self, right) + + def _add_(self, right): + if isinstance(right, Symbolic_object): + try: + return Symbolic_object(self._obj + right._obj) + except TypeError: + pass + return SymbolicExpression._add_(self, right) + + def _sub_(self, right): + if isinstance(right, Symbolic_object): + try: + return Symbolic_object(self._obj - right._obj) + except TypeError: + pass + return SymbolicExpression._sub_(self, right) + + def _mul_(self, right): + if isinstance(right, Symbolic_object): + try: + return Symbolic_object(self._obj * right._obj) + except TypeError: + pass + return SymbolicExpression._mul_(self, right) + + def _div_(self, right): + if isinstance(right, Symbolic_object): + try: + return Symbolic_object(self._obj / right._obj) + except TypeError: + pass + return SymbolicExpression._div_(self, right) def str(self, bits=None): @@ -2224,144 +520,28 @@ return str(R(self._obj)) - def _maxima_init_(self): - return maxima_init(self._obj) - - def _sys_init_(self, system): - return sys_init(self._obj, system) - -def maxima_init(x): - try: - return x._maxima_init_() - except AttributeError: - return repr(x) - -def sys_init(x, system): - try: - return x.__getattribute__('_%s_init_'%system)() - except AttributeError: - try: - return x._system_init_(system) + def _maxima_(self, maxima=maxima): + try: + return self.__maxima except AttributeError: - return repr(x) - -class SymbolicConstant(Symbolic_object): - def __init__(self, x): - Symbolic_object.__init__(self, x) - - def _is_atomic(self): - try: - return self._atomic - except AttributeError: - if isinstance(self, Rational): - self._atomic = False - else: - self._atomic = True - return self._atomic - - def _recursive_sub(self, kwds): - """ - EXAMPLES: - sage: a = SR(5/6) - sage: type(a) - - sage: a(x=3) - 5/6 - """ - return self - - def _recursive_sub_over_ring(self, kwds, ring): - return ring(self) - - + m = maxima(self._obj) + self.__maxima = m + return m class SymbolicPolynomial(Symbolic_object): - """ - An element of a polynomial ring as a formal symbolic expression. - - EXAMPLES: - A single variate polynomial: - sage: R. = QQ[] - sage: f = SR(x^3 + x) - sage: f(y=7) - x^3 + x - sage: f(x=5) - 130 - sage: f.integral(x) - x^4/4 + x^2/2 - sage: f(x=y) - y^3 + y - - A multivariate polynomial: - - sage: R. = ZZ[] - sage: f = SR(x^3 + x + y + theta^2); f - theta^2 + y + x + x^3 - sage: f(x=y, theta=y) - y^3 + y^2 + 2*y - sage: f(x=5) - y + theta^2 + 130 - - The polynomial must be over a field of characteristic 0. - sage: R. = GF(7)[] - sage: f = SR(w^3 + 1) - Traceback (most recent call last): - ... - TypeError: polynomial must be over a field of characteristic 0. - """ - # for now we do nothing except pass the info on to the superconstructor. It's + "An element of a polynomial ring as a formal symbolic expression." + + # for now we do nothing except pass the info on to the supercontructor. It's # not clear to me why we need anything else in this class -Bobby def __init__(self, p): - if p.parent().base_ring().characteristic() != 0: - raise TypeError, "polynomial must be over a field of characteristic 0." - Symbolic_object.__init__(self, p) - - def _recursive_sub(self, kwds): - return self._recursive_sub_over_ring(kwds, ring) - - def _recursive_sub_over_ring(self, kwds, ring): - f = self._obj - if is_Polynomial(f): - # Single variable case - v = f.parent().variable_name() - if kwds.has_key(v): - return ring(f(kwds[v])) - else: - if not ring is SR: - return ring(self) - else: - # Multivariable case - t = [] - for g in f.parent().gens(): - s = repr(g) - if kwds.has_key(s): - t.append(kwds[s]) - else: - t.append(g) - return ring(f(*t)) - - def polynomial(self, base_ring): - """ - Return self as a polynomial over the given base ring, if possible. - - INPUT: - base_ring -- a ring - - EXAMPLES: - sage: R. = QQ[] - sage: f = SR(x^2 -2/3*x + 1) - sage: f.polynomial(QQ) - x^2 - 2/3*x + 1 - sage: f.polynomial(GF(19)) - x^2 + 12*x + 1 - """ - f = self._obj - if base_ring is f.base_ring(): - return f - return f.change_ring(base_ring) - -################################################################## - - -zero_constant = SymbolicConstant(Integer(0)) + Symbolic_object.__init__(self, p) + +class SymbolicConstant(SymbolicExpression): + def __call__(self, x): + return self + +class Constant_object(SymbolicConstant, Symbolic_object): + pass + +zero_constant = Constant_object(Integer(0)) class SymbolicOperation(SymbolicExpression): @@ -2371,40 +551,33 @@ def __init__(self, operands): SymbolicExpression.__init__(self) - self._operands = operands # don't even make a copy -- ok, since immutable. - - def variables(self, vars=tuple([])): - """ - Return sorted list of variables that occur in the simplified - form of self. The ordering is alphabetic. - - EXAMPLES: - sage: x,y,z,w = var('x,y,z,w') - sage: f = (x - x) + y^2 - z/z + (w^2-1)/(w+1); f - y^2 + (w^2 - 1)/(w + 1) - 1 - sage: f.variables() - (w, y) - - sage: (x + y + z + a + b + c).variables() - (a, b, c, x, y, z) - - sage: (x^2 + x).variables() - (x,) - """ - if not self.is_simplified(): - return self.simplify().variables(vars) - - try: - return self.__variables + self._operands = [SER(op) for op in operands] + +class SymbolicComposition(SymbolicOperation): + r''' + Represents the symbolic composition of $f \circ g$. + ''' + def __init__(self, f, g): + SymbolicOperation.__init__(self, [f,g]) + + def _is_atomic(self): + return True + + def _repr_(self): + ops = self._operands + return "%s(%s)"% (ops[0]._repr_(), ops[1]._repr_()) + + def _latex_(self): + ops = self._operands + return r"%s \left( %s \right)"% (latex(ops[0]), latex(ops[1])) + + def _maxima_(self, maxima=maxima): + try: + return self.__maxima except AttributeError: - pass - vars = list(set(sum([list(op.variables()) for op in self._operands], list(vars)))) - - vars.sort(var_cmp) - vars = tuple(vars) - self.__variables = vars - return vars - -def var_cmp(x,y): - return cmp(repr(x), repr(y)) + ops = self._operands + m = ops[0]._maxima_(maxima)(ops[1]._maxima_(maxima)) + self.__maxima = m + return m + symbols = {operator.add:' + ', operator.sub:' - ', operator.mul:'*', @@ -2413,178 +586,82 @@ class SymbolicArithmetic(SymbolicOperation): - r""" + r''' Represents the result of an arithemtic operation on $f$ and $g$. - """ + ''' + def __init__(self, operands, op): SymbolicOperation.__init__(self, operands) + + if op is operator.add: + self._atomic = False + elif op is operator.sub: + self._atomic = False + elif op is operator.mul: + self._atomic = True + elif op is operator.div: + self._atomic = False + elif op is operator.pow: + self._atomic = True + elif op is operator.neg: + self._atomic = False + self._operator = op - def _recursive_sub(self, kwds): - """ - EXAMPLES: - sage: f = (x - x) + y^2 - z/z + (w^2-1)/(w+1); f - y^2 + (w^2 - 1)/(w + 1) - 1 - sage: f(y=10) - (w^2 - 1)/(w + 1) + 99 - sage: f(w=1,y=10) - 99 - sage: f(y=w,w=y) - (y^2 - 1)/(y + 1) + w^2 - 1 - - sage: f = y^5 - sqrt(2) - sage: f(10) - 100000 - sqrt(2) - """ - ops = self._operands - new_ops = [SR(op._recursive_sub(kwds)) for op in ops] - return self._operator(*new_ops) - - def _recursive_sub_over_ring(self, kwds, ring): - ops = self._operands - if self._operator == operator.pow: - new_ops = [ops[0]._recursive_sub_over_ring(kwds, ring=ring), Integer(ops[1])] - else: - new_ops = [op._recursive_sub_over_ring(kwds, ring=ring) for op in ops] - return ring(self._operator(*new_ops)) - - def __float__(self): - fops = [float(op) for op in self._operands] - return self._operator(*fops) - - def __complex__(self): - fops = [complex(op) for op in self._operands] - return self._operator(*fops) - - def _mpfr_(self, field): - if not self.is_simplified(): - return self.simplify()._mpfr_(field) - rops = [op._mpfr_(field) for op in self._operands] - return self._operator(*rops) - - def _complex_mpfr_field_(self, field): - if not self.is_simplified(): - return self.simplify()._complex_mpfr_field_(field) - rops = [op._complex_mpfr_field_(field) for op in self._operands] - return self._operator(*rops) - - def _complex_double_(self, field): - if not self.is_simplified(): - return self.simplify()._complex_double_(field) - rops = [op._complex_double_(field) for op in self._operands] - return self._operator(*rops) - - def _real_double_(self, field): - if not self.is_simplified(): - return self.simplify()._real_double_(field) - rops = [op._real_double_(field) for op in self._operands] - return self._operator(*rops) - - def _real_rqdf_(self, field): - if not self.is_simplified(): - return self.simplify()._real_rqdf_(field) - rops = [op._real_rqdf_(field) for op in self._operands] - return self._operator(*rops) - def _is_atomic(self): - try: - return self._atomic - except AttributeError: - op = self._operator - # todo: optimize with a dictionary - if op is operator.add: - self._atomic = False - elif op is operator.sub: - self._atomic = False - elif op is operator.mul: - self._atomic = True - elif op is operator.div: - self._atomic = False - elif op is operator.pow: - self._atomic = True - elif op is operator.neg: - self._atomic = False - return self._atomic - - def _repr_(self, simplify=True): - """ - TESTS: - sage: a = (1-1/r)^(-1); a - 1/(1 - (1/r)) - sage: a.derivative(r) - -1/((1 - (1/r))^2*r^2) - - sage: reset('a,b') - sage: s = 0*(1/a) + -b*(1/a)*(1 + -1*0*(1/a))*(1/(a*b + -1*b*(1/a))) - sage: s - -b/(a*(a*b - (b/a))) - sage: s(a=2,b=3) - -1/3 - sage: -3/(2*(2*3-(3/2))) - -1/3 - """ - if simplify: - if hasattr(self, '_simp'): - return self._simp._repr_(simplify=False) - else: - return self.simplify()._repr_(simplify=False) + return self._atomic + + def _repr_(self): ops = self._operands op = self._operator - - s = [o._repr_(simplify=False) for o in ops] - + + s = [str(o) for o in ops] # for the left operand, we need to surround it in parens when the # operator is mul/div/pow, and when the left operand contains an # operation of lower precedence - if op in [operator.mul, operator.div]: + if op in [operator.mul, operator.div, operator.pow]: if ops[0]._has_op(operator.add) or ops[0]._has_op(operator.sub): if not ops[0]._is_atomic(): s[0] = '(%s)' % s[0] - else: - try: - if isinstance(ops[0]._obj, Rational): - s[0] = '(%s)' % s[0] - except AttributeError: - pass - try: - if isinstance(ops[1]._obj, Rational): - s[1] = '(%s)' % s[1] - except AttributeError: - pass - - # for the right operand, we need to surround it in parens when - # the operation is mul/div/sub, and when the right operand - # contains a + or -. - if op in [operator.mul, operator.sub]: + # for the right operand, we need to surround it in parens when the + # operation is mul/div/sub and the, and when the right operand contains + # a + or -. + if op in [operator.mul, operator.div, operator.sub]: + if ops[1]._has_op(operator.add) or ops[1]._has_op(operator.sub): # avoid drawing parens if s1 an atomic operation if not ops[1]._is_atomic(): s[1] = '(%s)' % s[1] - - elif op is operator.div: - if not ops[1]._is_atomic() or ops[1]._has_op(operator.mul): + # if we have a compound expression on the right, then we need parens on + # the right... but in TeX, this is expressed by font size and position + elif op is operator.pow: + if ops[1]._has_op(operator.add) or \ + ops[1]._has_op(operator.sub) or \ + ops[1]._has_op(operator.mul) or \ + ops[1]._has_op(operator.div) or \ + ops[1]._has_op(operator.pow): s[1] = '(%s)' % s[1] - - elif op is operator.pow: - if not ops[0]._is_atomic(): - s[0] = '(%s)'% s[0] - if not ops[1]._is_atomic() or ('/' in s[1] or '*' in s[1]): - s[1] = '(%s)'% s[1] - if op is operator.neg: - if ops[0]._is_atomic(): - return '-%s' % s[0] - else: - return '-(%s)'%s[0] + return '-%s' % s[0] else: return '%s%s%s' % (s[0], symbols[op], s[1]) def _latex_(self): - # if we are not simplified, return the latex of a simplified version - if not self.is_simplified(): - return self.simplify()._latex_() op = self._operator ops = self._operands - s = [x._latex_() for x in self._operands] + ops_tex = [x._latex_() for x in self._operands] + + s = [o._latex_() for o in ops] + + # follow a very similar logic to _repr_, but we don't parenthesize + # exponents + if op in [operator.mul, operator.div, operator.pow]: + if ops[0]._has_op(operator.add) or ops[0]._has_op(operator.sub): + if not ops[0]._is_atomic(): + s[0] = '\\left( %s \\right)' % s[0] + if op in [operator.mul, operator.div, operator.sub]: + if ops[1]._has_op(operator.add) or ops[1]._has_op(operator.sub): + if not ops[1]._is_atomic(): + s[1] = '\\left( %s \\right)' % s[1] if op is operator.add: @@ -2593,37 +670,24 @@ return '%s - %s' % (s[0], s[1]) elif op is operator.mul: - if ops[0]._has_op(operator.add) or ops[0]._has_op(operator.sub): - s[0] = r'\left( %s \right)' %s[0] - if ops[1]._has_op(operator.add) or ops[1]._has_op(operator.sub): - s[1] = r'\left( %s \right)' %s[1] return '{%s \\cdot %s}' % (s[0], s[1]) elif op is operator.div: return '\\frac{%s}{%s}' % (s[0], s[1]) elif op is operator.pow: - if ops[0]._has_op(operator.add) or ops[0]._has_op(operator.sub) \ - or ops[0]._has_op(operator.mul) or ops[0]._has_op(operator.div) \ - or ops[0]._has_op(operator.pow): - s[0] = r'\left( %s \right)' % s[0] return '{%s}^{%s} ' % (s[0], s[1]) elif op is operator.neg: return '-%s' % s[0] - def _maxima_init_(self): - ops = self._operands - if self._operator is operator.neg: - return '-(%s)' % ops[0]._maxima_init_() - else: - return '(%s) %s (%s)' % (ops[0]._maxima_init_(), - infixops[self._operator], - ops[1]._maxima_init_()) - - def _sys_init_(self, system): - ops = self._operands - if self._operator is operator.neg: - return '-(%s)' % sys_init(ops[0], system) - else: - return '(%s) %s (%s)' % (sys_init(ops[0], system), - infixops[self._operator], - sys_init(ops[1], system)) + def _maxima_(self, maxima=maxima): + try: + return self.__maxima + except AttributeError: + ops = self._operands + if self._operator is operator.neg: + m = self._operator(ops[0]._maxima_(maxima)) + else: + m = self._operator(ops[0]._maxima_(maxima), + ops[1]._maxima_(maxima)) + self.__maxima = m + return m import re @@ -2636,37 +700,8 @@ raise ValueError, "variable name must be nonempty" - def __hash__(self): - return hash(self._name) - - def _recursive_sub(self, kwds): - # do the replacement if needed - if kwds.has_key(self): - return kwds[self] - else: - return self - - def _recursive_sub_over_ring(self, kwds, ring): - if kwds.has_key(self): - return ring(kwds[self]) - else: - return ring(self) - - def variables(self, vars=tuple([])): - """ - Return sorted list of variables that occur in the simplified - form of self. - """ - return (self, ) - - def __cmp__(self, right): - if isinstance(right, SymbolicVariable): - return cmp(self._repr_(), right._repr_()) - else: - return SymbolicExpression.__cmp__(self, right) - - def _repr_(self, simplify=True): - return self._name - - def __str__(self): + def __call__(self, x): + raise AttributeError, "A symbolic variable is not callable." + + def _repr_(self): return self._name @@ -2688,9 +723,11 @@ return a - def _maxima_init_(self): - return self._name - - def _sys_init_(self, system): - return self._name + def _maxima_(self, maxima=maxima): + try: + return self.__maxima + except AttributeError: + m = maxima(self._name) + self.__maxima = m + return m common_varnames = ['alpha', @@ -2729,1832 +766,111 @@ def tex_varify(a): - if a in common_varnames: + if a in common_varnames: return "\\" + a - elif len(a) == 1: - return a else: return '\\mbox{%s}'%a _vars = {} -def var(s, create=True): - r""" - Create a symbolic variable with the name \emph{s}. - - EXAMPLES: - sage: var('xx') - xx - """ - if isinstance(s, SymbolicVariable): - return s - s = str(s) - if ',' in s: - return tuple([var(x.strip()) for x in s.split(',')]) - elif ' ' in s: - return tuple([var(x.strip()) for x in s.split()]) +def var(s): + r''' Create a symbolic variable with the name \emph{s}''' try: - v = _vars[s] - _syms[s] = v + return _vars[s] + except KeyError: + v = SymbolicVariable(s) + _vars[s] = v return v - except KeyError: - if not create: - raise ValueError, "the variable '%s' has not been defined"%var - pass - v = SymbolicVariable(s) - _vars[s] = v - _syms[s] = v - return v - - -######################################################################################### -# Callable functions -######################################################################################### -def is_CallableSymbolicExpressionRing(x): - """ - EXAMPLES: - sage: is_CallableSymbolicExpressionRing(QQ) - False - sage: is_CallableSymbolicExpressionRing(CallableSymbolicExpressionRing((x,y,z))) - True - """ - return isinstance(x, CallableSymbolicExpressionRing_class) - -class CallableSymbolicExpressionRing_class(CommutativeRing): - def __init__(self, args): - self._default_precision = 53 # default precision bits - self._args = args - ParentWithBase.__init__(self, RR) - - def __call__(self, x): - """ - - TESTS: - sage: f(x) = x+1; g(y) = y+1 - sage: f.parent()(g) - x |--> y + 1 - sage: g.parent()(f) - y |--> x + 1 - sage: f(x) = x+2*y; g(y) = y+3*x - sage: f.parent()(g) - x |--> y + 3*x - sage: g.parent()(f) - y |--> 2*y + x - """ - return self._coerce_impl(x) - - def _coerce_impl(self, x): - if isinstance(x, SymbolicExpression): - if isinstance(x, CallableSymbolicExpression): - x = x._expr - return CallableSymbolicExpression(self, x) - return self._coerce_try(x, [SR]) - - def _repr_(self): - return "Callable function ring with arguments %s"%(self._args,) - - def args(self): - return self._args - - def arguments(self): - return self.args() - - def zero_element(self): - try: - return self.__zero_element - except AttributeError: - z = CallableSymbolicExpression(SR.zero_element(), self._args) - self.__zero_element = z - return z - - def _an_element_impl(self): - return self.zero_element() - - -_cfr_cache = {} -def CallableSymbolicExpressionRing(args, check=True): - if check: - if len(args) == 1 and isinstance(args[0], (list, tuple)): - args = args[0] - for arg in args: - if not isinstance(arg, SymbolicVariable): - raise TypeError, "Must construct a function with a tuple (or list) of" \ - +" SymbolicVariables." - args = tuple(args) - if _cfr_cache.has_key(args): - R = _cfr_cache[args]() - if not R is None: - return R - R = CallableSymbolicExpressionRing_class(args) - _cfr_cache[args] = weakref.ref(R) - return R - -def is_CallableSymbolicExpression(x): - """ - Returns true if x is a callable symbolic expression. - - EXAMPLES: - sage: f(x,y) = a + 2*x + 3*y + z - sage: is_CallableSymbolicExpression(f) - True - sage: is_CallableSymbolicExpression(a+2*x) - False - sage: def foo(n): return n^2 - ... - sage: is_CallableSymbolicExpression(foo) - False - """ - return isinstance(x, CallableSymbolicExpression) - -class CallableSymbolicExpression(SymbolicExpression): - r""" - A callable symbolic expression that knows the ordered list of - variables on which it depends. - - EXAMPLES: - sage: f(x,y) = a + 2*x + 3*y + z - sage: f - (x, y) |--> z + 3*y + 2*x + a - sage: f(1,2) - z + a + 8 - """ - def __init__(self, parent, expr): - RingElement.__init__(self, parent) - self._expr = expr - - def variables(self): - """ - EXAMPLES: - sage: g(x) = sin(x) + a - sage: g.variables() - (a, x) - sage: g.args() - (x,) - sage: g(y,x,z) = sin(x) + a - a - sage: g - (y, x, z) |--> sin(x) - sage: g.args() - (y, x, z) - """ - return self._expr.variables() - - def integral(self, x=None, a=None, b=None): - """ - Returns an integral of self. - """ - if a is None: - return SymbolicExpression.integral(self, x, None, None) - # if l. endpoint is None, compute an indefinite integral - else: - if x is None: - x = self.default_variable() - if not isinstance(x, SymbolicVariable): - x = var(repr(x)) - # if we supplied an endpoint, then we want to return a number. - return SR(self._maxima_().integrate(x, a, b)) - - integrate = integral - - def expression(self): - """ - Return the underlying symbolic expression (i.e., forget the - extra map structure). - """ - return self._expr - - def args(self): - return self.parent().args() - - def arguments(self): - return self.args() - - def _maxima_init_(self): - return self._expr._maxima_init_() - - def __float__(self): - return float(self._expr) - - def __complex__(self): - return complex(self._expr) - - def _mpfr_(self, field): - """ - Coerce to a multiprecision real number. - - EXAMPLES: - sage: RealField(100)(SR(10)) - 10.000000000000000000000000000 - """ - return (self._expr)._mpfr_(field) - - def _complex_mpfr_field_(self, field): - return field(self._expr) - - def _complex_double_(self, C): - return C(self._expr) - - def _real_double_(self, R): - return R(self._expr) - - def _real_rqdf_(self, R): - return R(self._expr) - - # TODO: should len(args) == len(vars)? - def __call__(self, *args): - vars = self.args() - dct = dict( (vars[i], args[i]) for i in range(len(args)) ) - return self._expr.substitute(dct) - - def _repr_(self, simplify=True): - args = self.args() - if len(args) == 1: - return "%s |--> %s" % (args[0], self._expr._repr_(simplify=simplify)) - else: - args = ", ".join(map(str, args)) - return "(%s) |--> %s" % (args, self._expr._repr_(simplify=simplify)) - - def _latex_(self): - args = self.args() - if len(args) == 1: - return "%s \\ {\mapsto}\\ %s" % (args[0], - self._expr._latex_()) - else: - vars = ", ".join(map(latex, args)) - # the weird TeX is to workaround an apparent JsMath bug - return "\\left(%s \\right)\\ {\\mapsto}\\ %s" % (args, self._expr._latex_()) - - def _neg_(self): - return CallableSymbolicExpression(self.parent(), -self._expr) - - def __add__(self, right): - """ - EXAMPLES: - sage: f(x,n,y) = x^n + y^m; g(x,n,m,z) = x^n +z^m - sage: f + g - (x, n, m, y, z) |--> z^m + y^m + 2*x^n - sage: g + f - (x, n, m, y, z) |--> z^m + y^m + 2*x^n - """ - if isinstance(right, CallableSymbolicExpression): - if self.parent() is right.parent(): - return self._add_(right) - else: - args = self._unify_args(right) - R = CallableSymbolicExpressionRing(args) - return R(self) + R(right) - else: - return RingElement.__add__(self, right) - - def _add_(self, right): - return CallableSymbolicExpression(self.parent(), self._expr + right._expr) - - def __sub__(self, right): - """ - EXAMPLES: - sage: f(x,n,y) = x^n + y^m; g(x,n,m,z) = x^n +z^m - sage: f - g - (x, n, m, y, z) |--> y^m - z^m - sage: g - f - (x, n, m, y, z) |--> z^m - y^m - """ - if isinstance(right, CallableSymbolicExpression): - if self.parent() is right.parent(): - return self._sub_(right) - else: - args = self._unify_args(right) - R = CallableSymbolicExpressionRing(args) - return R(self) - R(right) - else: - return RingElement.__sub__(self, right) - - def _sub_(self, right): - return CallableSymbolicExpression(self.parent(), self._expr - right._expr) - - def __mul__(self, right): - """ - EXAMPLES: - sage: f(x) = x+2*y; g(y) = y+3*x - sage: f*(2/3) - x |--> 2*(2*y + x)/3 - sage: f*g - (x, y) |--> (y + 3*x)*(2*y + x) - sage: (2/3)*f - x |--> 2*(2*y + x)/3 - - sage: f(x,y,z,a,b) = x+y+z-a-b; f - (x, y, z, a, b) |--> z + y + x - b - a - sage: f * (b*c) - (x, y, z, a, b) |--> b*c*(z + y + x - b - a) - sage: g(x,y,w,t) = x*y*w*t - sage: f*g - (x, y, a, b, t, w, z) |--> t*w*x*y*(z + y + x - b - a) - sage: (f*g)(2,3) - 6*t*w*(z - b - a + 5) - - sage: f(x,n,y) = x^n + y^m; g(x,n,m,z) = x^n +z^m - sage: f * g - (x, n, m, y, z) |--> (y^m + x^n)*(z^m + x^n) - sage: g * f - (x, n, m, y, z) |--> (y^m + x^n)*(z^m + x^n) - """ - if isinstance(right, CallableSymbolicExpression): - if self.parent() is right.parent(): - return self._mul_(right) - else: - args = self._unify_args(right) - R = CallableSymbolicExpressionRing(args) - return R(self)*R(right) - else: - return RingElement.__mul__(self, right) - - def _mul_(self, right): - return CallableSymbolicExpression(self.parent(), self._expr * right._expr) - - def __div__(self, right): - """ - EXAMPLES: - - sage: f(x,n,y) = x^n + y^m; g(x,n,m,z) = x^n +z^m - sage: f / g - (x, n, m, y, z) |--> (y^m + x^n)/(z^m + x^n) - sage: g / f - (x, n, m, y, z) |--> (z^m + x^n)/(y^m + x^n) - """ - if isinstance(right, CallableSymbolicExpression): - if self.parent() is right.parent(): - return self._div_(right) - else: - args = self._unify_args(right) - R = CallableSymbolicExpressionRing(args) - return R(self) / R(right) - else: - return RingElement.__div__(self, right) - - def _div_(self, right): - return CallableSymbolicExpression(self.parent(), self._expr / right._expr) - - def __pow__(self, right): - return CallableSymbolicExpression(self.parent(), self._expr ** right) - - def _unify_args(self, x): - r""" - Takes the variable list from another CallableSymbolicExpression object and - compares it with the current CallableSymbolicExpression object's variable list, - combining them according to the following rules: - - Let \code{a} be \code{self}'s variable list, let \code{b} be - \code{y}'s variable list. - - \begin{enumerate} - - \item If \code{a == b}, then the variable lists are identical, - so return that variable list. - - \item If \code{a} $\neq$ \code{b}, then check if the first $n$ - items in \code{a} are the first $n$ items in \code{b}, - or vice-versa. If so, return a list with these $n$ - items, followed by the remaining items in \code{a} and - \code{b} sorted together in alphabetical order. - - \end{enumerate} - - Note: When used for arithmetic between CallableSymbolicExpressions, - these rules ensure that the set of CallableSymbolicExpressions will have - certain properties. In particular, it ensures that the set is - a \emph{commutative} ring, i.e., the order of the input - variables is the same no matter in which order arithmetic is - done. - - INPUT: - x -- A CallableSymbolicExpression - - OUTPUT: - A tuple of variables. - - EXAMPLES: - sage: f(x, y, z) = sin(x+y+z) - sage: f - (x, y, z) |--> sin(z + y + x) - sage: g(x, y) = y + 2*x - sage: g - (x, y) |--> y + 2*x - sage: f._unify_args(g) - (x, y, z) - sage: g._unify_args(f) - (x, y, z) - - sage: f(x, y, z) = sin(x+y+z) - sage: g(w, t) = cos(w - t) - sage: g - (w, t) |--> cos(w - t) - sage: f._unify_args(g) - (t, w, x, y, z) - - sage: f(x, y, t) = y*(x^2-t) - sage: f - (x, y, t) |--> (x^2 - t)*y - sage: g(x, y, w) = x + y - cos(w) - sage: f._unify_args(g) - (x, y, t, w) - sage: g._unify_args(f) - (x, y, t, w) - sage: f*g - (x, y, t, w) |--> (x^2 - t)*y*(y + x - cos(w)) - - sage: f(x,y, t) = x+y - sage: g(x, y, w) = w + t - sage: f._unify_args(g) - (x, y, t, w) - sage: g._unify_args(f) - (x, y, t, w) - sage: f + g - (x, y, t, w) |--> y + x + w + t - - AUTHORS: - -- Bobby Moretti, thanks to William Stein for the rules - - """ - a = self.args() - b = x.args() - - # Rule #1 - if [str(x) for x in a] == [str(x) for x in b]: - return a - - # Rule #2 - new_list = [] - done = False - i = 0 - while not done and i < min(len(a), len(b)): - if var_cmp(a[i], b[i]) == 0: - new_list.append(a[i]) - i += 1 - else: - done = True - - temp = set([]) - # Sorting remaining variables. - for j in range(i, len(a)): - if not a[j] in temp: - temp.add(a[j]) - - for j in range(i, len(b)): - if not b[j] in temp: - temp.add(b[j]) - - temp = list(temp) - temp.sort(var_cmp) - new_list.extend(temp) - return tuple(new_list) - - -######################################################################################### -# End callable functions -######################################################################################### - -class SymbolicComposition(SymbolicOperation): - r""" - Represents the symbolic composition of $f \circ g$. - """ - def __init__(self, f, g): - """ - INPUT: - f, g -- both must be in the symbolic expression ring. - """ - SymbolicOperation.__init__(self, [f,g]) - - def _recursive_sub(self, kwds): - ops = self._operands - return ops[0](ops[1]._recursive_sub(kwds)) - - def _recursive_sub_over_ring(self, kwds, ring): - ops = self._operands - return ring(ops[0](ops[1]._recursive_sub_over_ring(kwds, ring=ring))) + +_syms = {} + +t = var('t') +x = var('x') +y = var('y') +z = var('z') +w = var('w') + +class Function_sin(PrimitiveFunction): + ''' + The sine function + ''' + def _repr_(self): + return "sin" + + def _latex_(self): + return "\\sin" def _is_atomic(self): return True - def _repr_(self, simplify=True): - if simplify: - if hasattr(self, '_simp'): - return self._simp._repr_(simplify=False) - else: - return self.simplify()._repr_(simplify=False) - ops = self._operands - try: - return ops[0]._repr_evaled_(ops[1]._repr_(simplify=False)) - except AttributeError: - return "%s(%s)"% (ops[0]._repr_(simplify=False), ops[1]._repr_(simplify=False)) - - def _maxima_init_(self): - ops = self._operands - try: - return ops[0]._maxima_init_evaled_(ops[1]._maxima_init_()) - except AttributeError: - return '%s(%s)' % (ops[0]._maxima_init_(), ops[1]._maxima_init_()) - - def _latex_(self): - if not self.is_simplified(): - return self.simplify()._latex_() - ops = self._operands - # certain functions (such as \sqrt) need braces in LaTeX - if (ops[0]).tex_needs_braces(): - return r"%s{ %s }" % ( (ops[0])._latex_(), (ops[1])._latex_()) - # ... while others (such as \cos) don't - return r"%s \left( %s \right)"%((ops[0])._latex_(),(ops[1])._latex_()) - - def _sys_init_(self, system): - ops = self._operands - return '%s(%s)' % (sys_init(ops[0],system), sys_init(ops[1],system)) - - def _mathematica_init_(self): - system = 'mathematica' - ops = self._operands - return '%s[%s]' % (sys_init(ops[0],system).capitalize(), sys_init(ops[1],system)) - - def __float__(self): - f = self._operands[0] - g = self._operands[1] - return float(f._approx_(float(g))) - - def __complex__(self): - f = self._operands[0] - g = self._operands[1] - return complex(f._approx_(float(g))) - - def _mpfr_(self, field): - """ - Coerce to a multiprecision real number. - - EXAMPLES: - sage: RealField(100)(sin(2)+cos(2)) - 0.49315059027853930839845163641 - - sage: RR(sin(pi)) - 0.000000000000000 - - sage: type(RR(sqrt(163)*pi)) - - - sage: RR(coth(pi)) - 1.00374187319732 - sage: RealField(100)(coth(pi)) - 1.0037418731973212882015526912 - sage: RealField(200)(acos(1/10)) - 1.4706289056333368228857985121870581235299087274579233690964 - """ - if not self.is_simplified(): - return self.simplify()._mpfr_(field) - f = self._operands[0] - g = self._operands[1] - x = f(g._mpfr_(field)) - if isinstance(x, SymbolicExpression): - if field.prec() <= 53: - return field(float(x)) - else: - raise TypeError, "precision loss" - else: - return x - - - def _complex_mpfr_field_(self, field): - """ - Coerce to a multiprecision complex number. - - EXAMPLES: - sage: ComplexField(100)(sin(2)+cos(2)+I) - 0.49315059027853930839845163641 + 1.0000000000000000000000000000*I - - """ - if not self.is_simplified(): - return self.simplify()._complex_mpfr_field_(field) - f = self._operands[0] - g = self._operands[1] - x = f(g._complex_mpfr_field_(field)) - if isinstance(x, SymbolicExpression): - if field.prec() <= 53: - return field(complex(x)) - else: - raise TypeError, "precision loss" - else: - return x - - def _complex_double_(self, field): - """ - Coerce to a complex double. - - EXAMPLES: - sage: CDF(sin(2)+cos(2)+I) - 0.493150590279 + 1.0*I - sage: CDF(coth(pi)) - 1.0037418732 - """ - if not self.is_simplified(): - return self.simplify()._complex_double_(field) - f = self._operands[0] - g = self._operands[1] - z = f(g._complex_double_(field)) - if isinstance(z, SymbolicExpression): - return field(complex(z)) - return z - - def _real_double_(self, field): - """ - Coerce to a real double. - - EXAMPLES: - sage: RDF(sin(2)+cos(2)) - 0.493150590279 - """ - if not self.is_simplified(): - return self.simplify()._real_double_(field) - f = self._operands[0] - g = self._operands[1] - z = f(g._real_double_(field)) - if isinstance(z, SymbolicExpression): - return field(float(z)) - return z - - def _real_rqdf_(self, field): - """ - Coerce to a real qdrf. - - EXAMPLES: - - """ - if not self.is_simplified(): - return self.simplify()._real_rqdf_(field) - f = self._operands[0] - g = self._operands[1] - z = f(g._real_rqdf_(field)) - if isinstance(z, SymbolicExpression): - raise TypeError, "precision loss" - else: - return z - -class PrimitiveFunction(SymbolicExpression): - def __init__(self, needs_braces=False): - SymbolicExpression.__init__(self) - self._tex_needs_braces = needs_braces - - def _recursive_sub(self, kwds): - if kwds.has_key(self): - return kwds[self] - return self - - def _recursive_sub_over_ring(self, kwds, ring): - if kwds.has_key(self): - return kwds[self] - return self - - def plot(self, *args, **kwds): - f = self(var('x')) - return SymbolicExpression.plot(f, *args, **kwds) +sin = Function_sin() +_syms['sin'] = sin + +class Function_cos(PrimitiveFunction): + ''' + The cosine function + ''' + def _repr_(self): + return "cos" + + def _latex_(self): + return "\\cos" def _is_atomic(self): return True - - def tex_needs_braces(self): - return self._tex_needs_braces - - def __call__(self, x, *args): - if isinstance(x, float): - return self._approx_(x) - try: - return getattr(x, self._repr_())(*args) - except AttributeError: - return SymbolicComposition(self, SR(x)) - - def _approx_(self, x): # must *always* be called with a float x as input. - s = '%s(%s), numer'%(self._repr_(), float(x)) - return float(maxima.eval(s)) - -_syms = {} - -class Function_erf(PrimitiveFunction): - r""" - The error function, defined as $\text{erf}(x) = - \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt$. - """ - - def _repr_(self, simplify=True): - return "erf" - - def _latex_(self): - return "\\text{erf}" - - def _approx_(self, x): - return float(1 - pari(float(x)).erfc()) - -erf = Function_erf() -_syms['erf'] = erf - -class Function_abs(PrimitiveFunction): - """ - The absolute value function. - - EXAMPLES: - sage: abs(x) - abs(x) - sage: abs(x^2 + y^2) - y^2 + x^2 - sage: abs(-2) - 2 - sage: sqrt(x^2) - abs(x) - sage: abs(sqrt(x)) - sqrt(x) - """ - def _repr_(self, simplify=True): - return "abs" - - def _latex_(self): - return "\\abs" - - def _approx_(self, x): - return float(x.__abs__()) - - def __call__(self, x): # special case - return SymbolicComposition(self, SR(x)) - -abs_symbolic = Function_abs() -_syms['abs'] = abs_symbolic - - - -class Function_ceil(PrimitiveFunction): - """ - The ceiling function. - - EXAMPLES: - sage: a = ceil(2/5 + x) - sage: a - ceil(x + 2/5) - sage: a(4) - 5 - sage: a(4.0) - 5 - sage: ZZ(a(3)) - 4 - sage: a = ceil(x^3 + x + 5/2) - sage: a - ceil(x^3 + x + 1/2) + 2 - sage: a(x=2) - 13 - - sage: ceil(5.4) - 6 - sage: type(ceil(5.4)) - - """ - def _repr_(self, simplify=True): - return "ceil" - - def _latex_(self): - return "\\text{ceil}" - - def _maxima_init_(self): - return "ceiling" - - _approx_ = math.ceil - - def __call__(self, x): - try: - return x.ceil() - except AttributeError: - if isinstance(x, float): - return math.ceil(x) - return SymbolicComposition(self, SR(x)) - -ceil = Function_ceil() -_syms['ceiling'] = ceil # spelled ceiling in maxima - - -class Function_floor(PrimitiveFunction): - """ - The floor function. - - EXAMPLES: - sage: floor(5.4) - 5 - sage: type(floor(5.4)) - - sage: var('x') - x - sage: a = floor(5.4 + x); a - floor(x + 0.4000000000000004) + 5 - sage: a(2) - 7 - """ - def _repr_(self, simplify=True): - return "floor" - - def _latex_(self): - return "\\text{floor}" - - def _maxima_init_(self): - return "floor" - - _approx_ = math.floor - - def __call__(self, x): - try: - return x.floor() - except AttributeError: - if isinstance(x, float): - return math.floor(x) - return SymbolicComposition(self, SR(x)) - -floor = Function_floor() -_syms['floor'] = floor # spelled ceiling in maxima - - -class Function_sin(PrimitiveFunction): - """ - The sine function - """ - def _repr_(self, simplify=True): - return "sin" - - def _latex_(self): - return "\\sin" - - _approx_ = math.sin - - def __call__(self, x): - try: - return x.sin() - except AttributeError: - if isinstance(x, float): - return math.sin(x) - return SymbolicComposition(self, SR(x)) - -sin = Function_sin() -_syms['sin'] = sin - -class Function_cos(PrimitiveFunction): - """ - The cosine function - """ - def _repr_(self, simplify=True): - return "cos" - - def _latex_(self): - return "\\cos" - - _approx_ = math.cos - - def __call__(self, x): - try: - return x.cos() - except AttributeError: - if isinstance(x, float): - return math.cos(x) - return SymbolicComposition(self, SR(x)) - - cos = Function_cos() _syms['cos'] = cos class Function_sec(PrimitiveFunction): - """ + ''' The secant function - - EXAMPLES: - sage: sec(pi/4) - sqrt(2) - sage: RR(sec(pi/4)) - 1.41421356237310 - sage: sec(1/2) - sec(1/2) - sage: sec(0.5) - 1.139493927324549 - """ - def _repr_(self, simplify=True): + ''' + def _repr_(self): return "sec" def _latex_(self): return "\\sec" - - def _approx_(self, x): - return 1/math.cos(x) sec = Function_sec() _syms['sec'] = sec -class Function_tan(PrimitiveFunction): - """ - The tangent function - - EXAMPLES: - sage: tan(pi) - 0 - sage: tan(3.1415) - -0.0000926535900581913 - sage: tan(3.1415/4) - 0.999953674278156 - sage: tan(pi/4) - 1 - sage: tan(1/2) - tan(1/2) - sage: RR(tan(1/2)) - 0.546302489843790 - """ - def _repr_(self, simplify=True): - return "tan" - - def _latex_(self): - return "\\tan" - - def _approx_(self, x): - return math.tan(x) - -tan = Function_tan() -_syms['tan'] = tan - -def atan2(y, x): - return atan(y/x) - -_syms['atan2'] = atan2 - -class Function_asin(PrimitiveFunction): - """ - The arcsine function - - EXAMPLES: - sage: asin(0.5) - 0.523598775598299 - sage: asin(1/2) - pi/6 - sage: asin(1 + I*1.0) - 1.061275061905036*I + 0.6662394324925153 - """ - def _repr_(self, simplify=True): - return "asin" - - def _latex_(self): - return "\\sin^{-1}" - - def _approx_(self, x): - return math.asin(x) - -asin = Function_asin() -_syms['asin'] = asin - -class Function_acos(PrimitiveFunction): - """ - The arccosine function - - EXAMPLES: - sage: acos(0.5) - 1.04719755119660 - sage: acos(1/2) - pi/3 - sage: acos(1 + I*1.0) - 0.9045568943023813 - 1.061275061905036*I - """ - def _repr_(self, simplify=True): - return "acos" - - def _latex_(self): - return "\\cos^{-1}" - - def _approx_(self, x): - return math.acos(x) - -acos = Function_acos() -_syms['acos'] = acos - - -class Function_atan(PrimitiveFunction): - """ - The arctangent function. - - EXAMPLES: - sage: atan(1/2) - atan(1/2) - sage: RDF(atan(1/2)) - 0.463647609001 - sage: atan(1 + I) - atan(I + 1) - """ - def _repr_(self, simplify=True): - return "atan" - - def _latex_(self): - return "\\tan^{-1}" - - def _approx_(self, x): - return math.atan(x) - -atan = Function_atan() -_syms['atan'] = atan - - -####### -# Hyperbolic functions -####### - -#tanh -class Function_tanh(PrimitiveFunction): - """ - The hyperbolic tangent function. - - EXAMPLES: - sage: tanh(pi) - tanh(pi) - sage: tanh(3.1415) - 0.996271386633702 - sage: float(tanh(pi)) # random low-order bits - 0.99627207622074987 - sage: tan(3.1415/4) - 0.999953674278156 - sage: tanh(pi/4) - tanh(pi/4) - sage: RR(tanh(1/2)) - 0.462117157260010 - - sage: CC(tanh(pi + I*e)) - 0.997524731976164 - 0.00279068768100315*I - sage: ComplexField(100)(tanh(pi + I*e)) - 0.99752473197616361034204366446 - 0.0027906876810031453884245163923*I - sage: CDF(tanh(pi + I*e)) - 0.997524731976 - 0.002790687681*I - """ - def _repr_(self, simplify=True): - return "tanh" - - def _latex_(self): - return "\\tanh" - - def _approx_(self, x): - return math.tanh(x) - -tanh = Function_tanh() -_syms['tanh'] = tanh - -#sinh -class Function_sinh(PrimitiveFunction): - """ - The hyperbolic sine function. - - EXAMPLES: - sage: sinh(pi) - sinh(pi) - sage: sinh(3.1415) - 11.5476653707437 - sage: float(sinh(pi)) # random low-order bits - 11.548739357257748 - sage: RR(sinh(pi)) - 11.5487393572577 - """ - def _repr_(self, simplify=True): - return "sinh" - - def _latex_(self): - return "\\sinh" - - def _approx_(self, x): - return math.sinh(x) - -sinh = Function_sinh() -_syms['sinh'] = sinh - -#cosh -class Function_cosh(PrimitiveFunction): - """ - The hyperbolic cosine function. - - EXAMPLES: - sage: cosh(pi) - cosh(pi) - sage: cosh(3.1415) - 11.5908832931176 - sage: float(cosh(pi)) # random low order bits - 11.591953275521519 - sage: RR(cosh(1/2)) - 1.12762596520638 - """ - def _repr_(self, simplify=True): - return "cosh" - - def _latex_(self): - return "\\cosh" - - def _approx_(self, x): - return math.cosh(x) - -cosh = Function_cosh() -_syms['cosh'] = cosh - -#coth -class Function_coth(PrimitiveFunction): - """ - The hyperbolic cotangent function. - - EXAMPLES: - sage: coth(pi) - coth(pi) - sage: coth(3.1415) - 1.00374256795520 - sage: float(coth(pi)) - 1.0037418731973213 - sage: RR(coth(pi)) - 1.00374187319732 - """ - def _repr_(self, simplify=True): - return "coth" - - def _latex_(self): - return "\\coth" - - def _approx_(self, x): - return 1/math.tanh(x) - -coth = Function_coth() -_syms['coth'] = coth - -#sech -class Function_sech(PrimitiveFunction): - """ - The hyperbolic secant function. - - EXAMPLES: - sage: sech(pi) - sech(pi) - sage: sech(3.1415) - 0.0862747018248192 - sage: float(sech(pi)) # random low order bits - 0.086266738334054432 - sage: RR(sech(pi)) - 0.0862667383340544 - """ - def _repr_(self, simplify=True): - return "sech" - - def _latex_(self): - return "\\sech" - - def _approx_(self, x): - return 1/math.cosh(x) - -sech = Function_sech() -_syms['sech'] = sech - - -#csch -class Function_csch(PrimitiveFunction): - """ - The hyperbolic cosecant function. - - EXAMPLES: - sage: csch(pi) - csch(pi) - sage: csch(3.1415) - 0.0865975907592133 - sage: float(csch(pi)) # random low-order bits - 0.086589537530046945 - sage: RR(csch(pi)) - 0.0865895375300470 - """ - def _repr_(self, simplify=True): - return "csch" - - def _latex_(self): - return "\\csch" - - def _approx_(self, x): - return 1/math.sinh(x) - -csch = Function_csch() -_syms['csch'] = csch - -############# -# log -############# - class Function_log(PrimitiveFunction): - """ - The log function. - - EXAMPLES: - sage: log(e^2) - 2 - sage: log(1024, 2) # the following is ugly (for now) - log(1024)/log(2) - sage: log(10, 4) - log(10)/log(4) - - sage: RDF(log(10,2)) - 3.32192809489 - sage: RDF(log(8, 2)) - 3.0 - sage: log(RDF(10)) - 2.30258509299 - sage: log(2.718) - 0.999896315728952 - """ - def __init__(self): - PrimitiveFunction.__init__(self) - - def _repr_(self, simplify=True): + ''' + The log function + ''' + def _repr_(self): return "log" - + def _latex_(self): return "\\log" - _approx_ = math.log - -function_log = Function_log() -ln = function_log - -def log(x, base=None): - if base is None: - try: - return x.log() - except AttributeError: - return function_log(x) - else: - try: - return x.log(base) - except AttributeError: - return function_log(x) / function_log(base) - -##################### -# The polylogarithm -##################### - - -class Function_polylog(PrimitiveFunction): - r""" - The polylog function $\text{Li}_n(z) = \sum_{k=1}^{\infty} z^k / k^n$. - - INPUT: - n -- object - z -- object - - EXAMPLES: - - """ - def __init__(self, n): - PrimitiveFunction.__init__(self) - self._n = n - - def _repr_(self, simplify=True): - return "polylog(%s)"%self._n - - def _repr_evaled_(self, args): - return 'polylog(%s, %s)'%(self._n, args) - - def _maxima_init_(self): - if self._n in [1,2,3]: - return 'li[%s]'%self._n - else: - return 'polylog(%s)'%self._n - - def _maxima_init_evaled_(self, args): - if self._n in [1,2,3]: - return 'li[%s](%s)'%(self._n, args) - else: - return 'polylog(%s, %s)'%(self._n, args) - - def n(self): - return self._n - - def _latex_(self): - return "\\text{Li}" - - def _approx_(self, x): - try: - return float(pari(x).polylog(self._n)) - except PariError: - raise TypeError, 'unable to coerce polylogarithm to float' - - -def polylog(n, z): - r""" - The polylog function $\text{Li}_n(z) = \sum_{k=1}^{\infty} z^k / k^n$. - - EXAMPLES: - sage: polylog(2,1) - pi^2/6 - sage: polylog(2,x^2+1) - polylog(2, x^2 + 1) - sage: polylog(4,0.5) - polylog(4, 0.500000000000000) - sage: float(polylog(4,0.5)) - 0.51747906167389934 - sage: polylog(2,z).taylor(z, 1/2, 3) - (-(6*log(2)^2 - pi^2))/12 + 2*log(2)*(z - 1/2) + (-2*log(2) + 2)*(z - 1/2)^2 + (8*log(2) - 4)*(z - 1/2)^3/3 - """ - return Function_polylog(n)(z) - -_syms['polylog2'] = Function_polylog(2) -_syms['polylog3'] = Function_polylog(3) - -############################## -# square root -############################## - -class Function_sqrt(PrimitiveFunction): - """ - The square root function. This is a symbolic square root. - - EXAMPLES: - sage: sqrt(-1) - I - sage: sqrt(2) - sqrt(2) - sage: sqrt(x^2) - abs(x) - """ - def __init__(self): - PrimitiveFunction.__init__(self, needs_braces=True) - - def _repr_(self, simplify=True): - return "sqrt" - - def _latex_(self): - return "\\sqrt" - - def _do_sqrt(self, x, prec=None, extend=True, all=False): - if prec: - return ComplexField(prec)(x).sqrt(all=all) - z = SymbolicComposition(self, SR(x)) - if all: - return [z, -z] - return z - - def __call__(self, x, *args, **kwds): - """ - - POSSIBLE INPUTS INCLUDE: - x -- a number - prec -- integer (default: None): if None, returns an exact - square root; otherwise returns a numerical square - root if necessary, to the given bits of precision. - extend -- bool (default: True); if True, return a square - root in an extension ring, if necessary. Otherwise, - raise a ValueError if the square is not in the base - ring. - all -- bool (default: False); if True, return all square - roots of self, instead of just one. - """ - if isinstance(x, float): - return math.sqrt(x) - if not isinstance(x, (Integer, Rational)): - try: - return x.sqrt(*args, **kwds) - except AttributeError: - pass - return self._do_sqrt(x, *args, **kwds) - - def _approx_(self, x): - return math.sqrt(x) - -sqrt = Function_sqrt() -_syms['sqrt'] = sqrt - -class Function_exp(PrimitiveFunction): - r""" - The exponential function, $\exp(x) = e^x$. - - EXAMPLES: - sage: exp(-1) - e^-1 - sage: exp(2) - e^2 - sage: exp(x^2 + log(x)) - x*e^x^2 - sage: exp(2.5) - 12.1824939607035 - sage: exp(float(2.5)) # random low order bits - 12.182493960703473 - sage: exp(RDF('2.5')) - 12.1824939607 - """ - def __init__(self): - PrimitiveFunction.__init__(self, needs_braces=True) - - def _repr_(self, simplify=True): - return "exp" - - def _latex_(self): - return "\\exp" - - def __call__(self, x): - # if x is an integer or rational, never call the sqrt method - if isinstance(x, float): - return self._approx_(x) - if not isinstance(x, (Integer, Rational)): - try: - return x.exp() - except AttributeError: - pass - return SymbolicComposition(self, SR(x)) - - - def _approx_(self, x): - return math.exp(x) - -exp = Function_exp() -_syms['exp'] = exp - +log = Function_log() +_syms['log'] = log ####################################################### -# Symbolic functions -####################################################### - -class SymbolicFunction(PrimitiveFunction): - """ - A formal symbolic function. - - EXAMPLES: - sage: f = function('foo') - sage: g = f(x,y,z) - sage: g - foo(x, y, z) - sage: g(x=var('theta')) - foo(theta, y, z) - """ - def __init__(self, name): - PrimitiveFunction.__init__(self, needs_braces=True) - self._name = str(name) - - def __hash__(self): - return hash(self._name) - - def _repr_(self, simplify=True): - return self._name - - - def _is_atomic(self): - return True - - def _latex_(self): - return "{\\rm %s}"%self._name - - def _maxima_init_(self): - return "'%s"%self._name - - def _approx_(self, x): - raise TypeError - - def __call__(self, *args, **kwds): - return SymbolicFunctionEvaluation(self, [SR(x) for x in args]) - - - -class SymbolicFunction_delayed(SymbolicFunction): - def simplify(self): - return self - - def is_simplified(self): - return True - - def _maxima_init_(self): - return "%s"%self._name - - def __call__(self, *args): - return SymbolicFunctionEvaluation_delayed(self, [SR(x) for x in args]) - - - -class SymbolicFunctionEvaluation(SymbolicExpression): - """ - The result of evaluating a formal symbolic function. - - EXAMPLES: - sage: h = function('gfun', x); h - gfun(x) - sage: k = h.integral(x); k - integrate(gfun(x), x) - sage: k(gfun=sin) - -cos(x) - sage: k(gfun=cos) - sin(x) - sage: k.diff(x) - gfun(x) - """ - def __init__(self, f, args=None, kwds=None): - """ - INPUT: - f -- symbolic function - args -- a tuple or list of symbolic expressions, at which - f is formally evaluated. - """ - SymbolicExpression.__init__(self) - self._f = f - if not args is None: - if not isinstance(args, tuple): - args = tuple(args) - self._args = args - self._kwds = kwds - - def _is_atomic(self): - return True - - def arguments(self): - return tuple(self._args) - - def keyword_arguments(self): - return self._kwds - - def _repr_(self, simplify=True): - if simplify: - return self.simplify()._repr_(simplify=False) - else: - args = ', '.join([x._repr_(simplify=simplify) for x in - self._args]) - if not self._kwds is None: - kwds = ', '.join(["%s=%s" %(x, y) for x,y in self._kwds.iteritems()]) - return '%s(%s, %s)' % (self._f._name, args, kwds) - else: - return '%s(%s)' % (self._f._name, args) - - def _latex_(self): - return "{\\rm %s}(%s)"%(self._f._name, ', '.join([x._latex_() for - x in self._args])) - - def _maxima_init_(self): - try: - return self.__maxima_init - except AttributeError: - n = self._f._name - if not (n in ['integrate', 'diff']): - n = "'" + n - s = "%s(%s)"%(n, ', '.join([x._maxima_init_() - for x in self._args])) - self.__maxima_init = s - return s - - def _recursive_sub(self, kwds): - """ - EXAMPLES: - sage: f = function('foo',x); f - foo(x) - sage: f(foo=sin) - sin(x) - sage: f(x+y) - foo(y + x) - sage: a = f(pi) - sage: a.substitute(foo = sin) - 0 - sage: a = f(pi/2) - sage: a.substitute(foo = sin) - 1 - sage: b = f(pi/3) + x + y - sage: b - y + x + foo(pi/3) - sage: b(foo = sin) - y + x + sqrt(3)/2 - sage: b(foo = cos) - y + x + 1/2 - sage: b(foo = cos, x=y) - 2*y + 1/2 - """ - function_sub = False - for x in kwds.keys(): - if repr(x) == self._f._name: - g = kwds[x] - function_sub = True - break - if function_sub: - # Very important to make a copy, since we are mutating a dictionary - # that will get used again by the calling function! - kwds = dict(kwds) - del kwds[x] - - arg = tuple([SR(x._recursive_sub(kwds)) for x in self._args]) - - if function_sub: - return g(*arg) - else: - return self.__class__(self._f, arg) - - def _recursive_sub_over_ring(self, kwds, ring): - raise TypeError, "no way to coerce the formal function to ring." - - def variables(self): - """ - Return the variables appearing in the simplified form of self. - - EXAMPLES: - sage: foo = function('foo') - sage: w = foo(x,y,a,b,z) + t - sage: w - foo(x, y, a, b, z) + t - sage: w.variables() - (a, b, t, x, y, z) - """ - try: - return self.__variables - except AttributeError: - pass - vars = sum([list(op.variables()) for op in self._args], []) - vars.sort(var_cmp) - vars = tuple(vars) - self.__variables = vars - return vars - -class SymbolicFunctionEvaluation_delayed(SymbolicFunctionEvaluation): - def simplify(self): - return self - - def is_simplified(self): - return True - - def __float__(self): - return float(self._maxima_()) - - def __complex__(self): - return complex(self._maxima_()) - - def _real_double_(self, R): - return R(float(self)) - - def _real_rqdf_(self, R): - raise TypeError - - def _complex_double_(self, C): - return C(float(self)) - - def _mpfr_(self, field): - if field.prec() <= 53: - return field(float(self)) - raise TypeError - - def _complex_mpfr_field_(self, field): - if field.prec() <= 53: - return field(float(self)) - raise TypeError - - def _maxima_init_(self): - try: - return self.__maxima_init - except AttributeError: - n = self._f._name - s = "%s(%s)"%(n, ', '.join([x._maxima_init_() - for x in self._args])) - self.__maxima_init = s - return s - - - -_functions = {} -def function(s, *args): - """ - Create a formal symbolic function with the name \emph{s}. - - EXAMPLES: - sage: f = function('cr', a) - sage: g = f.diff(a).integral(b) - sage: g - diff(cr(a), a, 1)*b - sage: g(cr=cos) - -sin(a)*b - sage: g(cr=sin(x) + cos(x)) - (cos(a) - sin(a))*b - - Basic arithmetic: - sage: x = var('x') - sage: h = function('f',x) - sage: 2*f - 2*f - sage: 2*h - 2*f(x) - """ - if len(args) > 0: - return function(s)(*args) - if isinstance(s, SymbolicFunction): - return s - s = str(s) - if ',' in s: - return tuple([function(x.strip()) for x in s.split(',')]) - elif ' ' in s: - return tuple([function(x.strip()) for x in s.split()]) - try: - f = _functions[s] - _syms[s] = f - return f - except KeyError: - pass - v = SymbolicFunction(s) - _functions[s] = v - _syms[s] = v - return v - -####################################################### -# This is buggy as is: -# sage: a = lim(exp(x^2)*(1-erf(x)), x=infinity) -# sage: maxima(a) -# 'limit(%e^x^2-%e^x^2*erf(x)) -# ??? -- what happened to the infinity in the above. -# With the old version one gets -# sage: maxima(a) -# 'limit(%e^x^2-%e^x^2*erf(x),x,inf) -# Which is right, since: -# (%i3) limit(%e^x^2-%e^x^2*erf(x),x,inf); -# (%o3) 'limit(%e^x^2-%e^x^2*erf(x),x,inf) -#a -def dummy_limit(*args): - s = str(args[1]) - return SymbolicFunctionEvaluation(function('limit'), args=(args[0],), kwds={s: args[2]}) - -######################################i################ - - - - -####################################################### - -symtable = {'%pi':'pi', '%e': 'e', '%i':'I', '%gamma':'euler_gamma', - 'li[2]':'polylog2', 'li[3]':'polylog3'} - -from sage.rings.infinity import infinity, minus_infinity - -_syms['inf'] = infinity -_syms['minf'] = minus_infinity - -from sage.misc.multireplace import multiple_replace - -maxima_tick = re.compile("'[a-z|A-Z|0-9|_]*") - -maxima_qp = re.compile("\?\%[a-z|A-Z|0-9|_]*") # e.g., ?%jacobi_cd - -maxima_var = re.compile("\%[a-z|A-Z|0-9|_]*") # e.g., ?%jacobi_cd - -def symbolic_expression_from_maxima_string(x, equals_sub=False, maxima=maxima): - syms = dict(_syms) - - if len(x) == 0: - raise RuntimeError, "invalid symbolic expression -- ''" - maxima.set('_tmp_',x) - - # This is inefficient since it so rarely is needed: - #r = maxima._eval_line('listofvars(_tmp_);')[1:-1] - - s = maxima._eval_line('_tmp_;') - - formal_functions = maxima_tick.findall(s) - if len(formal_functions) > 0: - for X in formal_functions: - syms[X[1:]] = function(X[1:]) - # You might think there is a potential very subtle bug if 'foo is in a string literal -- - # but string literals should *never* ever be part of a symbolic expression. - s = s.replace("'","") - - delayed_functions = maxima_qp.findall(s) - if len(delayed_functions) > 0: - for X in delayed_functions: - syms[X[2:]] = SymbolicFunction_delayed(X[2:]) - s = s.replace("?%","") - - s = multiple_replace(symtable, s) - s = s.replace("%","") - - if equals_sub: - s = s.replace('=','==') - - # have to do this here, otherwise maxima_tick catches it - syms['limit'] = dummy_limit - - global is_simplified - try: - # use a global flag so all expressions obtained via - # evaluation of maxima code are assumed pre-simplified - is_simplified = True - last_msg = '' - while True: - try: - w = sage_eval(s, syms) - except NameError, msg: - if msg == last_msg: - raise NameError, msg - msg = str(msg) - last_msg = msg - i = msg.find("'") - j = msg.rfind("'") - nm = msg[i+1:j] - syms[nm] = var(nm) - else: - break - if isinstance(w, (list, tuple)): - return w - else: - x = SR(w) - return x - except SyntaxError: - raise TypeError, "unable to make sense of Maxima expression '%s' in SAGE"%s - finally: - is_simplified = False +symtable = {'%pi':'_Pi_', '%e': '_E_', '%i':'_I_'} +import sage.functions.constants as c +_syms['_Pi_'] = SER(c.Pi) +_syms['_E_'] = SER(c.E) +_syms['_I_'] = SER(CC.gen(0)) # change when we create a symbolic I. + +def symbolic_expression_from_maxima_string(x): + global _syms + maxima.eval('listdummyvars: false') + maxima.eval('_tmp_: %s'%x) + r = maxima.eval('listofvars(_tmp_)')[1:-1] + if len(r) > 0: + # Now r is a list of all the indeterminate variables that + # appear in the expression x. + v = r.split(',') + for a in v: + _syms[a] = var(a) + s = maxima.eval('_tmp_') + for x, y in symtable.iteritems(): + s = s.replace(x, y) + #print s + #print _syms + return SymbolicExpressionRing(sage_eval(s, _syms)) def symbolic_expression_from_maxima_element(x): return symbolic_expression_from_maxima_string(x.name()) -def evaled_symbolic_expression_from_maxima_string(x): - return symbolic_expression_from_maxima_string(maxima.eval(x)) - - -# External access used by restore -syms_cur = _syms -syms_default = dict(syms_cur) Index: age/calculus/equations.py =================================================================== --- sage/calculus/equations.py (revision 4316) +++ (revision ) @@ -1,388 +1,0 @@ -r""" -Symbolic Equations and Inequalities. - -SAGE can solving symbolic equations and expressing inequalities. -For example, we derive the quadratic formula as follows: - - sage: qe = (a*x^2 + b*x + c == 0) - sage: print qe - 2 - a x + b x + c == 0 - sage: print solve(qe, x) - [ - 2 - - sqrt(b - 4 a c) - b - x == ---------------------- - 2 a, - 2 - sqrt(b - 4 a c) - b - x == -------------------- - 2 a - ] - -AUTHORS: - -- Bobby Moretti: initial version - -- William Stein: second version - -EXAMPLES: - sage: f = x^2 + y^2 == 1 - sage: f.solve(x) - [x == (-sqrt(1 - y^2)), x == sqrt(1 - y^2)] - - sage: f = x^5 + a - sage: solve(f==0,x) - [x == -e^(2*I*pi/5)*a^(1/5), x == -e^(4*I*pi/5)*a^(1/5), x == -e^(-(4*I*pi/5))*a^(1/5), x == -e^(-(2*I*pi/5))*a^(1/5), x == (-a^(1/5))] -""" - -_assumptions = [] - -from sage.structure.sage_object import SageObject -from sage.structure.sequence import Sequence -from sage.misc.sage_eval import sage_eval - -from calculus import maxima - -import operator - -symbols = {operator.lt:' < ', operator.le:' <= ', operator.eq:' == ', - operator.ne:' != ', - operator.ge:' >= ', operator.gt:' > '} - -maxima_symbols = dict(symbols) -maxima_symbols[operator.eq] = '=' -maxima_symbols[operator.ne] = '#' - - -latex_symbols = {operator.lt:' < ', operator.le:' \\leq ', operator.eq:' = ', - operator.ne:' \\neq ', - operator.ge:' \\geq ', operator.gt:' > '} - -comparisons = {operator.lt:set([-1]), operator.le:set([-1,0]), operator.eq:set([0]), - operator.ne:set([-1,1]), operator.ge:set([0,1]), operator.gt:set([1])} - -def var_cmp(x,y): - return cmp(str(x), str(y)) - -def paren(x): - if x._is_atomic(): - return repr(x) - else: - return '(%s)'%repr(x) - -class SymbolicEquation(SageObject): - def __init__(self, left, right, op): - self._left = left - self._right = right - self._op = op - - def __call__(self, *args, **argv): - return self._op(self._left(*args, **argv), self._right(*args,**argv)) - - # The maxima one is special: - def _maxima_(self, session=None): - if session is None: - return SageObject._maxima_(self, sage.calculus.calculus.maxima) - else: - return SageObject._maxima_(self, session) - - def substitute(self, *args, **kwds): - return self.__call__(*args, **kwds) - - subs = substitute - - def __cmp__(self, right): - """ - EXAMPLES: - sage: (x>0) == (x>0) - True - sage: (x>0) == (x>1) - False - sage: (x>0) != (x>1) - True - """ - if not isinstance(right, SymbolicEquation): - return cmp(type(self), type(right)) - c = cmp(self._op, right._op) - if c: return c - i = bool(self._left == right._left) - if not i: return -1 - i = bool(self._right == right._right) - if not i: return 1 - return 0 - - def _repr_(self): - return "%s%s%s" %(paren(self._left), symbols[self._op], paren(self._right)) - - def variables(self): - """ - EXAMPLES: - sage: f = (x+y+w) == (x^2 - y^2 - z^3); f - (y + x + w) == (-z^3 - y^2 + x^2) - sage: f.variables() - [w, x, y, z] - """ - try: - return self.__variables - except AttributeError: - v = list(set(list(self._left.variables()) + list(self._right.variables()))) - v.sort(var_cmp) - self.__variables = v - return v - - def operator(self): - return self._op - - def left(self): - return self._left - lhs = left - left_hand_side = left - - def right(self): - return self._right - rhs = right - right_hand_side = right - - def __str__(self): - """ - EXAMPLES: - sage: f = (x^2 - x == 0) - sage: f - (x^2 - x) == 0 - sage: print f - 2 - x - x == 0 - """ - s = self._maxima_().display2d(onscreen=False) - s = s.replace('%pi','pi').replace('%i',' I').replace('%e', ' e').replace(' = ',' == ') - return s - - def _latex_(self): - return "%s %s %s" %(self._left._latex_(), latex_symbols[self._op], - self._right._latex_()) - - # this is an excellent idea by Robert Bradshaw - #def __nonzero__(self): - # result = self._left.__cmp__(self._right) - # return result in comparisons[self._op] - def __nonzero__(self): - """ - Return True if this equality is definitely true. Return False - if it is false or the algorithm for testing equality is - inconclusive. - """ - m = self._maxima_() - try: - s = m.parent()._eval_line('is (%s)'%m.name()) - except TypeError, msg: - #raise ValueError, "unable to evaluate the predicate '%s'"%repr(self) - return cmp(self._left._maxima_() , self._right._maxima_()) == 0 - return s == 'true' - - def _maxima_init_(self, maxima=maxima, assume=False): - l = self._left._maxima_init_() - r = self._right._maxima_init_() - if assume and self._op == operator.eq: - return 'equal(%s, %s)'%(l, r) - return '(%s)%s(%s)' % (l, maxima_symbols[self._op], r) - - def assume(self): - if not self in _assumptions: - m = self._maxima_init_(assume=True) - maxima.assume(m) - _assumptions.append(self) - - def forget(self): - """ - Forget the given constraint. - """ - m = self._maxima_() - m.parent().forget(m) - try: - _assumptions.remove(self) - except ValueError: - pass - - def solve(self, x=None, multiplicities=False): - """ - Symbolically solve for the given variable. - - WARNING: In many cases, only one solution is computed. - - INPUT: - x -- a SymbolicVariable object (if not given, the first in the expression is used) - multiplicities -- (default: False) if True, also returns the multiplicities - of each solution, in order. - - OUTPUT: - A list of SymbolicEquations with the variable to solve for on the - left hand side. - - EXAMPLES: - sage: S = solve(x^3 - 1 == 0, x) - sage: S - [x == ((sqrt(3)*I - 1)/2), x == ((-sqrt(3)*I - 1)/2), x == 1] - sage: S[0] - x == ((sqrt(3)*I - 1)/2) - sage: S[0].right() - (sqrt(3)*I - 1)/2 - - We illustrate finding multiplicities of solutions: - sage: f = (x-1)^5*(x^2+1) - sage: solve(f == 0, x) - [x == -1*I, x == I, x == 1] - sage: solve(f == 0, x, multiplicities=True) - ([x == -1*I, x == I, x == 1], [1, 1, 5]) - sage: solve(g == 0, x) - [] - """ - if not self._op is operator.eq: - raise ValueError, "solving only implemented for equalities" - if x is None: - v = self.variables() - if len(v) == 0: - if multiplicities: - return [], [] - else: - return [] - x = v[0] - - m = self._maxima_() - P = m.parent() - s = m.solve(x).str() - - X = string_to_list_of_solutions(s) - if multiplicities: - if len(X) == 0: - return X, [] - else: - return X, sage_eval(P.get('multiplicities')) - else: - return X - - - -def assume(*args): - """ - Make the given assumptions. - - INPUT: - *args -- assumptions - - EXAMPLES: - sage: assume(x > 0) - sage: bool(sqrt(x^2) == x) - True - sage: forget() - sage: bool(sqrt(x^2) == x) - False - """ - for x in args: - if isinstance(x, (tuple, list)): - for y in x: - assume(y) - else: - try: - x.assume() - except KeyError: - raise TypeError, "assume not defined for objects of type '%s'"%type(x) - -def forget(*args): - """ - Forget the given assumption, or call with no arguments to forget - all assumptions. - - Here an assumption is some sort of symbolic constraint. - - INPUT: - *args -- assumptions (default: forget all assumptions) - - EXAMPLES: - We define and forget multiple assumptions: - sage: assume(x>0, y>0, z == 1, y>0) - sage: assumptions() - [x > 0, y > 0, z == 1] - sage: forget(x>0, z==1) - sage: assumptions() - [y > 0] - """ - if len(args) == 0: - forget_all() - return - for x in args: - if isinstance(x, (tuple, list)): - for y in x: - assume(y) - else: - try: - x.forget() - except KeyError: - raise TypeError, "forget not defined for objects of type '%s'"%type(x) - -def assumptions(): - """ - List all current symbolic assumptions. - - EXAMPLES: - sage: forget() - sage: assume(x^2+y^2 > 0) - sage: assumptions() - [(y^2 + x^2) > 0] - sage: forget(x^2+y^2 > 0) - sage: assumptions() - [] - sage: assume(x > y) - sage: assume(z > w) - sage: assumptions() - [x > y, z > w] - sage: forget() - sage: assumptions() - [] - """ - return list(_assumptions) - -def forget_all(): - global _assumptions - if len(_assumptions) == 0: - return - maxima._eval_line('forget(facts());') - #maxima._eval_line('forget([%s]);'%(','.join([x._maxima_init_() for x in _assumptions]))) - _assumptions = [] - -def solve(f, *args, **kwds): - """ - Algebraically solve an equation of system of equations for given variables. - - INPUT: - f -- equation or system of equations (given by a list or tuple) - *args -- variables to solve for. - - EXAMPLES: - sage: solve([x+y==6, x-y==4], x, y) - [[x == 5, y == 1]] - sage: solve([x^2+y^2 == 1, y^2 == x^3 + x + 1], x, y) - [[x == ((-sqrt(3)*I - 1)/2), y == ((-sqrt(3 - sqrt(3)*I))/sqrt(2))], - [x == ((-sqrt(3)*I - 1)/2), y == (sqrt(3 - sqrt(3)*I)/sqrt(2))], - [x == ((sqrt(3)*I - 1)/2), y == ((-sqrt(sqrt(3)*I + 3))/sqrt(2))], - [x == ((sqrt(3)*I - 1)/2), y == (sqrt(sqrt(3)*I + 3)/sqrt(2))], - [x == 0, y == -1], - [x == 0, y == 1]] - """ - if isinstance(f, (list, tuple)): - m = maxima(list(f)) - try: - s = m.solve(args) - except: - raise ValueError, "Unable to solve %s for %s"%(f, args) - a = repr(s) - return string_to_list_of_solutions(a) - else: - return f.solve(*args, **kwds) - -import sage.categories.all -objs = sage.categories.all.Objects() - -def string_to_list_of_solutions(s): - from sage.calculus.calculus import symbolic_expression_from_maxima_string - v = symbolic_expression_from_maxima_string(s, equals_sub=True) - return Sequence(v, universe=objs) - Index: sage/calculus/functional.py =================================================================== --- sage/calculus/functional.py (revision 4294) +++ sage/calculus/functional.py (revision 2327) @@ -1,326 +1,10 @@ -""" -Functional notation support for common calculus methods. -""" +from calculus import SER, SymbolicExpression -from calculus import SR, SymbolicExpression, CallableSymbolicExpression +def diff(f, *args): + if not isinstance(f, SymbolicExpression): + f = SER(f) + return f.derivative(*args) -def simplify(f): - """ - Simplify the expression f. - """ - try: - return f.simplify() - except AttributeError: - return f +derivative = diff -def derivative(f, *args, **kwds): - """ - The derivative of f. - - EXAMPLES: - We differentiate a callable symbolic function: - sage: f(x,y) = x*y + sin(x^2) + e^(-x) - sage: f - (x, y) |--> x*y + sin(x^2) + e^(-x) - sage: derivative(f, x) - (x, y) |--> y + 2*x*cos(x^2) - e^(-x) - sage: derivative(f, y) - (x, y) |--> x - - We differentiate a polynomial: - sage: t = polygen(QQ, 't') - sage: f = (1-t)^5; f - -t^5 + 5*t^4 - 10*t^3 + 10*t^2 - 5*t + 1 - sage: derivative(f) - -5*t^4 + 20*t^3 - 30*t^2 + 20*t - 5 - - We differentiate a symbolic expression: - sage: f = exp(sin(a - x^2))/x - sage: diff(f, x) - -2*cos(x^2 - a)*e^(-sin(x^2 - a)) - (e^(-sin(x^2 - a))/x^2) - sage: diff(f, a) - cos(x^2 - a)*e^(-sin(x^2 - a))/x - """ - try: - return f.derivative(*args, **kwds) - except AttributeError: - pass - if not isinstance(f, SymbolicExpression): - f = SR(f) - return f.derivative(*args, **kwds) - -diff = derivative -differentiate = derivative - -def integral(f, *args, **kwds): - """ - The integral of f. - - EXAMPLES: - sage: integral(sin(x), x) - -cos(x) - sage: integral(sin(x)^2, x, pi, 123*pi/2) - 121*pi/4 - sage: integral( sin(x), x, 0, pi) - 2 - - We integrate a symbolic function: - sage: f(x,y) = x*y/z + sin(z) - sage: integral(f, z) - (x, y) |--> x*y*log(z) - cos(z) - - sage: assume(b-a>0) - sage: integral( sin(x), x, a, b) - cos(a) - cos(b) - sage: forget() - - sage: print integral(x/(x^3-1), x) - 2 x + 1 - 2 atan(-------) - log(x + x + 1) sqrt(3) log(x - 1) - - --------------- + ------------- + ---------- - 6 sqrt(3) 3 - - sage: print integral( exp(-x^2), x ) - sqrt( pi) erf(x) - ---------------- - 2 - - You can have SAGE calculate multiple integrals. For example, - consider the function $exp(y^2)$ on the region between the lines - $x=y$, $x=1$, and $y=0$. We find the value of the integral on - this region using the command: - sage: area = integral(integral(exp(y^2),x,0,y),y,0,1); area - e/2 - 1/2 - sage: float(area) - 0.85914091422952255 - - We compute the line integral of sin(x) along the arc of the curve - $x=y^4$ from $(1,-1)$ to $(1,1)$: - sage: (x,y) = (t^4,t) - sage: (dx,dy) = (diff(x,t), diff(y,t)) - sage: integral(sin(x)*dx, t,-1, 1) - 0 - sage: restore('x,y') # restore the symbolic variables x and y - - SAGE is unable to do anything with the following integral: - sage: print integral( exp(-x^2)*log(x), x ) - / 2 - [ - x - I e log(x) dx - ] - / - - SAGE does not know how to compute this integral either. - sage: print integral( exp(-x^2)*ln(x), x, 0, oo) - inf - / 2 - [ - x - I e log(x) dx - ] - / - 0 - - This definite integral is easy: - sage: integral( ln(x)/x, x, 1, 2) - log(2)^2/2 - - SAGE can't do this elliptic integral (yet): - sage: integral(1/sqrt(2*t^4 - 3*t^2 - 2), t, 2, 3) - integrate(1/(sqrt(2*t^4 - 3*t^2 - 2)), t, 2, 3) - - A double integral: - sage: integral(integral(x*y^2, x, 0, y), y, -2, 2) - 32/5 - - This illustrates using assumptions: - sage: integral(abs(x), x, 0, 5) - 25/2 - sage: integral(abs(x), x, 0, a) - integrate(abs(x), x, 0, a) - sage: assume(a>0) - sage: integral(abs(x), x, 0, a) - a^2/2 - sage: forget() # forget the assumptions. - - We integrate and differentiate a huge mess: - sage: f = (x^2-1+3*(1+x^2)^(1/3))/(1+x^2)^(2/3)*x/(x^2+2)^2 - sage: g = integral(f, x) - sage: h = f - diff(g, x) - - Numerically h is 0, but the symbolic equality checker - unfortunately can't tell for sure: - sage: [float(h(i)) for i in range(5)] # random low-order bits - [0.0, -1.1102230246251565e-16, -8.3266726846886741e-17, -4.163336342344337e-17, -6.9388939039072284e-17] - sage: bool(h == 0) - False - """ - try: - return f.integral(*args, **kwds) - except AttributeError: - pass - if not isinstance(f, SymbolicExpression): - f = SR(f) - return f.integral(*args, **kwds) - -integrate = integral - -def limit(f, dir=None, **argv): - r""" - Return the limit as the variable v approaches a from the - given direction. - - \begin{verbatim} - limit(expr, x = a) - limit(expr, x = a, dir='above') - \end{verbatim} - - INPUT: - dir -- (default: None); dir may have the value `plus' (or 'above') - for a limit from above, `minus' (or 'below') for a limit from - below, or may be omitted (implying a two-sided - limit is to be computed). - **argv -- 1 named parameter - - ALIAS: You can also use lim instead of limit. - - EXAMPLES: - sage: limit(sin(x)/x, x=0) - 1 - sage: limit(exp(x), x=oo) - +Infinity - sage: lim(exp(x), x=-oo) - 0 - sage: lim(1/x, x=0) - und - - SAGE does not know how to do this limit (which is 0), - so it returns it unevaluated: - sage: lim(exp(x^2)*(1-erf(x)), x=infinity) - limit(e^x^2 - e^x^2*erf(x), x=+Infinity) - """ - if not isinstance(f, SymbolicExpression): - f = SR(f) - return f.limit(dir=dir, **argv) - -lim = limit - -def taylor(f, v, a, n): - """ - Expands self in a truncated Taylor or Laurent series in the - variable v around the point a, containing terms through $(x - a)^n$. - - INPUT: - v -- variable - a -- number - n -- integer - - EXAMPLES: - sage: taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6) - 1 - (k^2*x^2/2) - ((3*k^4 - 4*k^2)*x^4/24) - ((45*k^6 - 60*k^4 + 16*k^2)*x^6/720) - sage: taylor ((x + 1)^n, x, 0, 4) - 1 + n*x + (n^2 - n)*x^2/2 + (n^3 - 3*n^2 + 2*n)*x^3/6 + (n^4 - 6*n^3 + 11*n^2 - 6*n)*x^4/24 - - """ - if not isinstance(f, SymbolicExpression): - f = SR(f) - return f.taylor(v=v,a=a,n=n) - - -def expand(x, *args, **kwds): - """ - EXAMPLES: - sage: a = (1+I)*(2-sqrt(3)*I); a - (I + 1)*(2 - sqrt(3)*I) - sage: expand(a) - -sqrt(3)*I + 2*I + sqrt(3) + 2 - sage: a = (x-1)*(x^2 - 1); a - (x - 1)*(x^2 - 1) - sage: expand(a) - x^3 - x^2 - x + 1 - - You can also use expand on polynomial, integer, and other - factorizations: - sage: x = polygen(ZZ) - sage: F = factor(x^12 - 1); F - (x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + 1) * (x^2 + x + 1) * (x^4 - x^2 + 1) - sage: expand(F) - x^12 - 1 - sage: F.expand() - x^12 - 1 - sage: F = factor(2007); F - 3^2 * 223 - sage: expand(F) - 2007 - - Note: If you want to compute the expanded form of a polynomial - arithmetic operation quickly and the coefficients of the polynomial - all lie in some ring, e.g., the integers, it is vastly faster to - create a polynomial ring and do the arithmetic there. - - sage: x = polygen(ZZ) # polynomial over a given base ring. - sage: f = sum(x^n for n in range(5)) - sage: f*f # much faster, even if the degree is huge - x^8 + 2*x^7 + 3*x^6 + 4*x^5 + 5*x^4 + 4*x^3 + 3*x^2 + 2*x + 1 - - """ - try: - return x.expand(*args, **kwds) - except AttributeError: - return x - - -def laplace(f, t, s): - r""" - Attempts to compute and return the Laplace transform of self - with respect to the variable t and transform parameter s. If - Laplace cannot find a solution, a delayed function is returned. - - The function that is returned maybe be viewed as a function of s. - - EXAMPLES: - sage: f = exp (2*t + a) * sin(t) * t; f - t*e^(2*t + a)*sin(t) - sage: L = laplace(f, t, s); L - e^a*(2*s - 4)/((s^2 - 4*s + 5)^2) - sage: inverse_laplace(L, s, t) - t*e^(2*t + a)*sin(t) - - Unable to compute solution: - sage: laplace(1/s, s, t) - laplace(1/s, s, t) - """ - if not isinstance(f, SymbolicExpression): - f = SR(f) - return f.laplace(t,s) - -def inverse_laplace(f, t, s): - r""" - Attempts to compute and return the inverse Laplace transform of - self with respect to the variable t and transform parameter s. If - Laplace cannot find a solution, a delayed function is returned, which - is called \code{ilt}. - - EXAMPLES: - sage: f = t*cos(t) - sage: L = laplace(f, t, s); L - 2*s^2/(s^2 + 1)^2 - (1/(s^2 + 1)) - sage: inverse_laplace(L, s, t) - t*cos(t) - sage: print inverse_laplace(1/(s^3+1), s, t) - sqrt(3) t sqrt(3) t - sin(---------) cos(---------) - t - t/2 2 2 e - e (-------------- - --------------) + ----- - sqrt(3) 3 3 - - No explicit inverse Laplace transform, so one is returned formally as a function ilt. - sage: inverse_laplace(cos(s), s, t) - ilt(cos(s), s, t) - """ - if not isinstance(f, SymbolicExpression): - f = SR(f) - return f.inverse_laplace(t,s) - - Index: age/calculus/notes.txt =================================================================== --- sage/calculus/notes.txt (revision 4057) +++ (revision ) @@ -1,43 +1,0 @@ -On 4/22/07, Ondrej Certik wrote: -> As to the extensibility - I think it would be quite difficult to -> extend for example Maxima's limits facility (there are some limits -> that Maxima cannot do, but SymPy can), or Maxima's differential -> equations solver module. Either it would have to be done in LISP, or -> rewrite the whole module to python, neither of which I find easy. Or -> is there a better approach? - -I had in mind the following iterative approach: - - (1) Implement the basic limit formulas for products, sums, quotients, etc., - in Python. - (2) This reduces computing limits of symbolic expressions to computing - the limits of the leaves in the tree, i.e., symbolic variables, constants, and - primitive functions (like sinh, exp, log, erf). - (3) Limits of symbolic variables and constants are trivial. - (4) For some special functions one writes an optional method _limit_ - that computes the limit of that special function at a point from a given - direction. The default _limit_ method in the base class computes the - limit using maxima. So for each function for which you want better - speed or a different behavior from maxima, you just fill in the _limit_ method. - -Exactly the steps above would also work for symbolic differentiation. -(Integration is a completely different story.) - -Ondrej doesn't this will work (I totally disagree): - -"I don't think that would work for limits (it should work for -differentiation though) except some simple cases. As an example, let's -take this limit, that Maxima cannot do: - -limit((5**x+3**x)**(1/x), x, infty) - -(if you type this in SymPy, you will get 5, because we use the Gruntz -algorithm, as described here: -http://sympy.googlecode.com/svn/trunk/doc/gruntz.pdf). The only thing -that occurs to me how to fix maxima is to match the expression and -return 5 immediatelly, or implement the whole algorithm from scratch, -but then you will just write the same code as we did for SymPy. - -Integration is quite easy - just match the expression and return a -table integral. The general Risch algorithm is difficult, but I think -Axiom can do it." Index: age/calculus/predefined.py =================================================================== --- sage/calculus/predefined.py (revision 4270) +++ (revision ) @@ -1,52 +1,0 @@ -from calculus import var - -a = var('a') -b = var('b') -c = var('c') -d = var('d') -f = var('f') -g = var('g') -h = var('h') -j = var('j') -k = var('k') -l = var('l') -m = var('m') -n = var('n') -o = var('o') -p = var('p') -q = var('q') -r = var('r') -s = var('s') -t = var('t') -u = var('u') -v = var('v') -w = var('w') -x = var('x') -y = var('y') -z = var('z') -A = var('A') -B = var('B') -C = var('C') -D = var('D') -E = var('E') -F = var('F') -G = var('G') -H = var('H') -J = var('J') -K = var('K') -L = var('L') -M = var('M') -N = var('N') -O = var('O') -P = var('P') -Q = var('Q') -R = var('R') -S = var('S') -T = var('T') -U = var('U') -V = var('V') -W = var('W') -X = var('X') -Y = var('Y') -Z = var('Z') - Index: age/calculus/tests.py =================================================================== --- sage/calculus/tests.py (revision 4359) +++ (revision ) @@ -1,196 +1,0 @@ -""" -Further Examples. - -Compute the Christoffel symbol. - - sage: var('r theta phi') - (r, theta, phi) - sage: m = matrix(SR, [[(1-1/r),0,0,0],[0,-(1-1/r)^(-1),0,0],[0,0,-r^2,0],[0,0,0,-r^2*(sin(theta))^2]]) - sage: print m - [ 1 - (1/r) 0 0 0] - [ 0 -1/(1 - (1/r)) 0 0] - [ 0 0 -r^2 0] - [ 0 0 0 -r^2*sin(theta)^2] - - sage: def christoffel(i,j,k,vars,g): - ... s = 0 - ... ginv = g^(-1) - ... for l in range(g.nrows()): - ... s = s + (1/2)*ginv[k,l]*(g[j,l].diff(vars[i])+g[i,l].diff(vars[j])-g[i,j].diff(vars[l])) - ... return s - - sage: christoffel(3,3,2, [t,r,theta,phi], m) - -cos(theta)*sin(theta) - sage: X = christoffel(1,1,1,[t,r,theta,phi],m) - sage: print X - 1 - - - 1 - r - ------------- - 1 2 2 - 2 (1 - -) r - r - sage: print X.rational_simplify() - 1 - - ---------- - 2 - 2 r - 2 r - -Some basic things: - - sage: f(x,y) = x^3 + sinh(1/y) - sage: f - (x, y) |--> sinh(1/y) + x^3 - sage: f^3 - (x, y) |--> (sinh(1/y) + x^3)^3 - sage: (f^3).expand() - (x, y) |--> sinh(1/y)^3 + 3*x^3*sinh(1/y)^2 + 3*x^6*sinh(1/y) + x^9 - -A polynomial over a symbolic base ring: - sage: R = SR[x] - sage: f = R([1/sqrt(2), 1/(4*sqrt(2))]) - sage: f - 1/(4*sqrt(2))*x + 1/sqrt(2) - sage: -f - (-1/(4*sqrt(2)))*x + -1/sqrt(2) - sage: (-f).degree() - 1 - -Something that was a printing bug. This tests that we print -the simplified version using ASCII art: - sage: A = exp(I*pi/5) - sage: print A*A*A*A*A*A*A*A*A*A - 1 - -We check a statement made at the beginning of Friedlander and Joshi's book -on Distributions: - sage: f = sin(x^2) - sage: g = cos(x) + x^3 - sage: u = f(x+t) + g(x-t) - sage: u - sin((x + t)^2) + cos(x - t) + (x - t)^3 - sage: u.diff(t,2) - u.diff(x,2) - 0 - -Restoring variables after they have been turned into functions: - sage: x = function('x') - sage: sin(x).variables() - () - sage: restore('x') - sage: sin(x).variables() - (x,) - -MATHEMATICA: -Some examples of integration and differentiation taken from some -Mathematica docs: - sage: diff(x^n, x) - n*x^(n - 1) - sage: diff(x^2 * log(x+a), x) - 2*x*log(x + a) + x^2/(x + a) - sage: derivative(atan(x), x) - 1/(x^2 + 1) - sage: derivative(x^n, x, 3) - (n - 2)*(n - 1)*n*x^(n - 3) - sage: derivative( function('f')(x), x) - diff(f(x), x, 1) - sage: diff( 2*x*f(x^2), x) - 2*x*diff(f(x^2), x, 1) + 2*f(x^2) - sage: print integrate( 1/(x^4 - a^4), x) - x - atan(-) - log(x + a) log(x - a) a - - ---------- + ---------- - ------- - 3 3 3 - 4 a 4 a 2 a - sage: expand(integrate(log(1-x^2), x)) - x*log(1 - x^2) + log(x + 1) - log(x - 1) - 2*x - sage: integrate(log(1-x^2)/x, x) - log(x)*log(1 - x^2) + polylog(2, 1 - x^2)/2 - sage: integrate(exp(1-x^2),x) - sqrt(pi)*e*erf(x)/2 - sage: integrate(sin(x^2),x) - sqrt(pi)*((sqrt(2)*I + sqrt(2))*erf((sqrt(2)*I + sqrt(2))*x/2) + (sqrt(2)*I - sqrt(2))*erf((sqrt(2)*I - sqrt(2))*x/2))/8 - sage: integrate((1-x^2)^n,x) - integrate((1 - x^2)^n, x) - sage: integrate(x^x,x) - integrate(x^x, x) - sage: print integrate(1/(x^3+1),x) - 2 x - 1 - 2 atan(-------) - log(x - x + 1) sqrt(3) log(x + 1) - - --------------- + ------------- + ---------- - 6 sqrt(3) 3 - sage: integrate(1/(x^3+1), x, 0, 1) - (6*log(2) + sqrt(3)*pi)/18 + sqrt(3)*pi/18 - - sage: forget(); assume(c > 0) - sage: integrate(exp(-c*x^2), x, -oo, oo) - sqrt(pi)/sqrt(c) - sage: forget() - -The following are a bunch of examples of integrals that Mathematica -can do, but SAGE currently can't do: - sage: integrate(sqrt(x + sqrt(x)), x) # todo -- mathematica can do this - integrate(sqrt(x + sqrt(x)), x) - sage: integrate(log(x)*exp(-x^2)) # todo -- mathematica can do this - integrate(e^(-x^2)*log(x), x) - - sage.: # Todo -- Mathematica can do this and gets pi^2/15 - sage.: # integrate(log(1+sqrt(1+4*x)/2)/x, x, 0, 1) - [boom!] - Integral is divergent - - sage: integrate(ceil(x^2 + floor(x)), x, 0, 5) # todo: mathematica can do this - integrate(ceil(x^2) + floor(x), x, 0, 5) - - -MAPLE: -The basic differentiation and integration examples in the -Maple documentation: - - sage: diff(sin(x), x) - cos(x) - sage: diff(sin(x), y) - 0 - sage: diff(sin(x), x, 3) - -cos(x) - sage: diff(x*sin(cos(x)), x) - sin(cos(x)) - x*sin(x)*cos(cos(x)) - sage: diff(tan(x), x) - sec(x)^2 - sage: f = function('f'); f - f - sage: diff(f(x), x) - diff(f(x), x, 1) - sage: diff(f(x,y), x, y) - diff(f(x, y), x, 1, y, 1) - sage: diff(f(x,y), x, y) - diff(f(x,y), y, x) - 0 - sage: g = function('g') - sage: diff(g(x,y,z), x,z,z) - diff(g(x, y, z), x, 1, z, 2) - sage: integrate(sin(x), x) - -cos(x) - sage: integrate(sin(x), x, 0, pi) - 2 - sage: assume(b-a>0) # annoying -- maple doesn't require this... - sage: print integrate(sin(x), x, a, b) - cos(a) - cos(b) - sage: forget() - - - sage: integrate( x/(x^3-1), x) - (-log(x^2 + x + 1))/6 + atan((2*x + 1)/sqrt(3))/sqrt(3) + log(x - 1)/3 - sage: integrate(exp(-x^2), x) - sqrt(pi)*erf(x)/2 - sage: integrate(exp(-x^2)*log(x), x) # todo: maple can compute this exactly. - integrate(e^(-x^2)*log(x), x) - sage: f = exp(-x^2)*log(x) - sage: f.nintegral(x, 0, 999) - (-0.87005772672831549, 7.5584116743243612e-10, 567, 0) - sage: integral(1/sqrt(2*t^4 - 3*t^2 - 2), t, 2, 3) # todo: maple can do this - integrate(1/(sqrt(2*t^4 - 3*t^2 - 2)), t, 2, 3) - sage: integral(integral(x*y^2, x, 0, y), y, -2, 2) - 32/5 - -""" Index: age/calculus/todo.txt =================================================================== --- sage/calculus/todo.txt (revision 4270) +++ (revision ) @@ -1,3 +1,0 @@ -[ ] Limits. See the comments around dummy_limit in calculus.py - -[ ] Index: age/calculus/var.pyx =================================================================== --- sage/calculus/var.pyx (revision 4270) +++ (revision ) @@ -1,111 +1,0 @@ -import calculus - -def var(s): - """ - Create a symbolic variable with the name \emph{s}. - - INPUT: - s -- a string, either a single variable name, - or a space or comma separated list of - variable names. - - NOTE: The new variable is both returned and automatically injected - into the global namespace. If you use var in library code, it is - better to use sage.calculus.calculus.var, since it won't touch the - global namespace. - - EXAMPLES: - We define three variables: - sage: var('xx yy zz') - (xx, yy, zz) - - Then we make an algebraic expression out of them. - sage: f = xx^n + yy^n + zz^n; f - zz^n + yy^n + xx^n - - We can substitute a new variable name for n. - sage: f(n = var('sigma')) - zz^sigma + yy^sigma + xx^sigma - - If you make an important builtin variable into a symbolic variable, - you can get back the original value using restore: - sage: var('QQ RR') - (QQ, RR) - sage: QQ - QQ - sage: restore('QQ') - sage: QQ - Rational Field - - We make two new variables separated by commas: - sage: var('theta, gamma') - (theta, gamma) - sage: theta^2 + gamma^3 - gamma^3 + theta^2 - - The new variables are of type SymbolicVariable, and belong - to the symbolic expression ring: - sage: type(theta) - - sage: parent(theta) - Symbolic Ring - """ - G = globals() # this is the reason the code must be in SageX. - v = calculus.var(s) - if isinstance(v, tuple): - for x in v: - G[repr(x)] = x - else: - G[repr(v)] = v - return v - -def function(s, *args): - """ - Create a formal symbolic function with the name \emph{s}. - - INPUT: - s -- a string, either a single variable name, - or a space or comma separated list of - variable names. - - NOTE: The new function is both returned and automatically injected - into the global namespace. If you use var in library code, it is - better to use sage.calculus.calculus.function, since it won't - touch the global namespace. - - EXAMPLES: - We create a formal function called supersin. - sage: f = function('supersin', x) - sage: f - supersin(x) - - We can immediately use supersin in symbolic expressions: - sage: supersin(y+z) + A^3 - A^3 + supersin(z + y) - - We can define other functions in terms of supersin. - sage: g(x,y) = supersin(x)^2 + sin(y/2) - sage: g - (x, y) |--> sin(y/2) + supersin(x)^2 - sage: g.diff(y) - (x, y) |--> cos(y/2)/2 - sage: g.diff(x) - (x, y) |--> 2*supersin(x)*diff(supersin(x), x, 1) - sage: k = g.diff(x); k - (x, y) |--> 2*supersin(x)*diff(supersin(x), x, 1) - sage: k.substitute(supersin=sin) - 2*cos(x)*sin(x) - """ - if len(args) > 0: - return function(s)(*args) - - G = globals() # this is the reason the code must be in SageX. - v = calculus.function(s) - if isinstance(v, tuple): - for x in v: - G[repr(x)] = x - else: - G[repr(v)] = v - return v - - Index: age/calculus/wester.py =================================================================== --- sage/calculus/wester.py (revision 4359) +++ (revision ) @@ -1,540 +1,0 @@ -""" -Further examples from Wester's paper - -These are all the problems at - http://yacas.sourceforge.net/essaysmanual.html - -They come from the 1994 paper "Review of CAS mathematical capabilities", -by Michael Wester, who put forward 123 problems that a reasonable computer -algebra system should be able to solve and tested the then current -versions of various commercial CAS on this list. SAGE can do most of -the problems natively now, i.e., with no explicit calls to maxima or -other systems. - - -sage: factorial(50) -30414093201713378043612608166064768844377641568960512000000000000 -sage: factor(factorial(50)) -2^47 * 3^22 * 5^12 * 7^8 * 11^4 * 13^3 * 17^2 * 19^2 * 23^2 * 29 * 31 * 37 * 41 * 43 * 47 - -sage: # 1/2+...+1/10 = 4861/2520 -sage: sum(1/n for n in range(2,10+1)) == 4861/2520 -True - -sage: # Evaluate e^(Pi*Sqrt(163)) to 50 decimal digits -sage: a = e^(pi*sqrt(163)); a -e^(sqrt(163)*pi) -sage: print RealField(150)(a) -262537412640768743.99999999999925007259719819 - -sage: # Evaluate the Bessel function J[2] numerically at z=1+I. -sage: # NOTE -- we get a different answer than yacas -sage: bessel_J(2.0,1.0+I) -0.874211097673326 - 0.222469792478650*I - -sage: # Obtain period of decimal fraction 1/7=0.(142857). -sage: a = 1/7 -sage: print a -1/7 -sage: print a.period() -6 - -sage: # Continued fraction of 3.1415926535 -sage: a = 3.1415926535 -sage: continued_fraction(a) -[3, 7, 15, 1, 292, 1, 1, 6, 2, 13, 4] - -sage: # (not exactly ok) Sqrt(2*Sqrt(3)+4)=1+Sqrt(3). -sage: # The maxima backend equality checker fails this; maybe it *should*, since -sage: # the equality only holds for one choice of sign. -sage: a = sqrt(2*sqrt(3) + 4); b = 1 + sqrt(3) -sage: print float(a-b) -0.0 -sage: print bool(a == b) -False -sage: # We can, of course, do this in a quadratic field -sage: k. = QuadraticField(3) -sage: asqr = 2*sqrt3 + 4 -sage: b = 1+sqrt3 -sage: asqr == b^2 -True - -sage: # (not exactly ok) Sqrt(14+3*Sqrt(3+2*Sqrt(5-12*Sqrt(3-2*Sqrt(2)))))=3+Sqrt(2). -sage: a = sqrt(14+3*sqrt(3+2*sqrt(5-12*sqrt(3-2*sqrt(2))))) -sage: b = 3+sqrt(2) -sage: print a, b -sqrt(3 sqrt(2 sqrt(5 - 12 sqrt(3 - 2 sqrt(2))) + 3) + 14) sqrt(2) + 3 -sage: print bool(a==b) -False -sage: print float(a-b) -1.7763568394e-15 -sage: # 2*Infinity-3=Infinity. -sage: 2*infinity-3 == infinity -True - -sage: # (YES) Standard deviation of the sample (1, 2, 3, 4, 5). -sage: v = vector(RDF, 5, [1,2,3,4,5]) -sage: v.standard_deviation() -1.5811388300841898 - -sage: # (NO) Hypothesis testing with t-distribution. -sage: # (NO) Hypothesis testing with chi^2 distribution - -sage: # (YES) (x^2-4)/(x^2+4*x+4)=(x-2)/(x+2). -sage: R. = QQ[] -sage: (x^2-4)/(x^2+4*x+4) == (x-2)/(x+2) -True -sage: restore('x') - -sage: # (NO -- Maxima doesn't consider them equal.) -sage: # (Exp(x)-1)/(Exp(x/2)+1)=Exp(x/2)-1. -sage: f = (exp(x)-1)/(exp(x/2)+1) -sage: g = exp(x/2)-1 -sage: f -(e^x - 1)/(e^(x/2) + 1) -sage: g -e^(x/2) - 1 -sage: print f(10.0), g(10.0) -147.4131591025766 147.4131591025766 -sage: print bool(f == g) -False - -sage: # (YES) Expand (1+x)^20, take derivative and factorize. -sage: # first do it is using algebraic polys -sage: R. = QQ[] -sage: f = (1+x)^20 -sage: print f -x^20 + 20*x^19 + 190*x^18 + 1140*x^17 + 4845*x^16 + 15504*x^15 + 38760*x^14 + 77520*x^13 + 125970*x^12 + 167960*x^11 + 184756*x^10 + 167960*x^9 + 125970*x^8 + 77520*x^7 + 38760*x^6 + 15504*x^5 + 4845*x^4 + 1140*x^3 + 190*x^2 + 20*x + 1 -sage: print f.factor() -(x + 1)^20 -sage: # next do it symbolically -sage: restore('x') -sage: f = (1+x)^20; f -(x + 1)^20 -sage: g = f.expand(); g -x^20 + 20*x^19 + 190*x^18 + 1140*x^17 + 4845*x^16 + 15504*x^15 + 38760*x^14 + 77520*x^13 + 125970*x^12 + 167960*x^11 + 184756*x^10 + 167960*x^9 + 125970*x^8 + 77520*x^7 + 38760*x^6 + 15504*x^5 + 4845*x^4 + 1140*x^3 + 190*x^2 + 20*x + 1 -sage: g.factor() -(x + 1)^20 - - -sage: # (YES) Factorize x^100-1. -sage: factor(x^100-1) -(x - 1)*(x + 1)*(x^2 + 1)*(x^4 - x^3 + x^2 - x + 1)*(x^4 + x^3 + x^2 + x + 1)*(x^8 - x^6 + x^4 - x^2 + 1)*(x^20 - x^15 + x^10 - x^5 + 1)*(x^20 + x^15 + x^10 + x^5 + 1)*(x^40 - x^30 + x^20 - x^10 + 1) -sage: # Also, algebraically -sage: x = polygen(QQ) -sage: factor(x^100 - 1) -(x - 1) * (x + 1) * (x^2 + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1) * (x^8 - x^6 + x^4 - x^2 + 1) * (x^20 - x^15 + x^10 - x^5 + 1) * (x^20 + x^15 + x^10 + x^5 + 1) * (x^40 - x^30 + x^20 - x^10 + 1) -sage: restore('x') - - -sage: # (YES) Factorize x^4-3*x^2+1 in the field of rational numbers extended by roots of x^2-x-1. -sage: k.< a> = NumberField(x^2 - x -1) -sage: R.< y> = k[] -sage: f = y^4 - 3*y^2 + 1 -sage: f -y^4 + (-3)*y^2 + 1 -sage: factor(f) -(y + -a) * (y + -a + 1) * (y + a - 1) * (y + a) - -sage: # (YES) Factorize x^4-3*x^2+1 mod 5. -sage: k.< x > = GF(5) [ ] -sage: f = x^4 - 3*x^2 + 1 -sage: f.factor() -(x + 2)^2 * (x + 3)^2 -sage: # Alternatively, from symbol x as follows: -sage: reset('x') -sage: f = x^4 - 3*x^2 + 1 -sage: f.polynomial(GF(5)).factor() -(x + 2)^2 * (x + 3)^2 - -sage: # (YES) Partial fraction decomposition of (x^2+2*x+3)/(x^3+4*x^2+5*x+2) -sage: f = (x^2+2*x+3)/(x^3+4*x^2+5*x+2) -sage: print f - 2 - x + 2 x + 3 - ------------------- - 3 2 - x + 4 x + 5 x + 2 - -sage: print f.partial_fraction() - 3 2 2 - ----- - ----- + -------- - x + 2 x + 1 2 - (x + 1) - - -sage: # (BUG?) Assuming x>=y, y>=z, z>=x, deduce x=z. -sage: # Maxima doesn't agree that x==z is a conclusion... -sage: forget() -sage: restore('x,y,z') -sage: assume(x>=y, y>=z,z>=x) -sage: print bool(x==z) -False - -sage: # (YES) Assuming x>y, y>0, deduce 2*x^2>2*y^2. -sage: forget() -sage: assume(x>y, y>0) -sage: print assumptions() -[x > y, y > 0] -sage: print bool(2*x^2 > 2*y^2) -True -sage: forget() -sage: print assumptions() -[] -sage: # Solve the inequality Abs(x-1)>2. - -sage: # (NO) Maxima doesn't solve inequalities: -sage: eqn = abs(x-1) > 2 -sage: print eqn - abs(x - 1) > 2 - -sage: # (NO) Solve the inequality (x-1)*...*(x-5)<0. -sage: eqn = prod(x-i for i in range(1,5 +1)) < 0 -sage: # but don't know how to solve -sage: print eqn - (x - 5) (x - 4) (x - 3) (x - 2) (x - 1) < 0 - - -sage: # (YES) Cos(3*x)/Cos(x)=Cos(x)^2-3*Sin(x)^2 or similar equivalent combination. -sage: f = cos(3*x)/cos(x) -sage: g = cos(x)^2 - 3*sin(x)^2 -sage: h = f-g -sage: print h.trig_simplify() - 0 - -sage: # (YES) Cos(3*x)/Cos(x)=2*Cos(2*x)-1. -sage: f = cos(3*x)/cos(x) -sage: g = 2*cos(2*x) - 1 -sage: h = f-g -sage: print h.trig_simplify() - 0 -sage: # (NO) Define rewrite rules to match Cos(3*x)/Cos(x)=Cos(x)^2-3*Sin(x)^2. -sage: # SAGE has no notion of "rewrite rules". - - -sage: # (YES) Sqrt(997)-(997^3)^(1/6)=0 -sage: a = sqrt(997) - (997^3)^(1/6) -sage: print a - 0 -sage: print bool(a == 0) -True - -sage: # (NO) Sqrt(99983)-99983^3^(1/6)=0 -sage: # For some reason Maxima decides its not zero because of the factorization. -sage: a = sqrt(99983) - (99983^3)^(1/6) -sage: print a - sqrt(99983) - sqrt(13) sqrt(7691) -sage: print bool(a==0) -False -sage: float(a) # random low order bits -1.13686837722e-13 -sage: print 13*7691 -99983 - -sage: # (YES) (2^(1/3) + 4^(1/3))^3 - 6*(2^(1/3) + 4^(1/3))-6 = 0 -sage: ## same issue as above -- can only do using number fields -sage: a = (2^(1/3) + 4^(1/3))^3 - 6*(2^(1/3) + 4^(1/3)) - 6 -sage: print a - 1/3 1/3 3 1/3 1/3 - (4 + 2 ) - 6 (4 + 2 ) - 6 -sage: print bool(a==0) -False -sage: print float(a) -3.5527136788e-15 -sage: ## but we can do it using number fields. -sage: reset('x') -sage: k. = NumberField(x^3-2) -sage: a = (b + b^2)^3 - 6*(b + b^2) - 6 -sage: print a -0 - -sage: # (YES) Ln(Tan(x/2+Pi/4))-ArcSinh(Tan(x))=0 -sage: # Yes, in that the thing is clearly not equal to 0! -sage: f = log(tan(x/2 + pi/4)) - asin(tan(x)) -sage: bool(f == 0) -False -sage: [float(f(i/10)) for i in range(1,5)] # random low order bits -[-0.00033670040754082975, -0.0027778004096620235, -0.0098909940914040928, -0.025411145508414501] - -sage: # (YES) Numerically, the expression Ln(Tan(x/2+Pi/4))-ArcSinh(Tan(x))=0 and its derivative at x=0 are zero. -sage: g = f.derivative() -sage: abs(float(f(0))) < 1e-10 -True -sage: abs(float(g(0))) < 1e-10 -True -sage: print g - 2 x pi - sec (- + ---) 2 - 2 4 sec (x) - -------------- - ----------------- - x pi 2 - 2 tan(- + ---) sqrt(1 - tan (x)) - 2 4 - - -sage: # (NO?) Ln((2*Sqrt(r) + 1)/Sqrt(4*r 4*Sqrt(r) 1))=0. -sage: f = log( (2*sqrt(r) + 1) / sqrt(4*r + 4*sqrt(r) + 1)) -sage: print f - 2 sqrt(r) + 1 - log(-------------------------) - sqrt(4 r + 4 sqrt(r) + 1) -sage: print bool(f == 0) -False -sage: print [float(f(i)) for i in [0.1,0.3,0.5]] -[0.0, 0.0, 0.0] - - -sage: # (NO, except numerically) -sage: # (4*r+4*Sqrt(r)+1)^(Sqrt(r)/(2*Sqrt(r)+1))*(2*Sqrt(r)+1)^(2*Sqrt(r)+1)^(-1)-2*Sqrt(r)-1=0, assuming r>0. -sage: assume(r>0) -sage: f = (4*r+4*sqrt(r)+1)^(sqrt(r)/(2*sqrt(r)+1))*(2*sqrt(r)+1)^(2*sqrt(r)+1)^(-1)-2*sqrt(r)-1 -sage: print f - 1 sqrt(r) - ------------- ------------- - 2 sqrt(r) + 1 2 sqrt(r) + 1 - (2 sqrt(r) + 1) (4 r + 4 sqrt(r) + 1) - - 2 sqrt(r) - 1 -sage: bool(f == 0) -False -sage: [float(f(i)) for i in [0.1,0.3,0.5]] # random low-order bits -[0.0, 0.0, -2.2204460492503131e-16] - - -sage: # (YES) Obtain real and imaginary parts of Ln(3+4*I). -sage: a = log(3+4*I) -sage: print a - log(4 I + 3) -sage: print a.real() - log(5) -sage: print a.imag() - 4 - atan(-) - 3 - -sage: # (YES) Obtain real and imaginary parts of Tan(x+I*y) -sage: a = tan(x + I*y) -sage: print a - tan( I y + x) -sage: print a.real() - sin(2 x) - -------------------- - cosh(2 y) + cos(2 x) -sage: print a.imag() - sinh(2 y) - -------------------- - cosh(2 y) + cos(2 x) - -sage: # (YES) Simplify Ln(Exp(z)) to z for -Pi0, Re(y)>0, deduce x^(1/n)*y^(1/n)-(x*y)^(1/n)=0. -sage: assume(real(x) > 0, real(y) > 0) -sage: f = x^(1/n)*y^(1/n)-(x*y)^(1/n) -sage: print f - 0 -sage: forget() - -sage: # (??) Transform equations, (x==2)/2+(1==1)=>x/2+1==2. -sage: # This doesn't make any sense, in my opinion. Adding equations - -sage: # (SOMEWHAT) Solve Exp(x)=1 and get all solutions. -sage: solve(exp(x) == 1) -[x == 0] - -sage: # (SOMEWHAT) Solve Tan(x)=1 and get all solutions. -sage: solve(tan(x) == 1) -[x == (pi/4)] - -sage: # (YES) Solve a degenerate 3x3 linear system. -sage: # x+y+z==6,2*x+y+2*z==10,x+3*y+z==10 -sage: # First symbolically: -sage: solve([x+y+z==6, 2*x+y+2*z==10, x+3*y+z==10], x,y,z) -[[x == (4 - r1), y == 2, z == r1]] - -sage: # (YES) Invert a 2x2 symbolic matrix. -sage: # [[a,b],[1,a*b]] -sage: # Using multivariate poly ring -- much nicer -sage: R. = QQ[] -sage: m = matrix(2,2,[a,b, 1, a*b]) -sage: zz = m^(-1) -sage: print zz -[ a/(-1 + a^2) (-1)/(-1 + a^2)] -[(-1)/(-1*b + a^2*b) a/(-1*b + a^2*b)] - -sage: # (YES) Compute and factor the determinant of the 4x4 Vandermonde matrix in a, b, c, d. -sage: restore('a,b,c,d') -sage: m = matrix(SR, 4, 4, [[z^i for i in range(4)] for z in [a,b,c,d]]) -sage: print m - [ 1 a a^2 a^3] - [ 1 b b^2 b^3] - [ 1 c c^2 c^3] - [ 1 d d^2 d^3] -sage: d = m.determinant() -sage: print d.factor() - (b - a) (c - a) (c - b) (d - a) (d - b) (d - c) - -sage: # (YES) Compute and factor the determinant of the 4x4 Vandermonde matrix in a, b, c, d. -sage: # Do it instead in a multivariate ring -sage: R. = QQ[] -sage: m = matrix(R, 4, 4, [[z^i for i in range(4)] for z in [a,b,c,d]]) -sage: print m -[ 1 a a^2 a^3] -[ 1 b b^2 b^3] -[ 1 c c^2 c^3] -[ 1 d d^2 d^3] -sage: d = m.determinant() -sage: print d -b*c^2*d^3 - b*c^3*d^2 - b^2*c*d^3 + b^2*c^3*d + b^3*c*d^2 - b^3*c^2*d - a*c^2*d^3 + a*c^3*d^2 + a*b^2*d^3 - a*b^2*c^3 - a*b^3*d^2 + a*b^3*c^2 + a^2*c*d^3 - a^2*c^3*d - a^2*b*d^3 + a^2*b*c^3 + a^2*b^3*d - a^2*b^3*c - a^3*c*d^2 + a^3*c^2*d + a^3*b*d^2 - a^3*b*c^2 - a^3*b^2*d + a^3*b^2*c -sage: print d.factor() -(-1) * (-1*d + c) * (-1*d + b) * (-1*c + b) * (b - a) * (-1*d + a) * (-1*c + a) - -sage: # Find the eigenvalues of a 3x3 integer matrix. -sage: m = matrix(QQ, 3, [5,-3,-7, -2,1,2, 2,-3,-4]) -sage: m.eigenspaces() -[ -(3, [ -(1, 0, -1) -]), -(1, [ -(1, 1, -1) -]), -(-2, [ -(0, 1, 1) -]) -] - -sage: # OK Verify some standard limits found by L'Hopital's rule: -sage: # Verify(Limit(x,Infinity) (1+1/x)^x, Exp(1)); -sage: # Verify(Limit(x,0) (1-Cos(x))/x^2, 1/2); -sage: print limit( (1+1/x)^x, x = oo) -e -sage: print limit( (1-cos(x))/(x^2), x = 1/2) - 1 - 4 - 4 cos(-) - 2 - -sage: # (OK-ish) D(x)Abs(x) -sage: # Verify(D(x) Abs(x), Sign(x)); -sage: diff(abs(x)) -x/abs(x) - -sage: # (NO) (Integrate(x)Abs(x))=Abs(x)*x/2 -sage: integral(abs(x), x) -integrate(abs(x), x) - -sage: # (YES) Compute derivative of Abs(x), piecewise defined. -sage: # Verify(D(x)if(x<0) (-x) else x, -sage: # Simplify(if(x<0) -1 else 1)) -Piecewise defined function with 2 parts, [[(-10, 0), -1], [(0, 10), 1]] -sage: # (NOT really) Integrate Abs(x), piecewise defined. -sage: # Verify(Simplify(Integrate(x) -sage: # if(x<0) (-x) else x), -sage: # Simplify(if(x<0) (-x^2/2) else x^2/2)); -sage: f = piecewise([ [[-10,0], -x], [[0,10], x]]) -sage: f.integral() -100 - -sage: # (YES) Taylor series of 1/Sqrt(1-v^2/c^2) at v=0. -sage: restore('v,c') -sage: taylor(1/sqrt(1-v^2/c^2), v, 0, 7) -1 + v^2/(2*c^2) + 3*v^4/(8*c^4) + 5*v^6/(16*c^6) - -sage: # (OK-ish) (Taylor expansion of Sin(x))/(Taylor expansion of Cos(x)) = (Taylor expansion of Tan(x)). -sage: # TestYacas(Taylor(x,0,5)(Taylor(x,0,5)Sin(x))/ -sage: # (Taylor(x,0,5)Cos(x)), Taylor(x,0,5)Tan(x)); -sage: f = taylor(sin(x), x, 0, 8) -sage: g = taylor(cos(x), x, 0, 8) -sage: h = taylor(tan(x), x, 0, 8) -sage: f = f.power_series(QQ) -sage: g = g.power_series(QQ) -sage: h = h.power_series(QQ) -sage: f - g*h -O(x^8) - -sage: # (YES) Taylor expansion of Ln(x)^a*Exp(-b*x) at x=1. -sage: taylor(log(x)^a * exp(-b*x), x, 1, 3) -(x - 1)^a/e^b - (((x - 1)^a*a + 2*b*(x - 1)^a)*(x - 1)/(2*e^b)) + (3*(x - 1)^a*a^2 + (12*b + 5)*(x - 1)^a*a + 12*b^2*(x - 1)^a)*(x - 1)^2/(24*e^b) - (((x - 1)^a*a^3 + (6*b + 5)*(x - 1)^a*a^2 + (12*b^2 + 10*b + 6)*(x - 1)^a*a + 8*b^3*(x - 1)^a)*(x - 1)^3/(48*e^b)) - -sage: # (YES) Taylor expansion of Ln(Sin(x)/x) at x=0. -sage: taylor(log(sin(x)/x), x, 0, 10) --x^2/6 - (x^4/180) - (x^6/2835) - (x^8/37800) - (x^10/467775) - -sage: # (NO) Compute n-th term of the Taylor series of Ln(Sin(x)/x) at x=0. -sage: # need formal functions - -sage: # (NO) Compute n-th term of the Taylor series of Exp(-x)*Sin(x) at x=0. -sage: # (Sort of, with some work) -sage: # Solve x=Sin(y)+Cos(y) for y as Taylor series in x at x=1. -sage: # TestYacas(InverseTaylor(y,0,4) Sin(y)+Cos(y), -sage: # (y-1)+(y-1)^2/2+2*(y-1)^3/3+(y-1)^4); -sage: # Note that InverseTaylor does not give the series in terms of x but in terms of y which is semantically -sage: # wrong. But other CAS do the same. -sage: f = sin(y) + cos(y) -sage: g = f.taylor(y, 0, 10) -sage: h = g.power_series(QQ) -sage: k = (h - 1).reversion() -sage: print k -y + 1/2*y^2 + 2/3*y^3 + y^4 + 17/10*y^5 + 37/12*y^6 + 41/7*y^7 + 23/2*y^8 + 1667/72*y^9 + 3803/80*y^10 + O(y^11) - -sage: # [OK] Compute Legendre polynomials directly from Rodrigues's formula, P[n]=1/(2^n*n!) *(Deriv(x,n)(x^2-1)^n). -sage: # P(n,x) := Simplify( 1/(2*n)!! * -sage: # Deriv(x,n) (x^2-1)^n ); -sage: # TestYacas(P(4,x), (35*x^4)/8+(-15*x^2)/4+3/8); -sage: def P(n,x): -... return simplify(diff((x^2-1)^n,x,n) / (2^n * factorial(n))) -... -sage: print P(4,x).expand() - 4 2 - 35 x 15 x 3 - ----- - ----- + - - 8 4 8 - -sage: # (YES) Define the polynomial p=Sum(i,1,5,a[i]*x^i). -sage: # symbolically -sage: ps = sum(var('a%s'%i)*x^i for i in range(1,6)) -sage: print 'symbolic\n',ps -symbolic - 5 4 3 2 - a5 x + a4 x + a3 x + a2 x + a1 x -sage: # algebraically -sage: R = PolynomialRing(QQ,5,names='a') -sage: S. = PolynomialRing(R) -sage: p = S(list(R.gens()))*x -sage: print 'algebraic\n',p -algebraic -a4*x^5 + a3*x^4 + a2*x^3 + a1*x^2 + a0*x - -sage: # (YES) Convert the above to Horner's form. -sage: # Verify(Horner(p, x), ((((a[5]*x+a[4])*x -sage: # +a[3])*x+a[2])*x+a[1])*x); -sage: # We use the trick of evaluating the algebraic poly at a symbolic variable: -sage: restore('x') -sage: p(x) -x*(x*(x*(x*(a4*x + a3) + a2) + a1) + a0) - -sage: # (NO) Convert the result of problem 127 to Fortran syntax. -sage: # CForm(Horner(p, x)); - -sage: # (YES) Verify that True And False=False. -sage: (True and False) == False -True - -sage: # (YES) Prove x Or Not x. -sage: for x in [True, False]: -... print x or (not x) -True -True - -sage: # (YES) Prove x Or y Or x And y=>x Or y. -sage: for x in [True, False]: -... for y in [True, False]: -... if x or y or x and y: -... if not (x or y): -... print "failed!" -""" Index: sage/catalogue/catalogue.py =================================================================== --- sage/catalogue/catalogue.py (revision 3983) +++ sage/catalogue/catalogue.py (revision 2320) @@ -120,6 +120,7 @@ elementary = Elementary() function.elementary = elementary -import sage.calculus.all -elementary.sin = sage.calculus.all.sin -elementary.cos = sage.calculus.all.cos -elementary.log = sage.calculus.all.log +import sage.functions.all +elementary.sin = sage.functions.all.sin +elementary.cos = sage.functions.all.cos + + Index: sage/coding/all.py =================================================================== --- sage/coding/all.py (revision 3972) +++ sage/coding/all.py (revision 2132) @@ -24,19 +24,3 @@ from ag_code import ag_code -from code_bounds import (codesize_upper_bound, - dimension_upper_bound, - volume_hamming, - gilbert_lower_bound, - plotkin_upper_bound, - griesmer_upper_bound, - elias_upper_bound, - hamming_upper_bound, - singleton_upper_bound, - gv_info_rate, - entropy, - gv_bound_asymp, - hamming_bound_asymp, - singleton_bound_asymp, - plotkin_bound_asymp, - elias_bound_asymp, - mrrw1_bound_asymp) +from code_bounds import * Index: sage/coding/code_bounds.py =================================================================== --- sage/coding/code_bounds.py (revision 4062) +++ sage/coding/code_bounds.py (revision 3290) @@ -136,5 +136,5 @@ from sage.interfaces.all import gap -from sage.rings.all import QQ, RR, ZZ, RDF +from sage.rings.all import QQ, RR, ZZ from sage.rings.arith import factorial from sage.misc.functional import log, sqrt @@ -373,8 +373,8 @@ EXAMPLES: sage: elias_bound_asymp(1/4,2) - 0.399123963308 + 0.399123963307144 """ r = 1-1/q - return RDF((1-entropy(r-sqrt(r*(r-delta)), q))) + return (1-entropy(r-sqrt(r*(r-delta)), q)) def mrrw1_bound_asymp(delta,q): @@ -385,11 +385,11 @@ EXAMPLES: sage: mrrw1_bound_asymp(1/4,2) - 0.354578902665 - - """ - return RDF(entropy((q-1-delta*(q-2)-2*sqrt((q-1)*delta*(1-delta)))/q,q)) - - - - - + 0.354578902665270 + + """ + return entropy((q-1-delta*(q-2)-2*sqrt((q-1)*delta*(1-delta)))/q,q) + + + + + Index: sage/coding/linear_code.py =================================================================== --- sage/coding/linear_code.py (revision 4063) +++ sage/coding/linear_code.py (revision 2779) @@ -142,11 +142,4 @@ added LinearCode_from_vectorspace, extended_code, zeta_function -- Nick Alexander (2006-12-10): factor GUAVA code to guava.py - -TESTS: - sage: MS = MatrixSpace(GF(2),4,7) - sage: G = MS([[1,1,1,0,0,0,0], [1,0,0,1,1,0,0], [0,1,0,1,0,1,0], [1,1,0,1,0,0,1]]) - sage: C = LinearCode(G) - sage: C == loads(dumps(C)) - True """ @@ -634,7 +627,5 @@ def __cmp__(self, right): - if not isinstance(right, LinearCode): - return cmp(type(self), type(right)) - return cmp(self.__gen_mat, right.__gen_mat) + raise NotImplementedError def decode(self, right): @@ -785,4 +776,5 @@ EXAMPLES: + """ slength = self.length() @@ -793,9 +785,9 @@ rF = right.base_ring() if slength != rlength: - return False + return 0 if sdim != rdim: - return False + return 0 if sF != rF: - return False + return 0 V = VectorSpace(sF,sdim) sbasis = self.gens() @@ -804,10 +796,10 @@ rcheck = right.check_mat() for c in sbasis: - if rcheck*c: - return False + if rcheck*c != V(0): + return 0 for c in rbasis: - if scheck*c: - return False - return True + if scheck*c != V(0): + return 0 + return 1 def is_permutation_automorphism(self,g): Index: sage/combinat/combinat.py =================================================================== --- sage/combinat/combinat.py (revision 4022) +++ sage/combinat/combinat.py (revision 2664) @@ -16,108 +16,101 @@ \item Bell numbers, \code{bell_number} -\item Bernoulli numbers, \code{bernoulli_number} (though PARI's - bernoulli is better) - -\item Catalan numbers, \code{catalan_number} (not to be confused with - the Catalan constant) +\item Bernoulli numbers, \code{bernoulli_number} (though PARI's bernoulli is + better) + +\item Catalan numbers, \code{catalan_number} (not to be confused with the + Catalan constant) \item Eulerian/Euler numbers, \code{euler_number} (Maxima) -\item Fibonacci numbers, \code{fibonacci} (PARI) and - \code{fibonacci_number} (GAP) The PARI version is better. +\item Fibonacci numbers, \code{fibonacci} (PARI) and \code{fibonacci_number} (GAP) + The PARI version is better. \item Lucas numbers, \code{lucas_number1}, \code{lucas_number2}. -\item Stirling numbers, \code{stirling_number1}, -\code{stirling_number2}. - +\item Stirling numbers, \code{stirling_number1}, \code{stirling_number2}. \end{itemize} Set-theoretic constructions: \begin{itemize} - -\item Combinations of a multiset, \code{combinations}, -\code{combinations_iterator}, and \code{number_of_combinations}. These -are unordered selections without repetition of k objects from a -multiset S. - -\item Arrangements of a multiset, \code{arrangements} and - \code{number_of_arrangements} These are ordered selections without - repetition of k objects from a multiset S. - -\item Derangements of a multiset, \code{derangements} and -\code{number_of_derangements}. +\item Combinations of a multiset, \code{combinations}, \code{combinations_iterator}, +and \code{number_of_combinations}. These are unordered selections without +repetition of k objects from a multiset S. + +\item Arrangements of a multiset, \code{arrangements} and \code{number_of_arrangements} + These are ordered selections without repetition of k objects from a + multiset S. + +\item Derangements of a multiset, \code{derangements} and \code{number_of_derangements}. \item Tuples of a multiset, \code{tuples} and \code{number_of_tuples}. - An ordered tuple of length k of set S is a ordered selection with - repetitions of S and is represented by a sorted list of length k - containing elements from S. - -\item Unordered tuples of a set, \code{unordered_tuple} and - \code{number_of_unordered_tuples}. An unordered tuple of length k - of set S is a unordered selection with repetitions of S and is - represented by a sorted list of length k containing elements from S. - -\item Permutations of a multiset, \code{permutations}, -\code{permutations_iterator}, \code{number_of_permutations}. A -permutation is a list that contains exactly the same elements but + An ordered tuple of length k of set S is a ordered selection with + repetitions of S and is represented by a sorted list of length k + containing elements from S. + +\item Unordered tuples of a set, \code{unordered_tuple} and \code{number_of_unordered_tuples}. + An unordered tuple of length k of set S is a unordered selection with + repetitions of S and is represented by a sorted list of length k + containing elements from S. + +\item Permutations of a multiset, \code{permutations}, \code{permutations_iterator}, +\code{number_of_permutations}. A permutation is a list that contains exactly the same elements but possibly in different order. - \end{itemize} Partitions: \begin{itemize} - -\item Partitions of a set, \code{partitions_set}, - \code{number_of_partitions_set}. An unordered partition of set S is - a set of pairwise disjoint nonempty sets with union S and is - represented by a sorted list of such sets. - -\item Partitions of an integer, \code{partitions_list}, - \code{number_of_partitions_list}. An unordered partition of n is an - unordered sum $n = p_1+p_2 +\ldots+ p_k$ of positive integers and is - represented by the list $p = [p_1,p_2,\ldots,p_k]$, in nonincreasing - order, i.e., $p1\geq p_2 ...\geq p_k$. - -\item Ordered partitions of an integer, \code{ordered_partitions}, - \code{number_of_ordered_partitions}. An ordered partition of n is - an ordered sum $n = p_1+p_2 +\ldots+ p_k$ of positive integers and - is represented by the list $p = [p_1,p_2,\ldots,p_k]$, in - nonincreasing order, i.e., $p1\geq p_2 ...\geq p_k$. - -\item Restricted partitions of an integer, - \code{partitions_restricted}, - \code{number_of_partitions_restricted}. An unordered restricted - partition of n is an unordered sum $n = p_1+p_2 +\ldots+ p_k$ of - positive integers $p_i$ belonging to a given set $S$, and is - represented by the list $p = [p_1,p_2,\ldots,p_k]$, in nonincreasing - order, i.e., $p1\geq p_2 ...\geq p_k$. - -\item \code{partitions_greatest} implements a special type of - restricted partition. - -\item \code{partitions_greatest_eq} is another type of restricted -partition. +\item Partitions of a set, \code{partitions_set}, \code{number_of_partitions_set}. + An unordered partition of set S is a set of pairwise disjoint + nonempty sets with union S and is represented by a sorted list of + such sets. + +\item Partitions of an integer, \code{partitions_list}, \code{number_of_partitions_list}. + An unordered partition of n is an unordered sum + $n = p_1+p_2 +\ldots+ p_k$ of positive integers and is represented by + the list $p = [p_1,p_2,\ldots,p_k]$, in nonincreasing order, i.e., + $p1\geq p_2 ...\geq p_k$. + +\item Ordered partitions of an integer, \code{ordered_partitions}, + \code{number_of_ordered_partitions}. + An ordered partition of n is an ordered sum $n = p_1+p_2 +\ldots+ p_k$ + of positive integers and is represented by + the list $p = [p_1,p_2,\ldots,p_k]$, in nonincreasing order, i.e., + $p1\geq p_2 ...\geq p_k$. + +\item Restricted partitions of an integer, \code{partitions_restricted}, + \code{number_of_partitions_restricted}. + An unordered restricted partition of n is an unordered sum + $n = p_1+p_2 +\ldots+ p_k$ of positive integers $p_i$ belonging to a + given set $S$, and is represented by the list $p = [p_1,p_2,\ldots,p_k]$, + in nonincreasing order, i.e., $p1\geq p_2 ...\geq p_k$. + +\item \code{partitions_greatest} + implements a special type of restricted partition. + +\item \code{partitions_greatest_eq} is another type of restricted partition. \item Tuples of partitions, \code{partition_tuples}, - \code{number_of_partition_tuples}. A $k$-tuple of partitions is - represented by a list of all $k$-tuples of partitions which together - form a partition of $n$. - -\item Powers of a partition, \code{partition_power(pi, k)}. The power - of a partition corresponds to the $k$-th power of a permutation with - cycle structure $\pi$. - -\item Sign of a partition, \code{partition_sign( pi ) } This means the - sign of a permutation with cycle structure given by the partition - pi. - -\item Associated partition, \code{partition_associated( pi )} The - ``associated'' (also called ``conjugate'' in the literature) - partition of the partition pi which is obtained by transposing the + \code{number_of_partition_tuples}. + A $k$-tuple of partitions is represented by a list of all $k$-tuples + of partitions which together form a partition of $n$. + +\item Powers of a partition, \code{partition_power(pi, k)}. + The power of a partition corresponds to the $k$-th power of a + permutation with cycle structure $\pi$. + +\item Sign of a partition, \code{partition_sign( pi ) } + This means the sign of a permutation with cycle structure given by the + partition pi. + +\item Associated partition, \code{partition_associated( pi )} + The ``associated'' (also called ``conjugate'' in the literature) + partition of the partition pi which is obtained by transposing the corresponding Ferrers diagram. -\item Ferrers diagram, \code{ferrers_diagram}. Analogous to the Young - diagram of an irredicible representation of $S_n$. \end{itemize} +\item Ferrers diagram, \code{ferrers_diagram}. + Analogous to the Young diagram of an irredicible representation + of $S_n$. + \end{itemize} Related functions: @@ -266,5 +259,5 @@ [1, 1, 2, 5, 14, 42, 132] sage: maxima.eval("-(1/2)*taylor (sqrt (1-4*x^2), x, 0, 15)") - '-1/2+x^2+x^4+2*x^6+5*x^8+14*x^10+42*x^12+132*x^14' + '-1/2 + x^2 + x^4 + 2*x^6 + 5*x^8 + 14*x^10 + 42*x^12 + 132*x^14' sage: [catalan_number(i) for i in range(-7,7) if i != -1] [0, 0, 0, 0, 0, 0, 1, 1, 2, 5, 14, 42, 132] @@ -294,5 +287,5 @@ [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0] sage: maxima.eval("taylor (2/(exp(x)+exp(-x)), x, 0, 10)") - '1-x^2/2+5*x^4/24-61*x^6/720+277*x^8/8064-50521*x^10/3628800' + '1 - x^2/2 + 5*x^4/24 - 61*x^6/720 + 277*x^8/8064 - 50521*x^10/3628800' sage: [euler_number(i)/factorial(i) for i in range(11)] [1, 0, -1/2, 0, 5/24, 0, -61/720, 0, 277/8064, 0, -50521/3628800] Index: sage/combinat/sloane_functions.py =================================================================== --- sage/combinat/sloane_functions.py (revision 4064) +++ sage/combinat/sloane_functions.py (revision 3537) @@ -35,9 +35,4 @@ sage: a The integer sequence tau(n), which is the number of divisors of n. - -TESTS: - sage: a = sloane.A000001; - sage: a == loads(dumps(a)) - True AUTHORS: @@ -99,5 +94,4 @@ from sage.misc.misc import srange from sage.rings.integer_ring import ZZ -import sage.calculus.all as calculus Integer = ZZ @@ -117,9 +111,4 @@ def _repr_(self): raise NotImplementedError - - def __cmp__(self, other): - if not isinstance(other, SloaneSequence): - return cmp(type(self), type(other)) - return cmp(repr(self), repr(other)) def __call__(self, n): @@ -3397,5 +3386,5 @@ def _eval(self, n): - return arith.binomial(n, int(calculus.floor(n//2))) + return arith.binomial(n,arith.floor(n//2)) class A000292(SloaneSequence): Index: sage/crypto/all.py =================================================================== --- sage/crypto/all.py (revision 4359) +++ sage/crypto/all.py (revision 2699) @@ -11,2 +11,4 @@ lfsr_connection_polynomial) +from rsa import rsa_encrypt, rsa_decrypt + Index: sage/crypto/cipher.py =================================================================== --- sage/crypto/cipher.py (revision 4359) +++ sage/crypto/cipher.py (revision 2447) @@ -1,6 +1,2 @@ -""" -Ciphers. -""" - #***************************************************************************** # Copyright (C) 2007 David Kohel @@ -32,7 +28,4 @@ self._key = key - def __eq__(self, right): - return type(self) == type(right) and self._parent == right._parent and self._key == right._key - def __repr__(self): return str(self._key) Index: sage/crypto/classical.py =================================================================== --- sage/crypto/classical.py (revision 4359) +++ sage/crypto/classical.py (revision 2784) @@ -1,6 +1,2 @@ -""" -Classical Cryptosystems -""" - #***************************************************************************** # Copyright (C) 2007 David Kohel @@ -48,11 +44,4 @@ sage: e(m) GSVXZGRMGSVSZG - - TESTS: - sage: M = AlphabeticStrings() - sage: E = SubstitutionCryptosystem(M) - sage: E == loads(dumps(E)) - True - """ if not isinstance(S, StringMonoid_class): @@ -152,11 +141,4 @@ sage: e(S("THECATINTHEHAT")) TAHEHTNITACEHT - - EXAMPLES: - sage: S = AlphabeticStrings() - sage: E = TranspositionCryptosystem(S,14) - sage: E == loads(dumps(E)) - True - """ if not isinstance(S, StringMonoid_class): @@ -253,11 +235,4 @@ sage: e(S("THECATINTHEHAT")) TIGFEYOUBQOSMG - - TESTS: - sage: S = AlphabeticStrings() - sage: E = VigenereCryptosystem(S,14) - sage: E == loads(dumps(E)) - True - """ if not isinstance(S, StringMonoid_class): Index: sage/crypto/classical_cipher.py =================================================================== --- sage/crypto/classical_cipher.py (revision 4359) +++ sage/crypto/classical_cipher.py (revision 2688) @@ -1,6 +1,2 @@ -""" -Classical Ciphers. -""" - #***************************************************************************** # Copyright (C) 2007 David Kohel @@ -36,15 +32,6 @@ sage: e(m) GSVXZGRMGSVSZG - - TESTS: - sage: S = AlphabeticStrings() - sage: E = SubstitutionCryptosystem(S) - sage: E == loads(dumps(E)) - True """ SymmetricKeyCipher.__init__(self, parent, key) - - def __eq__(self, right): - return type(self) == type(right) and self.parent() == right.parent() and self.key() == right.key() def __call__(self, M): @@ -97,12 +84,4 @@ sage: e(m) EHTTACDNAEHTTAH - - - TESTS: - sage: S = AlphabeticStrings() - sage: E = TranspositionCryptosystem(S,14) - sage: E == loads(dumps(E)) - True - """ n = parent.block_length() @@ -152,11 +131,4 @@ sage: e(m) LOEMELXRTYIZHT - - TESTS: - sage: S = AlphabeticStrings() - sage: E = VigenereCryptosystem(S,11) - sage: E == loads(dumps(E)) - True - """ SymmetricKeyCipher.__init__(self, parent, key) Index: sage/crypto/cryptosystem.py =================================================================== --- sage/crypto/cryptosystem.py (revision 4359) +++ sage/crypto/cryptosystem.py (revision 2688) @@ -1,6 +1,2 @@ -""" -Cryptosystems. -""" - #***************************************************************************** # Copyright (C) 2007 David Kohel @@ -14,5 +10,5 @@ class Cryptosystem(Set_generic): - r""" + """ A cryptosystem is a pair of maps $$ @@ -46,13 +42,4 @@ self._block_length = block_length self._period = period - - def __eq__(self,right): - return type(self) == type(right) and \ - self._cipher_domain == right._cipher_domain and \ - self._cipher_codomain == right._cipher_codomain and \ - self._key_space == right._key_space and \ - self._block_length == right._block_length and \ - self._period == right._period - def plaintext_space(self): Index: sage/crypto/lfsr.py =================================================================== --- sage/crypto/lfsr.py (revision 4359) +++ sage/crypto/lfsr.py (revision 1097) @@ -1,4 +1,5 @@ -r""" -Linear feedback shift register (LFSR) sequence commands. +""" +Linear feedback shift register (LFSR) sequence commands + Stream ciphers have been used for Index: sage/crypto/rsa.py =================================================================== --- sage/crypto/rsa.py (revision 2055) +++ sage/crypto/rsa.py (revision 2055) @@ -0,0 +1,63 @@ +""" +Basic RSA commands + +Created 11-24-2005 by wdj. Last updated 12-02-2005. +""" + +########################################################################### +# Copyright (C) 2006 David Joyner and William Stein +# +# Distributed under the terms of the GNU General Public License (GPL) +# +# http://www.gnu.org/licenses/ +########################################################################### + +def rsa_encrypt(m, N, e): + """ + INPUT: + N -- typically a product of distinct primes. + e -- element of Z/phiN Z. + m -- integer relatively prime to e. + + OUTPUT: + RSA encryption $m^e \pmod N$. + + EXAMPLES: + sage: p = 53 + sage: q = 61 + sage: N = p*q + sage: phiN = euler_phi(N) + sage: R = IntegerModRing(int(phiN)) + sage: e = R(17) + sage: rsa_encrypt(123,N,e) + 855 + + AUTHOR: David Joyner (11-2005) + """ + return (m**e)%N + +def rsa_decrypt(c,N,e): + """ + Inverse function to rsa_encrypt. + + INPUT: + N -- typically a product of distinct primes. + e -- in Z/phiN Z. + c -- an integer relatively prime to e. + + EXAMPLES: + sage: p = 53 + sage: q = 61 + sage: N = p*q + sage: phiN = euler_phi(N) + sage: R = IntegerModRing(int(phiN)) + sage: e = R(17) + sage: rsa_decrypt(855,N,e) + 123 + + AUTHOR: + -- David Joyner (11-2005) + """ + d = 1/e; + return (c**d)%N + Index: sage/crypto/stream.py =================================================================== --- sage/crypto/stream.py (revision 4359) +++ sage/crypto/stream.py (revision 2771) @@ -1,6 +1,2 @@ -""" -Stream Cryptosystems. -""" - #***************************************************************************** # Copyright (C) 2007 David Kohel @@ -37,10 +33,4 @@ LFSR cryptosystem over Finite Field of size 2 - TESTS: - sage: E = LFSRCryptosystem(FiniteField(2)) - sage: E == loads(dumps(E)) - True - - TODO: Implement LFSR cryptosytem for arbitrary rings. The current implementation if limitated to the finite field of 2 elements only @@ -55,7 +45,4 @@ SymmetricKeyCryptosystem.__init__(self, S, S, None) self._field = field - - def __eq__(self,right): - return type(self) == type(right) and self._field == right._field def __call__(self, key): Index: sage/crypto/stream_cipher.py =================================================================== --- sage/crypto/stream_cipher.py (revision 4359) +++ sage/crypto/stream_cipher.py (revision 2710) @@ -44,12 +44,4 @@ sage: m == e(e(m)) True - - TESTS: - sage: FF = FiniteField(2) - sage: P. = PolynomialRing(FF) - sage: E = LFSRCryptosystem(FF) - sage: E == loads(dumps(E)) - True - """ SymmetricKeyCipher.__init__(self, parent, key = (poly, IS)) @@ -121,5 +113,5 @@ def __call__(self, M, mode = "ECB"): - r""" + """ EXAMPLES: sage: FF = FiniteField(2) Index: sage/databases/conway.py =================================================================== --- sage/databases/conway.py (revision 4182) +++ sage/databases/conway.py (revision 3702) @@ -66,5 +66,5 @@ import sage.misc.misc import sage.databases.db # very important that this be fully qualified -_CONWAYDATA = "%s/conway_polynomials/"%sage.databases.db.DB_HOME +_CONWAYDATA = "%s/data/src/conway/conway_table.py.bz2"%sage.misc.misc.SAGE_ROOT @@ -84,8 +84,7 @@ def _init(self): if not os.path.exists(_CONWAYDATA): - raise RuntimeError, "In order to initialize the database, the directory must exist."%_CONWAYDATA - os.chdir(_CONWAYDATA) - if os.system("bunzip2 -k conway_table.py.bz2"): - raise RuntimeError, "error decompressing table" + raise RuntimeError, "In order to initialize the database, the file %s must exist."%_CONWAYDATA + os.system("cp %s ."%_CONWAYDATA) + os.system("bunzip2 conway_table.py.bz2") from conway_table import conway_polynomials for X in conway_polynomials: @@ -95,6 +94,5 @@ self[p][n] = v self[p] = self[p] # so database knows it was changed - os.unlink("conway_table.pyc") - os.unlink("conway_table.py") + os.system("bzip2 conway_table.py; rm conway_table.pyc; cp conway_table.py %s"%_CONWAYDATA) self.commit() Index: sage/databases/cremona.py =================================================================== --- sage/databases/cremona.py (revision 4053) +++ sage/databases/cremona.py (revision 1097) @@ -256,5 +256,4 @@ sage: c.allcurves(11) {'a1': [[0, -1, 1, -10, -20], 0, 5], 'a3': [[0, -1, 1, 0, 0], 0, 5], 'a2': [[0, -1, 1, -7820, -263580], 0, 1]} - """ def __init__(self, read_only=True): Index: sage/dsage/database/jobdb.py =================================================================== --- sage/dsage/database/jobdb.py (revision 4353) +++ sage/dsage/database/jobdb.py (revision 4371) @@ -476,4 +476,5 @@ failures FROM jobs WHERE job_id = ?""" + cur = self.con.cursor() cur.execute(query, (job_id,)) @@ -597,5 +598,7 @@ """ - query = "SELECT * from jobs where killed = 1 AND status <> 'completed'" + query = """SELECT * from jobs + WHERE killed = 1 + AND status <> 'completed'""" cur = self.con.cursor() cur.execute(query) Index: sage/dsage/database/monitordb.py =================================================================== --- sage/dsage/database/monitordb.py (revision 4353) +++ sage/dsage/database/monitordb.py (revision 4371) @@ -131,6 +131,7 @@ def get_monitor(self, uuid): - query = """SELECT uuid, hostname, ip, anonymous, sage_version, os FROM monitors - WHERE uuid=?""" + query = """SELECT uuid, hostname, ip, anonymous, sage_version, os + FROM monitors + WHERE uuid=?""" cur = self.con.cursor() cur.execute(query, (uuid,)) @@ -152,6 +153,7 @@ """ - query = """SELECT uuid, hostname, ip, anonymous, sage_version, os FROM monitors - WHERE connected""" + query = """SELECT uuid, hostname, ip, anonymous, sage_version, os + FROM monitors + WHERE connected""" cur = self.con.cursor() cur.execute(query) Index: sage/dsage/dsage.py =================================================================== --- sage/dsage/dsage.py (revision 4352) +++ sage/dsage/dsage.py (revision 4339) @@ -143,4 +143,5 @@ INPUT: + server -- (string, default: None) the server you want to connect to if None, connects to the server Index: sage/dsage/scripts/dsage_server.py =================================================================== --- sage/dsage/scripts/dsage_server.py (revision 4353) +++ sage/dsage/scripts/dsage_server.py (revision 4371) @@ -74,5 +74,14 @@ print 'Error writing stats: %s' % (msg) return - + +def create_manhole(): + from twisted.manhole import telnet + factory = telnet.ShellFactory() + factory.username = 'yqiang' + factory.password = 'foo' + port = reactor.listenTCP(2000, factory) + + return port + def startLogging(log_file): """ @@ -193,5 +202,5 @@ # from sage.dsage.misc.countrefs import logInThread # logInThread(n=15) - + reactor.callWhenRunning(create_manhole) reactor.run(installSignalHandlers=1) Index: sage/dsage/server/server.py =================================================================== --- sage/dsage/server/server.py (revision 4353) +++ sage/dsage/server/server.py (revision 4371) @@ -103,5 +103,5 @@ uuid = jdict['monitor_id'] jdict['worker_info'] = self.monitordb.get_monitor(uuid) - + return jdict @@ -369,6 +369,6 @@ def generate_xml_stats(self): """ - This method returns a an XML document to be consumed by the Dashboard - widget + This method returns a an XML document to be consumed by the + Mac OS X Dashboard widget """ Index: sage/dsage/twisted/pb.py =================================================================== --- sage/dsage/twisted/pb.py (revision 4244) +++ sage/dsage/twisted/pb.py (revision 4371) @@ -291,4 +291,5 @@ """ + def __init__(self, DSageServer, avatarID): DefaultPerspective.__init__(self, DSageServer, avatarID) Index: sage/ext/cdefs.pxi =================================================================== --- sage/ext/cdefs.pxi (revision 4171) +++ sage/ext/cdefs.pxi (revision 3922) @@ -43,5 +43,4 @@ void mpz_abs (mpz_t rop, mpz_t op) void mpz_add (mpz_t rop, mpz_t op1, mpz_t op2) - void mpz_add_ui (mpz_t rop, mpz_t op1, unsigned long int op2) void mpz_addmul (mpz_t rop, mpz_t op1, mpz_t op2) void mpz_and (mpz_t rop, mpz_t op1, mpz_t op2) @@ -49,5 +48,4 @@ void mpz_clear(mpz_t integer) int mpz_cmp(mpz_t op1, mpz_t op2) - int mpz_cmpabs(mpz_t op1, mpz_t op2) int mpz_cmp_si(mpz_t op1, signed long int op2) int mpz_cmp_ui(mpz_t op1, unsigned long int op2) @@ -64,5 +62,4 @@ void mpz_tdiv_qr (mpz_t q, mpz_t r, mpz_t n, mpz_t d) double mpz_get_d (mpz_t op) - double mpz_get_d_2exp (long int *exp, mpz_t op) unsigned long int mpz_fdiv_ui (mpz_t n, unsigned long int d) unsigned long int mpz_fdiv_q_ui(mpz_t q, mpz_t n, unsigned long int d) @@ -94,8 +91,6 @@ void mpz_set_ui(mpz_t integer, unsigned long int n) int mpz_set_str(mpz_t rop, char *str, int base) - int mpz_set_d(mpz_t rop, double d) int mpz_sgn(mpz_t op) void mpz_sqrt (mpz_t rop, mpz_t op) - void mpz_sqrtrem (mpz_t rop, mpz_t rem, mpz_t op) void mpz_sub (mpz_t rop, mpz_t op1, mpz_t op2) void mpz_sub_ui(mpz_t rop, mpz_t op1, unsigned long int op2) Index: sage/ext/interactive_constructors_c.pyx =================================================================== --- sage/ext/interactive_constructors_c.pyx (revision 4077) +++ sage/ext/interactive_constructors_c.pyx (revision 1854) @@ -34,11 +34,13 @@ sage: GF(9,'c') Finite Field in c of size 3^2 - sage: c^3 - c^3 + sage: c + Traceback (most recent call last): + ... + NameError: name 'c' is not defined sage: inject_on(verbose=False) sage: GF(9,'c') Finite Field in c of size 3^2 - sage: c^3 - 2*c + 1 + sage: c + c ROLL YOUR OWN: If a constructor you would like to auto inject @@ -194,5 +196,5 @@ sage: S = quo(R, (x^3, x^2 + y^2), 'a,b') sage: S - Quotient of Polynomial Ring in x, y over Rational Field by the ideal (x^3, y^2 + x^2) + Quotient of Polynomial Ring in x, y over Rational Field by the ideal (y^2 + x^2, x^3) sage: a^2 -1*b^2 Index: sage/functions/all.py =================================================================== --- sage/functions/all.py (revision 4270) +++ sage/functions/all.py (revision 2106) @@ -1,6 +1,4 @@ from piecewise import Piecewise - -piecewise = Piecewise from transcendental import (exponential_integral_1, @@ -8,4 +6,5 @@ zeta, zeta_symmetric) +# This is too terrible code. #from elementary import (cosine, sine, exponential, # ElementaryFunction, @@ -13,9 +12,10 @@ from special import (bessel_I, bessel_J, bessel_K, bessel_Y, - hypergeometric_U, + hypergeometric_U, incomplete_gamma, spherical_bessel_J, spherical_bessel_Y, spherical_hankel1, spherical_hankel2, spherical_harmonic, jacobi, - inverse_jacobi, dilog, + inverse_jacobi, sinh, cosh, + tanh, coth, sech, csch, dilog, lngamma, exp_int, error_fcn) @@ -36,6 +36,4 @@ from constants import (pi, e, NaN, golden_ratio, log2, euler_gamma, catalan, - khinchin, twinprime, merten, brun, I) + khinchin, twinprime, merten, brun) -i = I # alias - Index: sage/functions/constants.py =================================================================== --- sage/functions/constants.py (revision 4316) +++ sage/functions/constants.py (revision 3290) @@ -19,13 +19,7 @@ euler_gamma sage: catalan # the Catalon constant - catalan + K sage: khinchin # Khinchin's constant khinchin - sage: twinprime - twinprime - sage: merten - merten - sage: brun - brun Support for coercion into the various systems means that if, e.g., @@ -60,13 +54,13 @@ can be coerced into other systems or evaluated. sage: a = pi + e*4/5; a - pi + 4*e/5 + (pi + ((e*4)/5)) sage: maxima(a) - %pi+4*%e/5 - sage: RealField(15)(a) # 15 *bits* of precision - 5.316 - sage: gp(a) - 5.316218116357029426750873360 # 32-bit - 5.3162181163570294267508733603616328824 # 64-bit - sage: print mathematica(a) # optional + %pi + 4*%e/5 + sage: a.str(15) # 15 *bits* of precision + '5.316' + sage: gp(a) + 5.316218116357029426750873360 # 32-bit + 5.3162181163570294267508733603616328824 # 64-bit + sage: mathematica(a) # optional 4 E --- + Pi @@ -108,16 +102,12 @@ EXAMPLES: Arithmetic with constants - sage: f = I*(e+1); f - (e + 1)*I - sage: f^2 - -(e + 1)^2 sage: pp = pi+pi; pp - 2*pi + (pi + pi) sage: R(pp) 6.2831853071795864769252867665590057683943387987502116419499 - sage: s = (1 + e^pi); s - e^pi + 1 + sage: s = (1 + e^pi);s + (1 + (e^pi)) sage: R(s) 24.140692632779269005729086367948547380266106242600211993445 @@ -125,6 +115,6 @@ 23.140692632779269005729086367948547380266106242600211993445 - sage: l = (1-log2)/(1+log2); l - (1 - log(2))/(log(2) + 1) + sage: l = (1-log2)/(1+log2);l + ((1 - log2)/(1 + log2)) sage: R(l) 0.18123221829928249948761381864650311423330609774776013488056 @@ -142,60 +132,4 @@ -- William Stein -- Alex Clemesha \& William Stein (2006-02-20): added new constants; removed todos - -- didier deshommes (2007-03-27): added constants from RQDF - -TESTS: - Coercion of each constant to the RQDF: - sage: RQDF(e) - 2.718281828459045235360287471352662497757247093699959574966967630 - sage: RQDF(pi) - 3.141592653589793238462643383279502884197169399375105820974944590 - sage: RQDF(e) - 2.718281828459045235360287471352662497757247093699959574966967630 - sage: RQDF(I) - Traceback (most recent call last): - ... - TypeError - sage: RQDF(golden_ratio) - 1.618033988749894848204586834365638117720309179805762862135448620 - sage: RQDF(log2) - 0.693147180559945309417232121458176568075500134360255254120680009 - sage: RQDF(euler_gamma) - 0.577215664901532860606512090082402431042159335939923598805767234 - sage: RQDF(catalan) - 0.915965594177219015054603514932384110774149374281672134266498119 - sage: RQDF(khinchin) - 2.685452001065306445309714835481795693820382293994462953051152345 - sage: RQDF(twinprime) - 0.660161815846869573927812110014555778432623360284733413319448422 - sage: RQDF(merten) - 0.261497212847642783755426838608695859051566648261199206192064212 - sage: RQDF(brun) - Traceback (most recent call last): - ... - TypeError: Brun's constant only available up to 41 bits - - -Coercing the sum of a bunch of the constants to many different floating point rings: - - sage: a = pi + e + golden_ratio + log2 + euler_gamma + catalan + khinchin + twinprime + merten; a - twinprime + merten + khinchin + euler_gamma + catalan + log(2) + pi + e + (sqrt(5) + 1)/2 - sage: parent(a) - Symbolic Ring - sage: RQDF(a) - 13.27134794019724931009881919957581394087110682000307481783297119 - sage: RR(a) - 13.2713479401972 - sage: RealField(212)(a) - 13.2713479401972493100988191995758139408711068200030748178329712 - sage: RealField(230)(a) - 13.271347940197249310098819199575813940871106820003074817832971189555 - sage: CC(a) - 13.2713479401972 - sage: CDF(a) - 13.2713479402 - sage: ComplexField(230)(a) - 13.271347940197249310098819199575813940871106820003074817832971189555 - sage: RDF(a) - 13.2713479402 """ @@ -222,31 +156,30 @@ from functions import Function_gen, Function_arith, Function, FunctionRing_class - ###################### # Ring of Constants ###################### -## class ConstantRing_class(FunctionRing_class): -## def _repr_(self): -## return "Ring of Real Mathematical Constants" - -## def __cmp__(self, right): -## if isinstance(right, ConstantRing_class): -## return 0 -## return -1 +class ConstantRing_class(FunctionRing_class): + def _repr_(self): + return "Ring of Real Mathematical Constants" + + def __cmp__(self, right): + if isinstance(right, ConstantRing_class): + return 0 + return -1 -## def __call__(self, x): -## try: -## return self._coerce_(x) -## except TypeError: -## return Constant_gen(x) - -## def _coerce_impl(self, x): -## if isinstance(x, (sage.rings.integer.Integer, -## sage.rings.rational.Rational, int, long)): -## return Constant_gen(x) -## raise TypeError, 'no canonical coercion of element into self.' - -## ConstantRing = ConstantRing_class() + def __call__(self, x): + try: + return self._coerce_(x) + except TypeError: + return Constant_gen(x) + + def _coerce_impl(self, x): + if isinstance(x, (sage.rings.integer.Integer, + sage.rings.rational.Rational, int, long)): + return Constant_gen(x) + raise TypeError, 'no canonical coercion of element into self.' + +ConstantRing = ConstantRing_class() @@ -254,42 +187,9 @@ # Constant functions ###################### -import sage.calculus.calculus class Constant(Function): def __init__(self, conversions={}): self._conversions = conversions - RingElement.__init__(self, sage.calculus.calculus.SR) - - # The maxima one is special: - def _maxima_(self, session=None): - if session is None: - return RingElement._maxima_(self, maxima) - else: - return RingElement._maxima_(self, session) - - def _has_op(self, x): - return False - - def substitute(self, *args, **kwds): - return self - - def _recursive_sub(self, kwds): - return self - - def _recursive_sub(self, kwds): - return self - - def _recursive_sub_over_ring(self, kwds, ring): - return ring(self) - - def variables(self): - return [] - - def _ser(self): - try: - return self.__ser - except AttributeError: - self.__ser = sage.calculus.calculus.SR._coerce_impl(self) - return self.__ser + RingElement.__init__(self, ConstantRing) def _neg_(self): @@ -302,31 +202,23 @@ return Integer(int(float(self))) - def _latex_(self): - return '\\text{%s}'%self - - def _complex_mpfr_field_(self, R): - return R(self._mpfr_(R._real_field())) - - def _real_double_(self, R): - return R(float(self)) - - def _complex_double_(self, R): - return R(float(self)) - # The following adds formal arithmetic support for generic constant def _add_(self, right): - return self._ser() + right + return Constant_arith(self, right, operator.add) def _sub_(self, right): - return self._ser() - right + return Constant_arith(self, right, operator.sub) def _mul_(self, right): - return self._ser() * right + return Constant_arith(self, right, operator.mul) def _div_(self, right): - return self._ser() / right + return Constant_arith(self, right, operator.div) def __pow__(self, right): - return self._ser() ** right + try: + right = self.parent()._coerce_(right) + except TypeError: + raise TypeError, "computation of %s^%s not defined"%(self, right) + return Constant_arith(self, right, operator.pow) def _interface_is_cached_(self): @@ -368,5 +260,6 @@ Function_arith.__init__(self, x, y, op) Constant.__init__(self) - + + class Pi(Constant): """ @@ -388,5 +281,5 @@ 3.1415926535897932384626433832795028841971693993751058209749 sage: pp = pi+pi; pp - 2*pi + (pi + pi) sage: R(pp) 6.2831853071795864769252867665590057683943387987502116419499 @@ -402,5 +295,5 @@ 'matlab':'pi','maple':'Pi','octave':'pi','pari':'Pi'}) - def _repr_(self, simplify=True): + def _repr_(self): return "pi" @@ -417,10 +310,7 @@ return R.pi() - def _real_double_(self,R): - return R.pi() - - def _real_rqdf_(self, R): - return R.pi() - + def _real_double_(self): + return sage.rings.all.RD.pi() + def __abs__(self): if self.str()[0] != '-': @@ -439,75 +329,4 @@ pi = Pi() - -class I_class(Constant): - """ - The formal square root of -1. - - EXAMPLES: - sage: I - I - sage: I^2 - -1 - sage: float(I) - Traceback (most recent call last): - ... - TypeError - sage: gp(I) - I - sage: RR(I) - Traceback (most recent call last): - ... - TypeError - sage: C = ComplexField(200); C - Complex Field with 200 bits of precision - sage: C(I) - 1.0000000000000000000000000000000000000000000000000000000000*I - sage: z = I + I; z - 2*I - sage: C(z) - 2.0000000000000000000000000000000000000000000000000000000000*I - sage: maxima(2*I) - 2*%i - """ - def __init__(self): - Constant.__init__(self, - {'axiom':'%i', - 'maxima':'%i','gp':'I','mathematica':'I', - 'matlab':'i','maple':'I','octave':'i','pari':'I'}) - - def _repr_(self, simplify=True): - return "I" - - def _latex_(self): - return "i" - - def _mathml_(self): - return "&i;" - - def __float__(self): - raise TypeError - - def _mpfr_(self, R): - raise TypeError - - def _real_rqdf_(self, R): - raise TypeError - - def _complex_mpfr_field_(self, R): - return R.gen() - - def _complex_double_(self, C): - return C.gen() - - def _real_double_(self, R): - raise TypeError - - def __abs__(self): - return Integer(1) - - def floor(self): - raise TypeError - -I = I_class() class E(Constant): @@ -523,5 +342,5 @@ 2.7182818284590452353602874713526624977572470936999595749670 sage: em = 1 + e^(1-e); em - e^(1 - e) + 1 + (1 + (e^(1 - e))) sage: R(em) 1.1793740787340171819619895873183164984596816017589156131574 @@ -545,5 +364,5 @@ 'octave':'e'}) - def _repr_(self, simplify=True): + def _repr_(self): return 'e' @@ -560,16 +379,7 @@ return Integer(2) - def _real_double_(self, R): - return R(1).exp() - - def _real_rqdf_(self, R): - """ - EXAMPLES: - sage: RQDF = RealQuadDoubleField () - sage: RQDF.e() - 2.718281828459045235360287471352662497757247093699959574966967630 - """ - return R.e() - + def _real_double_(self): + return sage.rings.all.RD.e() + # This just gives a string in singular anyways, and it's # *REALLY* slow! @@ -581,4 +391,6 @@ ee = e + + class NotANumber(Constant): """ @@ -589,5 +401,5 @@ {'matlab':'NaN'}) - def _repr_(self, simplify=True): + def _repr_(self): return 'NaN' @@ -595,15 +407,7 @@ return R('NaN') #??? nan in mpfr: void mpfr_set_nan (mpfr_t x) - def _real_double_(self, R): - return R.nan() - - def _real_rqdf_(self, R): - """ - EXAMPLES: - sage: RQDF (NaN) - 'NaN' - """ - return R.NaN() - + def _real_double_(self): + return sage.rings.all.RD.nan() + NaN = NotANumber() @@ -620,7 +424,7 @@ 1.6180339887498948482045868343656381177203091798057628621354 sage: grm = maxima(golden_ratio);grm - (sqrt(5)+1)/2 + (sqrt(5) + 1)/2 sage: grm + grm - sqrt(5)+1 + sqrt(5) + 1 sage: float(grm + grm) 3.2360679774997898 @@ -631,5 +435,5 @@ 'pari':'(1+sqrt(5))/2','octave':'(1+sqrt(5))/2', 'kash':'(1+Sqrt(5))/2'}) - def _repr_(self, simplify=True): + def _repr_(self): return 'golden_ratio' @@ -640,5 +444,5 @@ return float(0.5)*(float(1.0)+math.sqrt(float(5.0))) - def _real_double_(self, R): + def _real_double_(self): """ EXAMPLES: @@ -646,9 +450,6 @@ 1.61803398875 """ - return R('1.61803398874989484820458') - - def _real_rqdf_(self, R): - return R('1.61803398874989484820458683436563811772030917980576286213544862') - + return sage.rings.all.RDF(1.61803398874989484820458) + def _mpfr_(self,R): #is this OK for _mpfr_ ? return (R(1)+R(5).sqrt())*R(0.5) @@ -671,6 +472,6 @@ sage: R(log2) 0.69314718055994530941723212145817656807550013436025525412068 - sage: l = (1-log2)/(1+log2); l - (1 - log(2))/(log(2) + 1) + sage: l = (1-log2)/(1+log2);l + ((1 - log2)/(1 + log2)) sage: R(l) 0.18123221829928249948761381864650311423330609774776013488056 @@ -691,5 +492,5 @@ 'pari':'log(2)','octave':'log(2)'}) - def _repr_(self, simplify=True): + def _repr_(self): return 'log2' @@ -700,5 +501,5 @@ return math.log(2) - def _real_double_(self, R): + def _real_double_(self): """ EXAMPLES: @@ -706,14 +507,6 @@ 0.69314718056 """ - return R.log2() - - def _real_rqdf_(self, R): - """ - EXAMPLES: - sage: RQDF(log2) - 0.693147180559945309417232121458176568075500134360255254120680009 - """ - return R.log2() - + return sage.rings.all.RD.log2() + def _mpfr_(self,R): return R.log2() @@ -736,6 +529,6 @@ sage: R(euler_gamma) 0.57721566490153286060651209008240243104215933593992359880577 - sage: eg = euler_gamma + euler_gamma; eg - 2*euler_gamma + sage: eg = euler_gamma + euler_gamma;eg + (euler_gamma + euler_gamma) sage: R(eg) 1.1544313298030657212130241801648048620843186718798471976115 @@ -744,8 +537,7 @@ Constant.__init__(self, {'kash':'EulerGamma(R)','maple':'gamma', - 'mathematica':'EulerGamma','pari':'Euler', - 'maxima':'%gamma', 'maxima':'euler_gamma'}) - - def _repr_(self, simplify=True): + 'mathematica':'EulerGamma','pari':'Euler'}) + + def _repr_(self): return 'euler_gamma' @@ -756,5 +548,5 @@ return R.euler_constant() - def _real_double_(self, R): + def _real_double_(self): """ EXAMPLES: @@ -762,9 +554,6 @@ 0.577215664902 """ - return R.euler_constant() - - def _real_rqdf_(self, R): - return R('0.577215664901532860606512090082402431042159335939923598805767235') - + return sage.rings.all.RD.euler() + def floor(self): return Integer(0) @@ -778,19 +567,20 @@ EXAMPLES: - sage: catalan^2 + merten - merten + catalan^2 """ def __init__(self): Constant.__init__(self, {'mathematica':'Catalan','kash':'Catalan(R)', #kash: R is default prec - 'maple':'Catalan', 'maxima':'catalan'}) + 'maple':'Catalan'}) - def _repr_(self, simplify=True): - return 'catalan' + def _repr_(self): + return 'K' + + def _latex_(self): + return 'K' def _mpfr_(self, R): return R.catalan_constant() - def _real_double_(self, R): + def _real_double_(self): """ EXAMPLES: @@ -799,9 +589,6 @@ 0.915965594177 """ - return R('0.91596559417721901505460351493252') - - def _real_rqdf_(self, R): - return R('0.915965594177219015054603514932384110774149374281672134266498120') - + return sage.rings.all.RDF(0.91596559417721901505460351493252) + def __float__(self): """ @@ -830,10 +617,14 @@ Khinchin sage: m.N(200) # optional - 2.6854520010653064453097148354817956938203822939944629530511523455572188595371520028011411749318476979951534659052880900828976777164109630517925334832596683818523154213321194996260393285220448194096180686641664289 # 32-bit - 2.6854520010653064453097148354817956938203822939944629530511523455572188595371520028011411749318476979951534659052880900828976777164109630517925334832596683818523154213321194996260393285220448194096181 # 64-bit + 2.685452001065306445309714835481795693820382293994462953051152345557 # 32-bit + > 218859537152002801141174931847697995153465905288090082897677716410963051 # 32-bit + > 7925334832596683818523154213321194996260393285220448194096181 # 32-bit + 2.685452001065306445309714835481795693820382293994462953051152345557 # 64-bit + > 218859537152002801141174931847697995153465905288090082897677716410963051 # 64-bit + > 7925334832596683818523154213321194996260393285220448194096181 # 64-bit """ def __init__(self): Constant.__init__(self, - {'maxima':'khinchin', 'mathematica':'Khinchin'}) #Khinchin is only implemented in Mathematica + {'mathematica':'Khinchin'}) #Khinchin is only implemented in Mathematica # digits come from http://pi.lacim.uqam.ca/piDATA/khintchine.txt @@ -841,5 +632,5 @@ self.__bits = len(self.__value)*3-1 # underestimate - def _repr_(self, simplify=True): + def _repr_(self): return 'khinchin' @@ -849,5 +640,5 @@ raise NotImplementedError, "Khinchin's constant only available up to %s bits"%self.__bits - def _real_double_(self, R): + def _real_double_(self): """ EXAMPLES: @@ -855,9 +646,7 @@ 2.68545200107 """ - return R('2.685452001065306445309714835481795693820') - - def _real_rqdf_(self, R): - return R(self.__value[:65]) - + return sage.rings.all.RDF(2.685452001065306445309714835481795693820) + + def __float__(self): return 2.685452001065306445309714835481795693820 @@ -883,5 +672,5 @@ """ def __init__(self): - Constant.__init__(self,{'maxima':'twinprime'}) #Twin prime is not implemented in any other algebra systems. + Constant.__init__(self,{}) #Twin prime is not implemented in any other algebra systems. #digits come from http://www.gn-50uma.de/alula/essays/Moree/Moree-details.en.shtml @@ -894,5 +683,5 @@ return Integer(0) - def _repr_(self, simplify=True): + def _repr_(self): return 'twinprime' @@ -902,5 +691,5 @@ raise NotImplementedError, "Twin Prime constant only available up to %s bits"%self.__bits - def _real_double_(self, R): + def _real_double_(self): """ EXAMPLES: @@ -908,9 +697,6 @@ 0.660161815847 """ - return R('0.660161815846869573927812110014555778432') - - def _real_rqdf_(self, R): - return R(self.__value[:65]) - + return sage.rings.all.RDF(0.660161815846869573927812110014555778432) + def __float__(self): """ @@ -938,5 +724,5 @@ """ def __init__(self): - Constant.__init__(self,{'maxima':'merten'}) #Merten's constant is not implemented in any other algebra systems. + Constant.__init__(self,{}) #Merten's constant is not implemented in any other algebra systems. # digits come from Sloane's tables at http://www.research.att.com/~njas/sequences/table?a=77761&fmt=0 @@ -949,5 +735,5 @@ return Integer(0) - def _repr_(self, simplify=True): + def _repr_(self): return 'merten' @@ -966,5 +752,5 @@ raise NotImplementedError, "Merten's constant only available up to %s bits"%self.__bits - def _real_double_(self, R): + def _real_double_(self): """ EXAMPLES: @@ -972,9 +758,6 @@ 0.261497212848 """ - return R('0.261497212847642783755426838608695859051') - - def _real_rqdf_(self, R): - return R(self.__value[:65]) - + return sage.rings.all.RDF(0.261497212847642783755426838608695859051) + def __float__(self): """ @@ -984,5 +767,4 @@ """ return 0.261497212847642783755426838608695859051 - merten = Merten() @@ -1005,5 +787,5 @@ """ def __init__(self): - Constant.__init__(self,{'maxima':"brun"}) #Brun's constant is not implemented in any other algebra systems. + Constant.__init__(self,{}) #Brun's constant is not implemented in any other algebra systems. # digits come from Sloane's tables at http://www.research.att.com/~njas/sequences/table?a=65421&fmt=0 @@ -1016,5 +798,5 @@ return Integer(1) - def _repr_(self, simplify=True): + def _repr_(self): return 'brun' @@ -1025,5 +807,5 @@ Traceback (most recent call last): ... - TypeError: Brun's constant only available up to 41 bits + NotImplementedError: Brun's constant only available up to 41 bits sage: RealField(41)(brun) 1.90216058310 @@ -1031,7 +813,7 @@ if R.precision() <= self.__bits: return R(self.__value) - raise TypeError, "Brun's constant only available up to %s bits"%self.__bits - - def _real_double_(self, R): + raise NotImplementedError, "Brun's constant only available up to %s bits"%self.__bits + + def _real_double_(self): """ EXAMPLES: @@ -1039,9 +821,6 @@ 1.9021605831 """ - return R('1.9021605831040') - - def _real_rqdf_(self, R): - raise TypeError, "Brun's constant only available up to %s bits"%self.__bits - + return sage.rings.all.RDF(1.9021605831040) + def __float__(self): """ @@ -1054,2 +833,49 @@ brun=Brun() +class UniversalPolynomialElement(Constant): + """ + A universal indeterminate. + + EXAMPLES: + sage: x + x + sage: x.parent() + Univariate Polynomial Ring in x over Rational Field + """ + def __init__(self, name): + self._name = name + Constant.__init__(self, + {'axiom':name, + 'maxima':name, + 'gp':name,'kash':name, + 'mathematica':name, + 'matlab':name, + 'maple':name, + 'octave':name, + 'pari':name}) + + def _repr_(self): + return self._name + + def _latex_(self): + return self._name + + def _mathml_(self): + return "%s"%self._name + + def __float__(self): + raise TypeError + + def _mpfr_(self, R): + raise TypeError + + def _real_double_(self): + raise TypeError + + def __abs__(self): + raise TypeError + + def floor(self): + raise TypeError + +x = UniversalPolynomialElement('x') Index: sage/functions/functions.py =================================================================== --- sage/functions/functions.py (revision 4316) +++ sage/functions/functions.py (revision 3290) @@ -2,7 +2,14 @@ SAGE Functions Class -AUTHORS: - -- William Stein - -- didier deshommes (2007-03-26): added support for RQDF +EXAMPLES: + sage: f = 5*sin(x) + sage: f + (5*sin(x)) + sage: f(2) + (5*sin(2)) + sage: f(pi) + (5*sin(((1*pi) + 0))) + sage: float(f(pi)) + 6.1232339957367663e-16 """ import weakref @@ -19,5 +26,5 @@ import operator from sage.misc.latex import latex -from sage.calculus.calculus import maxima +from sage.interfaces.maxima import maxima, MaximaFunction import sage.functions.special as special from sage.libs.all import pari @@ -106,4 +113,6 @@ def _coerce_impl(self, x): + if is_Polynomial(x): + return self(x) return self(x) @@ -209,6 +218,11 @@ """ EXAMPLES: - sage: s = singular(e); s + sage: singular(e) 2.71828182845905 + sage: P = e.parent() + sage: old_prec = P.set_precision(200) + sage: singular(e) + 2.7182818284590452353602874713526624977572470936999595749670 + sage: _ = P.set_precision(old_prec) """ try: @@ -250,16 +264,16 @@ EXAMPLES: sage: s = e + pi - sage: bool(s == 0) + sage: s == 0 False sage: t = e^2 +pi - sage: bool(s == t) + sage: s == t False - sage: bool(s == s) + sage: s == s True - sage: bool(t == t) + sage: t == t True - sage: bool(s < t) + sage: s < t True - sage: bool(t < s) + sage: t < s False """ @@ -321,18 +335,18 @@ sage: s = (pi + pi) * e + e sage: s - 2*e*pi + e + (((pi + pi)*e) + e) sage: RR(s) 19.7977502738062 sage: maxima(s) - 2*%e*%pi+%e + 2*%e*%pi + %e sage: t = e^2 + pi + 2/3; t - pi + e^2 + 2/3 + (((e^2) + pi) + 2/3) sage: RR(t) 11.1973154191871 sage: maxima(t) - %pi+%e^2+2/3 + %pi + %e^2 + 2/3 sage: t^e - (pi + e^2 + 2/3)^e + ((((e^2) + pi) + 2/3)^e) sage: RR(t^e) 710.865247688858 @@ -346,16 +360,9 @@ self.__op = op # a binary operator, e.g., operator.sub - def _maxima_init_(self): - x = self.__x - y = self.__y - op = self.__op - return '(%s) %s (%s)'%(x._maxima_init_(), symbols[op] ,y._maxima_init_()) - - def _repr_(self): """ EXAMPLES: sage: log2 * e + pi^2/2 - e*log(2) + pi^2/2 + ((log2*e) + ((pi^2)/2)) """ return '(%s%s%s)'%(self.__x, symbols[self.__op], self.__y) @@ -365,9 +372,9 @@ EXAMPLES: sage: latex(log2 * e + pi^2/2) - {e \cdot \log \left( 2 \right)} + \frac{{\pi}^{2} }{2} + \log(2) \cdot e + \frac{\pi^{2}}{2} sage: latex(NaN^3 + 1/golden_ratio) - {\text{NaN}}^{3} + \frac{2}{\sqrt{ 5 } + 1} + \text{NaN}^{3} + \frac{1}{\phi} sage: latex(log2 + euler_gamma + catalan + khinchin + twinprime + merten + brun) - \text{twinprime} + \text{merten} + \text{khinchin} + \gamma + \text{catalan} + \text{brun} + \log \left( 2 \right) + \log(2) + \gamma + K + \text{khinchin} + \text{twinprime} + \text{merten} + \text{brun} """ if self.__op == operator.div: @@ -393,5 +400,5 @@ EXAMPLES: sage: gap(e + pi) - "pi + e" + "5.85987448204884" """ return '"%s"'%self.str() @@ -434,5 +441,5 @@ EXAMPLES: sage: maxima(e + pi) - %pi+%e + %pi + %e """ return self.__op(self.__x._maxima_(maxima), self.__y._maxima_(maxima)) @@ -462,5 +469,5 @@ EXAMPLES: sage: singular(e + pi) - pi + e + 5.85987448204884 """ return '"%s"'%self.str() @@ -474,6 +481,4 @@ return self.__op(self.__x._mpfr_(R), self.__y._mpfr_(R)) - def _real_rqdf_(self, R): - return self.__op(self.__x._real_rqdf_(R), self.__y._real_rqdf_(R)) class Function_gen(Function): @@ -485,7 +490,7 @@ sage: a = pi/2 + e sage: a - pi/2 + e + ((pi/2) + e) sage: maxima(a) - %pi/2+%e + %pi/2 + %e sage: RR(a) 4.28907815525394 @@ -495,5 +500,5 @@ sage: b = e + 5/7 sage: maxima(b) - %e+5/7 + %e + 5/7 sage: RR(b) 3.43256754274476 @@ -515,7 +520,4 @@ return R(self.__x) - def _real_rqdf_(self, R): - return R(self.__x) - def str(self, bits=None): if bits is None: @@ -598,8 +600,4 @@ return self.__f(x) - def _real_rqdf_(self, R): - x = R(self.__x) - return self.__f(x) - def _maxima_init_(self): try: @@ -767,4 +765,6 @@ class Function_maxima(Function): def __init__(self, var, defn, repr, latex): + #if not isinstance(x, MaximaFunction): + # raise TypeError, "x (=%s) must be a MaximaFunction" Function.__init__(self, {'maxima':defn}) self.__var = var Index: sage/functions/orthogonal_polys.py =================================================================== --- sage/functions/orthogonal_polys.py (revision 4016) +++ sage/functions/orthogonal_polys.py (revision 3301) @@ -386,9 +386,9 @@ sage: t = PolynomialRing(QQ, "t").gen() sage: gen_legendre_P(2,0,t) - '3*(1-t)^2/2-3*(1-t)+1' + '3*(1 - t)^2/2 - 3*(1 - t) + 1' sage: legendre_P(2,t) 3/2*t^2 - 1/2 sage: gen_legendre_P(3,1,t) - '-6*(5*(1-t)^2/4-5*(1-t)/2+1)*sqrt(1-t^2)' + '-6*(5*(1 - t)^2/4 - 5*(1 - t)/2 + 1)*sqrt(1 - t^2)' """ _init() @@ -412,7 +412,7 @@ sage: t = PolynomialRing(QQ, "t").gen() sage: gen_legendre_Q(2,0,t) - '(3*log(-(t+1)/(t-1))*t^2-6*t-log(-(t+1)/(t-1)))/4' + '(3*log( - (t + 1)/(t - 1))*t^2 - 6*t - log( - (t + 1)/(t - 1)))/4' sage: legendre_Q(2,t) - '(3*log(-(t+1)/(t-1))*t^2-6*t-log(-(t+1)/(t-1)))/4' + '(3*log( - (t + 1)/(t - 1))*t^2 - 6*t - log( - (t + 1)/(t - 1)))/4' sage: gen_legendre_Q(3,1,0.5) 2.49185259171 @@ -523,5 +523,5 @@ sage: t = PolynomialRing(QQ, 't').gen() sage: legendre_Q(2,t) - '(3*log(-(t+1)/(t-1))*t^2-6*t-log(-(t+1)/(t-1)))/4' + '(3*log( - (t + 1)/(t - 1))*t^2 - 6*t - log( - (t + 1)/(t - 1)))/4' sage: legendre_Q(3,0.5) -0.198654771479 Index: sage/functions/piecewise.py =================================================================== --- sage/functions/piecewise.py (revision 4316) +++ sage/functions/piecewise.py (revision 3636) @@ -2,7 +2,10 @@ Piecewise-defined Functions. -\sage implements a very simple class of piecewise-defined functions. Functions -may be any type of symbolic expression. Infinite intervals are not supported. -The endpoints of each interval must line up. +\sage implements a very simple class of piecewise-defined functions. +Functions must be piecewise polynomial, though some methods apply more +generally. Only compactly supported functions are currently +implemented. Moreover, the coefficients should be rational and the +support should be 'connected'. The intervals of polynomial support can +be in terms of rationals and $\pi$, or in terms of floats. Implemented methods: @@ -55,4 +58,8 @@ way is to use an element class and only define _mul_, and have a parent, so anything gets coerced properly. + +(For more general non-polynomial piecewise functions, it appears +a new class of functions (for example, 'ElementaryFunctionRing') is +needed. This a preliminary 'todo'.) AUTHOR: David Joyner (2006-04) -- initial version @@ -92,11 +99,4 @@ from sage.rings.all import QQ, RR, Integer, Rational -from sage.calculus.calculus import SR, var, maxima - -def meval(x): - from sage.calculus.calculus import symbolic_expression_from_maxima_element - return symbolic_expression_from_maxima_element(maxima(x)) - - class PiecewisePolynomial: def __init__(self, list_of_pairs): @@ -125,42 +125,33 @@ self.length(),self.list()) - def _latex_(self): - r""" + def latex(self): + """ EXAMPLES: - sage: f1(x) = 1 - sage: f2(x) = 1 - x + sage: f1 = lambda x:1 + sage: f2 = lambda x:1-x sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) - sage: latex(f) - \begin{cases} - x \ {\mapsto}\ 1 &\text{on $(0, 1)$}\cr - x \ {\mapsto}\ 1 - x &\text{on $(1, 2)$} - \end{cases} - - sage: f(x) = sin(x*pi/2) - sage: g(x) = 1-(x-1)^2 - sage: h(x) = -x - sage: P = Piecewise([[(0,1), f], [(1,3),g], [(3,5), h]]) - sage: latex(P) - \begin{cases} - x \ {\mapsto}\ \sin \left( \frac{{\pi \cdot x}}{2} \right) &\text{on $(0, 1)$}\cr - x \ {\mapsto}\ 1 - {\left( x - 1 \right)}^{2} &\text{on $(1, 3)$}\cr - x \ {\mapsto}\ -x &\text{on $(3, 5)$} - \end{cases} + sage: f.latex() + '\\begin{array}{ll} \\left\\{ 1,& 0 < x < 1 ,\\right. \\end{array}' + """ - intervals = self.intervals() - funcs = [p[1] for p in self.list()] - tex = ['\\begin{cases}\n'] - for i in range(len(funcs)): - f = funcs[i] - # left endpoint - a = intervals[i][0] - # right endpoint - b = intervals[i][1] - tex.append(r'%s &\text{on $(%s, %s)$}' % (f._latex_(), a, b)) - tex.append('\\cr\n') - # remove the last TeX newline - tex = tex[:-1] - tex.append('\n\\end{cases}') - return ''.join(tex) + x = PolynomialRing(QQ,'x').gen() + intvls = self.intervals() + fcn_list = [p[1] for p in self.list()] + tex = ["\\begin{array}{ll} \left\\{"] + for i in range(len(fcn_list)): + f = fcn_list[i] + a = intvls[i][0] + b = intvls[i][1] + tex.append(str(f(x))) + tex.append(",& %s < x < %s ,\\"%(a,b)) + tex = tex[:-2] + tex.append("\right\. \end{array}") + ltex = "" + for i in range(len(tex)-1): + ltex = ltex + tex[i] + ltex = ltex + str(tex[len(tex)-1]).replace("%","") + ltex = ltex.replace("\\\right\\.","\\right.") + ltex = ltex.replace("\\left\\{","\\left\{ ") + return ltex def intervals(self): @@ -169,8 +160,8 @@ EXAMPLES: - sage: f1(x) = 1 - sage: f2(x) = 1-x - sage: f3(x) = exp(x) - sage: f4(x) = sin(2*x) + sage: f1 = lambda x:1 + sage: f2 = lambda x:1-x + sage: f3 = lambda x:exp(x) + sage: f4 = lambda x:sin(2*x) sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]]) sage: f.intervals() @@ -185,13 +176,4 @@ """ Returns the list of functions (the "pieces"). - - EXAMPLES: - sage: f1(x) = 1 - sage: f2(x) = 1-x - sage: f3(x) = exp(x) - sage: f4(x) = sin(2*x) - sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]]) - sage: f.functions() - [x |--> 1, x |--> 1 - x, x |--> e^x, x |--> sin(2*x)] """ return self._functions @@ -262,18 +244,18 @@ Riemann sums in numerical integration based on a subdivision into N subintervals. - Set mode="midpoint" for the height of the rectangles to be - determined by the midpoint of the subinterval; + Set mode="midpoint" for the height of the rectangles to be + determined by the midpoint of the subinterval; set mode="right" for the height of the rectangles to be - determined by the right-hand endpoint of the subinterval; - the default is mode="left" (the height of the rectangles to be - determined by the leftt-hand endpoint of the subinterval). - - EXAMPLES: - sage: f1 = x^2 ## example 1 - sage: f2 = 5-x^2 + determined by the right-hand endpoint of the subinterval; + the default is mode="left" (the height of the rectangles to be + determined by the leftt-hand endpoint of the subinterval). + + EXAMPLES: + sage: f1 = lambda x:x^2 ## example 1 + sage: f2 = lambda x:5-x^2 sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) sage: f.riemann_sum_integral_approximation(6) 17/6 - sage: f.riemann_sum_integral_approximation(6,mode="right") + sage: f.riemann_sum_integral_approximation(6,mode="right") 19/6 sage: f.riemann_sum_integral_approximation(6,mode="midpoint") @@ -321,9 +303,10 @@ EXAMPLES: - sage: f1(x) = x^2 - sage: f2(x) = 5-x^2 + sage: f1 = lambda x:x^2 ## example 1 + sage: f2 = lambda x:5-x^2 sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) sage: f.riemann_sum(6,mode="midpoint") Piecewise defined function with 6 parts, [[(0, 1/3), 1/36], [(1/3, 2/3), 1/4], [(2/3, 1), 25/36], [(1, 4/3), 131/36], [(4/3, 5/3), 11/4], [(5/3, 2), 59/36]] + sage: x = PolynomialRing(QQ,'x').gen() ## example 2 sage: f = Piecewise([[(-1,1),1-x^2]]) sage: rsf = f.riemann_sum(7) @@ -411,6 +394,6 @@ EXAMPLES: - sage: f1(x) = x^2 ## example 1 - sage: f2(x) = 1-(1-x)^2 + sage: f1 = lambda x:x^2 ## example 1 + sage: f2 = lambda x:1-(1-x)^2 sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) sage: P = f.plot(rgbcolor=(0.7,0.1,0.5), plot_points=40) @@ -444,19 +427,18 @@ def critical_points(self): """ - Return the critical points of this piecewise function. - - WARNINGS: Uses maxima, which prints the warning to use results - with caution. Only works for piecewise functions whose parts - are polynomials with real critical not occurring on the - interval endpoints. + Function to return the critical points. Uses maxima, which + prints the warning to use results with caution. Only works for + piecewise functions whose parts are polynomials with real + critical not occurring on the interval endpoints. EXAMPLES: sage: x = PolynomialRing(QQ, 'x').0 sage: f1 = x^0 - sage: f2 = 10*x - x^2 - sage: f3 = 3*x^4 - 156*x^3 + 3036*x^2 - 26208*x - sage: f = Piecewise([[(0,3),f1],[(3,10),f2],[(10,20),f3]]) + sage: f2 = 1-x + sage: f3 = 2*x + sage: f4 = 10*x-x^2 + sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]]) sage: f.critical_points() - [5.0, 12.000000000000171, 12.9999999999996, 14.000000000000229] + [5.0] """ maxima = sage.interfaces.all.maxima @@ -465,20 +447,13 @@ N = len(fcns) crit_pts = [] - I = self.intervals() for i in range(N): maxima.eval("eqn:diff(%s,x)=0"%fcns[i]) ans = maxima.eval("allroots(eqn)") - while True: - start = ans.find('x=') - if start == -1: - break - ans = ans[start+2:] - end = ans.find(',') - if end == -1: - end = ans.find(']') - r = float(ans[:end]) - if I[i][0] < r < I[i][1]: + if "[x =" in ans: + i1 = ans.index("[x =") + i2 = ans.index("]") + r = eval(ans[i1+4:i2]) + if self.intervals()[i][0] < r < self.intervals()[i][1]: crit_pts.append(r) - ans = ans[end+1:] return crit_pts @@ -500,17 +475,17 @@ """ Evaluates self at x0. Returns the average value of the jump if x0 is - an interior endpoint of one of the intervals of self and the - usual value otherwise. + an interior endpoint of one of the intervals of self and the + usual value otherwise. EXAMPLES: - sage: f1(x) = 1 - sage: f2(x) = 1-x - sage: f3(x) = exp(x) - sage: f4(x) = sin(2*x) + sage: f1 = lambda x:1 + sage: f2 = lambda x:1-x + sage: f3 = lambda x:exp(x) + sage: f4 = lambda x:sin(2*x) sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]]) sage: f(0.5) 1 sage: f(2.5) - 12.18249396070347 + 12.1824939607035 sage: f(1) 1/2 @@ -536,11 +511,12 @@ EXAMPLES: - sage: f1 = z + sage: x = PolynomialRing(QQ,'x').gen() + sage: f1 = lambda z:1 sage: f2 = 1-x - sage: f3 = exp(y) - sage: f4 = sin(2*t) + sage: f3 = lambda y:exp(y) + sage: f4 = lambda t:sin(2*t) sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]]) sage: f.which_function(3/2) - 1 - x + -x + 1 """ n = self.length() @@ -558,5 +534,5 @@ raise ValueError,"Function not defined outside of domain." - def integral(self, x=None): + def integral(self): r""" Returns the definite integral (as computed by maxima) @@ -565,24 +541,20 @@ EXAMPLES: - sage: f1(x) = 1 - sage: f2(x) = 1-x + sage: f1 = lambda x:1 + sage: f2 = lambda x:1-x sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) sage: f.integral() 1/2 - sage: f1(x) = -1 - sage: f2(x) = 2 + sage: f1 = lambda x:-1 + sage: f2 = lambda x:2 sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]]) sage: f.integral() - pi/2 - """ - #maxima = sage.interfaces.all.maxima - #x = PolynomialRing(QQ,'x').gen() - #ints = [maxima('%s'%p[1](x)).integral('x', p[0][0], p[0][1]) \ - # for p in self.list()] - #RETURN MEVAl(repr(sum(ints))) - funcs = self.functions() - invs = self.intervals() - n = len(funcs) - return sum([funcs[i].integral(x,invs[i][0],invs[i][1]) for i in range(n)]) + (pi/2) + """ + maxima = sage.interfaces.all.maxima + x = PolynomialRing(QQ,'x').gen() + ints = [maxima('%s'%p[1](x)).integral('x', p[0][0], p[0][1]) \ + for p in self.list()] + return sage_eval(str(sum(ints)).replace("%","")) def convolution(self,other): @@ -624,5 +596,5 @@ R1 = f0.parent() xx = R1.gen() - var = repr(xx) + var = str(xx) if len(f.intervals())==1 and len(g.intervals())==1: f = f.unextend() @@ -632,6 +604,6 @@ b1 = g.intervals()[0][0] b2 = g.intervals()[0][1] - i1 = repr(f0).replace(var,repr(uu)) - i2 = repr(g0).replace(var,"("+repr(tt-uu)+")") + i1 = str(f0).replace(var,str(uu)) + i2 = str(g0).replace(var,"("+str(tt-uu)+")") cmd1 = "integrate((%s)*(%s),%s,%s,%s)"%(i1,i2, uu, a1, tt-b1) ## if a1+b1 < tt < a2+b1 cmd2 = "integrate((%s)*(%s),%s,%s,%s)"%(i1,i2, uu, tt-b2, tt-b1) ## if a1+b2 < tt < a2+b1 @@ -642,10 +614,8 @@ conv3 = maxima.eval(cmd3) conv4 = maxima.eval(cmd4) - # this is a very, very, very ugly hack - x = PolynomialRing(QQ,'x').gen() - fg1 = sage_eval(conv1.replace("tt",var), {'x':x}) ## should be = R2(conv1) - fg2 = sage_eval(conv2.replace("tt",var), {'x':x}) ## should be = R2(conv2) - fg3 = sage_eval(conv3.replace("tt",var), {'x':x}) ## should be = R2(conv3) - fg4 = sage_eval(conv4.replace("tt",var), {'x':x}) ## should be = R2(conv4) + fg1 = sage_eval(conv1.replace("tt",var)) ## should be = R2(conv1) + fg2 = sage_eval(conv2.replace("tt",var)) ## should be = R2(conv2) + fg3 = sage_eval(conv3.replace("tt",var)) ## should be = R2(conv3) + fg4 = sage_eval(conv4.replace("tt",var)) ## should be = R2(conv4) if a1-b11 or len(g.intervals())>1: z = Piecewise([[(-3*abs(N-M),3*abs(N-M)),0*xx**0]]) @@ -690,5 +660,5 @@ sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]]) sage: f.derivative() - Piecewise defined function with 2 parts, [[(0, pi/2), 0], [(pi/2, pi), 0]] + Piecewise defined function with 2 parts, [[(0, (pi/2)), 0], [((pi/2), pi), 0]] """ maxima = sage.interfaces.all.maxima @@ -697,7 +667,5 @@ diffs = [maxima('%s'%p[1](x)).diff('x') \ for p in self.list()] - dlist = [[(p[0][0], p[0][1]), - R(sage_eval(repr(maxima('%s'%p[1](x)).diff('x')).replace("%",""), - {'x':x}))] for p in self.list()] + dlist = [[(p[0][0], p[0][1]), R(sage_eval(str(maxima('%s'%p[1](x)).diff('x')).replace("%","")))] for p in self.list()] return Piecewise(dlist) @@ -724,5 +692,5 @@ return Piecewise(dlist) - def plot(self, *args, **kwds): + def plot(self, **kwds): """ Returns the plot of self. @@ -744,5 +712,5 @@ """ plot = sage.plot.plot.plot - return sum([plot(p[1], p[0][0], p[0][1], *args, **kwds ) for p in self.list()]) + return sum([plot(p[1], p[0][0], p[0][1], **kwds ) for p in self.list()]) def fourier_series_cosine_coefficient(self,n,L): @@ -762,5 +730,5 @@ sage: f = Piecewise([[(-1,1),f]]) sage: f.fourier_series_cosine_coefficient(2,1) - 1/pi^2 + (1/(pi^2)) sage: f = lambda x:x^2 sage: f = Piecewise([[(-pi,pi),f]]) @@ -771,5 +739,5 @@ sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]]) sage: f.fourier_series_cosine_coefficient(5,pi) - -3/(5*pi) + (-3/(5*pi)) """ maxima = sage.interfaces.all.maxima @@ -779,11 +747,11 @@ fcn = '(%s)*cos('%p[1](x) + 'pi*x*%s/%s)/%s'%(n,L,L) fcn = fcn.replace("pi","%"+"pi") - a = repr(p[0][0]).replace("pi","%"+"pi") - b = repr(p[0][1]).replace("pi","%"+"pi") + a = str(p[0][0]).replace("pi","%"+"pi") + b = str(p[0][1]).replace("pi","%"+"pi") cmd = "integrate("+fcn+", x, %s, %s )"%(a, b) int = maxima(cmd).trigsimp() ints.append(int) ans = sum(ints) - return meval(repr(ans)) + return sage_eval(str(ans).replace("%","")) def fourier_series_sine_coefficient(self,n,L): @@ -811,11 +779,11 @@ fcn = '(%s)*sin('%p[1](x) + 'pi*x*%s/%s)/%s'%(n,L,L) fcn = fcn.replace("pi","%"+"pi") - a = repr(p[0][0]).replace("pi","%"+"pi") - b = repr(p[0][1]).replace("pi","%"+"pi") + a = str(p[0][0]).replace("pi","%"+"pi") + b = str(p[0][1]).replace("pi","%"+"pi") cmd = "integrate("+fcn+", x, %s, %s )"%(a, b) int = maxima(cmd).trigsimp() ints.append(int) ans = sum(ints) - return meval(repr(ans)) + return sage_eval(str(ans).replace("%","")) def fourier_series_partial_sum(self,N,L): @@ -832,19 +800,20 @@ sage: f = Piecewise([[(-1,1),f]]) sage: f.fourier_series_partial_sum(3,1) - cos(2*pi*x)/pi^2 - (4*cos(pi*x)/pi^2) + 1/3 + '1/3 + ((-4/(pi^2))*cos(1*pi*x/1) + 0*sin(1*pi*x/1)) + ((1/(pi^2))*cos(2*pi*x/1) + 0*sin(2*pi*x/1))' sage: f1 = lambda x:-1 sage: f2 = lambda x:2 sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]]) sage: f.fourier_series_partial_sum(3,pi) - -3*sin(2*x)/pi + sin(x)/pi - (3*cos(x)/pi) + 1/4 + '1/4 + ((-3/pi)*cos(1*pi*x/pi) + (1/pi)*sin(1*pi*x/pi)) + (0*cos(2*pi*x/pi) + (-3/pi)*sin(2*pi*x/pi))' + """ a0 = self.fourier_series_cosine_coefficient(0,L) - A = [repr(self.fourier_series_cosine_coefficient(n,L))+"*cos(%s*pi*x/%s)"%(n,L) for n in range(1,N)] - B = [repr(self.fourier_series_sine_coefficient(n,L))+"*sin(%s*pi*x/%s)"%(n,L) for n in range(1,N)] + A = [str(self.fourier_series_cosine_coefficient(n,L))+"*cos(%s*pi*x/%s)"%(n,L) for n in range(1,N)] + B = [str(self.fourier_series_sine_coefficient(n,L))+"*sin(%s*pi*x/%s)"%(n,L) for n in range(1,N)] FS = ["("+A[i] +" + " + B[i]+")" for i in range(0,N-1)] - sumFS = repr(a0/2)+" + " + sumFS = str(a0/2)+" + " for s in FS: sumFS = sumFS+s+ " + " - return meval(sumFS[:-3]) + return sumFS[:-3] def fourier_series_partial_sum_cesaro(self,N,L): @@ -861,20 +830,20 @@ sage: f = Piecewise([[(-1,1),f]]) sage: f.fourier_series_partial_sum_cesaro(3,1) - cos(2*pi*x)/(3*pi^2) - (8*cos(pi*x)/(3*pi^2)) + 1/3 + '1/3 + ((2/3*(-4/(pi^2)))*cos(1*pi*x/1) + 0*sin(1*pi*x/1)) + ((1/3*(1/(pi^2)))*cos(2*pi*x/1) + 0*sin(2*pi*x/1))' sage: f1 = lambda x:-1 sage: f2 = lambda x:2 sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]]) sage: f.fourier_series_partial_sum_cesaro(3,pi) - -sin(2*x)/pi + 2*sin(x)/(3*pi) - (2*cos(x)/pi) + 1/4 + '1/4 + ((2/3*(-3/pi))*cos(1*pi*x/pi) + (2/3*(1/pi))*sin(1*pi*x/pi)) + (0*cos(2*pi*x/pi) + (1/3*(-3/pi))*sin(2*pi*x/pi))' """ a0 = self.fourier_series_cosine_coefficient(0,L) - A = [repr((1-n/N)*self.fourier_series_cosine_coefficient(n,L))+"*cos(%s*pi*x/%s)"%(n,L) for n in range(1,N)] - B = [repr((1-n/N)*self.fourier_series_sine_coefficient(n,L))+"*sin(%s*pi*x/%s)"%(n,L) for n in range(1,N)] + A = [str((1-n/N)*self.fourier_series_cosine_coefficient(n,L))+"*cos(%s*pi*x/%s)"%(n,L) for n in range(1,N)] + B = [str((1-n/N)*self.fourier_series_sine_coefficient(n,L))+"*sin(%s*pi*x/%s)"%(n,L) for n in range(1,N)] FS = ["("+A[i] +" + " + B[i]+")" for i in range(0,N-1)] - sumFS = repr(a0/2)+" + " + sumFS = str(a0/2)+" + " for s in FS: sumFS = sumFS+s+ " + " - return meval(sumFS[:-3]) + return sumFS[:-3] def fourier_series_partial_sum_hann(self,N,L): @@ -891,19 +860,20 @@ sage: f = Piecewise([[(-1,1),f]]) sage: f.fourier_series_partial_sum_hann(3,1) - 0.500000000000000*(cos(2*pi/3) + 1)*cos(2*pi*x)/pi^2 - (2.00000000000000*(cos(pi/3) + 1)*cos(pi*x)/pi^2) + 1/3 + '1/3 + ((1+cos(pi*1/3))*(0.5*(-4/(pi^2)))*cos(1*pi*x/1) + (1+cos(pi*1/3))*0.0*sin(1*pi*x/1)) + ((1+cos(pi*2/3))*(0.5*(1/(pi^2)))*cos(2*pi*x/1) + (1+cos(pi*2/3))*0.0*sin(2*pi*x/1))' sage: f1 = lambda x:-1 sage: f2 = lambda x:2 sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]]) sage: f.fourier_series_partial_sum_hann(3,pi) - -1.50000000000000*(cos(2*pi/3) + 1)*sin(2*x)/pi + 0.500000000000000*(cos(pi/3) + 1)*sin(x)/pi - (1.50000000000000*(cos(pi/3) + 1)*cos(x)/pi) + 1/4 + '1/4 + ((1+cos(pi*1/3))*(0.5*(-3/pi))*cos(1*pi*x/pi) + (1+cos(pi*1/3))*(0.5*(1/pi))*sin(1*pi*x/pi)) + ((1+cos(pi*2/3))*0.0*cos(2*pi*x/pi) + (1+cos(pi*2/3))*(0.5*(-3/pi))*sin(2*pi*x/pi))' + """ a0 = self.fourier_series_cosine_coefficient(0,L) - A = ["(1+cos(pi*%s/%s))*"%(n,N)+repr((0.5)*self.fourier_series_cosine_coefficient(n,L))+"*cos(%s*pi*x/%s)"%(n,L) for n in range(1,N)] - B = ["(1+cos(pi*%s/%s))*"%(n,N)+repr((0.5)*self.fourier_series_sine_coefficient(n,L))+"*sin(%s*pi*x/%s)"%(n,L) for n in range(1,N)] + A = ["(1+cos(pi*%s/%s))*"%(n,N)+str((0.5)*self.fourier_series_cosine_coefficient(n,L))+"*cos(%s*pi*x/%s)"%(n,L) for n in range(1,N)] + B = ["(1+cos(pi*%s/%s))*"%(n,N)+str((0.5)*self.fourier_series_sine_coefficient(n,L))+"*sin(%s*pi*x/%s)"%(n,L) for n in range(1,N)] FS = ["("+A[i] +" + " + B[i]+")" for i in range(0,N-1)] - sumFS = repr(a0/2)+" + " + sumFS = str(a0/2)+" + " for s in FS: sumFS = sumFS+s+ " + " - return meval(sumFS[:-3]) + return sumFS[:-3] def fourier_series_partial_sum_filtered(self,N,L,F): @@ -921,19 +891,20 @@ sage: f = Piecewise([[(-1,1),f]]) sage: f.fourier_series_partial_sum_filtered(3,1,[1,1,1]) - cos(2*pi*x)/pi^2 - (4*cos(pi*x)/pi^2) + 1/3 + '1/3 + ((1*(-4/(pi^2)))*cos(1*pi*x/1) + 0*sin(1*pi*x/1)) + ((1*(1/(pi^2)))*cos(2*pi*x/1) + 0*sin(2*pi*x/1))' sage: f1 = lambda x:-1 sage: f2 = lambda x:2 sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]]) sage: f.fourier_series_partial_sum_filtered(3,pi,[1,1,1]) - -3*sin(2*x)/pi + sin(x)/pi - (3*cos(x)/pi) + 1/4 + '1/4 + ((1*(-3/pi))*cos(1*pi*x/pi) + (1*(1/pi))*sin(1*pi*x/pi)) + (0*cos(2*pi*x/pi) + (1*(-3/pi))*sin(2*pi*x/pi))' + """ a0 = self.fourier_series_cosine_coefficient(0,L) - A = [repr((F[n])*self.fourier_series_cosine_coefficient(n,L))+"*cos(%s*pi*x/%s)"%(n,L) for n in range(1,N)] - B = [repr((F[n])*self.fourier_series_sine_coefficient(n,L))+"*sin(%s*pi*x/%s)"%(n,L) for n in range(1,N)] + A = [str((F[n])*self.fourier_series_cosine_coefficient(n,L))+"*cos(%s*pi*x/%s)"%(n,L) for n in range(1,N)] + B = [str((F[n])*self.fourier_series_sine_coefficient(n,L))+"*sin(%s*pi*x/%s)"%(n,L) for n in range(1,N)] FS = ["("+A[i] +" + " + B[i]+")" for i in range(0,N-1)] - sumFS = repr(a0/2)+" + " + sumFS = str(a0/2)+" + " for s in FS: sumFS = sumFS+s+ " + " - return meval(sumFS[:-3]) + return sumFS[:-3] def plot_fourier_series_partial_sum(self,N,L,xmin,xmax, **kwds): @@ -970,6 +941,6 @@ pi = 3.14159265 xi = xmin + i*h - yi = ff.replace("pi",repr(RR(pi))) - yi = sage_eval(yi.replace("x",repr(xi))) + yi = ff.replace("pi",str(RR(pi))) + yi = sage_eval(yi.replace("x",str(xi))) pts.append([xi,yi]) return line(pts, **kwds) @@ -1008,6 +979,6 @@ pi = 3.14159265 xi = xmin + i*h - yi = ff.replace("pi",repr(RR(pi))) - yi = sage_eval(yi.replace("x",repr(xi))) + yi = ff.replace("pi",str(RR(pi))) + yi = sage_eval(yi.replace("x",str(xi))) pts.append([xi,yi]) return line(pts, **kwds) @@ -1046,6 +1017,6 @@ pi = 3.14159265 xi = xmin + i*h - yi = ff.replace("pi",repr(RR(pi))) - yi = sage_eval(yi.replace("x",repr(xi))) + yi = ff.replace("pi",str(RR(pi))) + yi = sage_eval(yi.replace("x",str(xi))) pts.append([xi,yi]) return line(pts, **kwds) @@ -1086,6 +1057,6 @@ pi = 3.14159265 xi = xmin + i*h - yi = ff.replace("pi",repr(RR(pi))) - yi = sage_eval(yi.replace("x",repr(xi))) + yi = ff.replace("pi",str(RR(pi))) + yi = sage_eval(yi.replace("x",str(xi))) pts.append([xi,yi]) return line(pts, **kwds) @@ -1168,5 +1139,5 @@ 0 sage: f.cosine_series_coefficient(3,1) - -4/(9*pi^2) + (-4/(9*(pi^2))) sage: f1 = lambda x:-1 sage: f2 = lambda x:2 @@ -1175,7 +1146,7 @@ 0 sage: f.cosine_series_coefficient(3,pi) - 2/pi + (2/pi) sage: f.cosine_series_coefficient(111,pi) - 2/(37*pi) + (2/(37*pi)) """ @@ -1186,12 +1157,11 @@ fcn = '2*(%s)*cos('%p[1](x) + 'pi*x*%s/%s)/%s'%(n,L,L) fcn = fcn.replace("pi","%"+"pi") - a = repr(p[0][0]).replace("pi","%"+"pi") - b = repr(p[0][1]).replace("pi","%"+"pi") + a = str(p[0][0]).replace("pi","%"+"pi") + b = str(p[0][1]).replace("pi","%"+"pi") cmd = "integrate("+fcn+", x, %s, %s )"%(a, b) I = maxima(cmd).trigsimp() ints.append(I) ans = sum(ints) - return meval(repr(ans)) - + return sage_eval(str(ans).replace("%","")) def sine_series_coefficient(self,n,L): @@ -1217,5 +1187,6 @@ 0 sage: f.sine_series_coefficient(3,1) - 4/(3*pi) + (4/(3*pi)) + """ maxima = sage.interfaces.all.maxima @@ -1225,48 +1196,54 @@ fcn = '2*(%s)*sin('%p[1](x) + 'pi*x*%s/%s)/%s'%(n,L,L) fcn = fcn.replace("pi","%"+"pi") - a = repr(p[0][0]).replace("pi","%"+"pi") - b = repr(p[0][1]).replace("pi","%"+"pi") + a = str(p[0][0]).replace("pi","%"+"pi") + b = str(p[0][1]).replace("pi","%"+"pi") cmd = "integrate("+fcn+", x, %s, %s )"%(a, b) I = maxima(cmd).trigsimp() ints.append(I) ans = sum(ints) - return meval(repr(ans)) - - def laplace(self, x, s): - r""" - Returns the Laplace transform of self with respect to the - variable var. - - INPUT: - x -- variable of self - s -- variable of Laplace transform. - + return sage_eval(str(ans).replace("%","")) + + def laplace_transform(self,var = "s",latex_output=0): + r""" + Returns the laplace transform of self, as a function of var. We assume that a piecewise function is 0 outside of its domain and that the left-most endpoint of the domain is 0. EXAMPLES: - sage: f = Piecewise([[(0,1),1],[(1,2), 1-x]]) - sage: f.laplace(x, s) - -e^(-s)/s - (e^(-s)/s^2) + (s + 1)*e^(-2*s)/s^2 + 1/s - sage: f.laplace(x, w) - -e^(-w)/w - (e^(-w)/w^2) + (w + 1)*e^(-2*w)/w^2 + 1/w - - sage: f = Piecewise([[(1,2), 1-y]]) - sage: f.laplace(y, t) - (t + 1)*e^(-2*t)/t^2 - (e^(-t)/t^2) - """ - x = var(x) - s = var(s) + sage: f1 = lambda x:1 + sage: f2 = lambda x:1-x + sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) + sage: f.laplace_transform() + '1/s - e^-s/s + (s + 1)*e^-(2*s)/s^2 - e^-s/s^2' + sage: f.laplace_transform("w",latex_output=1) + ' - {{e^{ - w}}\\over{w}} - {{e^{ - w}}\\over{w^2}} + {{\\left(w + 1\\right)\\,e^{ - 2\\,w}}\\over{w^2}} + {{1}\\over{w}}' + sage: f.laplace_transform("w",True) + ' - {{e^{ - w}}\\over{w}} - {{e^{ - w}}\\over{w^2}} + {{\\left(w + 1\\right)\\,e^{ - 2\\,w}}\\over{w^2}} + {{1}\\over{w}}' + sage: f.laplace_transform("w") + '1/w - e^-w/w + (w + 1)*e^-(2*w)/w^2 - e^-w/w^2' + + """ + maxima = sage.interfaces.all.maxima + x = PolynomialRing(QQ,'x').gen() ints = [] for p in self.list(): - g = SR(p[1]) - fcn = maxima('(%s)*exp(-%s*%s)'%(g._maxima_init_(), s, x)) - ints.append(fcn.integral(x, p[0][0], p[0][1])) + fcn = '(%s)*exp(-%s*x)'%(p[1](x),var) + ints.append(maxima(fcn).integral('x', p[0][0], p[0][1])) ans = "" + ans_latex = "" for i in range(len(ints)-1): - ans = ans+repr(ints[i]) + " + " - ans = ans+repr(ints[len(ints)-1]) - return meval(ans) - + ans = ans+str(ints[i]).replace("%","")+" + " + ans_latex = ans_latex+str(ints[i])+" + " + ans = ans+str(ints[len(ints)-1]).replace("%","") + ans_latex = ans_latex + str(ints[len(ints)-1]) + + if latex_output == 0: + return ans + if latex_output == 1: + ans0 = maxima.eval("tex("+ans_latex+")") + ans0 = ans0.replace("$$","") + ans0 = ans0.replace("false","") + return ans0 + def __add__(self,other): """ Index: sage/functions/special.py =================================================================== --- sage/functions/special.py (revision 4053) +++ sage/functions/special.py (revision 2641) @@ -1,3 +1,3 @@ -r""" +r"""nodoctest -- TODO remove Special Functions @@ -5,6 +5,5 @@ -- David Joyner (2006-13-06) -This module provides easy access to many of Maxima and PARI's -special functions. +Some of Maxima's and Pari's special functions are wrapped. Maxima's special functions package (which includes spherical harmonic @@ -332,17 +331,12 @@ from sage.rings.rational_field import RationalField from sage.rings.real_mpfr import RealField -from sage.rings.complex_field import ComplexField from sage.misc.sage_eval import sage_eval -from sage.rings.all import ZZ, QQ, RR +from sage.rings.all import QQ, RR import sage.rings.commutative_ring as commutative_ring import sage.rings.ring as ring -from sage.interfaces.maxima import maxima - -def meval(x): - from sage.calculus.calculus import symbolic_expression_from_maxima_element - return symbolic_expression_from_maxima_element(maxima(x)) - from functions import * + +from sage.misc.functional import exp _done = False @@ -361,9 +355,4 @@ return RR, a -def _setup_CC(prec): - from sage.libs.pari.all import pari - CC = ComplexField(prec) - a = pari.set_real_precision(int(prec/3.3)+1) ## 3.3 < log(10,2) - return CC, a def bessel_I(nu,z,alg = "pari",prec=53): @@ -410,7 +399,7 @@ EXAMPLES: sage: bessel_I(1,1,"pari",500) - 0.565159103992485027207696027609863307328899621621092009480294489479255640964371134092664997766814410064677886055526302676857637684917179812041131208121 + 0.56515910399248502720769602760986330732889962162109200948029448947925564096437113409266499776681441006467788605552630267685763768491717981204113120812118 sage: bessel_I(1,1) - 0.565159103992485 + 0.56515910399248503 sage: bessel_I(2,1.1,"maxima") # last few digits are random 0.16708949925104899 @@ -427,5 +416,5 @@ return b else: - return sage_eval(maxima.eval("bessel_i(%s,%s)"%(float(nu),float(z)))) + return eval(maxima.eval("bessel_i(%s,%s)"%(float(nu),float(z)))) def bessel_J(nu,z,alg="pari",prec=53): @@ -484,12 +473,10 @@ if alg=="pari": from sage.libs.pari.all import pari - K,a = _setup(prec) - if not (z in K): - K, a = _setup_CC(prec) - b = K(pari(z).besselj(nu)) + R,a = _setup(prec) + b = R(pari(z).besselj(nu)) pari.set_real_precision(a) return b else: - return meval("bessel_j(%s,%s)"%(nu, z)) + return RR(maxima.eval("bessel_j(%s,%s)"%(float(nu),float(z)))) def bessel_K(nu,z,prec=53): @@ -513,7 +500,7 @@ EXAMPLES: sage: bessel_K(1,1) - 0.601907230197235 + 0.60190723019723458 sage: bessel_K(1,1,500) - 0.601907230197234574737540001535617339261586889968106456017767959168553582946237840168863706958258215354644099783140050908469292813493294605655726961996 + 0.60190723019723457473754000153561733926158688996810645601776795916855358294623784016886370695825821535464409978314005090846929281349329460565572696199608 """ from sage.libs.pari.all import pari @@ -564,7 +551,7 @@ EXAMPLES: sage: hypergeometric_U(1,1,1) - 0.596347362323194 + 0.59634736232319407 sage: hypergeometric_U(1,1,1,70) - 0.59634736232319407434 + 0.59634736232319407434152 """ from sage.libs.pari.all import pari @@ -574,23 +561,59 @@ return b -def spherical_bessel_J(n, var): +def incomplete_gamma(s,x,prec=53): + r""" + Implements the incomplete Gamma function. + + INPUT: + s, x -- ocmplex numbers. + prec -- bits of precision. + + The argument x and s are complex numbers + (x must be a positive real number if s = 0). + The result returned is $\int_x^\infty e^{-t}t^{s-1}dt$. + + EXAMPLES: + sage: incomplete_gamma(0.1,6,200) + 119.99999984701215693242493354706493878953914933130704861011488 + sage: incomplete_gamma(0,6,200) + 120.00000000000000000000000000000000000000000000000000000000000 + sage: incomplete_gamma(0.3,6,200) + 119.99990598341125737887779259683594390225182610507857843438320 + sage: incomplete_gamma(0.3,6) + 119.99990598341125 + sage: incomplete_gamma(0.5,6) + 119.99830020752132 + sage: incomplete_gamma(0.5,6,100) + 119.99830020752131890111421425093 + """ + from sage.libs.pari.all import pari + R,a = _setup(prec) + b = R(pari(x).incgam(s)) + pari.set_real_precision(a) + return b + +def spherical_bessel_J(n,x): r""" Returns the spherical Bessel function of the first kind for integers n > -1. - Reference: A&S 10.1.8 page 437 and A&S 10.1.15 page 439. EXAMPLES: + sage: x = PolynomialRing(QQ, 'x').gen() sage: spherical_bessel_J(2,x) - ((-(1 - (24/(8*x^2))))*sin(x) - (3*cos(x)/x))/x + '( - (1 - 24/(8*x^2))*sin(x) - 3*cos(x)/x)/x' + + Here I = sqrt(-1). + """ _init() - return meval("spherical_bessel_j(%s,%s)"%(ZZ(n),var)) - -def spherical_bessel_Y(n,var): + R = x.parent() + y = R.gen() + return maxima.eval("spherical_bessel_j(%s,%s)"%(n,y)).replace("%i","I") + +def spherical_bessel_Y(n,x): r""" Returns the spherical Bessel function of the second kind for integers n > -1. - Reference: A&S 10.1.9 page 437 and A&S 10.1.15 page 439. @@ -598,10 +621,15 @@ sage: x = PolynomialRing(QQ, 'x').gen() sage: spherical_bessel_Y(2,x) - (-(3*sin(x)/x - (1 - (24/(8*x^2)))*cos(x)))/x + '-(3*sin(x)/x - (1 - 24/(8*x^2))*cos(x))/x' + + Here I = sqrt(-1). + """ _init() - return meval("spherical_bessel_y(%s,%s)"%(ZZ(n),var)) - -def spherical_hankel1(n,var): + R = x.parent() + y = R.gen() + return maxima.eval("spherical_bessel_y(%s,%s)"%(n,y)).replace("%i","I") + +def spherical_hankel1(n,x): r""" Returns the spherical Hankel function of the first @@ -611,8 +639,12 @@ EXAMPLES: sage: spherical_hankel1(2,'x') - -3*I*(-x^2/3 - I*x + 1)*e^(I*x)/x^3 + '-3*I*( - x^2/3 - I*x + 1)*%e^(I*x)/x^3' + + Here I = sqrt(-1). + """ _init() - return meval("spherical_hankel1(%s,%s)"%(ZZ(n),var)) + y = str(x) + return maxima.eval("spherical_hankel1(%s,%s)"%(n,y)).replace("%i","I") def spherical_hankel2(n,x): @@ -624,5 +656,5 @@ EXAMPLES: sage: spherical_hankel2(2,'x') - '3*I*(-x^2/3+I*x+1)*%e^-(I*x)/x^3' + '3*I*( - x^2/3 + I*x + 1)*%e^-(I*x)/x^3' Here I = sqrt(-1). @@ -640,11 +672,30 @@ EXAMPLES: + sage: x = PolynomialRing(QQ, 'x').gen() + sage: y = PolynomialRing(QQ, 'y').gen() sage: spherical_harmonic(3,2,x,y) - 15*sqrt(7)*cos(x)*sin(x)^2*e^(2*I*y)/(4*sqrt(30)*sqrt(pi)) + '15*sqrt(7)*cos(x)*sin(x)^2*e^(2*I*y)/(4*sqrt(30)*sqrt(pi))' sage: spherical_harmonic(3,2,1,2) - 15*sqrt(7)*e^(4*I)*cos(1)*sin(1)^2/(4*sqrt(30)*sqrt(pi)) + -0.25556469795208248 - 0.29589824630616246*I + + Here I = sqrt(-1). + """ _init() - return meval("spherical_harmonic(%s,%s,%s,%s)"%(ZZ(m),ZZ(n),x,y)) + if not(is_Polynomial(x) and is_Polynomial(y)): + s1 = maxima.eval("spherical_harmonic(%s,%s,%s,%s)"%(m,n,x,y)) + s2 = s1.replace("%i","I") + s3 = s2.replace("%pi","pi") + s4 = s3.replace("%e","CC(e)") + return sage_eval(s4) + R = x.parent() + x1 = R.gen() + R = y.parent() + y1 = R.gen() + s1 = maxima.eval("spherical_harmonic(%s,%s,%s,%s)"%(m,n,x,y)) + s2 = s1.replace("%i","I") + s3 = s2.replace("%pi","pi") + s4 = s3.replace("%e","e") + return s4 ####### elliptic functions and integrals @@ -659,19 +710,18 @@ EXAMPLES: sage: jacobi("sn",1,1) - tanh(1) + 0.76159415595576485 sage: jacobi("cd",1,1/2) - jacobi_cd(1, 1/2) - sage: RDF(jacobi("cd",1,1/2)) - 0.724009721659 - sage: RDF(jacobi("cn",1,1/2)); RDF(jacobi("dn",1,1/2)); RDF(jacobi("cn",1,1/2)/jacobi("dn",1,1/2)) - 0.595976567672 - 0.823161001632 - 0.724009721659 - sage: jsn = jacobi("sn",x,1) - sage: P = plot(jsn,0,1) - sage.: P.show() - """ - _init() - return meval("jacobi_%s(%s,%s)"%(sym, x, m)) + 0.72400972165937116 + sage: jacobi("cn",1,1/2);jacobi("dn",1,1/2);jacobi("cn",1,1/2)/jacobi("dn",1,1/2) + 0.59597656767214113 + 0.82316100163159622 + 0.72400972165937116 + sage: jsn = lambda x: jacobi("sn",x,1) + sage: P= plot(jsn,0,1) + + Now to view this, just type show(P). + + """ + return eval(maxima.eval("jacobi_%s(%s,%s)"%(sym, float(x), float(m)))) def inverse_jacobi(sym,x,m): @@ -683,17 +733,15 @@ EXAMPLES: sage: jacobi("sn",1/2,1/2) - jacobi_sn(1/2, 1/2) - sage: float(jacobi("sn",1/2,1/2)) 0.4707504736556572 - sage: float(inverse_jacobi("sn",0.47,1/2)) + sage: inverse_jacobi("sn",0.47,1/2) 0.4990982313222197 - sage: float(inverse_jacobi("sn",0.4707504,0.5)) + sage: inverse_jacobi("sn",0.4707504,1/2) 0.49999991146655459 - sage: P = plot(inverse_jacobi('sn', x, 0.5), 0, 1, plot_points=20) + sage: ijsn = lambda x: inverse_jacobi("sn",x,1/2) + sage: P= plot(ijsn,0,1) Now to view this, just type show(P). """ - _init() - return meval("inverse_jacobi_%s(%s,%s)"%(sym, x,m)) + return eval(maxima.eval("inverse_jacobi_%s(%s,%s)"%(sym, float(x),float(m)))) #### elliptic integrals @@ -702,35 +750,39 @@ #### of Jacobi elliptic functions but faster to evaluate directly) -## def sinh(t): -## try: -## return t.sinh() -## except AttributeError: -## from sage.calculus.calculus import exp -## return (exp(t)-exp(-t))/2 - -## def cosh(t): -## try: -## return t.cosh() -## except AttributeError: -## from sage.calculus.calculus import exp -## return (exp(t)+exp(-t))/2 - -## def coth(t): -## try: -## return t.coth() -## except AttributeError: -## return 1/tanh(t) - -## def sech(t): -## try: -## return t.sech() -## except AttributeError: -## return 1/cosh(t) - -## def csch(t): -## try: -## return t.csch() -## except AttributeError: -## return 1/sinh(t) +def sinh(t): + try: + return t.sinh() + except AttributeError: + return (exp(t)-exp(-t))/2 + +def cosh(t): + try: + return t.cosh() + except AttributeError: + return (exp(t)+exp(-t))/2 + +def tanh(t): + try: + return t.tanh() + except AttributeError: + return sinh(t)/cosh(t) + +def coth(t): + try: + return t.coth() + except AttributeError: + return 1/tanh(t) + +def sech(t): + try: + return t.sech() + except AttributeError: + return 1/cosh(t) + +def csch(t): + try: + return t.csch() + except AttributeError: + return 1/sinh(t) def dilog(t): Index: sage/functions/transcendental.py =================================================================== --- sage/functions/transcendental.py (revision 4055) +++ sage/functions/transcendental.py (revision 3301) @@ -148,11 +148,16 @@ sage: zeta(RR(2)) 1.6449340668482264364724151666460251892189499012067984377356 - sage: zeta(I) - 0.00330022368532410 - 0.418155449141322*I """ try: return s.zeta() except AttributeError: - return ComplexField()(s).zeta() + return RealField()(s).zeta() + +## prec = s.prec() +## s = pari.new_with_prec(s, prec) +## z = s.zeta()._sage_() +## if z.prec() < prec: +## raise RuntimeError, "Error computing zeta(%s) -- precision loss."%s +## return z def zeta_symmetric(s): Index: sage/geometry/lattice_polytope.py =================================================================== --- sage/geometry/lattice_polytope.py (revision 4061) +++ sage/geometry/lattice_polytope.py (revision 2915) @@ -1098,5 +1098,5 @@ Return a string representation of this face. """ - return repr(self._vertices) + return str(self._vertices) def boundary_points(self): Index: sage/graphs/graph.py =================================================================== --- sage/graphs/graph.py (revision 4359) +++ sage/graphs/graph.py (revision 3741) @@ -8,5 +8,5 @@ NetworkX-0.33, fixed plotting bugs (2007-01-23): basic tutorial, edge labels, loops, - multiple edges and arcs + multiple edges & arcs (2007-02-07): graph6 and sparse6 formats, matrix input -- Emily Kirkmann (2007-02-11): added graph_border option to plot and show @@ -216,7 +216,4 @@ from sage.plot.plot import Graphics, GraphicPrimitive_NetworkXGraph import sage.graphs.graph_fast as graph_fast -from sage.rings.integer import Integer -from sage.rings.integer_ring import ZZ - class GenericGraph(SageObject): @@ -648,19 +645,15 @@ return self._nxg.nodes() - def relabel(self, perm, inplace=True, quick=False): + def relabel(self, perm, inplace=True): r""" Uses a dictionary or permutation to relabel the (di)graph. If perm is a dictionary, each old vertex v is a key in the dictionary, and its new label is d[v]. If perm is a list, - we think of it as a map $i \mapsto perm[i]$ (only for graphs - with $V = \{0,1,...,n-1\}$ ). If perm is a per mutation, the + we think of it as a map i \mapsto perm[i] (only for graphs + with V = {0,1,...,n-1} ). If perm is a per mutation, the permutation is simply applied to the graph, under the - assumption that $V = \{0,1,...,n-1\}$ is the vertex set, and - the permutation acts on the set $\{1,2,...,n\}$, where we think - of $n = 0$. - - INPUT: - quick -- if True, simply return the enumeration of the new graph - without constructing it. Requires that perm is of type list. + assumption that V = {0,1,...,n-1} is the vertex set, and + the permutation acts on the set {1,2,...,n}, where we think + of n = 0. EXAMPLES: @@ -696,15 +689,4 @@ """ if type(perm) == list: - if quick: - n = self.order() - numbr = 0 - if isinstance(self, Graph): - for i,j,l in self.edge_iterator(): - numbr += 1<<((n-(perm[i]+1))*n + n-(perm[j]+1)) - numbr += 1<<((n-(perm[j]+1))*n + n-(perm[i]+1)) - elif isinstance(self, DiGraph): - for i,j,l in self.arc_iterator(): - numbr += 1<<((n-(perm[i]+1))*n + n-(perm[j]+1)) - return numbr if isinstance(self, Graph): oldd = self._nxg.adj @@ -826,5 +808,5 @@ def plot(self, pos=None, layout=None, vertex_labels=True, edge_labels=False, node_size=200, graph_border=False, color_dict=None, partition=None, - edge_colors=None, scaling_term=0.05, xmin=None, xmax=None): # xmin and xmax are ignored + edge_colors=None, scaling_term=0.05): """ Returns a graphics object representing the (di)graph. @@ -1159,4 +1141,5 @@ if not isinstance(data, str): raise ValueError, 'If input format is graph6, then data must be a string' + from sage.rings.integer import Integer n = data.find('\n') if n == -1: @@ -1168,12 +1151,14 @@ k = 0 for i in range(n): - d[i] = {} for j in range(i): if m[k] == '1': - d[i][j] = None + if d.has_key(i): + d[i][j] = None + else: + d[i] = {j : None} k += 1 self._nxg = networkx.XGraph(d) elif format == 'sparse6': - from math import ceil, floor + from sage.rings.arith import ceil, floor from sage.misc.functional import log n = data.find('\n') @@ -1182,5 +1167,5 @@ s = data[:n] n, s = graph_fast.N_inverse(s[1:]) - k = int(ceil(log(n,2))) + k = ceil(log(n,2)) l = [graph_fast.binary(ord(i)-63) for i in s] for i in range(len(l)): @@ -1189,5 +1174,5 @@ b = [] x = [] - for i in range(int(floor(len(bits)/(k+1)))): + for i in range(floor(len(bits)/(k+1))): b.append(int(bits[(k+1)*i:(k+1)*i+1],2)) x.append(int(bits[(k+1)*i+1:(k+1)*i+k+1],2)) @@ -1925,7 +1910,7 @@ # encode bit vector - from math import ceil + from sage.rings.arith import ceil from sage.misc.functional import log - k = int(ceil(log(n,2))) + k = ceil(log(n,2)) v = 0 i = 0 @@ -2037,8 +2022,5 @@ ### Visualization - def plot3d(self, bgcolor=(1,1,1), - vertex_color=(1,0,0), vertex_size=0.06, - edge_color=(0,0,0), edge_size=0.02, - pos3d=None, **kwds): + def plot3d(self, bgcolor=(1,1,1), vertex_color=(1,0,0), edge_color=(0,0,0), pos3d=None): """ Plots the graph using Tachyon, and returns a Tachyon object containing @@ -2046,11 +2028,8 @@ INPUT: - bgcolor -- rgb tuple (default: (1,1,1)) - vertex_color -- rgb tuple (default: (1,0,0)) - vertex_size -- float (default: 0.06) - edge_color -- rgb tuple (default: (0,0,0)) - edge_size -- float (default: 0.02) + bgcolor + vertex_color + edge_color pos3d -- a position dictionary for the vertices - **kwds -- passed on to the Tachyon command EXAMPLES: @@ -2065,10 +2044,9 @@ sage: C.plot3d(edge_color=(0,1,0), vertex_color=(1,1,1), bgcolor=(0,0,0)).save('sage.png') # long time """ - TT, pos3d = tachyon_vertex_plot(self, bgcolor=bgcolor, vertex_color=vertex_color, - vertex_size=vertex_size, pos3d=pos3d, **kwds) + TT, pos3d = tachyon_vertex_plot(self, bgcolor=bgcolor, vertex_color=vertex_color, pos3d=pos3d) TT.texture('edge', ambient=0.1, diffuse=0.9, specular=0.03, opacity=1.0, color=edge_color) for u,v,l in self.edges(): TT.fcylinder( (pos3d[u][0],pos3d[u][1],pos3d[u][2]),\ - (pos3d[v][0],pos3d[v][1],pos3d[v][2]), edge_size,'edge') + (pos3d[v][0],pos3d[v][1],pos3d[v][2]), .02,'edge') return TT @@ -2227,8 +2205,5 @@ else: a = search_tree(self, partition, dict=False, lab=False, dig=self.loops()) - if len(a) != 0: - a = PermutationGroup([perm_group_elt(aa) for aa in a]) - else: - a = PermutationGroup([[]]) + a = PermutationGroup([perm_group_elt(aa) for aa in a]) if translation: return a,b @@ -3224,5 +3199,5 @@ Returns a copy of digraph with arcs reversed in direction. - TODO: results in error because of the following NetworkX bug (0.33) - trac 92 + TODO: results in error because of the following NetworkX bug (0.33) - trac #92 EXAMPLES: @@ -3273,9 +3248,5 @@ ### Visualization - def plot3d(self, bgcolor=(1,1,1), vertex_color=(1,0,0), - vertex_size=0.06, - arc_size=0.02, - arc_size2=0.0325, - arc_color=(0,0,0), pos3d=None, **kwds): + def plot3d(self, bgcolor=(1,1,1), vertex_color=(1,0,0), arc_color=(0,0,0), pos3d=None): """ Plots the graph using Tachyon, and returns a Tachyon object containing @@ -3283,13 +3254,9 @@ INPUT: - bgcolor -- rgb tuple (default: (1,1,1)) - vertex_color -- rgb tuple (default: (1,0,0)) - vertex_size -- float (default: 0.06) - arc_color -- rgb tuple (default: (0,0,0)) - arc_size -- float (default: 0.02) - arc_size2 -- float (default: 0.0325) + bgcolor + vertex_color + arc_color pos3d -- a position dictionary for the vertices - **kwds -- passed on to the Tachyon command - + NOTE: The weaknesses of the NetworkX spring layout are illustrated even further in the @@ -3298,30 +3265,29 @@ EXAMPLE: - This is a running example + # This is a running example - A directed version of the dodecahedron + # A directed version of the dodecahedron sage: D = DiGraph( { 0: [1, 10, 19], 1: [8, 2], 2: [3, 6], 3: [19, 4], 4: [17, 5], 5: [6, 15], 6: [7], 7: [8, 14], 8: [9], 9: [10, 13], 10: [11], 11: [12, 18], 12: [16, 13], 13: [14], 14: [15], 15: [16], 16: [17], 17: [18], 18: [19], 19: []} ) - If I use an undirected version of my graph, the output is as expected + # If I use an undirected version of my graph, the output is as expected sage: import networkx sage: pos3d=networkx.spring_layout(graphs.DodecahedralGraph()._nxg, dim=3) sage: D.plot3d(pos3d=pos3d).save('sage.png') # long time - However, if I use the directed version, everything gets skewed bizarrely: + # However, if I use the directed version, everything gets skewed bizarrely: sage: D.plot3d().save('sage.png') # long time """ - TT, pos3d = tachyon_vertex_plot(self, bgcolor=bgcolor, vertex_color=vertex_color, - vertex_size=vertex_size, pos3d=pos3d, **kwds) + TT, pos3d = tachyon_vertex_plot(self, bgcolor=bgcolor, vertex_color=vertex_color, pos3d=pos3d) TT.texture('arc', ambient=0.1, diffuse=0.9, specular=0.03, opacity=1.0, color=arc_color) for u,v,l in self.arcs(): TT.fcylinder( (pos3d[u][0],pos3d[u][1],pos3d[u][2]),\ - (pos3d[v][0],pos3d[v][1],pos3d[v][2]), arc_size,'arc') + (pos3d[v][0],pos3d[v][1],pos3d[v][2]), .02,'arc') TT.fcylinder( (0.25*pos3d[u][0] + 0.75*pos3d[v][0],\ 0.25*pos3d[u][1] + 0.75*pos3d[v][1],\ 0.25*pos3d[u][2] + 0.75*pos3d[v][2],), - (pos3d[v][0],pos3d[v][1],pos3d[v][2]), arc_size2,'arc') + (pos3d[v][0],pos3d[v][1],pos3d[v][2]), .0325,'arc') return TT - def show3d(self, bgcolor=(1,1,1), vertex_color=(1,0,0), arc_color=(0,0,0), pos3d=None, **kwds): + def show3d(self, bgcolor=(1,1,1), vertex_color=(1,0,0), edge_color=(0,0,0), pos3d=None, **kwds): """ Plots the graph using Tachyon, and shows the resulting plot. @@ -3339,15 +3305,15 @@ EXAMPLE: - This is a running example + # This is a running example - A directed version of the dodecahedron + # A directed version of the dodecahedron sage: D = DiGraph( { 0: [1, 10, 19], 1: [8, 2], 2: [3, 6], 3: [19, 4], 4: [17, 5], 5: [6, 15], 6: [7], 7: [8, 14], 8: [9], 9: [10, 13], 10: [11], 11: [12, 18], 12: [16, 13], 13: [14], 14: [15], 15: [16], 16: [17], 17: [18], 18: [19], 19: []} ) - If I use an undirected version of my graph, the output is as expected + # If I use an undirected version of my graph, the output is as expected sage: import networkx sage: pos3d=networkx.spring_layout(graphs.DodecahedralGraph()._nxg, dim=3) sage: D.plot3d(pos3d=pos3d).save('sage.png') # long time - However, if I use the directed version, everything gets skewed bizarrely: + # However, if I use the directed version, everything gets skewed bizarrely: sage: D.plot3d().save('sage.png') # long time """ @@ -3385,8 +3351,5 @@ else: a = search_tree(self, partition, dict=False, lab=False, dig=True) - if len(a) != 0: - a = PermutationGroup([perm_group_elt(aa) for aa in a]) - else: - a = PermutationGroup([[]]) + a = PermutationGroup([perm_group_elt(aa) for aa in a]) if translation: return a,b @@ -3420,5 +3383,5 @@ break return True, map - else: + else: return False, None else: @@ -3448,8 +3411,5 @@ return b -def tachyon_vertex_plot(g, bgcolor=(1,1,1), - vertex_color=(1,0,0), - vertex_size=0.06, - pos3d=None, **kwds): +def tachyon_vertex_plot(g, bgcolor=(1,1,1), vertex_color=(1,0,0), pos3d=None): import networkx from math import sqrt @@ -3482,5 +3442,5 @@ pos3d[v][1] = pos3d[v][1]/r pos3d[v][2] = pos3d[v][2]/r - TT = Tachyon(camera_center=(1.4,1.4,1.4), antialiasing=13, **kwds) + TT = Tachyon(camera_center=(1.4,1.4,1.4), antialiasing=13) TT.light((4,3,2), 0.02, (1,1,1)) TT.texture('node', ambient=0.1, diffuse=0.9, specular=0.03, opacity=1.0, color=vertex_color) @@ -3488,48 +3448,34 @@ TT.plane((-1.6,-1.6,-1.6), (1.6,1.6,1.6), 'bg') for v in verts: - TT.sphere((pos3d[v][0],pos3d[v][1],pos3d[v][2]), vertex_size, 'node') + TT.sphere((pos3d[v][0],pos3d[v][1],pos3d[v][2]), .06, 'node') return TT, pos3d -def enum(graph, quick=False): +def enum(graph): """ Used for isomorphism checking. - - INPUT: - quick -- now we know that the vertices are 0,1,...,n-1 EXAMPLES: sage: from sage.graphs.graph import enum sage: enum(graphs.DodecahedralGraph()) - 646827340296833569479885332381965103655612500627043016896502674924517797573929148319427466126170568392555309533861838850 + 646827340296833569479885332381965103655612500627043016896502674924517797573929148319427466126170568392555309533861838850L sage: enum(graphs.MoebiusKantorGraph()) - 29627597595494233374689380190219099810725571659745484382284031717525232288040 + 29627597595494233374689380190219099810725571659745484382284031717525232288040L sage: enum(graphs.FlowerSnark()) - 645682215283153372602620320081348424178216159521280462146968720908564261127120716040952785862033320307812724373694972050 + 645682215283153372602620320081348424178216159521280462146968720908564261127120716040952785862033320307812724373694972050L sage: enum(graphs.CubeGraph(3)) - 6100215452666565930 + 6100215452666565930L sage: enum(graphs.CubeGraph(4)) - 31323620658472264895128471376615338141839885567113523525061169966087480352810 + 31323620658472264895128471376615338141839885567113523525061169966087480352810L sage: enum(graphs.CubeGraph(5)) - 56178607138625465573345383656463935701397275938329921399526324254684498525419117323217491887221387354861371989089284563861938014744765036177184164647909535771592043875566488828479926184925998575521710064024379281086266290501476331004707336065735087197243607743454550839234461575558930808225081956823877550090 + 56178607138625465573345383656463935701397275938329921399526324254684498525419117323217491887221387354861371989089284563861938014744765036177184164647909535771592043875566488828479926184925998575521710064024379281086266290501476331004707336065735087197243607743454550839234461575558930808225081956823877550090L sage: enum(graphs.CubeGraph(6)) - 17009933328531023098235951265708015080189260525466600242007791872273951170067729430659625711869482140011822425402311004663919203785115296476561677814427201708237805402966561863692388687547518491537427897858240566495945005294876576523289206747123399572439707189803821880345487300688962557172856432472391025950779306221469432919735886988596366979797317084123956762362685536557279604675024249987913439836592296340787741671304722135394212035449285260308821361913500205796919484488876249630521666898413890977354122918711285458724686283296097840711521153201188450783978019001984591992381570913097193343212274205747843852376395748070926193308573472616983062165141386183945049871456376379041631456999916186868438148001405477879591035696239287238767746380404501285533026300096772164676955425088646172718295360584249310479706751274583871684827338312536787740914529353458829503642591918761588296961192261166874864565050490306157300749101788751129640698534818737753110920871293122429238702542726347017441416450649382146313791818349648006634724962025571237208317435310419071153813687071275479812184286929976456778629116002591936357623320676067640749567446551071011889378108453641887998273235139859889734259803684619153716302058849155208478850 + 17009933328531023098235951265708015080189260525466600242007791872273951170067729430659625711869482140011822425402311004663919203785115296476561677814427201708237805402966561863692388687547518491537427897858240566495945005294876576523289206747123399572439707189803821880345487300688962557172856432472391025950779306221469432919735886988596366979797317084123956762362685536557279604675024249987913439836592296340787741671304722135394212035449285260308821361913500205796919484488876249630521666898413890977354122918711285458724686283296097840711521153201188450783978019001984591992381570913097193343212274205747843852376395748070926193308573472616983062165141386183945049871456376379041631456999916186868438148001405477879591035696239287238767746380404501285533026300096772164676955425088646172718295360584249310479706751274583871684827338312536787740914529353458829503642591918761588296961192261166874864565050490306157300749101788751129640698534818737753110920871293122429238702542726347017441416450649382146313791818349648006634724962025571237208317435310419071153813687071275479812184286929976456778629116002591936357623320676067640749567446551071011889378108453641887998273235139859889734259803684619153716302058849155208478850L """ + M = graph.am() enumeration = 0 n = graph.order() - if quick: - if isinstance(graph, Graph): - for i, j, l in graph.edge_iterator(): - enumeration += 1 << ((n-(i+1))*n + n-(j+1)) - enumeration += 1 << ((n-(j+1))*n + n-(i+1)) - elif isinstance(graph, DiGraph): - for i, j, l in graph.arc_iterator(): - enumeration += 1 << ((n-(i+1))*n + n-(j+1)) - return enumeration - M = graph.am() for i, j in M.nonzero_positions(): enumeration += 1 << ((n-(i+1))*n + n-(j+1)) - return ZZ(enumeration) - - - - + return enumeration + + Index: sage/graphs/graph_database.py =================================================================== --- sage/graphs/graph_database.py (revision 4359) +++ sage/graphs/graph_database.py (revision 3734) @@ -156,23 +156,23 @@ EXAMPLES: - Obtain a list of graphs: + # Obtain a list of graphs: sage: glist = graphs_query.get_list_of_graphs(max_degree=4, min_degree=3) - Inspect and display graphs individually: + # Inspect and display graphs individually: sage: glist[0] Graph on 5 vertices sage.: glist[8].show(layout='circular') - Now we can use functions from the graphs_list. - Convert to graph6 format: + # Now we can use functions from the graphs_list. + # Convert to graph6 format: sage: graph6list = graphs_list.to_graph6(glist, output_list=True) - View the graph6 format of the first graph: + # View the graph6 format of the first graph: sage: graph6list[0] 'Dl{' - Convert to list of graphics arrays: + # Convert to list of graphics arrays: garray = graphs_list.to_graphics_arrays(glist) - Show the last graphics array in the list: + # Show the last graphics array in the list: sage.: garray[len(garray)-1].show() """ @@ -215,6 +215,6 @@ EXAMPLES: - Recursive searching: - First note the following sizes of output: + # Recursive searching: + # First note the following sizes of output: sage: graphs_query.number_of(nodes=5) 34 @@ -224,5 +224,5 @@ 6 - Now find all graphs with 5 vertices and 4 edges: + # Now find all graphs with 5 vertices and 4 edges: sage: data = graphs_query.get_data_set(nodes=5) sage: redata = graphs_query.get_data_set(data_dict=data, edges=4) @@ -230,5 +230,5 @@ 6 - Note that this is equivalent to searching both at once: + # Note that this is equivalent to searching both at once: sage: redata == graphs_query.get_data_set(nodes=5, edges=4) True @@ -280,22 +280,21 @@ EXAMPLES: - Can obtain a list of graphics arrays for all graphs with 7 or fewer nodes: - (pretty fast too -- time it) + # Can obtain a list of graphics arrays for all graphs with 7 or fewer nodes: + # (pretty fast too -- time it) sage: all_7 = graphs_query.get_list_of_graphics_arrays() - And notice that Networkx's ordering structure is preserved. - Display the first graphics array in the list: + # And notice that Networkx's ordering structure is preserved. + # Display the first graphics array in the list: sage.: all_7[0].show() - - And the last: + # And the last: sage.: all_7[len(all_7)-1].show() - And also notice that displaying properties will make the list longer: - (Twice as many graphics objects in the arrays) + # And also notice that displaying properties will make the list longer: + # (Twice as many graphics objects in the arrays) sage: all_7_with_props = graphs_query.get_list_of_graphics_arrays(with_properties=True) sage: len(all_7) < len(all_7_with_props) True - Properties are displayed to the right of each graph + # Properties are displayed to the right of each graph sage.: all_7_with_props[5].show() """ @@ -417,16 +416,15 @@ EXAMPLES: - Properties are displayed to the right of each graph: + # Properties are displayed to the right of each graph: sage.: graphs_query.show_graphs(nodes=3, with_properties=True) - Without properties, you can show up to 20 graphs. - But with properties, you are limited to 10. + # Without properties, you can show up to 20 graphs. + # But with properties, you are limited to 10. sage: graphs_query.number_of(nodes=4) 11 - Without displaying properties (default): + # Without displaying properties (default): sage.: graphs_query.show_graphs(nodes=4) - - But with properties, we will raise an exception: + # But with properties, we will raise an exception: sage: graphs_query.show_graphs(nodes=4, with_properties=True) Traceback (most recent call last): @@ -435,5 +433,5 @@ If more than 10 graphs, try get_list_of_graphics_arrays. - In this case, use get_list_of_graphics_arrays: + # In this case, use get_list_of_graphics_arrays: sage: garray = graphs_query.get_list_of_graphics_arrays(nodes=4, with_properties=True) sage.: garray[0].show() Index: sage/graphs/graph_fast.pyx =================================================================== --- sage/graphs/graph_fast.pyx (revision 4176) +++ sage/graphs/graph_fast.pyx (revision 3744) @@ -229,4 +229,7 @@ sage_free(disp) + + + def binary(n, length=None): """ Index: sage/graphs/graph_generators.py =================================================================== --- sage/graphs/graph_generators.py (revision 4359) +++ sage/graphs/graph_generators.py (revision 3734) @@ -178,10 +178,10 @@ EXAMPLES: - Construct and show a barbell graph - Bar = 4, Bells = 9 + # Construct and show a barbell graph + # Bar = 4, Bells = 9 sage: g = graphs.BarbellGraph(9,4) sage.: g.show() - Create several barbell graphs in a SAGE graphics array + # Create several barbell graphs in a SAGE graphics array sage: g = [] sage: j = [] @@ -237,5 +237,5 @@ EXAMPLES: - Construct and show a bull graph + # Construct and show a bull graph sage: g = graphs.BullGraph() sage.: g.show() @@ -268,9 +268,9 @@ EXAMPLES: - Construct and show a circular ladder graph with 26 nodes + # Construct and show a circular ladder graph with 26 nodes sage: g = graphs.CircularLadderGraph(13) sage.: g.show() - Create several circular ladder graphs in a SAGE graphics array + # Create several circular ladder graphs in a SAGE graphics array sage: g = [] sage: j = [] @@ -312,8 +312,8 @@ EXAMPLES: - Show a Claw graph + # Show a Claw graph sage.: (graphs.ClawGraph()).show() - Inspect a Claw graph + # Inspect a Claw graph sage: G = graphs.ClawGraph() sage: G @@ -347,14 +347,26 @@ Filling the position dictionary in advance adds O(n) to the - constructor. - - EXAMPLES: - Compare plotting using the predefined layout and networkx: + constructor. Feel free to race the constructors below in the + examples section. The much larger difference is the time added + by the spring-layout algorithm when plotting. (Also shown in the + example below). The spring model is typically described as O(n^3), + as appears to be the case in the NetworkX source code. + + EXAMPLES: + # Compare the constructors (results will vary) sage: import networkx + sage.: time n = networkx.cycle_graph(3989); spring3989 = Graph(n) + # CPU time: 0.05 s, Wall time: 0.07 s + sage.: time posdict3989 = graphs.CycleGraph(3989) + # CPU time: 5.18 s, Wall time: 6.17 s + + # Compare the plotting speeds (results will vary) sage: n = networkx.cycle_graph(23) sage: spring23 = Graph(n) sage: posdict23 = graphs.CycleGraph(23) - sage.: spring23.show() - sage.: posdict23.show() + sage.: time spring23.show() + # CPU time: 2.04 s, Wall time: 2.72 s + sage.: time posdict23.show() + # CPU time: 0.57 s, Wall time: 0.71 s We next view many cycle graphs as a SAGE graphics array. @@ -420,5 +432,5 @@ EXAMPLES: - Construct and show a diamond graph + # Construct and show a diamond graph sage: g = graphs.DiamondGraph() sage.: g.show() @@ -445,10 +457,10 @@ EXAMPLES: - Construct and show a Dodecahdedral graph + # Construct and show a Dodecahdedral graph sage: g = graphs.DodecahedralGraph() sage.: g.show() - Create several dodecahedral graphs in a SAGE graphics array - They will be drawn differently due to the use of the spring-layout algorithm + # Create several dodecahedral graphs in a SAGE graphics array + # They will be drawn differently due to the use of the spring-layout algorithm sage: g = [] sage: j = [] @@ -482,10 +494,10 @@ EXAMPLES: - Add one vertex to an empty graph and then show: + # Add one vertex to an empty graph and then show: sage: empty1 = graphs.EmptyGraph() sage: empty1.add_vertex() sage.: empty1.show() - Use for loops to build a graph from an empty graph: + # Use for loops to build a graph from an empty graph: sage: empty2 = graphs.EmptyGraph() sage: for i in range(5): @@ -520,6 +532,6 @@ EXAMPLES: - Construct and show a grid 2d graph - Rows = 5, Columns = 7 + # Construct and show a grid 2d graph + # Rows = 5, Columns = 7 sage: g = graphs.Grid2dGraph(5,7) sage.: g.show() @@ -555,5 +567,5 @@ EXAMPLES: - Construct and show a house graph + # Construct and show a house graph sage: g = graphs.HouseGraph() sage.: g.show() @@ -585,5 +597,5 @@ EXAMPLES: - Construct and show a house X graph + # Construct and show a house X graph sage: g = graphs.HouseXGraph() sage.: g.show() @@ -626,5 +638,5 @@ EXAMPLE: - Construct and show a Krackhardt kite graph + # Construct and show a Krackhardt kite graph sage: g = graphs.KrackhardtKiteGraph() sage.: g.show() @@ -652,9 +664,9 @@ EXAMPLES: - Construct and show a ladder graph with 14 nodes + # Construct and show a ladder graph with 14 nodes sage: g = graphs.LadderGraph(7) sage.: g.show() - Create several ladder graphs in a SAGE graphics array + # Create several ladder graphs in a SAGE graphics array sage: g = [] sage: j = [] @@ -699,10 +711,10 @@ EXAMPLES: - Construct and show a lollipop graph - Candy = 13, Stick = 4 + # Construct and show a lollipop graph + # Candy = 13, Stick = 4 sage: g = graphs.LollipopGraph(13,4) sage.: g.show() - Create several lollipop graphs in a SAGE graphics array + # Create several lollipop graphs in a SAGE graphics array sage: g = [] sage: j = [] @@ -753,10 +765,10 @@ EXAMPLES: - Construct and show an Octahedral graph + # Construct and show an Octahedral graph sage: g = graphs.OctahedralGraph() sage.: g.show() - Create several octahedral graphs in a SAGE graphics array - They will be drawn differently due to the use of the spring-layout algorithm + # Create several octahedral graphs in a SAGE graphics array + # They will be drawn differently due to the use of the spring-layout algorithm sage: g = [] sage: j = [] @@ -806,18 +818,18 @@ EXAMPLES: - Show default drawing by size: - 'line': n < 11 + # Show default drawing by size: + # 'line': n < 11 sage: p = graphs.PathGraph(10) sage.: p.show() - 'circle': 10 < n < 41 + # 'circle': 10 < n < 41 sage: q = graphs.PathGraph(25) sage.: q.show() - 'line': n > 40 + # 'line': n > 40 sage: r = graphs.PathGraph(55) sage.: r.show() - Override the default drawing: + # Override the default drawing: sage: s = graphs.PathGraph(5,'circle') sage.: s.show() @@ -890,14 +902,29 @@ other nodes away from the (0) node, and thus look very similar to this constructor's positioning. - + + Filling the position dictionary in advance adds O(n) to the + constructor. Feel free to race the constructors below in the + examples section. The much larger difference is the time added + by the spring-layout algorithm when plotting. (Also shown in the + example below). The spring model is typically described as O(n^3), + as appears to be the case in the NetworkX source code. + EXAMPLES: sage: import networkx - Compare the plots: + Compare the constructors (results will vary) + sage.: time n = networkx.star_graph(3989); spring3989 = Graph(n) + # CPU time: 0.08 s, Wall time: 0.10 s + sage.: time posdict3989 = graphs.StarGraph(3989) + # CPU time: 5.43 s, Wall time: 7.41 s + + Compare the plotting speeds (results will vary) sage: n = networkx.star_graph(23) sage: spring23 = Graph(n) sage: posdict23 = graphs.StarGraph(23) - sage.: spring23.show() - sage.: posdict23.show() + sage.: time spring23.show() + # CPU time: 2.31 s, Wall time: 3.14 s + sage.: time posdict23.show() + # CPU time: 0.68 s, Wall time: 0.80 s View many star graphs as a SAGE Graphics Array @@ -968,14 +995,14 @@ EXAMPLES: - Construct and show a Tetrahedral graph + # Construct and show a Tetrahedral graph sage: g = graphs.TetrahedralGraph() sage: g.save('sage.png') - The following example requires networkx: + # The following example requires networkx: sage: import networkx as NX - Compare this Tetrahedral, Wheel(4), Complete(4), and the - Tetrahedral plotted with the spring-layout algorithm below - in a SAGE graphics array: + # Compare this Tetrahedral, Wheel(4), Complete(4), and the + # Tetrahedral plotted with the spring-layout algorithm below + # in a SAGE graphics array: sage: tetra_pos = graphs.TetrahedralGraph() sage: tetra_spring = Graph(NX.tetrahedral_graph()) @@ -1016,4 +1043,11 @@ (See Graphics Array examples below). + Filling the position dictionary in advance adds O(n) to the + constructor. Feel free to race the constructors below in the + examples section. The much larger difference is the time added + by the spring-layout algorithm when plotting. (Also shown in the + example below). The spring model is typically described as O(n^3), + as appears to be the case in the NetworkX source code. + EXAMPLES: We view many wheel graphs with a SAGE Graphics Array, first @@ -1052,10 +1086,18 @@ sage.: G.show() - Compare the plotting: + Compare the constructors (results will vary): + sage.: time n = networkx.wheel_graph(3989); spring3989 = Graph(n) + # CPU time: 0.07 s, Wall time: 0.09 s + sage.: time posdict3989 = graphs.WheelGraph(3989) + # CPU time: 5.99 s, Wall time: 8.74 s + + # Compare the plotting speeds (results will vary) sage: n = networkx.wheel_graph(23) sage: spring23 = Graph(n) sage: posdict23 = graphs.WheelGraph(23) - sage.: spring23.show() - sage.: posdict23.show() + sage.: time spring23.show() + # CPU time: 2.24 s, Wall time: 3.00 s + sage.: time posdict23.show() + # CPU time: 0.68 s, Wall time: 1.14 s """ pos_dict = {} @@ -1091,5 +1133,5 @@ EXAMPLES: - Inspect a flower snark: + # Inspect a flower snark: sage: F = graphs.FlowerSnark() sage: F @@ -1098,5 +1140,5 @@ 'ShCGHC@?GGg@?@?Gp?K??C?CA?G?_G?Cc' - Now show it: + # Now show it: sage.: F.show() """ @@ -1329,4 +1371,11 @@ below). + Filling the position dictionary in advance adds O(n) to the + constructor. Feel free to race the constructors below in the + examples section. The much larger difference is the time added + by the spring-layout algorithm when plotting. (Also shown in the + example below). The spring model is typically described as O(n^3), + as appears to be the case in the NetworkX source code. + EXAMPLES: We view many Complete graphs with a SAGE Graphics Array, first @@ -1364,19 +1413,18 @@ sage.: G = sage.plot.plot.GraphicsArray(j) sage.: G.show() - - Compare the constructors (results will vary) - sage: import networkx + # Compare the constructors (results will vary) sage.: time n = networkx.complete_graph(1559); spring1559 = Graph(n) - CPU time: 6.85 s, Wall time: 9.71 s + # CPU time: 6.85 s, Wall time: 9.71 s sage.: time posdict1559 = graphs.CompleteGraph(1559) - CPU time: 9.67 s, Wall time: 11.75 s - - We compare plotting: - sage: import networkx + #CPU time: 9.67 s, Wall time: 11.75 s + + We compare the plotting speeds (results will vary): sage: n = networkx.complete_graph(23) sage: spring23 = Graph(n) sage: posdict23 = graphs.CompleteGraph(23) - sage.: spring23.show() - sage.: posdict23.show() + sage.: time spring23.show() + # CPU time: 3.51 s, Wall time: 4.29 s + sage.: time posdict23.show() + # CPU time: 0.82 s, Wall time: 0.96 s """ pos_dict = {} @@ -1422,5 +1470,5 @@ examples section. The much larger difference is the time added by the spring-layout algorithm when plotting. (Also shown in the - example below). The spring model is typically described as $O(n^3)$, + example below). The spring model is typically described as O(n^3), as appears to be the case in the NetworkX source code. @@ -1510,5 +1558,5 @@ EXAMPLES: - Plot several n-cubes in a SAGE Graphics Array + # Plot several n-cubes in a SAGE Graphics Array sage: g = [] sage: j = [] @@ -1526,5 +1574,5 @@ sage.: G.show(figsize=[6,4]) - Use the plot options to display larger n-cubes + # Use the plot options to display larger n-cubes sage: g = graphs.CubeGraph(9) sage.: g.show(figsize=[12,12],vertex_labels=False, node_size=20) @@ -1612,11 +1660,11 @@ time regular_sparse = graphs.RandomGNP(1559,.22) - CPU time: 31.79 s, Wall time: 38.78 s + # CPU time: 31.79 s, Wall time: 38.78 s time fast_sparse = graphs.RandomGNPFast(1559,.22) - CPU time: 21.72 s, Wall time: 26.44 s + # CPU time: 21.72 s, Wall time: 26.44 s time regular_dense = graphs.RandomGNP(1559,.88) - CPU time: 38.75 s, Wall time: 47.65 s + # CPU time: 38.75 s, Wall time: 47.65 s time fast_dense = graphs.RandomGNP(1559,.88) - CPU time: 39.15 s, Wall time: 48.22 s + # CPU time: 39.15 s, Wall time: 48.22 s """ import networkx @@ -1636,9 +1684,9 @@ EXAMPLES: - Plot a random graph on 12 nodes with p = .71 + # Plot a random graph on 12 nodes with p = .71 sage: fast = graphs.RandomGNPFast(12,.71) sage.: fast.show() - View many random graphs using a SAGE Graphics Array + # View many random graphs using a SAGE Graphics Array sage: g = [] sage: j = [] Index: sage/graphs/graph_isom.py =================================================================== --- sage/graphs/graph_isom.py (revision 4359) +++ sage/graphs/graph_isom.py (revision 3880) @@ -1,36 +1,14 @@ """ -N.I.C.E. - Nice (as in open source) Isomorphism Check Engine - -Automorphism group computation and isomorphism checking for graphs. - -This is an open source implementation of Brendan McKay's algorithm for graph -automorphism and isomorphism. McKay released a C version of his algorithm, -named nauty (No AUTomorphisms, Yes?) under a license that is not GPL -compatible. Although the program is open source, reading the source disallows -anyone from recreating anything similar and releasing it under the GPL. Also, -many people have complained that the code is difficult to understand. The -first main goal of NICE was to produce a genuinely open graph isomorphism -program, which has been accomplished. The second goal is for this code to be -understandable, so that computed results can be trusted and further derived -work will be possible. - -To determine the isomorphism type of a graph, it is convenient to define a -canonical label for each isomorphism class- essentially an equivalence class -representative. Loosely (albeit incorrectly), the canonical label is defined -by enumerating all labeled graphs, then picking the maximal one in each -isomorphism class. The NICE algorithm is essentially a backtrack search. It -searches through the rooted tree of partition nests (where each partition is -equitable) for implicit and explicit automorphisms, and uses this information -to eliminate large parts of the tree from further searching. Since the leaves -of the search tree are all discrete ordered partitions, each one of these -corresponds to an ordering of the vertices of the graph, i.e. another member -of the isomorphism class. Once the algorithm has finished searching the tree, -it will know which leaf corresponds to the canonical label. In the process, -generators for the automorphism group are also produced. +Automorphism group computation and isomorphism checking for graphs + +This is a new open source implementation of Brendan McKay's algorithm +for graph automorphism and isomorphism. This is not a derived work of +nauty (one of the other restrictively licensed implementations of +McKay's algorithm), and is completely open source (released under the +GPL). AUTHORS: - * Robert L. Miller -- (2007-03-20) initial version - * Tom Boothby -- (2007-03-20) help with indicator function - * Robert L. Miller -- (2007-04-07--17) optimizations + Robert L. Miller -- (2007-03-20) initial version + Tom Boothby -- (2007-03-20) help with indicator function REFERENCE: @@ -39,13 +17,17 @@ NOTE: - Often we assume that G is a graph on vertices {0,1,...,n-1}. + Often we assume that G is a graph on vertices {0,1,...,n-1}, and gamma is + an element of SymmetricGroup(n), considered as action on the set + {1,2,...,n} where we take 0 == n. """ -#***************************************************************************** -# Copyright (C) 2006 - 2007 Robert L. Miller +############################################################################## +# +# Copyright (C) 2007 Robert L. Miller # # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ -#***************************************************************************** +# +############################################################################## from sage.graphs.graph import Graph, DiGraph @@ -167,5 +149,5 @@ INPUT: list_perm -- if True, assumes gamma is a list representing the map - i \mapsto gamma[i]. + i \mapsto gamma[i]. EXAMPLES: @@ -224,5 +206,5 @@ else: return False -def degree(G, v, W, bool_matrix_format=False): +def degree(G, v, W): """ Returns the number of edges from v to vertices in W, i.e. the degree of v @@ -234,35 +216,11 @@ sage: from sage.graphs.graph_isom import degree sage: P = graphs.PetersenGraph() - sage: M = P.adjacency_matrix() sage: Pi1 = [[0,2,3,4],[5,7,8,1],[6,9]] sage: degree(P, 6, Pi1[1]) 2 - sage: degree(M, 6, Pi1[1], bool_matrix_format=True) - 2 To see what is going on: sage.: P.show(partition=Pi1) - - NOTE: - Regarding the bool_matrix_format option, see the following timings: - sage.: import sage.graphs.graph_isom - sage.: from sage.graphs.graph_isom import degree - sage.: P = graphs.PetersenGraph() - sage.: M = P.adjacency_matrix() - sage.: str(M).replace('\n',',').replace(' ',',').replace('1','True').replace('0','False') - '[False,True,False,False,True,True,False,False,False,False],[True,False,True,False,False,False,True,False,False,False],[False,True,False,True,False,False,False,True,False,False],[False,False,True,False,True,False,False,False,True,False],[True,False,False,True,False,False,False,False,False,True],[True,False,False,False,False,False,False,True,True,False],[False,True,False,False,False,False,False,False,True,True],[False,False,True,False,False,True,False,False,False,True],[False,False,False,True,False,True,True,False,False,False],[False,False,False,False,True,False,True,True,False,False]' - sage.: M = [[False,True,False,False,True,True,False,False,False,False],[True,False,True,False,False,False,True,False,False,False],[False,True,False,True,False,False,False,True,False,False],[False,False,True,False,True,False,False,False,True,False],[True,False,False,True,False,False,False,False,False,True],[True,False,False,False,False,False,False,True,True,False],[False,True,False,False,False,False,False,False,True,True],[False,False,True,False,False,True,False,False,False,True],[False,False,False,True,False,True,True,False,False,False],[False,False,False,False,True,False,True,True,False,False]] - sage.: Pi1 = [[0,2,3,4],[5,7,8,1],[6,9]] - sage.: timeit degree(P, 6, Pi1[1]) - 10000 loops, best of 3: 24.9 [micro]s per loop - sage.: timeit degree(M, 6, Pi1[1], bool_matrix_format=True) - 100000 loops, best of 3: 8.27 [micro]s per loop - """ - if bool_matrix_format: - i = 0 - for u in W: - if G[u][v]: - i += 1 - return i + """ i = 0 for u in W: @@ -335,5 +293,5 @@ return PiNew -def sort_by_degree(G, A, B, bool_matrix_format=False): +def sort_by_degree(G, A, B): """ Assuming A and B are subsets of the vertex set of G, returns an ordered @@ -353,5 +311,5 @@ ddict = {} for a in A: - dd = degree(G, a, B, bool_matrix_format=bool_matrix_format) + dd = degree(G, a, B) if ddict.has_key(dd): ddict[dd].append(a) @@ -360,5 +318,5 @@ return ddict.values() -def refine(G, Pi, alpha, bool_matrix_format=False): +def refine(G, Pi, alpha): """ The key refinement procedure. Given a graph G, a partition Pi of the @@ -414,5 +372,5 @@ r = len(PiT) while k < r: - X = sort_by_degree(G, PiT[k], W, bool_matrix_format=bool_matrix_format) + X = sort_by_degree(G, PiT[k], W) s = len(X) if s != 1: @@ -456,12 +414,12 @@ return replace_in(Pi, i, [[v], L]) -def perp(G, Pi, v, bool_matrix_format=False): +def perp(G, Pi, v): """ Refines the partition Pi by cutting out a vertex, then using refine, comparing against only that vertex. """ - return refine(G, comp(Pi, v), [[v]], bool_matrix_format=bool_matrix_format) - -def partition_nest(G, Pi, V, bool_matrix_format=False): + return refine(G, comp(Pi, v), [[v]]) + +def partition_nest(G, Pi, V): """ Given a sequence of vertices V = (v_1,...,v_{m-1}) of a graph G, and a @@ -476,7 +434,7 @@ C.f. 2.9 in [1]. """ - L = [refine(G, Pi, Pi, bool_matrix_format=bool_matrix_format)] + L = [refine(G, Pi, Pi)] for i in range(len(V)): - L.append(perp(G, L[i], V[i], bool_matrix_format=bool_matrix_format)) + L.append(perp(G, L[i], V[i])) return L @@ -495,8 +453,12 @@ return Pi[i] -def indicator(G, Pi, V, k=None, bool_matrix_format=False): +def indicator(G, Pi, V): """ Takes a labelled graph, an ordered partition, and a partition nest, and outputs an integer, which is the same under any fixed permutation. + + AUTHORS: + Tom Boothby -- sum then product method + Robert Miller -- vertex degree check EXAMPLES: @@ -531,23 +493,14 @@ sage: indicator(G, [range(93)], partition_nest(G, [range(93)], V)) == indicator(H, [range(93)], partition_nest(H, [range(93)], [g.list()[v-1] for v in V])) True - - AUTHORS: - * Tom Boothby -- sum then product method - * Robert Miller -- vertex degree check - """ - if k is not None: - V = V[:k] - if bool_matrix_format: - n = len(G) - else: - n = G.order() + """ from sage.misc.misc import prod - LL = [] + LL = [0]*G.order() for partition in V: a = len(partition) for k in range(a): - LL.append( len(partition[k])*(1 + \ - sum( [ degree(G, partition[k][0], partition[i], bool_matrix_format=bool_matrix_format) for i in range(a) ] ) ) ) - return prod(LL) + LL[k] += len(partition[k])*(1 + \ + sum( [ degree(G, partition[k][0], partition[i]) for i in range(len(partition)) ] ) \ + ) + return prod([l for l in LL if l!=0]) def get_permutation(eta, nu, list_perm=False): @@ -638,5 +591,5 @@ return H -def search_tree(G, Pi, lab=True, dig=False, dict=False, proof=False, verbosity=0): +def search_tree(G, Pi, lab=True, dig=False, dict=False, proof=False): """ Assumes that the vertex set of G is {0,1,...,n-1} for some n. @@ -1034,28 +987,6 @@ from copy import copy from sage.rings.infinity import Infinity - from sage.graphs.graph import enum n = G.order() Pi = copy(Pi) - - if n == 0: - if lab: - H = copy(G) - if dict: - ddd = {} - if proof: - dd = {} - if dict: - return [[]], ddd, H, dd - else: - return [[]], H, dd - if lab and dict: - return [[]], ddd, H - elif lab: - return [[]], H - elif dict: - return [[]], ddd - else: - return [[]] - if proof: @@ -1079,16 +1010,4 @@ Pi = Pi2 - # create the boolean matrix - M = [] - for _ in range(n): - M.append( [False]*n ) - if isinstance(G, Graph): - for i, j, l in G.edge_iterator(): - M[i][j] = True - M[j][i] = True - elif isinstance(G, DiGraph): - for i, j, l in G.arc_iterator(): - M[i][j] = True - #begin BDM's algorithm: L = 100 @@ -1107,27 +1026,25 @@ output = [] k = None - h = None - hh = None +# h = None +# hh = None hb = None - hzb = None - hzf = None - qzb = None +# hzb = None +# hzf = None +# qzb = None while not state is None: - if verbosity > 1: - print 'k: ' + str(k) - print 'nu: ' + str(nu) - print 'rho: ' + str(rho) - if verbosity > 3: - print 'e: ' + str(e) - print 'hh: ' + str(hh) - print 'hb: ' + str(hb) - print 'h: ' + str(h) - print 'eta: ' + str(eta) - print 'zb: ' + str(zb) - print 'hzb: ' + str(hzb) - print 'hzf: ' + str(hzf) - print 'qzb: ' + str(qzb) +# print 'e: ' + str(e) +# print 'k: ' + str(k) +# print 'hh: ' + str(hh) +# print 'hb: ' + str(hb) +# print 'h: ' + str(h) +# print 'nu: ' + str(nu) +# print 'eta: ' + str(eta) +# print 'rho: ' + str(rho) +# print 'zb: ' + str(zb) +# print 'hzb: ' + str(hzb) +# print 'hzf: ' + str(hzf) +# print 'qzb: ' + str(qzb) if state == 1: - if verbosity > 0: print 'state: 1' +# print 'state: 1' size = 1 k = 1 @@ -1137,5 +1054,5 @@ l = 0 Theta = [[i] for i in range(n)] - nu[1] = refine(M, Pi, Pi, bool_matrix_format=True) + nu[1] = refine(G, Pi, Pi) hh = 2 if not dig: @@ -1149,8 +1066,8 @@ state = 2 elif state == 2: - if verbosity > 0: print 'state: 2' +# print 'state: 2' k += 1 - nu[k] = perp(M, nu[k-1], v[k-1], bool_matrix_format=True) - Lambda[k] = indicator(M, Pi, nu.values(), k, bool_matrix_format=True) + nu[k] = perp(G, nu[k-1], v[k-1]) + Lambda[k] = indicator(G, Pi, nu.values()) if h == 0: state = 5 else: @@ -1160,14 +1077,17 @@ state = 3 else: - qzb = Lambda[k] - zb[k] + if zb[k] == Infinity: # This replaces the last line, + qzb = -1 # since MinusInfinity is not + else: # yet implemented. + qzb = Lambda[k] - zb[k] # if hzb == k-1 and qzb == 0: hzb = k if qzb > 0: zb[k] = Lambda[k] state = 3 elif state == 3: - if verbosity > 0: print 'state: 3' +# print 'state: 3' if hzf <= k or (lab and qzb >= 0): state = 4 ##changed hzb to hzf, == to <= else: state = 6 elif state == 4: - if verbosity > 0: print 'state: 4' +# print 'state: 4' if is_discrete(nu[k]): state = 7 else: @@ -1178,10 +1098,10 @@ state = 2 elif state == 5: - if verbosity > 0: print 'state: 5' +# print 'state: 5' zf[k] = Lambda[k] zb[k] = Lambda[k] state = 4 elif state == 6: - if verbosity > 0: print 'state: 6' +# print 'state: 6' kprime = k k = min([ hh-1, max(ht-1,hzb) ]) @@ -1194,16 +1114,16 @@ state = 12 elif state == 7: - if verbosity > 0: print 'state: 7' +# print 'state: 7' if h == 0: state = 18 elif k < hzf: state = 8 ## BDM had !=, broke at G = Graph({0:[],1:[],2:[]}), Pi = [[0,1,2]] else: gamma = get_permutation(eta.values(), nu.values(), list_perm=True) - if verbosity > 2: print gamma - if enum(G, quick=True) == G.relabel(gamma, inplace=False, quick=True): # if G^gamma == G: + # print gamma + if G == G.relabel(gamma, inplace=False): # if G^gamma == G: state = 10 else: state = 8 elif state == 8: - if verbosity > 0: print 'state: 8' +# print 'state: 8' if (not lab) or (qzb < 0): state = 6 elif (qzb > 0) or (k < len(rho)): state = 9 @@ -1212,8 +1132,8 @@ else: gamma = get_permutation(nu.values(), rho.values(), list_perm=True) - if verbosity > 2: print gamma + # print gamma state = 10 elif state == 9: - if verbosity > 0: print 'state: 9' +# print 'state: 9' rho = copy(nu) qzb = 0 @@ -1223,5 +1143,5 @@ state = 6 elif state == 10: - if verbosity > 0: print 'state: 10' +# print 'state: 10' l = min([l+1,L]) Omega[l] = min_cell_reps(orbit_partition(gamma, list_perm=True)) @@ -1238,14 +1158,14 @@ state = 13 elif state == 11: - if verbosity > 0: print 'state: 11' +# print 'state: 11' k = hb state = 12 elif state == 12: - if verbosity > 0: print 'state: 12' +# print 'state: 12' if e[k] == 1: W[k] = [v for v in W[k] if v in Omega[l]] state = 13 elif state == 13: - if verbosity > 0: print 'state: 13' +# print 'state: 13' if k == 0: state = None else: @@ -1258,5 +1178,5 @@ state = 14 elif state == 14: - if verbosity > 0: print 'state: 14' +# print 'state: 14' for cell in Theta: if v[k] in cell: @@ -1273,5 +1193,5 @@ else: state = 15 elif state == 15: - if verbosity > 0: print 'state: 15' +# print 'state: 15' hh = min(hh,k+1) hzf = min(hzf,k) @@ -1282,5 +1202,5 @@ state = 2 elif state == 16: - if verbosity > 0: print 'state: 16' +# print 'state: 16' if len(W[k]) == index and ht == k+1: ht = k size = size*index @@ -1289,7 +1209,7 @@ state = 13 elif state == 17: - if verbosity > 0: print 'state: 17' +# print 'state: 17' if e[k] == 0: - li = W[k] + l = W[k] for i in range(1,l+1): boo = True @@ -1299,5 +1219,5 @@ break if boo: - li = [v for v in li if v in Omega[i]] + l = [v for v in l if v in Omega[i]] W[k] = l e[k] = 1 @@ -1311,5 +1231,5 @@ state = 13 elif state == 18: - if verbosity > 0: print 'state: 18' +# print 'state: 18' h = k ht = k @@ -1321,6 +1241,6 @@ else: rho = copy(nu) - hzb = k ## BDM had k+1 - hb = k ## BDM had k+1 + hzb = k#+1 # ??? + hb = k#+1 # ??? zb[k+2] = Infinity qzb = 0 Index: sage/graphs/graph_list.py =================================================================== --- sage/graphs/graph_list.py (revision 4359) +++ sage/graphs/graph_list.py (revision 3734) @@ -58,5 +58,5 @@ sage: l = ['N@@?N@UGAGG?gGlKCMO','XsGGWOW?CC?C@HQKHqOjYKC_uHWGX?P?~TqIKA`OA@SAOEcEA??'] sage: graphs_list.from_graph6(l) - [Graph on 15 vertices, Graph on 25 vertices] + [Graph on 14 vertices, Graph on 25 vertices] """ from sage.graphs.graph import Graph @@ -217,8 +217,8 @@ sage: garray = graphs_list.to_graphics_arrays(glist) - Display the first graphics array in the list. + # Display the first graphics array in the list. sage.: garray[0].show() - Display the last graphics array in the list. + # Display the last graphics array in the list. sage.: garray[len(garray)-1].show() """ @@ -280,5 +280,5 @@ EXAMPLES: - Create a list of graphs: + # Create a list of graphs: sage: glist = [] sage: glist.append(graphs.CompleteGraph(6)) @@ -297,12 +297,12 @@ sage: glist.append(graphs.WheelGraph(9)) - Check that length is <= 20: + # Check that length is <= 20: sage: len(glist) 14 - Show the graphs in a graphics array: + # Show the graphs in a graphics array: sage.: graphs_list.show_graphs(glist) - But more than 20 graphs will raise an exception: + # But more than 20 graphs will raise an exception: sage: glist21 = [] sage: for i in range(21): @@ -314,5 +314,5 @@ ValueError: List is too long to display in a graphics array. Try using the to_graphics_arrays function. - In this case, use to_graphics_arrays instead: + # In this case, use to_graphics_arrays instead: sage: garrays = graphs_list.to_graphics_arrays(glist21) sage.: garrays[0].show() Index: sage/groups/abelian_gps/dual_abelian_group_element.py =================================================================== --- sage/groups/abelian_gps/dual_abelian_group_element.py (revision 4063) +++ sage/groups/abelian_gps/dual_abelian_group_element.py (revision 3290) @@ -270,6 +270,5 @@ if is_ComplexField(F): I = F.gen() - PI = F(pi) - ans = prod([exp(2*PI*I*expsX[i]*expsg[i]/invs[i]) for i in range(len(expsX))]) + ans = prod([exp(2*pi*I*expsX[i]*expsg[i]/invs[i]) for i in range(len(expsX))]) return ans ans = F(1) ## assumes F is the cyclotomic field Index: sage/groups/group.pyx =================================================================== --- sage/groups/group.pyx (revision 3977) +++ sage/groups/group.pyx (revision 2419) @@ -121,20 +121,4 @@ return True - def cayley_graph(self): - from sage.graphs.graph import DiGraph - arrows = {} - for g in self: - for s in self.gens(): - gs = g*s - if not gs == g: - try: - _ = arrows[g] - except KeyError: - arrows[g] = {} - - arrows[g][gs] = s - - return DiGraph(arrows) - cdef class AlgebraicGroup(Group): """ Index: sage/gsl/dft.py =================================================================== --- sage/gsl/dft.py (revision 4063) +++ sage/gsl/dft.py (revision 3627) @@ -67,6 +67,8 @@ from sage.rings.real_mpfr import RR from sage.rings.all import CC, I -from sage.calculus.all import sin, cos -from sage.functions.constants import pi +from math import sin +from math import cos +from sage.functions.constants import Pi +pi = Pi() from sage.gsl.fft import FastFourierTransform from sage.gsl.dwt import WaveletTransform @@ -317,7 +319,6 @@ J = self.index_object() ## must be = range(N) N = len(J) - S = self.list() - PI = F(pi) - FT = [sum([S[i]*cos(2*PI*i/N) for i in J]) for j in J] + S = self.list() + FT = [sum([S[i]*cos(2*pi*i/N) for i in J]) for j in J] return IndexedSequence(FT,J) @@ -328,5 +329,4 @@ EXAMPLES: sage: J = range(5) - sage: I = CC.0; pi = CC(pi) sage: A = [exp(-2*pi*i*I/5) for i in J] sage: s = IndexedSequence(A,J) @@ -338,7 +338,6 @@ J = self.index_object() ## must be = range(N) N = len(J) - S = self.list() - PI = F(pi) - FT = [sum([S[i]*sin(2*PI*i/N) for i in J]) for j in J] + S = self.list() + FT = [sum([S[i]*sin(2*pi*i/N) for i in J]) for j in J] return IndexedSequence(FT,J) Index: sage/gsl/integration.pyx =================================================================== --- sage/gsl/integration.pyx (revision 4062) +++ sage/gsl/integration.pyx (revision 3908) @@ -75,5 +75,5 @@ We check this with a symbolic integration: - sage: (sin(x)^3+sin(x)).integral(x,0,pi) + sage: maxima('sin(x)^3+sin(x)').integral(x,0,pi) 10/3 @@ -103,10 +103,4 @@ the answer and whose second component is an error estimate. - REMARK: - There is also a method \code{nintegral} on symbolic expressions - that implements numerical integration using Maxima. It is potentially - very useful for symbolic expressions. - - MORE EXAMPLES: If we want to change the error tolerances and gauss rule used @@ -144,16 +138,10 @@ One can integrate any real-valued callable function: - sage: numerical_integral(lambda x: abs(zeta(x)), [1.1,1.5]) # slightly random output + sage: numerical_integral(zeta, [1.1,1.5]) # slightly random output (1.8488570602160455, 2.052643677492633e-14) - - We can also numerically integrate symbolic expressions using either this - function (which uses GSL) or the native integration (which uses Maxima): - sage: exp(-1/x).nintegral(x, 1, 2) # via maxima - (0.50479221787318396, 5.6043194293440752e-15, 21, 0) - sage: numerical_integral(exp(-1/x), 1, 2) - (0.50479221787318407, 5.6043194293440744e-15) + IMPLEMENTATION NOTES: - Uses calls to the GSL -- the GNU Scientific Library -- C library. + Uses GSL -- the GNU Scientific Library AUTHORS: Index: sage/interfaces/cleaner.py =================================================================== --- sage/interfaces/cleaner.py (revision 4016) +++ sage/interfaces/cleaner.py (revision 3882) @@ -1,5 +1,2 @@ -"""nodoctest -""" - ############################################################################### # SAGE: System for Algebra and Geometry Experimentation Index: sage/interfaces/expect.py =================================================================== --- sage/interfaces/expect.py (revision 4296) +++ sage/interfaces/expect.py (revision 3704) @@ -345,7 +345,7 @@ self._expect.close = dummy except Exception, msg: - pass + print msg except Exception, msg: - pass + print msg def cputime(self): @@ -356,14 +356,4 @@ def quit(self, verbose=False): - """ - EXAMPLES: - sage: a = maxima('y') - sage: maxima.quit() - sage: a._check_valid() - Traceback (most recent call last): - ... - ValueError: The maxima session in which this object was defined is no longer running. - """ - self._session_number += 1 if self._expect is None: return @@ -469,28 +459,4 @@ raise KeyboardInterrupt, "Ctrl-c pressed while running %s"%self - def interrupt(self, tries=20, timeout=0.3): - E = self._expect - if E is None: - return True - success = False - try: - for i in range(tries): - E.sendline(self._quit_string()) - E.sendline(chr(3)) - try: - E.expect(self._prompt, timeout=timeout) - success= True - break - except (pexpect.TIMEOUT, pexpect.EOF), msg: - #print msg - pass - except Exception, msg: - pass - if success: - pass - else: - self.quit() - return success - def eval(self, code, strip=True): """ @@ -500,6 +466,5 @@ (ignored in the base class). """ - if not isinstance(code, str): - raise TypeError, 'input code must be a string.' + code = str(code) code = code.strip() try: @@ -508,5 +473,5 @@ self._keyboard_interrupt() except TypeError, s: - raise TypeError, 'error evaluating "%s":\n%s'%(code,s) + return 'error evaluating "%s":\n%s'%(code,s) def execute(self, *args, **kwds): @@ -779,5 +744,5 @@ def _reduce(self): - return repr(self) + return str(self) def _r_action(self, x): # used for coercion @@ -820,6 +785,6 @@ try: P = self.parent() - if P is None or P._session_number == BAD_SESSION or self._session_number == -1 or \ - P._session_number != self._session_number: + if P is None is None or P._session_number == BAD_SESSION or self._session_number == -1 or \ + (P._restart_on_ctrlc and P._session_number != self._session_number): raise ValueError, "The %s session in which this object was defined is no longer running."%P.name() except AttributeError: @@ -844,6 +809,5 @@ if not (P is None): P.clear(self._name) - - except (RuntimeError, pexpect.ExceptionPexpect), msg: # needed to avoid infinite loops in some rare cases + except RuntimeError, msg: # needed to avoid infinite loops in some rare cases #print msg pass @@ -856,5 +820,5 @@ def sage(self): - return sage.misc.sage_eval.sage_eval(repr(self)) + return sage.misc.sage_eval.sage_eval(str(self)) def __repr__(self): @@ -910,5 +874,5 @@ def __int__(self): - return int(repr(self)) + return int(str(self)) def bool(self): @@ -923,16 +887,16 @@ def __long__(self): - return long(repr(self)) + return long(str(self)) def __float____(self): - return float(repr(self)) + return float(str(self)) def _integer_(self): import sage.rings.all - return sage.rings.all.Integer(repr(self)) + return sage.rings.all.Integer(str(self)) def _rational_(self): import sage.rings.all - return sage.rings.all.Rational(repr(self)) + return sage.rings.all.Rational(str(self)) def name(self): Index: sage/interfaces/gap.py =================================================================== --- sage/interfaces/gap.py (revision 4059) +++ sage/interfaces/gap.py (revision 3634) @@ -592,7 +592,4 @@ return s - def __nonzero__(self): - return self.bool() - def __len__(self): """ Index: sage/interfaces/macaulay2.py =================================================================== --- sage/interfaces/macaulay2.py (revision 4059) +++ sage/interfaces/macaulay2.py (revision 3690) @@ -377,7 +377,7 @@ return self.parent().new('%s %% %s'%(self.name(), x.name())) - def __nonzero__(self): + def is_zero(self): P = self.parent() - return P.eval('%s == 0'%self.name()) == 'false' + return P.eval('%s == 0'%self.name()) == 'true' def sage_polystring(self): Index: sage/interfaces/mathematica.py =================================================================== --- sage/interfaces/mathematica.py (revision 4026) +++ sage/interfaces/mathematica.py (revision 3690) @@ -78,7 +78,7 @@ sage: c = m(5) - sage: print m('b + c x') + sage: m('b + c x') b + c x - sage: print m('b') + c*m('x') + sage: m('b') + c*m('x') b + 5 x @@ -182,5 +182,5 @@ sage: x = mathematica(pi/2) - sage: print x + sage: x Pi -- @@ -193,4 +193,10 @@ sage: loads(dumps(n)) == n True + +This example illustrates saving to a file in the local directory +and to one in the \sage database directory. + + sage.: n.save('n') + sage.: n.db('n') AUTHOR: @@ -436,9 +442,9 @@ def __repr__(self): P = self._check_valid() - return P.get(self._name, ascii_art=False).strip() + return P.get(self._name, ascii_art=True) def __str__(self): P = self._check_valid() - return P.get(self._name, ascii_art=True) + return P.get(self._name, ascii_art=False).strip() def str(self): Index: sage/interfaces/maxima.py =================================================================== --- sage/interfaces/maxima.py (revision 4320) +++ sage/interfaces/maxima.py (revision 3828) @@ -32,29 +32,37 @@ sage: F = maxima.factor('x^5 - y^5') sage: F - -(y-x)*(y^4+x*y^3+x^2*y^2+x^3*y+x^4) + -(y - x)*(y^4 + x*y^3 + x^2*y^2 + x^3*y + x^4) sage: type(F) Note that Maxima objects can also be displayed using ``ASCII art''; -to see a normal linear representation of any Maxima object x. -Just use the print command: +to see a normal linear representation of any Maxima object x, use \code{str(x)}. - sage: print F + sage: F.display2d() 4 3 2 2 3 4 - (y - x) (y + x y + x y + x y + x ) -You can always use \code{repr(x)} to obtain the linear representation -of an object. This can be useful for moving maxima data to other -systems. - sage: repr(F) - '-(y-x)*(y^4+x*y^3+x^2*y^2+x^3*y+x^4)' +We can make this the default: + sage: maxima.display2d(True) + sage: F + 4 3 2 2 3 4 + - (y - x) (y + x y + x y + x y + x ) + +You can always use \code{x.str()} to obtain the linear representation +of an object, even without changing the display2d flag. This can +be useful for moving maxima data to other systems. sage: F.str() - '-(y-x)*(y^4+x*y^3+x^2*y^2+x^3*y+x^4)' + '-(y - x)*(y^4 + x*y^3 + x^2*y^2 + x^3*y + x^4)' + sage: maxima.display2d(False) + sage: F + -(y - x)*(y^4 + x*y^3 + x^2*y^2 + x^3*y + x^4) + + The \code{maxima.eval} command evaluates an expression in maxima -and returns the result as a \emph{string} not a maxima object. +and returns the result as a string. sage: print maxima.eval('factor(x^5 - y^5)') - -(y-x)*(y^4+x*y^3+x^2*y^2+x^3*y+x^4) + -(y - x)*(y^4 + x*y^3 + x^2*y^2 + x^3*y + x^4) We can create the polynomial $f$ as a Maxima polynomial, then call @@ -63,7 +71,7 @@ sage: f = maxima('x^5 - y^5') sage: f^2 - (x^5-y^5)^2 + (x^5 - y^5)^2 sage: f.factor() - -(y-x)*(y^4+x*y^3+x^2*y^2+x^3*y+x^4) + -(y - x)*(y^4 + x*y^3 + x^2*y^2 + x^3*y + x^4) Control-C interruption works well with the maxima interface, @@ -82,7 +90,7 @@ sage: a = maxima('(1 + sqrt(2))^5'); a - (sqrt(2)+1)^5 + (sqrt(2) + 1)^5 sage: a.expand() - 29*sqrt(2)+41 + 29*sqrt(2) + 41 sage: a = maxima('(1 + sqrt(2))^5') @@ -102,22 +110,22 @@ sage: f = maxima('(x + 3*y + x^2*y)^3') sage: f.expand() - x^6*y^3+9*x^4*y^3+27*x^2*y^3+27*y^3+3*x^5*y^2+18*x^3*y^2+27*x*y^2+3*x^4*y+9*x^2*y+x^3 + x^6*y^3 + 9*x^4*y^3 + 27*x^2*y^3 + 27*y^3 + 3*x^5*y^2 + 18*x^3*y^2 + 27*x*y^2 + 3*x^4*y + 9*x^2*y + x^3 sage: f.subst('x=5/z') - (5/z+25*y/z^2+3*y)^3 + (5/z + 25*y/z^2 + 3*y)^3 sage: g = f.subst('x=5/z') sage: h = g.ratsimp(); h - (27*y^3*z^6+135*y^2*z^5+(675*y^3+225*y)*z^4+(2250*y^2+125)*z^3+(5625*y^3+1875*y)*z^2+9375*y^2*z+15625*y^3)/z^6 + (27*y^3*z^6 + 135*y^2*z^5 + (675*y^3 + 225*y)*z^4 + (2250*y^2 + 125)*z^3 + (5625*y^3 + 1875*y)*z^2 + 9375*y^2*z + 15625*y^3)/z^6 sage: h.factor() - (3*y*z^2+5*z+25*y)^3/z^6 + (3*y*z^2 + 5*z + 25*y)^3/z^6 sage: eqn = maxima(['a+b*c=1', 'b-a*c=0', 'a+b=5']) sage: s = eqn.solve('[a,b,c]'); s - [[a=(25*sqrt(79)*%i+25)/(6*sqrt(79)*%i-34),b=(5*sqrt(79)*%i+5)/(sqrt(79)*%i+11),c=(sqrt(79)*%i+1)/10],[a=(25*sqrt(79)*%i-25)/(6*sqrt(79)*%i+34),b=(5*sqrt(79)*%i-5)/(sqrt(79)*%i-11),c=-(sqrt(79)*%i-1)/10]] + [[a = (25*sqrt(79)*%i + 25)/(6*sqrt(79)*%i - 34),b = (5*sqrt(79)*%i + 5)/(sqrt(79)*%i + 11),c = (sqrt(79)*%i + 1)/10],[a = (25*sqrt(79)*%i - 25)/(6*sqrt(79)*%i + 34),b = (5*sqrt(79)*%i - 5)/(sqrt(79)*%i - 11),c = - (sqrt(79)*%i - 1)/10]] Here is an example of solving an algebraic equation: sage: maxima('x^2+y^2=1').solve('y') - [y=-sqrt(1-x^2),y=sqrt(1-x^2)] + [y = - sqrt(1 - x^2),y = sqrt(1 - x^2)] sage: maxima('x^2 + y^2 = (x^2 - y^2)/sqrt(x^2 + y^2)').solve('y') - [y=-sqrt((-y^2-x^2)*sqrt(y^2+x^2)+x^2),y=sqrt((-y^2-x^2)*sqrt(y^2+x^2)+x^2)] + [y = - sqrt(( - y^2 - x^2)*sqrt(y^2 + x^2) + x^2),y = sqrt(( - y^2 - x^2)*sqrt(y^2 + x^2) + x^2)] You can even nicely typeset the solution in latex: @@ -129,20 +137,20 @@ sage: e = maxima('sin(u + v) * cos(u)^3'); e - cos(u)^3*sin(v+u) + cos(u)^3*sin(v + u) sage: f = e.trigexpand(); f - cos(u)^3*(cos(u)*sin(v)+sin(u)*cos(v)) + cos(u)^3*(cos(u)*sin(v) + sin(u)*cos(v)) sage: f.trigreduce() - (sin(v+4*u)+sin(v-2*u))/8+(3*sin(v+2*u)+3*sin(v))/8 + (sin(v + 4*u) + sin(v - 2*u))/8 + (3*sin(v + 2*u) + 3*sin(v))/8 sage: w = maxima('3 + k*%i') sage: f = w^2 + maxima('%e')^w sage: f.realpart() - %e^3*cos(k)-k^2+9 + %e^3*cos(k) - k^2 + 9 sage: f = maxima('x^3 * %e^(k*x) * sin(w*x)'); f x^3*%e^(k*x)*sin(w*x) sage: f.diff('x') - k*x^3*%e^(k*x)*sin(w*x)+3*x^2*%e^(k*x)*sin(w*x)+w*x^3*%e^(k*x)*cos(w*x) + k*x^3*%e^(k*x)*sin(w*x) + 3*x^2*%e^(k*x)*sin(w*x) + w*x^3*%e^(k*x)*cos(w*x) sage: f.integrate('x') - (((k*w^6+3*k^3*w^4+3*k^5*w^2+k^7)*x^3+(3*w^6+3*k^2*w^4-3*k^4*w^2-3*k^6)*x^2+(-18*k*w^4-12*k^3*w^2+6*k^5)*x-6*w^4+36*k^2*w^2-6*k^4)*%e^(k*x)*sin(w*x)+((-w^7-3*k^2*w^5-3*k^4*w^3-k^6*w)*x^3+(6*k*w^5+12*k^3*w^3+6*k^5*w)*x^2+(6*w^5-12*k^2*w^3-18*k^4*w)*x-24*k*w^3+24*k^3*w)*%e^(k*x)*cos(w*x))/(w^8+4*k^2*w^6+6*k^4*w^4+4*k^6*w^2+k^8) + (((k*w^6 + 3*k^3*w^4 + 3*k^5*w^2 + k^7)*x^3 + (3*w^6 + 3*k^2*w^4 - 3*k^4*w^2 - 3*k^6)*x^2 + ( - 18*k*w^4 - 12*k^3*w^2 + 6*k^5)*x - 6*w^4 + 36*k^2*w^2 - 6*k^4)*%e^(k*x)*sin(w*x) + (( - w^7 - 3*k^2*w^5 - 3*k^4*w^3 - k^6*w)*x^3 + (6*k*w^5 + 12*k^3*w^3 + 6*k^5*w)*x^2 + (6*w^5 - 12*k^2*w^3 - 18*k^4*w)*x - 24*k*w^3 + 24*k^3*w)*%e^(k*x)*cos(w*x))/(w^8 + 4*k^2*w^6 + 6*k^4*w^4 + 4*k^6*w^2 + k^8) sage: f = maxima('1/x^2') @@ -151,8 +159,8 @@ sage: g = maxima('f/sinh(k*x)^4') sage: g.taylor('x', 0, 3) - f/(k^4*x^4)-2*f/(3*k^2*x^2)+11*f/45-62*k^2*f*x^2/945 + f/(k^4*x^4) - 2*f/(3*k^2*x^2) + 11*f/45 - 62*k^2*f*x^2/945 sage: maxima.taylor('asin(x)','x',0, 10) - x+x^3/6+3*x^5/40+5*x^7/112+35*x^9/1152 + x + x^3/6 + 3*x^5/40 + 5*x^7/112 + 35*x^9/1152 \subsection{Examples involving matrices} @@ -170,5 +178,5 @@ [[0,4],[3,1]] sage: A.eigenvectors() - [[[0,4],[3,1]],[1,0,0,-4],[0,1,0,-2],[0,0,1,-4/3],[1,2,3,4]] + [[[0,4],[3,1]],[1,0,0, - 4],[0,1,0, - 2],[0,0,1, - 4/3],[1,2,3,4]] We can also compute the echelon form in \sage: @@ -186,5 +194,5 @@ sage: _ = maxima.eval("f(t) := t*sin(t)") sage: maxima("laplace(f(t),t,s)") - 2*s/(s^2+1)^2 + 2*s/(s^2 + 1)^2 sage: maxima("laplace(delta(t-3),t,s)") #Dirac delta function @@ -193,10 +201,10 @@ sage: _ = maxima.eval("f(t) := exp(t)*sin(t)") sage: maxima("laplace(f(t),t,s)") - 1/(s^2-2*s+2) + 1/(s^2 - 2*s + 2) sage: _ = maxima.eval("f(t) := t^5*exp(t)*sin(t)") sage: maxima("laplace(f(t),t,s)") - 360*(2*s-2)/(s^2-2*s+2)^4-480*(2*s-2)^3/(s^2-2*s+2)^5+120*(2*s-2)^5/(s^2-2*s+2)^6 - sage: print maxima("laplace(f(t),t,s)") + 360*(2*s - 2)/(s^2 - 2*s + 2)^4 - 480*(2*s - 2)^3/(s^2 - 2*s + 2)^5 + 120*(2*s - 2)^5/(s^2 - 2*s + 2)^6 + sage: maxima("laplace(f(t),t,s)").display2d() 3 5 360 (2 s - 2) 480 (2 s - 2) 120 (2 s - 2) @@ -206,11 +214,11 @@ sage: maxima("laplace(diff(x(t),t),t,s)") - s*?%laplace(x(t),t,s)-x(0) + s*?%laplace(x(t),t,s) - x(0) sage: maxima("laplace(diff(x(t),t,2),t,s)") - -?%at('diff(x(t),t,1),t=0)+s^2*?%laplace(x(t),t,s)-x(0)*s + -?%at('diff(x(t),t,1),t = 0) + s^2*?%laplace(x(t),t,s) - x(0)*s It is difficult to read some of these without the 2d representation: - sage: print maxima("laplace(diff(x(t),t,2),t,s)") + sage.: maxima("laplace(diff(x(t),t,2),t,s)").display2d() ! d ! 2 @@ -254,5 +262,5 @@ sage: S = maxima('nusum(exp(1+2*i/n),i,1,n)') - sage: print S + sage.: S.display2d() 2/n + 3 2/n + 1 %e %e @@ -265,5 +273,5 @@ sage: T = S*maxima('2/n') sage: T.tlimit('n','inf') - %e^3-%e + %e^3 - %e \subsection{Miscellaneous} @@ -276,5 +284,5 @@ Defining functions in maxima: sage: maxima.eval('fun[a] := a^2') - 'fun[a]:=a^2' + 'fun[a] := a^2' sage: maxima('fun[10]') 100 @@ -290,8 +298,19 @@ \subsection{Latex Output} -To tex a maxima object do this: +The latex output of Maxima is not perfect. E.g., + + sage: maxima.eval('tex(sin(u) + sinh(v^2))') + '$$\\sinhv^2 + \\sinu$$false' + +Notice the lack of space after the sin macro, which is a latex syntax +error. In \sage this is automatically fixed via a substition for +trig functions, which may have potentially bad side effects: sage: latex(maxima('sin(u) + sinh(v^2)')) \sinh v^2+\sin u + +It would be nice if somebody would fix this problem. One way would +be to improve Maxima by making the fix to Maxima and giving this back +to the Maxima people. Here's another example: @@ -319,10 +338,4 @@ sage: maxima(sin(x)) sin(x) - -A long complicated input expression: - - sage: maxima._eval_line('((((((((((0) + ((1) / ((n0) ^ (0)))) + ((1) / ((n1) ^ (1)))) + ((1) / ((n2) ^ (2)))) + ((1) / ((n3) ^ (3)))) + ((1) / ((n4) ^ (4)))) + ((1) / ((n5) ^ (5)))) + ((1) / ((n6) ^ (6)))) + ((1) / ((n7) ^ (7)))) + ((1) / ((n8) ^ (8)))) + ((1) / ((n9) ^ (9)));') - '1/n9^9+1/n8^8+1/n7^7+1/n6^6+1/n5^5+1/n4^4+1/n3^3+1/n2^2+1/n1+1' - """ @@ -342,5 +355,5 @@ #***************************************************************************** -import os, re, sys +import os, re import pexpect cygwin = os.uname()[0][:6]=="CYGWIN" @@ -349,11 +362,15 @@ from pexpect import EOF -import random - import sage.rings.all +#import sage.rings.complex_number2 as complex_number from sage.misc.misc import verbose, DOT_SAGE, SAGE_ROOT from sage.misc.multireplace import multiple_replace + +SAGE_START = '_s_start_' +SAGE_END = '_s_stop_' +cnt = 0 +seq = 0 COMMANDS_CACHE = '%s/maxima_commandlist_cache.sobj'%DOT_SAGE @@ -372,5 +389,8 @@ def __call__(self, x): import sage.rings.all - return Expect.__call__(self, x) + if sage.rings.all.is_Infinite(x): + return Expect.__call__(self, 'inf') + else: + return Expect.__call__(self, x) def __init__(self, script_subdirectory=None, logfile=None, server=None): @@ -380,14 +400,11 @@ # TODO: Input and output prompts in maxima can be changed by # setting inchar and outchar.. - eval_using_file_cutoff = 256 + eval_using_file_cutoff = 200 self.__eval_using_file_cutoff = eval_using_file_cutoff - STARTUP = '%s/local/bin/sage-maxima.lisp'%SAGE_ROOT - if not os.path.exists(STARTUP): - raise RuntimeError, 'You must get the file local/bin/sage-maxima.lisp' Expect.__init__(self, name = 'maxima', prompt = '\(\%i[0-9]+\)', - command = 'maxima -p "%s"'%STARTUP, - maxread = 10000, + command = "maxima --disable-readline", + maxread = 1, # CRUCIAL to use less buffering for maxima (or get all kinds of hangs on OS X and 64-bit machines, etc! script_subdirectory = script_subdirectory, restart_on_ctrlc = False, @@ -398,11 +415,5 @@ logfile = logfile, eval_using_file_cutoff=eval_using_file_cutoff) - self._display_prompt = '' # must match what is in the file local/ibn/sage-maxima.lisp!! - self._output_prompt_re = re.compile('\(\%o[0-9]+\)') - self._ask = ['zero or nonzero?', 'an integer?', 'positive, negative, or zero?', 'positive or negative?'] - self._prompt_wait = [self._prompt] + [re.compile(x) for x in self._ask] - self._error_re = re.compile('(debugmode|Incorrect syntax|Maxima encountered a Lisp error)') self._display2d = False - def __getattr__(self, attrname): @@ -412,6 +423,28 @@ def _start(self): + # For some reason sending a single input line at startup avoids + # lots of weird timing issues when doing doctests. Expect._start(self) - self._eval_line('0;') + self(1) + + # this doesn't work. + #def x_start(self): + # Expect._start(self) + # self._expect.sendline('inchar:"__SAGE__";') + # self._change_prompt('__SAGE__[0-9]+\)') + # self.expect().expect('__SAGE__[0-9]+\)') + + def _eval_line_using_file(self, line, tmp): + F = open(tmp, 'w') + F.write(line) + F.close() + if self._expect is None: + self._start() + # For some reason this trivial comp + # keeps certain random freezes from occuring. Do not remove this. + # The space before the \n is also important. + self._expect.sendline('0;batchload("%s"); \n'%tmp) + self._expect.expect(self._prompt) + return '' def __reduce__(self): @@ -421,176 +454,84 @@ return 'quit();' - def _sendline(self, str): - self._sendstr(str) - os.write(self._expect.child_fd, os.linesep) - - def _sendstr(self, str): + def _eval_line(self, line, reformat=True, allow_use_file=False, + wait_for_prompt=True): if self._expect is None: self._start() - try: - os.write(self._expect.child_fd, str) - except OSError: - self._crash_msg() - self.quit() - self._sendstr(str) - - def _crash_msg(self): - print "Maxima crashed -- automatically restarting." - - def _expect_expr(self, expr=None, timeout=None): - if expr is None: - expr = self._prompt_wait + if not wait_for_prompt: + return Expect._eval_line(self, line) + line = line.rstrip().rstrip(';') + if line == '': + return '' + global seq + seq += 1 + start = SAGE_START + str(seq) + end = SAGE_END + str(seq) + line = '%s;\n%s; %s;'%(start, line, end) if self._expect is None: self._start() + if allow_use_file and self.__eval_using_file_cutoff and \ + len(line) > self.__eval_using_file_cutoff: + return self._eval_line_using_file(line, tmp) try: - if timeout: - i = self._expect.expect(expr,timeout=timeout) - else: - i = self._expect.expect(expr) - if i > 0: - v = self._expect.before - j = v.find('Is ') - v = v[j:] - msg = "Computation failed since Maxima requested additional constraints (use assume):\n" + v + self._ask[i-1] - self._sendstr(chr(3)) - self._sendstr(chr(3)) - self._expect_expr() - raise ValueError, msg - except KeyboardInterrupt, msg: - #print self._expect.before - i = 0 - while True: - try: - print "Control-C pressed. Interrupting Maxima. Please wait a few seconds..." - self._sendstr('quit;\n'+chr(3)) - self._sendstr('quit;\n'+chr(3)) - self.interrupt() - self.interrupt() - except KeyboardInterrupt: - i += 1 - if i > 10: - break - pass - else: - break - raise KeyboardInterrupt, msg - - def _interrupt(self): - for i in range(15): + E = self._expect + #print "in = '%s'"%line + E.sendline(line) + self._expect.expect(end) + # We have timeouts below, since getting the end above + # means the computation completed, but on some systems + # (Cygwin) the expect interface can sometimes hang getting + # the final prompts. try: - self._sendstr('quit;\n'+chr(3)) - self._expect_expr(timeout=2) + self._expect.expect(end, timeout=1) + out = self._expect.before + self._expect.expect(self._prompt, timeout=1) + out += self._expect.before except pexpect.TIMEOUT: - pass - except pexpect.EOF: - self._crash_msg() - self.quit() - else: - return - - def _before(self): - return self._expect.before - - def _batch(self, str, batchload=True): - F = open(tmp, 'w') - F.write(str) - F.close() - if batchload: - cmd = 'batchload("%s");'%tmp - else: - cmd = 'batch("%s");'%tmp - self._sendline(cmd) - self._expect_expr() - out = self._before() - self._error_check(str, out) - return out - - def _error_check(self, str, out): - r = self._error_re - m = r.search(out) - if not m is None: - self._error_msg(str, out) - - def _error_msg(self, str, out): - raise TypeError, "Error executing code in Maxima\nCODE:\n\t%s\nMaxima ERROR:\n\t%s"%(str, out.replace('-- an error. To debug this try debugmode(true);','')) - - def _eval_line(self, line, reformat=True, allow_use_file=False, - wait_for_prompt=True): - if len(line) == 0: + out = self._expect.before + if not '(%o' in out: + self._expect.expect(self._prompt) + + except EOF: + if self._quit_string() in line: + return '' + except KeyboardInterrupt: + self._keyboard_interrupt() return '' - line = line.rstrip() - if line[-1] != '$' and line[-1] != ';': - line += ';' - - self._synchronize() - - if len(line) > self.__eval_using_file_cutoff: - a = self(line) - return repr(a) - else: - self._sendline(line) - - if not wait_for_prompt: - return - - self._expect_expr(self._display_prompt) - self._expect_expr() - out = self._before() - self._error_check(line, out) - + + if 'Incorrect syntax:' in out: + raise RuntimeError, out + + import os + if cygwin: + # for reasons I can't deduce yet, maxima behaves somewhat + # differently under cygwin... + out = out.lstrip(';') + else: + i = out.rfind(start) + j = out.rfind(end) + out = out[i+len(start):j] + if not reformat: return out - - r = self._output_prompt_re - m = r.search(out) - if m is None: - o = out[:-2] - else: - o = out[m.end()+1:-2] - o = ''.join([x.strip() for x in o.split()]) - return o - - i = o.rfind('(%o') - return o[:i] - - - def _synchronize(self): - """ - Synchronize pexpect interface. - - This put a random integer (plus one!) into the output stream, - then waits for it, thus resynchronizing the stream. If the - random integer doesn't appear within 1 second, maxima is sent - interrupt signals. - - This way, even if you somehow left maxima in a busy state - computing, calling _synchronize gets everything fixed. - - EXAMPLES: - This makes Maxima start a calculation: - sage: maxima._sendstr('1/1'*500) - - When you type this command, this synchronize command is implicitly - called, and the above running calculation is interrupted: - sage: maxima('2+2') - 4 - """ - if self._expect is None: return - r = random.randrange(2147483647) - s = str(r+1) - cmd = "1+%s;\n"%r - self._sendstr(cmd) - try: - self._expect_expr(timeout=0.5) - if not s in self._before(): - self._expect_expr(s,timeout=0.5) - self._expect_expr(timeout=0.5) - except pexpect.TIMEOUT, msg: - self._interrupt() - except pexpect.EOF: - self._crash_msg() - self.quit() - - + if 'error' in out: + return out + out = out.lstrip() + i = out.find('(%o') + out0 = out[:i].strip() + i += out[i:].find(')') + out1 = out[i+1:].strip() + out = out0 + out1 + out = ''.join(out.split()) # no whitespace + i = out.rfind(';;') + if i != -1: + out = out[i+2:] + out = out.replace('-', ' - ').replace('+',' + ').replace('=',' = ').replace(': =',' :=').replace('^ - ','^-') + if out[:3] == ' - ': + out = '-' + out[3:] + out = out.replace('E - ', 'E-') + out = out.replace('%e - ', '%e-') + i = out.rfind('(%o') + return out[:i] + ########################################### @@ -685,5 +626,5 @@ return 'false' - def function(self, args, defn, rep=None, latex=None): + def function(self, args, defn, repr=None, latex=None): """ Return the Maxima function with given arguments @@ -694,5 +635,5 @@ defn -- a string (or Maxima expression) that defines a function of the arguments in Maxima. - rep -- an optional string; if given, this is how the function will print. + repr -- an optional string; if given, this is how the function will print. EXAMPLES: @@ -702,13 +643,12 @@ sage: f = maxima.function('x,y', 'sin(x)+cos(y)') sage: f(2,3.5) - sin(2)-.9364566872907963 + sin(2) - .9364566872907963 sage: f sin(x)+cos(y) - sage: g = f.integrate('z') - sage: g - (cos(y)+sin(x))*z + sage: g = f.integrate('z'); g + (cos(y) + sin(x))*z sage: g(1,2,3) - 3*(cos(2)+sin(1)) + 3*(cos(2) + sin(1)) The function definition can be a maxima object: @@ -718,24 +658,19 @@ gamma(x)*sin(x) sage: t(2) - sin(2) + sin(2) sage: float(t(2)) 0.90929742682568171 sage: loads(t.dumps()) gamma(x)*sin(x) + + """ name = self._next_var_name() - if isinstance(defn, MaximaElement): - defn = defn.str() - elif not isinstance(defn, str): - defn = str(defn) - if isinstance(args, MaximaElement): - args = args.str() - elif not isinstance(args, str): - args = str(args) - cmd = '%s(%s) := %s'%(name, args, defn) - maxima._eval_line(cmd) - if rep is None: - rep = defn - f = MaximaFunction(self, name, rep, args, latex) + defn = str(defn) + args = str(args) + maxima.eval('%s(%s) := %s'%(name, args, defn)) + if repr is None: + repr = defn + f = MaximaFunction(self, name, repr, args, latex) return f @@ -743,20 +678,11 @@ """ Set the variable var to the given value. - - INPUT: - var -- string - value -- string - """ - if not isinstance(value, str): - raise TypeError - cmd = '%s : %s$'%(var, value.rstrip(';')) - if len(cmd) > self.__eval_using_file_cutoff: - self._batch(cmd, batchload=True) - else: - self._eval_line(cmd) - #self._sendline(cmd) - #self._expect_expr() - #out = self._before() - #self._error_check(cmd, out) + """ + cmd = '%s : %s$"";'%(var, str(value).rstrip(';')) + out = self._eval_line(cmd, reformat=False, allow_use_file=True) + + if out.find("error") != -1: + raise TypeError, "Error executing code in Maxima\nCODE:\n\t%s\nMaxima ERROR:\n\t%s"%(cmd, out) + def get(self, var): @@ -764,15 +690,17 @@ Get the string value of the variable var. """ - s = self._eval_line('%s;'%var) + s = self._eval_line('%s'%var) return s def clear(self, var): - """ - Clear the variable named var. - """ - try: - self._eval_line('kill(%s)$'%var, reformat=False) - except TypeError: - pass + """ + Clear the variable named var. + """ + if self._expect is None: + return + try: + self._expect.sendline('kill(%s);\n'%var) + except: # program around weirdness in pexpect + pass def console(self): @@ -782,25 +710,25 @@ return maxima_version() -## def display2d(self, flag=True): -## """ -## Set the flag that determines whether Maxima objects are -## printed using their 2-d ASCII art representation. When the -## maxima interface starts the default is that objects are not -## represented in 2-d. - -## INPUT: -## flag -- bool (default: True) - -## EXAMPLES -## sage: maxima('1/2') -## 1/2 -## sage: maxima.display2d(True) -## sage: maxima('1/2') -## 1 -## - -## 2 -## sage: maxima.display2d(False) -## """ -## self._display2d = bool(flag) + def display2d(self, flag=True): + """ + Set the flag that determines whether Maxima objects are + printed using their 2-d ASCII art representation. When the + maxima interface starts the default is that objects are not + represented in 2-d. + + INPUT: + flag -- bool (default: True) + + EXAMPLES + sage: maxima('1/2') + 1/2 + sage: maxima.display2d(True) + sage: maxima('1/2') + 1 + - + 2 + sage: maxima.display2d(False) + """ + self._display2d = bool(flag) def plot2d(self, *args): @@ -934,12 +862,12 @@ EXAMPLES: - sage: maxima.de_solve('diff(y,x,2) + 3*x = y', ['x','y'], [1,1,1]) - y=3*x-2*%e^(x-1) - sage: maxima.de_solve('diff(y,x,2) + 3*x = y', ['x','y']) - y=%k1*%e^x+%k2*%e^-x+3*x - sage: maxima.de_solve('diff(y,x) + 3*x = y', ['x','y']) - y=(%c-3*(-x-1)*%e^-x)*%e^x - sage: maxima.de_solve('diff(y,x) + 3*x = y', ['x','y'],[1,1]) - y=-%e^-1*(5*%e^x-3*%e*x-3*%e) + sage.: maxima.de_solve('diff(y,x,2) + 3*x = y', ['x','y'], [1,1,1]) + y = 3*x - 2*%e^(x - 1) + sage.: maxima.de_solve('diff(y,x,2) + 3*x = y', ['x','y']) + y = %k1*%e^x + %k2*%e^ - x + 3*x + sage.: maxima.de_solve('diff(y,x) + 3*x = y', ['x','y']) + y = (%c - 3*( - x - 1)*%e^ - x)*%e^x + sage.: maxima.de_solve('diff(y,x) + 3*x = y', ['x','y'],[1,1]) + y = - %e^ - 1*(5*%e^x - 3*%e*x - 3*%e) """ if not isinstance(vars, str): @@ -972,13 +900,13 @@ EXAMPLES: - sage: maxima.clear('x'); maxima.clear('f') - sage: maxima.de_solve_laplace("diff(f(x),x,2) = 2*diff(f(x),x)-f(x)", ["x","f"], [0,1,2]) - f(x)=x*%e^x+%e^x + sage.: maxima.clear('x'); maxima.clear('f') + sage.: maxima.de_solve_laplace("diff(f(x),x,2) = 2*diff(f(x),x)-f(x)", ["x","f"], [0,1,2]) + f(x) = x*%e^x + %e^x - sage: maxima.clear('x'); maxima.clear('f') - sage: f = maxima.de_solve_laplace("diff(f(x),x,2) = 2*diff(f(x),x)-f(x)", ["x","f"]) - sage: f - f(x)=x*%e^x*(?%at('diff(f(x),x,1),x=0))-f(0)*x*%e^x+f(0)*%e^x - sage: print f + sage.: maxima.clear('x'); maxima.clear('f') + sage.: f = maxima.de_solve_laplace("diff(f(x),x,2) = 2*diff(f(x),x)-f(x)", ["x","f"]) + sage.: f + f(x) = x*%e^x*(at('diff(f(x),x,1),x = 0)) - f(0)*x*%e^x + f(0)*%e^x + sage.: f.display2d() ! x d ! x x @@ -1013,6 +941,6 @@ sage: eqns = ["x + z = y","2*a*x - y = 2*a^2","y - 2*z = 2"] sage: vars = ["x","y","z"] - sage: maxima.solve_linear(eqns, vars) - [x=a+1,y=2*a,z=a-1] + sage.: maxima.solve_linear(eqns, vars) + [x = a + 1,y = 2*a,z = a - 1] """ eqs = "[" @@ -1030,33 +958,33 @@ return self('linsolve(%s, %s)'%(eqs, vrs)) -## def unit_quadratic_integer(self, n): -## r""" -## Finds a unit of the ring of integers of the quadratic number -## field $\Q(\sqrt{n})$, $n>1$, using the qunit maxima command. - -## EXAMPLE: -## sage: u = maxima.unit_quadratic_integer(101) -## sage: u.parent() -## Number Field in a with defining polynomial x^2 - 101 -## sage: u -## a + 10 -## sage: u = maxima.unit_quadratic_integer(13) -## sage: u -## 5*a + 18 -## sage: u.parent() -## Number Field in a with defining polynomial x^2 - 13 -## """ -## from sage.rings.all import QuadraticField, Integer -## # Take square-free part so sqrt(n) doesn't get simplified further by maxima -## # (The original version of this function would yield wrong answers if -## # n is not squarefree.) -## n = Integer(n).square_free_part() -## if n < 1: -## raise ValueError, "n (=%s) must be >= 1"%n -## s = str(self('qunit(%s)'%n)).lower() -## r = re.compile('sqrt\(.*\)') -## s = r.sub('a', s) -## a = QuadraticField(n, 'a').gen() -## return eval(s) + def unit_quadratic_integer(self, n): + r""" + Finds a unit of the ring of integers of the quadratic number + field $\Q(\sqrt{n})$, $n>1$, using the qunit maxima command. + + EXAMPLE: + sage: u = maxima.unit_quadratic_integer(101) + sage: u.parent() + Number Field in a with defining polynomial x^2 - 101 + sage: u + a + 10 + sage: u = maxima.unit_quadratic_integer(13) + sage: u + 5*a + 18 + sage: u.parent() + Number Field in a with defining polynomial x^2 - 13 + """ + from sage.rings.all import QuadraticField, Integer + # Take square-free part so sqrt(n) doesn't get simplified further by maxima + # (The original version of this function would yield wrong answers if + # n is not squarefree.) + n = Integer(n).square_free_part() + if n < 1: + raise ValueError, "n (=%s) must be >= 1"%n + s = str(self('qunit(%s)'%n)).lower() + r = re.compile('sqrt\(.*\)') + s = r.sub('a', s) + a = QuadraticField(n, 'a').gen() + return eval(s) def plot_list(self, ptsx, ptsy, options=None): @@ -1077,6 +1005,6 @@ EXAMPLES: - sage: zeta_ptsx = [ (pari(1/2 + i*I/10).zeta().real()).precision(1) for i in range (70,150)] - sage: zeta_ptsy = [ (pari(1/2 + i*I/10).zeta().imag()).precision(1) for i in range (70,150)] + sage.: zeta_ptsx = [ (pari(1/2 + i*I/10).zeta().real()).precision(1) for i in range (70,150)] + sage.: zeta_ptsy = [ (pari(1/2 + i*I/10).zeta().imag()).precision(1) for i in range (70,150)] sage.: maxima.plot_list(zeta_ptsx, zeta_ptsy) sage.: opts='[gnuplot_preamble, "set nokey"], [gnuplot_term, ps], [gnuplot_out_file, "zeta.eps"]' @@ -1136,20 +1064,4 @@ return P('%s(%s)'%(self.name(), x)) - def __str__(self): - """ - Printing an object explicitly gives ASCII art: - - EXAMPLES: - sage: f = maxima('1/(x-1)^3'); f - 1/(x-1)^3 - sage: print f - 1 - -------- - 3 - (x - 1) - - """ - return self.display2d(onscreen=False) - def __cmp__(self, other): """ @@ -1175,21 +1087,13 @@ # thanks to David Joyner for telling me about using "is". P = self.parent() - try: - if P.eval("is (%s < %s)"%(self.name(), other.name())) == P._true_symbol(): - return -1 - elif P.eval("is (%s > %s)"%(self.name(), other.name())) == P._true_symbol(): - return 1 - elif P.eval("is (%s = %s)"%(self.name(), other.name())) == P._true_symbol(): - return 0 - except TypeError: - pass - return cmp(repr(self),repr(other)) - # everything is supposed to be comparable in Python, so we define - # the comparison thus when no comparable in interfaced system. - - def sage_object(self): - from sage.calculus.calculus import symbolic_expression_from_maxima_string - return symbolic_expression_from_maxima_string(self.name(), maxima=self.parent()) - + if P.eval("is (%s < %s)"%(self.name(), other.name())) == P._true_symbol(): + return -1 + elif P.eval("is (%s > %s)"%(self.name(), other.name())) == P._true_symbol(): + return 1 + elif P.eval("is (%s = %s)"%(self.name(), other.name())) == P._true_symbol(): + return 0 + else: + return -1 # everything is supposed to be comparable in Python, so we define + # the comparison thus when no comparable in interfaced system. def numer(self): return self.comma('numer') @@ -1214,16 +1118,15 @@ def str(self): - P = self._check_valid() + self._check_valid() + P = self.parent() return P.get(self._name) def __repr__(self): - P = self._check_valid() - try: - return self.__repr - except AttributeError: - pass - r = P.get(self._name) - self.__repr = r - return r + self._check_valid() + P = self.parent() + if P._display2d: + return self.display2d(onscreen=False) + else: + return P.get(self._name) def display2d(self, onscreen=True): @@ -1238,17 +1141,17 @@ P = self.parent() s = P._eval_line('display2d : true; %s'%self.name(), reformat=False) - P._eval_line('display2d : false;', reformat=False) - - r = P._output_prompt_re - - m = r.search(s) - s = s[m.start():] - i = s.find('\n') - s = s[i+1 + len(P._display_prompt):] - m = r.search(s) - if not m is None: - s = s[:m.start()] + ' '*(m.end() - m.start()) + s[m.end():].rstrip() - # if ever want to dedent, see - # http://mail.python.org/pipermail/python-list/2006-December/420033.html + P._eval_line('display2d : false', reformat=False) + if not cygwin: + i = s.find('true') + i += s[i:].find('\n') + s = s[i+1:] + i = s.find('true') + i += s[i:].find('\n') + j = s.rfind('(%o') + s = s[i:j-2] + i = s.find('(%o') + j = i + s[i:].find(')') + s = s[:i] + ' '*(j-i+1) + s[j+1:] + s = s.lstrip('\n') if onscreen: print s @@ -1275,10 +1178,10 @@ sage: f.diff('x', 2) 2 - sage: maxima('sin(x^2)').diff('x',4) - 16*x^4*sin(x^2)-12*sin(x^2)-48*x^2*cos(x^2) + sage: maxima('sin(x^2)').diff('x',4) + 16*x^4*sin(x^2) - 12*sin(x^2) - 48*x^2*cos(x^2) sage: f = maxima('x^2 + 17*y^2') sage: f.diff('x') - 34*y*'diff(y,x,1)+2*x + 2*x sage: f.diff('y') 34*y @@ -1349,12 +1252,12 @@ EXAMPLES: sage: maxima('x^2+1').integral() - x^3/3+x + x^3/3 + x sage: maxima('x^2+ 1 + y^2').integral('y') - y^3/3+x^2*y+y + y^3/3 + x^2*y + y sage: maxima('x / (x^2+1)').integral() - log(x^2+1)/2 + log(x^2 + 1)/2 sage: maxima('1/(x^2+1)').integral() atan(x) - sage: maxima('1/(x^2+1)').integral('x', 0, infinity) + sage.: maxima('1/(x^2+1)').integral('x', 0, infinity) %pi/2 sage: maxima('x/(x^2+1)').integral('x', -1, 1) @@ -1380,8 +1283,5 @@ def __float__(self): - try: - return float(repr(self.numer())) - except ValueError: - raise TypeError, "unable to coerce '%s' to float"%repr(self) + return float(str(self.numer())) def __len__(self): @@ -1424,16 +1324,5 @@ if n < 0 or n >= len(self): raise IndexError, "n = (%s) must be between %s and %s"%(n, 0, len(self)-1) - # If you change the n+1 to n below, better change __iter__ as well. return ExpectElement.__getitem__(self, n+1) - - def __iter__(self): - """ - EXAMPLE: - sage: v = maxima('create_list(i*x^i,i,0,5)') - sage: list(v) - [0, x, 2*x^2, 3*x^3, 4*x^4, 5*x^5] - """ - for i in range(len(self)): - yield self[i] def subst(self, val): @@ -1448,5 +1337,5 @@ self._check_valid() P = self.parent() - s = P._eval_line('tex(%s);'%self.name(), reformat=False) + s = P._eval_line('tex(%s)'%self.name(), reformat=False) if not '$$' in s: raise RuntimeError, "Error texing maxima object." @@ -1522,6 +1411,6 @@ sage: f = maxima('1/((1+x)*(x-1))') sage: f.partial_fraction_decomposition('x') - 1/(2*(x-1))-1/(2*(x+1)) - sage: print f.partial_fraction_decomposition('x') + 1/(2*(x - 1)) - 1/(2*(x + 1)) + sage: f.partial_fraction_decomposition('x').display2d() 1 1 --------- - --------- @@ -1547,7 +1436,4 @@ self.__args = args self.__latex = latex - - def __reduce__(self): - return reduce_load_Maxima_function, (self.parent(), self.__defn, self.__args, self.__latex) def __call__(self, *x): @@ -1578,5 +1464,5 @@ else: args = self.__args + ',' + var - return P.function(args, repr(f)) + return P.function(args, str(f)) @@ -1589,8 +1475,4 @@ def reduce_load_Maxima(): return maxima - -def reduce_load_Maxima_function(parent, defn, args, latex): - return parent.function(args, defn, defn, latex) - import os @@ -1604,4 +1486,2 @@ import sage.interfaces.quit sage.interfaces.quit.expect_quitall() - - Index: sage/interfaces/mwrank.py =================================================================== --- sage/interfaces/mwrank.py (revision 4062) +++ sage/interfaces/mwrank.py (revision 1097) @@ -68,5 +68,5 @@ def __call__(self, cmd): - return self.eval(str(cmd)) + return self.eval(cmd) def console(self): Index: sage/interfaces/qsieve.py =================================================================== --- sage/interfaces/qsieve.py (revision 4331) +++ sage/interfaces/qsieve.py (revision 2550) @@ -8,15 +8,4 @@ import sage.rings.integer - -from sage.misc.all import tmp_dir - -_tmp_dir = False -def tmpdir(): - global _tmp_dir - if _tmp_dir: - return - X = tmp_dir('qsieve') - os.environ['TMPDIR'] = X - _tmp_dir = True def qsieve(n, block=True, time=False, verbose=False): @@ -42,6 +31,5 @@ sieve computation is complete before using SAGE further. If False, SAGE will run while the sieve computation - runs in parallel. If q is the returned object, use - q.quit() to terminate a running factorization. + runs in parallel. time -- (default: False) if True, time the command using the UNIX "time" command (which you might have to install). @@ -87,5 +75,4 @@ else: t = '' - tmpdir() out = os.popen('echo "%s" | %s QuadraticSieve 2>&1'%(n,t)).read() z = data_to_list(out, n, time=time) @@ -165,5 +152,4 @@ else: cmd = 'QuadraticSieve' - tmpdir() self._p = pexpect.spawn(cmd) cleaner.cleaner(self._p.pid, 'QuadraticSieve') @@ -198,6 +184,4 @@ """ if self._done: - if hasattr(self, '_killed') and self._killed: - return "Factorization was terminated early." v = data_to_list(self._get(), self._n, self._do_time) if self._do_time: @@ -255,18 +239,6 @@ Terminate the QuadraticSieve process, in case you want to give up on computing this factorization. - - EXAMPLES: - sage: n = next_prime(2^110)*next_prime(2^100) - sage: qs = qsieve(n, block=False) - sage: qs - Proper factors so far: [] - sage: qs.quit() - sage: qs - Factorization was terminated early. - """ - pid = self.pid() - os.killpg(int(pid),9) - #self._p.close() - self._killed = True + """ + self._p.close() self._done = True Index: sage/interfaces/quit.py =================================================================== --- sage/interfaces/quit.py (revision 4022) +++ sage/interfaces/quit.py (revision 2920) @@ -1,5 +1,5 @@ """nodoctest""" -import os, time, pexpect +import os, time expect_objects = [] @@ -11,7 +11,7 @@ try: R.quit(verbose=verbose) - pass - except RuntimeError: - pass + except RuntimeError, msg: + if verbose: + print msg kill_spawned_jobs() @@ -24,12 +24,18 @@ pid = L[:i].strip() cmd = L[i+1:].strip() - try: - #s = "Killing spawned job %s"%pid - #if len(cmd) > 0: - # s += ' (%s)'%cmd - #print s - os.killpg(int(pid), 9) - except: - pass + j = 0 + while j < 3: + j += 1 + if not is_running(pid): + break + try: + os.killpg(int(pid), 9) + except OSError, msg: + pass + else: + j += 1 + if j > 5: + os.kill(int(pid), 9) + break def is_running(pid): Index: sage/interfaces/singular.py =================================================================== --- sage/interfaces/singular.py (revision 4269) +++ sage/interfaces/singular.py (revision 3036) @@ -262,5 +262,4 @@ import sage.misc.misc as misc -import sage.rings.integer @@ -547,16 +546,12 @@ return SingularElement(self, None, name, True) - def ring(self, char=0, vars='(x)', order='lp', check=True): + def ring(self, char=0, vars='(x)', order='lp'): r""" Create a Singular ring and makes it the current ring. INPUT: - char -- characteristic of the base ring (see examples below), - which must be either 0, prime (!), or one of - several special codes (see examples below). + char -- characteristic of the base ring (see examples below) vars -- a tuple or string that defines the variable names order -- string -- the monomial order (default: 'lp') - check -- if True, check primality of the characteristic - if it is an integer. OUTPUT: @@ -587,5 +582,5 @@ sage: R3 = singular.ring(7, '(x(1..10))', 'ds') - This is a polynomial ring over the transcendental extension $\Q(a)$ of $\Q$: + This is a polynomial ring over the transcendtal extension $\Q(a)$ of $\Q$: sage: R4 = singular.ring('(0,a)', '(mu,nu)', 'lp') @@ -612,9 +607,4 @@ except RuntimeError, TypeError: # error if variable is not already defined. pass - if check and isinstance(char, (int, long, sage.rings.integer.Integer)): - if char != 0: - n = sage.rings.integer.Integer(char) - if not n.is_prime(): - raise ValueError, "the characteristic must be 0 or prime" R = self('%s,%s,%s'%(char, vars, order), 'ring') self.eval('short=0') # make output include *'s for multiplication for *THIS* ring. @@ -804,7 +794,7 @@ P.eval('%s[%s] = %s'%(self.name(), n, value.name())) - def __nonzero__(self): + def is_zero(self): P = self.parent() - return P.eval('%s == 0'%self.name()) == '0' + return P.eval('%s == 0'%self.name()) == '1' def sage_polystring(self): Index: sage/lfunctions/dokchitser.py =================================================================== --- sage/lfunctions/dokchitser.py (revision 4287) +++ sage/lfunctions/dokchitser.py (revision 3709) @@ -366,31 +366,20 @@ sage: L.taylor_series(1) 0.305999773834052*z + 0.186547797268162*z^2 + -0.136791463097188*z^3 + 0.0161066468496401*z^4 + 0.0185955175398802*z^5 + O(z^6) - - We compute a Taylor series where each coefficient is to high precision. - sage: E = EllipticCurve('389a') - sage: L = E.Lseries_dokchitser(200) - sage: L.taylor_series(1,3) - 6.2239725530250970363983975962696997888173850098274602272589e-73 + (-3.5271062035449946049211903242820246129524508593200000161038e-73)*z + 0.75931650028842677023019260789472201907809751649492435158581*z^2 + O(z^3) # 32-bit - 6.2239725530250970363983942830358807065824196704980264311105e-73 + (-3.5271062035449946049211884466825561834034352383203420991340e-73)*z + 0.75931650028842677023019260789472201907809751649492435158581*z^2 + O(z^3) # 64-bit """ self.__check_init() a = self.__CC(a) k = Integer(k) - try: - z = self.gp()('Vec(Lseries(%s,,%s))'%(a,k-1)) - except TypeError, msg: - raise RuntimeError, "%s\nUnable to compute Taylor expansion (try lowering the number of terms)"%msg - r = repr(z) - if 'pole' in r: - raise ArithmeticError, r - elif 'Warning' in r: - i = r.rfind('\n') - msg = r[:i].replace('digits','decimal digits') + z = self.gp().eval('Lseries(%s,,%s)'%(a,k-1)) + if 'pole' in z: + raise ArithmeticError, z + elif 'Warning' in z: + i = z.rfind('\n') + msg = z[:i].replace('digits','decimal digits') verbose(msg, level=-1) - v = list(z) - K = self.__CC - v = [K(repr(x)) for x in v] - R = self.__CC[[var]] - return R(v,len(v)) + z = z[i+1:] + t = self.__CC[[var]].gen(0) + z = z.replace('S',var).replace('.E','.0E').replace(' ','') + f = sage_eval(z, locals={var:t}) + return f def check_functional_equation(self, T=1.2): Index: sage/libs/cf/cf.pyxe =================================================================== --- sage/libs/cf/cf.pyxe (revision 2755) +++ sage/libs/cf/cf.pyxe (revision 2755) @@ -0,0 +1,1528 @@ +r""" +Interface to the Factory library. + +AUTHORS: + -- Martin Albrecht (2006-05-01) + -- William Stein (2006-05-07): trivial reformating of long lines + +From the Factory manual: + + Factory is a C++ class library which implements a recursive + canonical form representation of polynomial data. The library + enables the user to use multivariate polynomials over different + base domains such as Z, Q, GF(q) and algebraic as well as + transcendental extensions of Q and GF(q). + + Factory was developed at the University of Kaiserslautern as an + independent part of the computer algebra system singular which + uses parts of Factory such as polynomial gcd's and polynomial + factorization. This manual describes the features of Factory and + how to use it. Since Factory uses inheritance, operator + overloading and templates, it will be helpful if the reader has + already some experience in C++ programming. + + Factory uses a recursive representation of multivariate polynomial + data. Each polynomial in Factory is represented as a univariate + polynomial with coefficients that can be polynomials too. To have + a unique representation of polynomials we have to fix an ordering + on the variables. This ordering is managed by Factory's level + system. The basic data type which is introduced by Factory is the + class CanonicalForm which handles polynomials as well as elements + of the base domain. + +This interface is mainly a one-to-one copy of the low level polynomial +arithmetic implemented in the Factory. It follows Factory's concept of +representing multivariate polynomials recursive as univariate +polynomials over polynomial rings. + +When using this library keep that in mind and avoid relying on +variable names (i.e. strings) when interfacing this code with the rest +of SAGE. Variables in the Factory are uniquely represented through +their level and not their name (which may only have one character +btw.) + +So far only integers, GF(p), and GF(p^n) have been inplemented as base +domains. + +EXAMPLES: + sage: F = GF(5) + sage: cf.setBaseDomain(F) + sage: R = MPolynomialRing(F,2, names = ['x','y']) + sage: x,y = R.gens() + sage: f = y**2-x**9-x + sage: f == (cf.CF(y)**2-cf.CF(x)**9 - x)._sage_(R) + True + sage: n = 3 + sage: astr = ['a'+str(i) for i in range(1,2*n)] + sage: R0 = MPolynomialRing(F,2*n+2,names = [str(x),str(y),'t']+astr) + sage: vars0 = R0.gens() + sage: t = vars0[2] + sage: xt = add([vars0[i]*t**(i-2) for i in range(3,2*n+2)]) + sage: yt = t + sage: ft = f(xt,yt) + sage: ft == cf.CF(f)([(x,cf.CF(xt)),(y,cf.CF(yt))])._sage_(R0) #fast! + True + +\note{This interface is not very SAGE-ish and is rather meant to +function as a backend for higher level classes like +MPolynomial_polydict for fast polynomial arithmetic. This libary is a +bit behind Singluar if we shuffle few data to the engine, do lot's of +computation and shuffle some data back. This loss of performance is +believed to be due to the clumsy way we deal with memory allocation +which can be addresses. If we shuffle around alot of data between SAGE +and Singular it's performance decreases alot as much IPC overhead is +involved, while Factory beats the native implementation.} + +TODO: Refactor to avoid memory allocations. +""" + +#***************************************************************************** +# +# SAGE: System for Algebra and Geometry Experimentation +# +# Copyright (C) 2006 William Stein +# +# Distributed under the terms of the GNU General Public License (GPL) +# +# This code is distributed in the hope that it will be useful, +# but WITHOUT ANY WARRANTY; without even the implied warranty of +# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU +# General Public License for more details. +# +# The full text of the GPL is available at: +# +# http://www.gnu.org/licenses/ +#***************************************************************************** + + +#header{ +#include +#include +#include +using namespace std; +#}header + +cdef extern from "stdlib.h": + void free(void *ptr) + +################################# +#cdef int _sig_on +#cdef int _sig_off +#cdef int _sig_check +include '../../../ext/stdsage.pxi' # path is a hack for pyrexembed +include '../../../ext/interrupt.pxi' # path is a hack for pyrexembed +################################# + +include 'misc.pxi' + + + +cdef class Variable: + + """ + The class Variable defines a programming interface to the level + system of Factory. Each object of class Variable refers to a + level and thus to a variable (determined by the level) of the + polynomial ring. Objects of class Variable must not only refer to + variables of the polynomial ring, but also to so called algebraic + and transcendental variables. + + Arithmetic with Variables can be pretty weird, consider this: + + sage: 1000*cf.Variable(1) + v_1*v_1000 + + This behavior is correct as 1000 is coerced into a Variable of + level 1000 whichs default name happens to be'v_1000' + """ + + cdef void *thisptr + + def __init__(self, name=None, level=None): + """ + The class Variable defines a programming interface to the + level system of Factory. Each object of class Variable refers + to a level and thus to a variable (determined by the level) of + the polynomial ring. Objects of class Variable must not only + refer to variables of the polynomial ring, but also to so + called algebraic and transcendental variables. + + INPUT: + + name -- The level of the variable is specified to be the + level of the variable which has name name. That means that + the generated object can not only be a polynomial variable + but also an algebraic or transcendental variable. In the + case that there is no variable with name name the + generated variable refers to the anonymous polynomial + variable which has the lowest level. After the + construction the name of the variable is set to name. + + If name is an integer it is interpreted as a level. + + level -- The constructor generates an object of class + Variable with level set to level. level has to be a level + of a polynomial variable or of a transcendental variable. + + The default constructor creates an object with level set to + LEVEL_BASEDOMAIN. Such a variable evaluates into the unit + element of the current base domain when converted to a + CanonicalForm + """ + #embed{ void *Variable_init() + return new Variable(); + #}embed + + #embed{ void *Variable_initn(char name) + return new Variable(name); + #}embed + + #embed{ void *Variable_initl(int level) + return new Variable(level); + #}embed + + #embed{ void *Variable_initln( int level, char name) + return new Variable(level,name); + #}embed + + cdef char *cNameStr + cdef char cName + + from sage.rings.integer import Integer + + if isinstance(name,int) or isinstance(name,Integer): + level=name + name =None + + if name != None and isinstance(name,str): + cNameStr = name + cName = cNameStr[0] + if level!=None: + self.thisptr = Variable_initln(int(level), cName) + else: + self.thisptr = Variable_initn(cName) + else: + if level!=None: + self.thisptr = Variable_initl(int(level)) + else: + self.thisptr = Variable_init() + + def __dealloc__(Variable self): + delete_Variable(self.thisptr) + + def level(self): + """ + Returns the level of this variable. + """ + #embed{ int Variable_level(void *obj) + return ((Variable *)obj)->level(); + #}embed + + return Variable_level(self.thisptr) + + def name(Variable self): + """ + Returns the name of this variable. Do not identify a Variable + via it's name() us it's level() instead. + """ + #embed{ char Variable_name(void *obj) + return ((Variable *)obj)->name(); + #}embed + + # this is slow, but I crashed the nativ way + return str(CanonicalForm(self)) + + def highest(self): + #embed{ void *Variable_highest(void *obj) + return new Variable(((Variable *)obj)->highest()); + #}embed + _sig_on + return make_Variable(Variable_highest(self.thisptr)) + + def next(self): + """ + This method returns an object of class Variable which refers + to the variable which has the same level as self increased by + one. + """ + #embed{ void *Variable_next(void *obj) + return new Variable(((Variable *)obj)->next()); + #}embed + _sig_on + return make_Variable(Variable_next(self.thisptr)) + + def __cmp__(self, Variable other): + """ + Comparison is performed by comparing the levels. + """ + #embed{ int Variable_cmp(void *obj,void *obj2) + if(*(Variable *)obj == *(Variable *)obj2) { + return 0; + } else if( *(Variable *)obj < *(Variable *)obj2) { + return -1; + } else if( *(Variable *)obj > *(Variable *)obj2) { + return 1; + } + #}embed + + return Variable_cmp(self.thisptr,other.thisptr) + + def __add__(self, other): + return CanonicalForm(self) + CanonicalForm(other) + + def __sub__(self, other): + return CanonicalForm(self) - CanonicalForm(other) + + def __mul__(self, other): + return CanonicalForm(self) * CanonicalForm(other) + + def __div__(self, other): + return CanonicalForm(self) / CanonicalForm(other) + + def __pow__(self, long other, ignored): + return CanonicalForm(self)**other + + def __mod__(self, other): + return CanonicalForm(self) % CanonicalForm(other) + + def __repr__(self): + #embed{ char *Variable_to_str(void* x) + ostringstream instore; + instore << *(Variable*)x ; + int n = strlen(instore.str().data()); + char* buf = (char*)malloc(n+1); + strcpy(buf, instore.str().data()); + #}embed + +## cdef char *ret +## _sig_on +## ret = Variable_to_str(self.thisptr) +## _sig_off +## return string(ret) + + #might be slow but it kept crashing otherwise + return str(CanonicalForm(self)) + + + + +#embed{ void delete_Variable( void *obj) +if(obj) { + delete ((Variable*)obj); +} +#}embed + +cdef make_Variable(void *x): + """ + """ + cdef Variable y + _sig_off + y = Variable() + delete_Variable(y.thisptr) + y.thisptr = x + return y + +cdef class CanonicalForm: + """ + CanonicalForm is the main class in the Factory + """ + cdef void *thisptr + cdef void *iterator + + def __init__(self,arg=None, arg2=None, kcache=None): + """ + If arg is a Variable and exp==None this constructor generates + a canonical form out of the variable arg. If the level of x + is the level of the base domain (LEVEL_BASEDOMAIN) then the + unit element of the current base domain is created. + + If arg is an integer this constructor generates an object of + class CanonicalForm out of this integer. In the case of a + finite base domain the canonical mapping of the integers into + the domain is used. + + If arg is a Variable and exp is an integer this constructor + generates an object of class CanonicalForm, which is the + polynomial arg^exp. In the case that x has level + LEVEL_BASEDOMAIN the unit element of the current base domain + is created. + + If arg is a SAGE polynomial this constructor generates an + object of class CanonicalForm, which is a representaion of arg + over currently active base domain. If arg is defined over a + finite extension field arg2 must be an algebraic Variable + representing this field. Those variables are returned by + setBaseDomain when called with a finite extension field as + parameter. + + If arg is a string it is parsed by the CF library. It is not + recommended to use string IO for communication with this + library as names are not unique in this library and the result + might not match the expectations. The best practice here is to + construct a SAGE ring, pass the string to that ring and pass + the newly constructed SAGE polynomial to this library. + + INPUT: + arg -- integer, Variable, Polynomial, or string + arg2 -- optional second argument in dependence of first one + kcache -- optional base field cache to use MPolynomial elements + + EXAMPLES: + sage: R=PolynomialRing(GF(127),2) + sage: f=R('3234*x0*x1+6575*x0^2+52') + sage: cf.setBaseDomain(GF(127)) + sage: cf.CF(f) + 59*v_1*v_2+98*v_1^2+52 + sage: cf.CF(f)._sage_(R) + 52 + 59*x0*x1 + 98*x0^2 + sage: cf.CF(f)._sage_(R)==f + True + """ + #embed{ void *CF_init() + return new CanonicalForm(); + #}embed + + #embed{ void *CF_initi(int arg) + return new CanonicalForm(arg); + #}embed + + #embed{ void *CF_initV(void *arg) + return new CanonicalForm(*(Variable *)arg); + #}embed + + #embed{ void *CF_initVn(void *arg, int exp) + return new CanonicalForm(*(Variable *)arg,exp); + #}embed + + #embed{ void *CF_copy(void *arg) + return new CanonicalForm(*(CanonicalForm *)arg); + #}embed + + #embed{ void *str_to_CF(char* s) + istringstream in; + in.str( s ); + CanonicalForm* y = new CanonicalForm(); + in >> *y; + return y; + #}embed + + #embed{ void CF_polygen(void *ret, int c, int level, long e) + *(CanonicalForm*)ret += c * CanonicalForm(Variable(level),e); + #}embed + + #embed{ void CF_mongen(void *ret, int level, long e) + if(e!=0) + *(CanonicalForm*)ret *= CanonicalForm(Variable(level),e); + #}embed + + cdef CanonicalForm cPoly + cdef CanonicalForm t + cdef int varlevel + + from sage.rings.multi_polynomial_element import is_MPolynomial + from sage.rings.field_element import is_FieldElement + from sage.rings.polynomial_element import is_Polynomial + + # maybe a switch-case dict might be faster here? + if is_FieldElement(arg) and arg.parent().degree()!=1: + cPoly = CF(0) + if not isinstance(arg2,Variable): + arg2 = setBaseDomain(arg.parent()) + varlevel = arg2.level() + f = arg._pari_().lift().lift() + if f.poldegree() == 0: + self.thisptr = CF_initi(int(f)) + return + for i from 0 <= i <= f.poldegree(): + CF_polygen(cPoly.thisptr,int(f[i]),varlevel,i) + self.thisptr = cPoly.thisptr + return + + elif is_MPolynomial(arg): + poly = CF(0) + for e,c in arg.element().dict().iteritems(): + t = CF(0) + if kcache!=None: + if not kcache.has_key(c): + kcache[c]=CF(c,arg2) + t = t + kcache[c] + else: + t = t + CF(c,arg2) + for i in e.nonzero_positions(): + CF_mongen(t.thisptr,i+1,e[i]) + poly = poly + t + cPoly = poly + self.thisptr = cPoly.thisptr + return + + elif is_Polynomial(arg): + cPoly = CF(0) + v = Variable(1) + #TODO: speed can be improved here + for i from 0 <= i <= arg.degree(): + cPoly = cPoly + CF(arg[i])*CF(v,i) + self.thisptr = cPoly.thisptr + + elif isinstance(arg,CanonicalForm): + self.thisptr = CF_copy((arg).thisptr) + + elif isinstance(arg,Variable) and arg2==None: + self.thisptr = CF_initV((arg).thisptr) + + elif isinstance(arg,Variable) and arg2!=None: + self.thisptr = CF_initVn((arg).thisptr,int(arg2)) + + elif isinstance(arg,str): + arg = "".join([arg,";"]) + self.thisptr = str_to_CF(arg) + + elif arg==None: + self.thisptr = CF_init() + + else: + try: + arg = int(arg) + self.thisptr = CF_initi(arg) + except: + raise TypeError, "Cannot create CanonicalForm with params %s and %s"%(arg, arg2) + + def __repr__(self):# + """ + """ + #embed{ char *CF_to_str(void* x) + ostringstream instore; + instore << *(CanonicalForm*)x ; + int n = strlen(instore.str().data()); + char* buf = (char*)malloc(n+1); + strcpy(buf, instore.str().data()); + return buf; + #}embed + + cdef char *ret + _sig_on + ret = CF_to_str(self.thisptr) + return string(ret) + + def __add__(left, right): + #embed{ void *CF_add(void *l, void *r) + return new CanonicalForm((*(CanonicalForm*)l) + (*(CanonicalForm*)r)); + #}embed + + if not isinstance(left,CanonicalForm): + left,right = right,left + if not isinstance(right,CanonicalForm): + right = CanonicalForm(right) + _sig_on + return make_CF(CF_add((left).thisptr,(right).thisptr)) + + def __sub__(left, right): + #embed{ void *CF_sub(void *l, void *r) + return new CanonicalForm((*(CanonicalForm*)l) - (*(CanonicalForm*)r)); + #}embed + cdef CanonicalForm cRight + cdef CanonicalForm cLeft + try: + cRight = CanonicalForm(right) + cLeft = CanonicalForm(left) + except TypeError: + raise ArithmeticError, "Cannot subtract %s from %s"%(right,left) + _sig_on + return make_CF(CF_sub(cLeft.thisptr,cRight.thisptr)) + + def __mul__(left, right): + #embed{ void *CF_mul(void *l, void *r) + return new CanonicalForm((*(CanonicalForm*)l) * (*(CanonicalForm*)r)); + #}embed + + cdef CanonicalForm cRight + cdef CanonicalForm cLeft + try: + cRight = CanonicalForm(right) + cLeft = CanonicalForm(left) + except TypeError: + raise ArithmeticError, "Cannot multiply %s and %s"%(left,right) + _sig_on + return make_CF(CF_mul(cLeft.thisptr,cRight.thisptr)) + + def __div__(left, right): + #embed{ void *CF_div(void *l, void *r) + return new CanonicalForm((*(CanonicalForm*)l) / (*(CanonicalForm*)r)); + #}embed + + cdef CanonicalForm cRight + cdef CanonicalForm cLeft + try: + cRight = CanonicalForm(right) + cLeft = CanonicalForm(left) + except TypeError: + raise ArithmeticError, "Cannot divide %s through %s"%(left,right) + _sig_on + return make_CF(CF_div(cLeft.thisptr,cRight.thisptr)) + + def __pow__(CanonicalForm self,long e, ignored): + #embed{ void *CF_pow(void *l, long e) + return new CanonicalForm( power((*(CanonicalForm*)l),e) ); + #}embed + _sig_on + return make_CF(CF_pow(self.thisptr, e)) + + def __neg__(CanonicalForm self): + #embed{ void *CF_neg(void *l) + return new CanonicalForm(-(*(CanonicalForm*)l)); + #}embed + _sig_on + return make_CF(CF_neg(self.thisptr)) + +## Commenting out all in place arithmetic as SAGE does not use in +## place arithmetic, e.g. when you e += 1 for an element which is +## also an element of a list the element in the list does not +## change in SAGE. + +## +## def __iadd__(CanonicalForm self,other): +## #embed{ void CF_iadd(void *l, void *r) +## (*(CanonicalForm*)l) += (*(CanonicalForm*)r); +## #}embed + +## cdef CanonicalForm cOther +## try: +## cOther = CanonicalForm(other) +## except TypeError: +## raise ArithmeticError, "Cannot add %s and %s"%(self,other) + +## CF_iadd(self.thisptr,cOther.thisptr) +## return self + +## def __isub__(CanonicalForm self, other): +## #embed{ void *CF_isub(void *l, void *r) +## (*(CanonicalForm*)l) -= (*(CanonicalForm*)r); +## #}embed +## cdef CanonicalForm cOther +## try: +## cOther = CanonicalForm(other) +## except TypeError: +## raise ArithmeticError, "Cannot subtract %s from %s"%(other,self) +## CF_isub(self.thisptr,cOther.thisptr) +## return self + +## def __imul__(CanonicalForm self, other): +## #embed{ void *CF_imul(void *l, void *r) +## (*(CanonicalForm*)l) *= (*(CanonicalForm*)r); +## #}embed + +## cdef CanonicalForm cOther +## try: +## cOther = CanonicalForm(other) +## except TypeError: +## raise ArithmeticError, "Cannot multiply %s and %s"%(self,other) + +## CF_imul(self.thisptr,cOther.thisptr) +## return self + +## def __idiv__(CanonicalForm self, other): +## #embed{ void CF_idiv(void *l, void *r) +## (*(CanonicalForm*)l) /= (*(CanonicalForm*)r); +## #}embed + +## cdef CanonicalForm cOther +## try: +## cOther = CanonicalForm(other) +## except TypeError: +## raise ArithmeticError, "Cannot divide %s through %s"%(self,other) + +## CF_idiv(self.thisptr,cOther.thisptr) +## return self + +## def __ipow__(CanonicalForm self,long e, ignored): +## #embed{ void CF_ipow(void *l, long e) +## (*(CanonicalForm*)l) = power((*(CanonicalForm*)l),e); +## #}embed +## CF_ipow(self.thisptr,e) +## return self + + + def __cmp__(CanonicalForm self,other): + """ + It is possible to compare two canonical forms. Therefore, you can use + this operator. The behaviour of < and > in the case of polynomials is + not fixed yet, but may deal with levels and degrees in a future + release of the Factory library. + """ + #embed{ int CF_cmp(void *obj,void *obj2) + if( *(CanonicalForm *)obj == *(CanonicalForm *)obj2 ) { + return 0; + } else if(*(CanonicalForm *)obj < *(CanonicalForm *)obj2) { + return -1; + } else if(*(CanonicalForm *)obj > *(CanonicalForm *)obj2) { + return 1; + } + #}embed + + cdef int res + cdef CanonicalForm cOther + if not isinstance(other, CanonicalForm): + try: + cOther = CanonicalForm(other) + except TypeError: + raise TypeError, "Cannot compare %s and %s"%(self,other) + else: + cOther = other + _sig_on + res = CF_cmp(self.thisptr,cOther.thisptr) + _sig_off + return res + + def __mod__(CanonicalForm self, other): + #embed{ void *CF_mod(void *l, void *r) + return new CanonicalForm((*(CanonicalForm*)l) % (*(CanonicalForm*)r)); + #}embed + + cdef CanonicalForm cOther + try: + cOther = CanonicalForm(other) + except TypeError: + raise ArithmeticError, "Cannot take %s modulo %s"%(self,other) + _sig_on + return make_CF(CF_mod(self.thisptr,cOther.thisptr)) + + def __getitem__(CanonicalForm self, i): + """ + operator [] - return i'th coefficient from self. + + Returns self if self is in a base domain and i equals zero. + Returns zero (from the current domain) if self is in a base + domain and i is larger than zero. Otherwise, returns the + coefficient to x^i in self (if x denotes the main variable of + self) or zero if self does not contain x^i. Elements in an + algebraic extension are considered polynomials. i should be + larger or equal zero. + + INPUT: + i -- exponent, this may be a tuple of exponents + + EXAMPLE: + sage: r=MPolynomialRing(ZZ,2,['x','y']) + sage: x,y = r.gens() + sage: cf.setBaseDomain(ZZ) + sage: f=cf.CF(3*x*y^2) + sage: f[1] + 0 + sage: f[2] + 3*v_1 + sage: f[2,1] + 3 + + NOTE: + Never use a loop like + + for i in range(f.degree( )): + print f[ i ] + + which is much slower than + + for c in f: + print c + + """ + #embed{ void *CF_getitem(void *e,int i) + if (((CanonicalForm*)e)->degree() < i) { + return new CanonicalForm(0); + } else { + return new CanonicalForm((*(CanonicalForm*)e)[i]); + } + #}embed + cdef void *ret + if isinstance(i,int): + _sig_on + ret = CF_getitem(self.thisptr,i) + return make_CF(ret) + elif isinstance(i,tuple): + _sig_on + ret = self.thisptr + for j in i: + ret = CF_getitem(ret,int(j)) + return make_CF(ret) + else: + raise TypeError, "Cannot return coefficent for %s"%i + + def __iter__(CanonicalForm self): + """ + Iterator with respect to the main variable. It returns + coeffients and expontents of terms. You may get the + variable these refer to by calling mvar() on self. + """ + return self + + def __next__(CanonicalForm self): + """ + Iterator implementation. + """ + #embed{ void *CF_iterinit(void *e) + return new CFIterator(*(CanonicalForm*)e); + #}embed + + #embed{ void *CF_iter(void *it, int *exp) + if(((CFIterator*)it)->hasTerms()) { + CanonicalForm *ret = new CanonicalForm(((CFIterator*)it)->coeff()); + *exp = ((CFIterator*)it)->exp(); + (*(CFIterator*)it)++; + return ret; + } else { + return NULL; + } + #}embed + + cdef void *ret + cdef int exp + + if self.is_zero(): + raise StopIteration + + if self.iterator == NULL: + self.iterator = CF_iterinit(self.thisptr) + + _sig_on + ret = CF_iter(self.iterator,&exp) + _sig_off + if ret!=NULL: + _sig_on + return (make_CF(ret),exp) + else: + self.iterator = CF_iterinit(self.thisptr) + raise StopIteration + + def __call__(CanonicalForm self, arg): + """ + Returns the current object with the substitution described + through arg performed. + + If arg is a list or tuple and it's elements are not tuples it + is interpreted as a list of values to substituted in at their + list index. + + If arg is a list or tuple and it's elements are tuples with + two elements then it is interpreted to hold the variable to be + substitute as the first element of those tuples and the value + to be substituted in as the second. + + If arg is a dict it's keys are interpreted to describe the + variables and the matching values the values to be substituted + in. + + INPUT: + arg -- tuple, list or dict to describe substitution + + EXAMPLES: + sage: f=cf.CF(3)*cf.Variable(1)*cf.Variable(2) + sage: f + 3*v_1*v_2 + sage: f((cf.Variable(1),'2')) + 6*v_1 + sage: f((3,2)) + 18 + + """ + #embed{ void CF_call(void *self, void *f, void *x) + (*(CanonicalForm*)self) = (*(CanonicalForm*)self)(*(CanonicalForm*)f,*(Variable*)x); + #}embed + + cdef void *r + cdef Variable x + cdef CanonicalForm f + + if isinstance(arg,tuple) or isinstance(arg,list): + if not isinstance(arg[0],tuple) and len(arg)==self.mvar().level(): + r = CF_copy(self.thisptr) + _sig_on + for i from 0 <= i < len(arg): + f = CanonicalForm(arg[i]) + x = Variable(i+1) + CF_call(r,f.thisptr,x.thisptr) + return make_CF(r) + elif len(arg[0])==2: + r = CF_copy(self.thisptr) + _sig_on + for var,val in arg: + f = CanonicalForm(val) + x = CanonicalForm(var).mvar() #safer than calling Variable + CF_call(r,f.thisptr,x.thisptr) + return make_CF(r) + elif isinstance(arg,dict): + r = CF_copy(self.thisptr) + _sig_on + for var,val in arg.iteritems(): + f = CanonicalForm(val) + x = CanonicalForm(var).mvar() #safer than calling Variable + CF_call(r,f.thisptr,x.thisptr) + return make_CF(r) + else: + raise TypeError, "Cannot substitute with parameter %s"%arg + + def is_one(CanonicalForm self): + """ + This predicate returns true if self represents the unit + element of the current base domain. There is also the + possibiliy to test if a CanonicalForm f is one via f == 1, but + f.is_one() is faster since there is no conversion of the + integer 1 into a CanonicalForm and then doing a comparison of + two objects of class CanonicalForm. + """ + #embed{ int CF_isOne(void *e) + return ((CanonicalForm*)e)->isOne(); + #}embed + return bool(CF_isOne(self.thisptr)) + + def is_zero(CanonicalForm self): + """ + This predicate returns true if self represents the zero + element of the current base domain. Like the predicate is_one + using the predicate f.is_zero() is also faster than a + comparison via f == 0. + """ + #embed{ int CF_isZero(void *e) + return ((CanonicalForm*)e)->isZero(); + #}embed + + return bool(CF_isZero(self.thisptr)) + + def is_imm(CanonicalForm self): + """ + This predicate returns true if self is represented as an + immediate object (e.g. is a small integer or an object of a + finite base field). See Chapter 14 [Internal Design], page + 21, for information about how immediate objects are + represented. + + The return value represents if self is represented as a + immediate value in the Factory and makes no statement about + this element in SAGE. + """ + #embed{ int CF_isImm(void *e) + return ((CanonicalForm*)e)->isImm(); + #}embed + return bool(CF_isImm(self.thisptr)) + + def _int_(CanonicalForm self): + """ + This selector returns the integer value of self. This is only + valid if the value of self is in the range of the machine + integers. A good way to find out if this is the case is to + test if the current object is an immediate object via + self.is_imm(). + """ + #embed{ int CF_intval(void *e) + return ((CanonicalForm*)e)->intval(); + #}embed + return CF_intval(self.thisptr) + + def lc(CanonicalForm self): + """ + Returns the leading coefficient of self with respect to + lexicographic ordering. Elements in an algebraic extension + are considered polynomials so lc() always returns a leading + coefficient in a base domain. This method is useful to get + the base domain over which self is defined. + + Returns self if self is in a base domain. + """ + #embed{ void *CF_lc(void *e) + return new CanonicalForm(((CanonicalForm*)e)->lc()); + #}embed + _sig_on + return make_CF(CF_lc(self.thisptr)) + + def Lc(CanonicalForm self): + """ + Returns the leading coefficient of self with respect to + lexicographic ordering. In contrast to lc() elements in an + algebraic extension are considered coefficients so Lc() always + returns a leading coefficient in a coefficient domain. + + Returns self if self is in a base domain. + """ + #embed{ void *CF_Lc(void *e) + return new CanonicalForm(((CanonicalForm*)e)->Lc()); + #}embed + _sig_on + return make_CF(CF_Lc(self.thisptr)) + + def LC(CanonicalForm self, v=None): + """ + LC( ) returns the leading coefficient of self where self is + considered a univariate polynomial in its main variable. An + element of an algebraic extension is considered an univariate + polynomial, too. + + LC( v ) returns the leading coefficient of CO where CO is + considered an univariate polynomial in the polynomial variable v. + + Returns self if self is in a base domain. + + INPUT: + v -- if set the leading coefficient with respect to this + variable is returned. + OUTPUT: + CanonicalForm representing the lead coefficient. + + + """ + #embed{ void *CF_LC(void *e,void *v) + if (v == NULL) { + return new CanonicalForm(((CanonicalForm*)e)->LC()); + } else { + return new CanonicalForm( ((CanonicalForm*)e)->LC( *((Variable*)v) ) ); + } + #}embed + + if v==None: + _sig_on + return make_CF(CF_LC(self.thisptr,NULL)) + + if not isinstance(v, Variable): + v = Variable(v) + _sig_on + return make_CF(CF_LC(self.thisptr,(v).thisptr)) + + def total_degree(CanonicalForm self): + """ + Returns the total degree of self. + """ + #embed{ int CF_total_degree(void *e) + return totaldegree(*(CanonicalForm*)e); + #}embed + return CF_total_degree(self.thisptr) + + def degree(CanonicalForm self, x=None): + """ + This selector returns the degree of self with respect to its + main variable or a provided variable. If self is an element + of a coefficient domain the returned degree is zero except + that the current object represents the zero of a base domain + in which case -1 is returned. + + INPUT: + variable -- if set the degree with respect to this + variable will be returned. + """ + + #embed{ int CF_degree(void *e,void *x) + if (x == NULL) { + return ((CanonicalForm*)e)->degree(); + } else { + return ((CanonicalForm*)e)->degree( *((Variable*)x) ); + } + #}embed + + if x==None: + return CF_degree(self.thisptr,NULL) + + if not isinstance(x,Variable): + variable = Variable(x) + return CF_degree(self.thisptr,(x).thisptr) + + def lt(CanonicalForm self): + """ + Returns the leading term with respect to the main variable. + + This is equivalent to head() in the Factory C++ API. + """ + #embed{ void *CF_head(void *e) + return new CanonicalForm(head(*(CanonicalForm*)e)); + #}embed + _sig_on + return make_CF(CF_head(self.thisptr)) + + def tail(CanonicalForm self): + """ + Returns f - lt(f) + """ + #embed{ void *CF_tail(void *e) + return new CanonicalForm((*(CanonicalForm*)e) - head(*(CanonicalForm*)e)); + #}embed + _sig_on + return make_CF(CF_tail(self.thisptr)) + + + def tailcoeff(CanonicalForm self): + """ + This selector returns the tail coefficient of self. That is + the non vanishing coefficient of the term with the lowest + exponent. If self represents the zero element of a base domain + then zero is returned. + """ + #embed{ void *CF_tailcoeff(void *e) + return new CanonicalForm(((CanonicalForm*)e)->tailcoeff()); + #}embed + _sig_on + return make_CF(CF_tailcoeff(self.thisptr)) + + def taildegree(CanonicalForm self): + """ + This selector returns the lowest exponent of the terms of self + that have non vanishing coefficients. If the self represents + the zero element of a base domain then None is returned. + """ + #embed{ int CF_taildegree(void *e) + return ((CanonicalForm*)e)->taildegree(); + #}embed + + cdef res + res = CF_taildegree(self.thisptr) + if res==-1: + return + else: + return res + + def mvar(CanonicalForm self): + """ + This selector returns the main varible of self. If self is an + element of a base domain then Variable(LEVEL_BASEDOMAIN) is + returned. + """ + #embed{ void *CF_mvar(void *e) + return new Variable(((CanonicalForm*)e)->mvar()); + #}embed + _sig_on + return make_Variable(CF_mvar(self.thisptr)) + + def level(CanonicalForm self): + """ + This selector returns the level of the main variable of self. + """ + + #embed{ int CF_level(void *e) + return ((CanonicalForm*)e)->level(); + #}embed + + return CF_level(self.thisptr) + + def mapinto(CanonicalForm self, mapMe = False): + """ + This method returns the mapping of the current object into the + current base domain. If the current object is a polynomial, + then the value returned by mapdomain is the polynomial mapped + into the polynomial ring over the current base domain. + + INPUT: + mapMe -- if True this object is mappend and nothing is returned + (default: False) + """ + + #embed{ void *CF_mapinto(void *self) + return new CanonicalForm(((CanonicalForm*)self)->mapinto()); + #}embed + #embed{ void CF_mapintos(void *self) + *(CanonicalForm*)self = ((CanonicalForm*)self)->mapinto(); + #}embed + + if mapMe: + CF_mapintos(self.thisptr) + else: + _sig_on + return make_CF(CF_mapinto(self.thisptr)) + + def inZ(CanonicalForm self): + """ + This method returns true if the object is a rational integer. + """ + #embed{ int CF_inZ(void *self) + return ((CanonicalForm*)self)->inZ(); + #}embed + return bool(CF_inZ(self.thisptr)) + + def inQ(CanonicalForm self): + """ + This method returns true if the object is a rational number. + """ + #embed{ int CF_inQ(void *self) + return ((CanonicalForm*)self)->inQ(); + #}embed + return bool(CF_inQ(self.thisptr)) + + def inFF(CanonicalForm self): + """ + This method returns true if the object is an element of Fp. + """ + #embed{ int CF_inFF(void *self) + return ((CanonicalForm*)self)->inFF(); + #}embed + return bool(CF_inFF(self.thisptr)) + + def inGF(CanonicalForm self): + """ + This method returns true if the object is an element of GF(q). + """ + #embed{ int CF_inGF(void *self) + return ((CanonicalForm*)self)->inGF(); + #}embed + return bool(CF_inGF(self.thisptr)) + + def inPP(CanonicalForm self): + """ + This method returns true if the object is in a prime power domain. + """ + #embed{ int CF_inPP(void *self) + return ((CanonicalForm*)self)->inPP(); + #}embed + return bool(CF_inPP(self.thisptr)) + + def inBaseDomain(CanonicalForm self): + """ + This method returns true if the object is an element of a base domain, + e.g. Z, Q, Z mod p, Fp or GF(q). + """ + #embed{ int CF_inBaseDomain(void *self) + return ((CanonicalForm*)self)->inBaseDomain(); + #}embed + return bool(CF_inBaseDomain(self.thisptr)) + + def inExtension(CanonicalForm self): + r""" + This method returns true if the object is an element of an algebraic + extension. + + \code{In Factory this means that the object is not an element + of GF(q) even if that would be true mathematically. All that + can be said when inExtension returns true is, that the object + contains an algebraic variable.} + """ + #embed{ int CF_inExtension(void *self) + return ((CanonicalForm*)self)->inExtension(); + #}embed + return bool(CF_inExtension(self.thisptr)) + + def inCoeffDomain(CanonicalForm self): + """ + This method returns true if the object is not a polynomial. + """ + #embed{ int CF_inCoeffDomain(void *self) + return ((CanonicalForm*)self)->inCoeffDomain(); + #}embed + return bool(CF_inCoeffDomain(self.thisptr)) + + def inPolyDomain(CanonicalForm self): + """ + This method returns true if the object is a polynomial. + """ + #embed{ int CF_inPolyDomain(void *self) + return ((CanonicalForm*)self)->inPolyDomain(); + #}embed + return bool(CF_inPolyDomain(self.thisptr)) + + + def isFFinGF(CanonicalForm self): + """ + """ + #embed{ int CF_isFFinGF(void *self) + return ((CanonicalForm*)self)->isFFinGF(); + #}embed + return bool(CF_isFFinGF(self.thisptr)) + + def is_univariate(CanonicalForm self): + """ + Returns True if self is univariate, False otherwise + """ + #embed{ int CF_isUnivariate(void *self) + return ((CanonicalForm*)self)->isUnivariate(); + #}embed + return bool(CF_isUnivariate(self.thisptr)) + + def is_homogeneous(CanonicalForm self): + """ + Returns True if self is homogeneous, False otherwise + """ + #embed{ int CF_isHomogeneous(void *self) + return ((CanonicalForm*)self)->isHomogeneous(); + #}embed + return bool(CF_isHomogeneous(self.thisptr)) + + def __cflist__(self, f, e): + level = f.level()-1 + if f.inCoeffDomain(): + return [] + [tuple([f,e])] + else: + retval = [] + for c,e2 in f: + new_e = e.copy() + if e2!=0: + new_e[level]=e2 + retval = retval + self.__cflist__(c,new_e) + return retval + + def _sage_(self, R, kcache=None): + """ + Coerces self into R. + + A very simple coercion strategy is used: The variable with + level=1 is mapped to the first variable in the ring, the + variable with level=2 to the second, etc. Elements of the + coefficient domain are coerced via strings. So in that case + variable names (e.g. of an algebraic variable) do matter in + contrast to the warning stated earlier. + + INPUT: + R -- ring to coerce to + kcache -- optional finite field cache + + EXAMPLES: + sage: cf.setBaseDomain(GF(127)) + sage: R=MPolynomialRing(GF(127),2,'x') + sage: f=cf.CF(R('3*x0+2')) + sage: f._sage_(R) + 2 + 3*x0 + sage: k = GF(2**8) + sage: v = cf.setBaseDomain(k) + sage: R=MPolynomialRing(k,2,'x') + sage: f=cf.CF(R('a^20*x0+a^10'),v);f + a_1^7*v_1+a_1^5*v_1+a_1^4*v_1+a_1^2*v_1+a_1^6+a_1^5+a_1^4+a_1^2 + sage: f._sage_(R) + a^6 + a^5 + a^4 + a^2 + (a^7 + a^5 + a^4 + a^2)*x0 + """ + from sage.rings.polynomial_ring import is_PolynomialRing + + self.mapinto(mapMe=True) + + if is_PolynomialRing(R): + l = [0]*(self.degree()+1) + base = R.base_ring() + for c,e in self: + cs = str(c) + if kcache!=None: + if not kcache.has_key(cs): + kcache[cs]=base(cs) + l[e]=kcache[cs] + else: + l[e]=base(cs) + return R(l) + else: + d={} + level = self.level()-1 + ngens = R.ngens() + + from sage.rings.polydict import PolyDict,ETuple + + kgen = str(R.base_ring().gen()) + + if self.is_zero(): + # obey binary compatibility + return R(PolyDict({})) + + if self.inCoeffDomain(): + # catch the constant case + cs = str(self) + if not self.inFF(): + cs = cs.replace(str(self.mvar()),kgen) + d[ETuple({},ngens)]=R.base_ring()(cs) + return R(PolyDict(d,force_int_exponents=False,force_etuples=False)) + + for c,e in self: + exp = {} + if e!=0: + exp[level]= e + l = self.__cflist__(c,exp) + for c2,e2 in l: + if c2.is_zero(): + continue + cs = str(c2) + if not c2.inFF(): + cs = cs.replace(str(c2.mvar()),kgen) + if kcache!=None: + if not kcache.has_key(cs): + kcache[cs]=R.base_ring()(cs) + d[ETuple(e2,ngens)]=kcache[cs] + else: + d[ETuple(e2,ngens)]=R.base_ring()(cs) + return R(PolyDict(d,force_int_exponents=False,force_etuples=False)) + + def lcm(CanonicalForm self,right): + #embed{ void *CF_lcm(void *l, void *r) + return new CanonicalForm(lcm(*(CanonicalForm*)l,(*(CanonicalForm*)r))); + #}embed + + if not isinstance(right,CanonicalForm): + right = CanonicalForm(right) + _sig_on + return make_CF(CF_lcm(self.thisptr,(right).thisptr)) + + + +CF = CanonicalForm + +#embed{ void delete_CF( void *obj) +if(obj) { + delete ((CanonicalForm*)obj); +} +#}embed + +cdef make_CF(void *x): + """ + """ + cdef CanonicalForm y + _sig_off + y = CanonicalForm() + delete_CF(y.thisptr) + y.thisptr = x + return y + + +def setCharacteristic(int p): + """ + Sets the characteristic for all following Factory operations to + p. We also accept SAGE finite fields here which will be + automatically coerced to a format the Factory can understand. If + the finite fieldis an extension field the algebraic element 'a' is + returned which is the root of the defining polynomial of the + extension field. + + INPUT: + p -- characteristic to use in Factory + + TODO: + The Factory also supports p^n as base chacteristic but that + would need the gftable lookup which is not provided in the + format needed by the Factory through Singular. Use rootOf() + if you need to to construct GF(p^n) which however - I guess - + is slower than the setCharacteristic(p,n) approach. If you + provide this function with a SAGE finite field this behavior + is implemented. + """ + #embed{ void setCharacteristicp(int p) + setCharacteristic(p); + #}embed + + #embed{ void setCharacteristicpn(int p, int n) + setCharacteristic(p,n); + #}embed + + #embed{ void setCharacteristicpnn(int p, int n, char name) + setCharacteristic(p,n,n); + #}embed + + setCharacteristicp(p) + +__domains__ = dict() + +def setBaseDomain(k,force=False): + """ + Sets the CF BaseDomain so it matches k. At the moment + GF(p), GF(p^n), and ZZ are supported. If an extension + field is provided the algebraic variable used to describe + this extension field is returned nothing otherwise. This + variable is needed to coerce SAGE polynomials to CF polynomials. + + This variable is also cached locally so if the same finite + extension field is provided twice the same variable will be + returned. This behavior can be overriden by setting force to True. + + Caching the algebraic variable locally not only is faster than + recreating one when needed it alsoe ensures coercion from SAGE + polynomials yields the same results evertime, even if one lost + track of the algebraic variable needed to coerce. + + INPUT: + force -- enforce creation of a new algebraic variable (default: False) + + """ + global __domains__ + + from sage.rings.finite_field import is_FiniteField + + setCharacteristicp(int(k.characteristic())) + + if is_FiniteField(k) and k.degree()!=1: + if not __domains__.has_key(k) or force==True: + setCharacteristic(0) #make sure nex step works + __domains__[k] = rootOf(k.polynomial()) + setCharacteristic(k.characteristic()) + return __domains__[k] + + + +def getCharacteristic(): + """ + Returns the currently set characteristic + """ + #embed{ int getCharacteristicp() + return getCharacteristic(); + #}embed + return getCharacteristicp() + +def rootOf(mipo, char *name = NULL): + r""" + This function returns the algebraic variable that is defined by + the minimal polynomial mipo. name has to be a name that is not + yet assigned to a variable. This is the only way to define an + algebraic variable. The level of the returned variable depends on + how many algebraic variables are defined so far. The variable + remains anonymous if the user does not specify a name. mipo has + to be an irreducible univariate polynomial over the current base + domain. + + INPUT: + mipo -- minimal polynomial to define the algebraic variable, + this must be a CanonicalForm or something coercable to + ca CanonicalForm via the CanonicalForm constructor + + name -- name to assign to the algebraic variable + + \note{Do not use this function to construct a base domain for + finite extension fields. Use setBaseDomain() instead.} + """ + #embed{ void *rootOfpn(void *mipo, char name) + return new Variable(rootOf(*(CanonicalForm*)mipo, name)); + #}embed + #embed{ void *rootOfp(void *mipo) + return new Variable(rootOf(*(CanonicalForm*)mipo)); + #}embed + + cdef char *cName + cdef char cChar + cdef CanonicalForm cMipo + + if not isinstance(mipo,CanonicalForm): + cMipo = CanonicalForm(mipo) + else: + cMipo = mipo + + if name == NULL or len(name)==0: + _sig_on + return make_Variable(rootOfp(cMipo.thisptr)) + else: + cName = name + cChar = cName[0] + _sig_on + return make_Variable(rootOfpn(cMipo.thisptr,cChar)) + +def getMipo( a, x=None ): + """ + This function returns the minimal polynomial that defines the + algebraic variable a. The minimal polynomial is returned as a + CanonicalForm of variable x. + + INPUT: + a -- algebraic variable + x -- variable the minimal polynomial is expressed in + """ + #embed{ void *CF_getMipo(void *a, void *x) + return new CanonicalForm(getMipo(*(Variable*)a,*(Variable*)x)); + #}embed + + cdef Variable cA + cdef Variable cX + + if x==None: + x='x' + + if not isinstance(a,Variable): + cA = Variable(a) + else: + cA = a + if not isinstance(x,Variable): + cX = Variable(x) + else: + cX = x + + _sig_on + return make_CF(CF_getMipo(cA.thisptr,cX.thisptr)) + + Index: sage/libs/cf/ftmpl_inst.cc =================================================================== --- sage/libs/cf/ftmpl_inst.cc (revision 1088) +++ sage/libs/cf/ftmpl_inst.cc (revision 1088) @@ -0,0 +1,80 @@ +/* emacs edit mode for this file is -*- C++ -*- */ +/* $Id: ftmpl_inst.cc,v 1.10 2006/05/15 09:03:05 Singular Exp $ */ + +//{{{ docu +// +// ftmpl_inst.cc - Factory's template instantiations. +// +// For a detailed description how to instantiate Factory's +// template classes and functions and how to add new +// instantiations see the `README' file. +// +//}}} + +#include + +#ifdef macintosh +#include <:templates:ftmpl_array.cc> +#include <:templates:ftmpl_factor.cc> +#include <:templates:ftmpl_list.cc> +#include <:templates:ftmpl_functions.h> +#include <:templates:ftmpl_matrix.cc> +#else +#include +#include +#include +#include +#include +#endif + +#include + + +//{{{ explicit template class instantiations +template class Factor; +template class List; +template class ListItem; +template class ListIterator; +template class List; +template class ListItem; +template class ListIterator; +template class Array; +template class List; +template class ListItem; +template class ListIterator; +template class Matrix; +template class SubMatrix; +template class Array; +//}}} + +//{{{ explicit template function instantiations +#ifndef NOSTREAMIO +template OSTREAM & operator << ( OSTREAM &, const List & ); +template OSTREAM & operator << ( OSTREAM &, const List & ); +template OSTREAM & operator << ( OSTREAM &, const List & ); +template OSTREAM & operator << ( OSTREAM &, const Array & ); +template OSTREAM & operator << ( OSTREAM &, const Factor & ); +template OSTREAM & operator << ( OSTREAM &, const Matrix & ); +template OSTREAM & operator << ( OSTREAM &, const Array & ); +#endif /* NOSTREAMIO */ + +template int operator == ( const Factor &, const Factor & ); + +template List Union ( const List &, const List & ); + +#if ! defined(WINNT) || defined(__GNUC__) +template CanonicalForm tmax ( const CanonicalForm &, const CanonicalForm & ); +template CanonicalForm tmin ( const CanonicalForm &, const CanonicalForm & ); + +template Variable tmax ( const Variable &, const Variable & ); +template Variable tmin ( const Variable &, const Variable & ); + +template int tmax ( const int &, const int & ); +template int tmin ( const int &, const int & ); +template int tabs ( const int & ); +#endif +//}}} + +// +// place here your own template stuff, not yet instantiated by factory +// Index: sage/libs/cf/misc.pxi =================================================================== --- sage/libs/cf/misc.pxi (revision 294) +++ sage/libs/cf/misc.pxi (revision 294) @@ -0,0 +1,11 @@ +# Unset the signal handler and create a string from the buffer, +# then free the memory in the buffer. +cdef object string(char* s): + _sig_off + # Makes a python string and deletes what is pointed to by s. + t = str(s) + free(s) + return t + +_INIT = None + Index: sage/libs/linbox/linbox.pyx =================================================================== --- sage/libs/linbox/linbox.pyx (revision 4367) +++ sage/libs/linbox/linbox.pyx (revision 3506) @@ -268,16 +268,15 @@ def smithform(self): - raise NotImplementedError - #cdef mpz_t* v - #linbox_integer_dense_smithform(&v, self.matrix, self.nrows, self.ncols) - #z = [] - #cdef Integer k - #cdef size_t n - #for n from 0 <= n < self.ncols: - # k = Integer() - # mpz_set(k.value, v[n]) - # mpz_clear(v[n]) - # z.append(k) - #linbox_integer_dense_delete_array(v) - #return z - + cdef mpz_t* v + linbox_integer_dense_smithform(&v, self.matrix, self.nrows, self.ncols) + z = [] + cdef Integer k + cdef size_t n + for n from 0 <= n < self.ncols: + k = Integer() + mpz_set(k.value, v[n]) + mpz_clear(v[n]) + z.append(k) + linbox_integer_dense_delete_array(v) + return z + Index: sage/libs/ntl/ntl.pyx =================================================================== --- sage/libs/ntl/ntl.pyx (revision 4368) +++ sage/libs/ntl/ntl.pyx (revision 3597) @@ -581,9 +581,5 @@ [1 0 0 2] """ - if ZZX_is_zero(self.x): - return False - return bool(ZZ_is_one(ZZX_leading_coefficient(self.x))) - - # return bool(ZZX_is_monic(self.x)) + return bool(ZZX_is_monic(self.x)) def __neg__(self): @@ -1632,16 +1628,6 @@ sage: g [1 0 0 2] - sage: f = ntl.ZZ_pX([1,2,0,3,0,2]) - sage: f.is_monic() - False - """ - # The following line is what we should have. However, strangely this is *broken* - # on PowerPC Intel in NTL, so we program around - # the problem. (William Stein) - #return bool(ZZ_pX_is_monic(self.x)) - - if ZZ_pX_is_zero(self.x): - return False - return bool(ZZ_p_is_one(ZZ_pX_leading_coefficient(self.x))) + """ + return bool(ZZ_pX_is_monic(self.x)) def __neg__(self): @@ -2789,7 +2775,7 @@ Finite Field in a of size 2^8 """ - from sage.rings.finite_field import FiniteField + from sage.rings.finite_field import FiniteField_ext_pari f = ntl_GF2E_modulus()._sage_() - return FiniteField(int(2)**GF2E_degree(),modulus=f,name=names) + return FiniteField_ext_pari(int(2)**GF2E_degree(),modulus=f,name=names) @@ -2919,12 +2905,4 @@ OUTPUT: FiniteFieldElement over k - - - EXAMPLE: - sage: ntl.GF2E_modulus([1,1,0,1,1,0,0,0,1]) - sage: e = ntl.GF2E([0,1]) - sage: a = e._sage_(); a - a - """ cdef int i @@ -2933,7 +2911,7 @@ if k==None: - from sage.rings.finite_field import FiniteField + from sage.rings.finite_field import FiniteField_ext_pari f = ntl_GF2E_modulus()._sage_() - k = FiniteField(2**deg,name='a',modulus=f) + k = FiniteField_ext_pari(2**deg,modulus=f) if cache != None: Index: age/libs/singular/singular-cdefs.pxi =================================================================== --- sage/libs/singular/singular-cdefs.pxi (revision 4364) +++ (revision ) @@ -1,311 +1,0 @@ -""" -C level declarations of symbols in the SINGULAR libary. - -AUTHOR: Martin Albrecht - -NOTE: our ring, poly etc. types are not the Singular ring, poly, -etc. types. They are deferences. So a Singular ring is a ring* here. - -""" -include "../../ext/cdefs.pxi" - -cdef extern from "stdsage.h": - ctypedef void PyObject - - # Global tuple -- useful optimization - void init_global_empty_tuple() - object PY_NEW(object t) - void* PY_TYPE(object o) - int PY_TYPE_CHECK(object o, object t) - object IS_INSTANCE(object o, object t) - - -# Shared Library Loading, -cdef extern from "dlfcn.h": - void *dlopen(char *, long) - char *dlerror() - -cdef extern from "stdlib.h": - void *calloc(size_t nmemb, size_t size) - void free(void *ptr) - -cdef extern from "libsingular.h": - - # This is the basis ring datatype of SINGULAR - # - # Please note: This is not the SINGULAR ring, it has one pointer - # layer less. - - ctypedef struct ring "ip_sring": - int *order # array of orderings - int *block0 # starting pos - int *block1 # ending pos - int **wvhdl - int OrdSgn - int ShortOut - int CanShortOut - char **names - short N - - # This is the basic polynomial datatype of SINGULAR - # - # Please note: This is not the SINGULAR poly, it has one pointer - # layer less. - - ctypedef struct poly "polyrec": - poly *next - - ctypedef struct ideal "ip_sideal": - poly **m - long rank - int nrows - int ncols - - # comment from the SINGULAR code: - # - # 'keiner (ausser obachman) darf das folgenden benutzen !!!' - # in English: 'nobody (except obachman) may use the following!!!' - # - # I feel so obachman today. - - ctypedef struct intvec: - int *(*ivGetVec)() - int (*rows)() - int (*cols)() - - # oMalloc Bins - ctypedef struct omBin "omBin_s" - - # numbers, i.e. coefficients (except modInt) - ctypedef struct number "snumber": - mpz_t z - mpz_t n - int s - - # SINGULAR Init - # ------------------------ - void feInitResources(char *name) - - void rChangeCurrRing(ring *r) - cdef ring *currRing - cdef omBin *rnumber_bin - cdef int (*pLDeg)(poly *p, int *l, ring *r) - - int siInit(char *) - - # OMalloc - void *omAlloc0(size_t size) - void *omAllocBin(omBin *bin) - char *omStrDup(char *) - void omFree(void *) - - cdef enum rRingOrder_t: - ringorder_no - ringorder_a - ringorder_a64 # for int64 weights - ringorder_c - ringorder_C - ringorder_M - ringorder_S - ringorder_s - ringorder_lp - ringorder_dp - ringorder_rp - ringorder_Dp - ringorder_wp - ringorder_Wp - ringorder_ls - ringorder_ds - ringorder_Ds - ringorder_ws - ringorder_Ws - ringorder_L - - # rDefault accepts the characteristic, the number of variables, - # and an array of variable names. - ring *rDefault(int char, int nvars, char **names) - - void rDelete(ring *r) # Destructor - int rChar(ring *r) # Returns the characteristic of the ring - char* rRingVar(short i, ring *r) # Returns the name of the i-th variable of the ring r. - - void rComplete(ring *r, int force) - void rUnComplete(ring *r) - - void rWrite(ring *r) - - int rRing_has_Comp(ring *r) - - ### Polynomials - ### The rule of thumb is: p_XXX accepts a ring as parameter, while - ### pXXX doesn't. It relies on currRing. - - ## Constructions / Destructors - ## ------------------------------- - poly *p_Init(ring *r) - void p_Delete(poly **p, ring *r) - - poly *p_ISet(int i, ring *r) - poly *p_NSet(number *n,ring *r) - - - int p_SetCoeff(poly *p, number *n, ring *r) - number *p_GetCoeff(poly *p, ring *r) - - int p_SetExp(poly *p, int v, int e, ring *r) - int p_GetExp(poly *p, int v, ring *r) - unsigned long p_SetComp(poly *p, unsigned long v, ring *r) - unsigned long p_GetComp(poly *p, ring *r) - - void pLcm(poly *a, poly *b, poly *m) - - - void p_Setm(poly *p, ring *r) - - poly *p_Copy(poly *p, ring *r) - - # homogenizes p by multiplying certain powers of the varnum-th variable - poly *pHomogen (poly *p, int varnum) - - int pIsHomogeneous(poly *p) - - ## IO - ## ------------------------------- - char *p_String(poly *p, ring *r, ring *r) - - char *p_Read(char *c, poly *p, ring *r) - poly *pmInit(char *c, int b) - - void writemon(poly *p, int ko, ring *r) - char *StringAppendS(char *) - void StringSetS(char *) - - ## Arithmetic - ## ------------------------------- - - # returns -p, p is destroyed - poly *p_Neg(poly *p, ring *r) - - # returns p*n, p is const (i.e. copied) - poly *pp_Mult_nn(poly *p, number *n, ring *r) - - # returns p*m, does neither destroy p nor m - poly *pp_Mult_mm(poly *p, poly *m, ring *r) - - # returns p+q, destroys p and q - poly *p_Add_q(poly *p, poly *q, ring *r) - - # return p - m*q, destroys p; const: q,m - poly *p_Minus_mm_Mult_qq(poly *p, poly *m, poly *q, ring *r) - - # returns p + m*q destroys p, const: q, m - poly *p_Plus_mm_Mult_qq(poly *p, poly *m, poly *q, ring *r) - - # returns p*q, does neither destroy p nor q - poly *pp_Mult_qq(poly *p, poly *q, ring *r) - - # returns p*q, destroys p and q - poly *p_Mult_q(poly *p, poly *q, ring *r) - - poly *pDivide(poly *,poly *) - - # returns p*Coeff(m) for such monomials pm of p, for which m is - # divisble by pm - poly *pp_Mult_Coeff_mm_DivSelect(poly *p, poly *m, ring *r) - - # returns merged p and q, assumes p and q have no monomials which are equal - poly *p_Merge_q(poly *p, poly c, ring *r) - - # sorts p using bucket sort: returns sorted poly assumes that - # monomials of p are all different reverses it first, if revert == - # TRUE, use this if input p is "almost" sorted correctly - poly *p_SortMerge(poly *p, ring *r, int revert) - - # like SortMerge, except that p may have equal monimals - poly *p_SortAdd(poly *p, ring *r, int revert) - - # returns the i-th power of p; p will be destroyed, requires global ring - poly *pPower(poly *p, int exp) - - # returns newly allocated copy of Lm(p), coef is copied, - # next=NULL, p might be NULL - poly *p_Head(poly *p, ring *r) - - # returns TRUE, if leading monom of a divides leading monom of b - # i.e., if there exists a expvector c > 0, s.t. b = a + c; - int p_DivisibleBy(poly *a, poly *b, ring *r) - - # like pDivisibleBy, except that it is assumed that a!=NULL, b!=NULL - int p_LmDivisibleBy(poly *a, poly *b, ring *r) - - - long pDeg(poly *p, ring *r) - long pTotaldegree(poly *p, ring *r) - - poly *pNext(poly *p) - - # pCmp: args may be NULL - # returns: (p2==NULL ? 1 : (p1 == NULL ? -1 : p_LmCmp(p1, p2))) - int p_Cmp(poly *l, poly *r, ring *r) - int p_ExpVectorEqual(poly *p, poly *m, ring *r) - - - int p_IsConstant(poly *, ring *) - int p_IsUnit(poly *, ring *) - - poly *pSubst(poly *, int varidx, poly *value) - - poly *pInvers(int n, poly *, intvec *) - - # gcd of f and g - poly *singclap_gcd ( poly *f, poly *g ) - - poly *singclap_resultant ( poly *f, poly *g , poly *x) - - int singclap_extgcd( poly *f, poly *g, poly *res, poly *pa, poly *pb ) - - poly *singclap_pdivide ( poly *f, poly *g ) - - void singclap_divide_content( poly *f ) - - ideal *singclap_factorize ( poly *f, intvec ** v , int with_exps) - - int singclap_isSqrFree(poly *f) - - poly *kNF(ideal *F, ideal *Q, poly *p) - - # this guy actually calculates with napoly not poly - #poly *singclap_alglcm(poly *f, poly *g) - - # Numbers - # --------------------------------- - - # General - number *n_Init(int n, ring *r) - void n_Delete(number **n, ring *r) - int nInt(number *n) - number *n_Div(number *a, number *b, ring *r) - int n_GreaterZero(number *a, ring *r) - int n_IsZero(number *a, ring *r) - number *n_Sub(number *a, number *b, ring *r) - - number *nInvers(number *n) - - cdef long SR_INT - long SR_TO_INT(number *) - long SR_HDL(mpz_t ) - - # QQ - number *nlInit2gmp(mpz_t i, mpz_t j) - number *nlInit2(int i, int j) - number *nlGetNom(number *n, ring *r) - number *nlGetDenom(number *n, ring *r) - - - # - # Ideals - # ----------------------------------- - - ideal *idInit(int size, int rank) - void id_Delete(ideal **, ring *) - ideal *fast_map(ideal *, ring *, ideal *, ring *) Index: age/libs/singular/singular.pxd =================================================================== --- sage/libs/singular/singular.pxd (revision 4095) +++ (revision ) @@ -1,12 +1,0 @@ -include "singular-cdefs.pxi" - -from sage.rings.rational cimport Rational -from sage.structure.element cimport Element - - -cdef class Conversion: - cdef extern Rational si2sa_QQ(self, number (*),ring (*)) - cdef extern number *sa2si_QQ(self, Rational ,ring (*)) - - cdef extern object si2sa(self, number *n, ring *_ring, object base) - cdef extern number *sa2si(self, Element elem, ring * _ring) Index: age/libs/singular/singular.pxi =================================================================== --- sage/libs/singular/singular.pxi (revision 4092) +++ (revision ) @@ -1,2 +1,0 @@ -cdef extern Rational (si2sa_QQ(number (*),ring (*))) -cdef extern number (*(sa2si_QQ(Rational ,ring (*)))) Index: age/libs/singular/singular.pyx =================================================================== --- sage/libs/singular/singular.pyx (revision 4095) +++ (revision ) @@ -1,113 +1,0 @@ -""" -Singular C function and class declaration - -AUTHOR: Martin Albrecht -""" - -################################################################################ -# -################################################################################ - -############################################################################### -# SAGE: System for Algebra and Geometry Experimentation -# Copyright (C) 2005, 2006 William Stein -# Distributed under the terms of the GNU General Public License (GPL) -# The full text of the GPL is available at: -# http://www.gnu.org/licenses/ -############################################################################### - -include "singular-cdefs.pxi" - -from sage.rings.rational_field import RationalField -from sage.rings.finite_field import FiniteField_prime_modn - - -cdef extern from "stdsage.h": - ctypedef void PyObject - object PY_NEW(object t) - int PY_TYPE_CHECK(object o, object t) - PyObject** FAST_SEQ_UNSAFE(object o) - void init_csage() - -cdef class Conversion: - """ - A work around class to export the contained methods/functions - """ - - cdef public Rational si2sa_QQ(self, number *n, ring *_ring): - """ - Converts a SINGULAR rational number to a SAGE rational number. - - INPUT: - n -- number - _ring -- singular ring, used to check type of n - - OUTPUT: - SAGE rational number matching n - """ - - #TYPECHECK HERE - - cdef number *nom - cdef number *denom - cdef mpq_t _z - - cdef mpz_t nom_z, denom_z - - cdef Rational z - - mpq_init(_z) - - ## Immediate integers handles carry the tag 'SR_INT', i.e. the last bit is 1. - ## This distuingishes immediate integers from other handles which point to - ## structures aligned on 4 byte boundaries and therefor have last bit zero. - ## (The second bit is reserved as tag to allow extensions of this scheme.) - ## Using immediates as pointers and dereferencing them gives address errors. - - nom = nlGetNom(n, _ring) - mpz_init(nom_z) - - if (SR_HDL(nom) & SR_INT): mpz_set_si(nom_z, SR_TO_INT(nom)) - else: mpz_set(nom_z,&nom.z) - - mpq_set_num(_z,nom_z) - n_Delete(&nom,_ring) - mpz_clear(nom_z) - - denom = nlGetDenom(n, _ring) - mpz_init(denom_z) - - if (SR_HDL(denom) & SR_INT): mpz_set_si(denom_z, SR_TO_INT(denom)) - else: mpz_set(denom_z,&denom.z) - - mpq_set_den(_z, denom_z) - n_Delete(&denom,_ring) - mpz_clear(denom_z) - - z = Rational() - z.set_from_mpq(_z) - return z - - cdef public number *sa2si_QQ(self, Rational r, ring *_ring): - """ - """ - cdef number *z - return nlInit2gmp( mpq_numref(r.value), mpq_denref(r.value) ) - - cdef public object si2sa(self, number *n, ring *_ring, object base): - if PY_TYPE_CHECK(base, FiniteField_prime_modn): - return base(nInt(n)) - - elif PY_TYPE_CHECK(base, RationalField): - return self.si2sa_QQ(n,_ring) - else: - raise ValueError, "cannot convert SINGULAR number" - - cdef public number *sa2si(self, Element elem, ring * _ring): - if PY_TYPE_CHECK(elem._parent, FiniteField_prime_modn): - return n_Init(int(elem),_ring) - - elif PY_TYPE_CHECK(elem._parent, RationalField): - return self.sa2si_QQ(elem, _ring) - else: - raise ValueError, "cannot convert SINGULAR number" Index: sage/matrix/benchmark.py =================================================================== --- sage/matrix/benchmark.py (revision 4336) +++ sage/matrix/benchmark.py (revision 3487) @@ -25,9 +25,6 @@ t = -timeout alarm(0) - w.append(float(t)) - if w[1] == 0: - w.append(0.0) - else: - w.append(w[0]/w[1]) + w.append(t) + w.append(w[0]/w[1]) w = tuple(w) @@ -518,12 +515,12 @@ raise ValueError, 'unknown system "%s"'%system -def det_GF(n=400, p=16411 , system='sage'): +def det_GF(n=400, p=16411 ,min=1, max=100, system='sage'): """ Dense integer determinant over GF. Given an n x n (with n=400) matrix A over GF with random entries - compute det(A). - """ - if system == 'sage': - A = random_matrix(GF(p), n, n) + between min=1 and max=100, inclusive, compute det(A). + """ + if system == 'sage': + A = random_matrix(GF(p), n, n, x=min, y=max+1) t = cputime() d = A.determinant() Index: sage/matrix/constructor.py =================================================================== --- sage/matrix/constructor.py (revision 3973) +++ sage/matrix/constructor.py (revision 3187) @@ -72,5 +72,4 @@ ring for all the entries (using the Sequence object): - sage: x = polygen(QQ) sage: m = matrix([[1/3,2+x],[3,4]]); m [ 1/3 x + 2] Index: sage/matrix/matrix0.pyx =================================================================== --- sage/matrix/matrix0.pyx (revision 4022) +++ sage/matrix/matrix0.pyx (revision 3543) @@ -717,5 +717,5 @@ S = [] for x in self.list(): - S.append(repr(x)) + S.append(str(x)) tmp = [] Index: sage/matrix/matrix1.pyx =================================================================== --- sage/matrix/matrix1.pyx (revision 4062) +++ sage/matrix/matrix1.pyx (revision 3477) @@ -82,5 +82,5 @@ matrix([0,1,2],[3,4,5],[6,7,8]) sage: a.charpoly('x').expand() - -x^3+12*x^2+18*x + -x^3 + 12*x^2 + 18*x sage: m.charpoly() x^3 - 12*x^2 - 18*x Index: sage/matrix/matrix2.pyx =================================================================== --- sage/matrix/matrix2.pyx (revision 4330) +++ sage/matrix/matrix2.pyx (revision 3665) @@ -18,8 +18,7 @@ from sage.structure.sequence import _combinations, Sequence -from sage.misc.misc import verbose, get_verbose, graphics_filename +from sage.misc.misc import verbose, get_verbose from sage.rings.number_field.all import is_NumberField from sage.rings.integer_ring import ZZ -from sage.rings.rational_field import QQ import sage.modules.free_module @@ -391,5 +390,5 @@ # TODO: find reasonable cutoff (Field specific, but seems to be quite large for Q[x]) # if R.is_integral_domain() and R.is_exact() and algorithm == "hessenberg": - if R.is_field() and R.is_exact() and algorithm == "hessenberg": + if R.is_field() and algorithm == "hessenberg": c = self.charpoly('x')[0] if self._nrows % 2: @@ -2094,5 +2093,5 @@ if PyErr_CheckSignals(): raise KeyboardInterrupt for r from start_row <= r < nr: - if A.get_unsafe(r, c): + if A.get_unsafe(r, c) != 0: pivots.append(c) a_inverse = ~A.get_unsafe(r,c) @@ -2101,5 +2100,5 @@ for i from 0 <= i < nr: if i != start_row: - if A.get_unsafe(i,c): + if A.get_unsafe(i,c) != 0: minus_b = -A.get_unsafe(i, c) A.add_multiple_of_row(i, start_row, minus_b, c) @@ -2308,116 +2307,6 @@ return True - def visualize_structure(self, filename=None, maxsize=512): - """ - Write a PNG image to 'filename' which visualizes self by putting - black pixels in those positions which have nonzero entries. - - White pixels are put at positions with zero entries. If 'maxsize' - is given, then the maximal dimension in either x or y direction is - set to 'maxsize' depending on which is bigger. If the image is - scaled, the darkness of the pixel reflects how many of the - represented entries are nonzero. So if e.g. one image pixel - actually represents a 2x2 submatrix, the dot is darker the more of - the four values are nonzero. - - INPUT: - filename -- either a path or None in which case a filename in - the current directory is chosen automatically - (default:None) - - maxsize -- maximal dimension in either x or y direction of the resulting - image. If None or a maxsize larger than - max(self.nrows(),self.ncols()) is given the image will have - the same pixelsize as the matrix dimensions (default: 512) - - """ - import gd - import os + - cdef int x, y, _x, _y, v, b2 - cdef int ir,ic - cdef float b, fct - - mr, mc = self.nrows(), self.ncols() - - if maxsize is None: - - ir = mc - ic = mr - b = 1.0 - - elif max(mr,mc) > maxsize: - - maxsize = float(maxsize) - ir = int(mc * maxsize/max(mr,mc)) - ic = int(mr * maxsize/max(mr,mc)) - b = max(mr,mc)/maxsize - - else: - - ir = mc - ic = mr - b = 1.0 - - b2 = b**2 - fct = 255.0/b2 - - im = gd.image((ir,ic),1) - white = im.colorExact((255,255,255)) - im.fill((0,0),white) - - # these speed things up a bit - colorExact = im.colorExact - setPixel = im.setPixel - - for x from 0 <= x < ic: - for y from 0 <= y < ir: - v = b2 - for _x from 0 <= _x < b: - for _y from 0 <= _y < b: - if not self.get_unsafe((x*b + _x), (y*b + _y)).is_zero(): - v-=1 #increase darkness - - v = int(v*fct) - val = colorExact((v,v,v)) - setPixel((y,x), val) - - if filename is None: - filename = graphics_filename() - - im.writePng(filename) - - def density(self): - """ - Return the density of self. - - By density we understand the ration of the number of nonzero - positions and the self.nrows() * self.ncols(), i.e. the number - of possible nonzero positions. - - EXAMPLE: - - First, note that the density parameter does not ensure - the density of a matrix, it is only an upper bound. - - sage: A = random_matrix(GF(127),200,200,density=0.3) - sage: A.density() # somewhat random - 643/2500 - - sage: A = matrix(QQ,3,3,[0,1,2,3,0,0,6,7,8]) - sage: A.density() - 2/3 - - """ - cdef int x,y,k - k = 0 - nr = self.nrows() - nc = self.ncols() - for x from 0 <= x < nr: - for y from 0 <= y < nc: - if not self.get_unsafe(x,y).is_zero(): - k+=1 - return QQ(k)/QQ(nr*nc) - cdef decomp_seq(v): Index: sage/matrix/matrix_complex_double_dense.pyx =================================================================== --- sage/matrix/matrix_complex_double_dense.pyx (revision 4062) +++ sage/matrix/matrix_complex_double_dense.pyx (revision 3616) @@ -328,5 +328,4 @@ EXAMPLES: - sage: I = CDF.gen() sage: m = I*Matrix(CDF, 3, range(9)) sage: vals, vecs = m.eigen_left() @@ -358,5 +357,5 @@ EXAMPLES: - sage: m = CDF.gen() * matrix(CDF, 3, range(9)) + sage: m = I*Matrix(CDF, 3, range(9)) sage: vals, vecs = m.eigen() sage: vecs*m # random lower order precision @@ -428,5 +427,4 @@ EXAMPLES: - sage: I = CDF.gen() sage: A =I*matrix(CDF, 3,3, [1,2,5,7.6,2.3,1,1,2,-1]) sage: A # slightly random output @@ -473,6 +471,5 @@ EXAMPLES: - sage: I = CDF.gen() - sage: A = I*matrix(CDF, 3,3, [1,2,5,7.6,2.3,1,1,2,-1]) + sage: A =I*matrix(CDF, 3,3, [1,2,5,7.6,2.3,1,1,2,-1]) sage: A # slightly random output [1.0*I 2.0*I 5.0*I] @@ -547,5 +544,5 @@ [ 0 1.0] [2.0 3.0] - sage: RDF(log(abs(m.determinant()))) + sage: log(abs(m.determinant())) 0.69314718056 sage: m.log_determinant() @@ -654,5 +651,4 @@ EXAMPLES: - sage: I = CDF.gen() sage: m = matrix(CDF,[[1,2],[3,4]]) sage: m = I*m Index: sage/matrix/matrix_integer_dense.pyx =================================================================== --- sage/matrix/matrix_integer_dense.pyx (revision 4367) +++ sage/matrix/matrix_integer_dense.pyx (revision 3665) @@ -497,5 +497,5 @@ # def _dict(self): - def __nonzero__(self): + def is_zero(self): cdef mpz_t *a, *b cdef Py_ssize_t i, j @@ -503,6 +503,6 @@ for i from 0 <= i < self._nrows * self._ncols: if mpz_cmp_si(self._entries[i], 0): - return True - return False + return False + return True def _multiply_linbox(self, Matrix right): @@ -1117,5 +1117,5 @@ #### Elementary divisors - def elementary_divisors(self, algorithm='pari'): + def elementary_divisors(self, algorithm='linbox'): """ Return the elementary divisors of self, in order. @@ -1133,7 +1133,7 @@ INPUT: self -- matrix - algorithm -- (default: 'pari') + algorithm -- (default: 'linbox') 'pari': works robustless, but is slower. - 'linbox' -- use linbox (currently off, broken) + 'linbox' -- use linbox OUTPUT: @@ -1158,5 +1158,4 @@ else: if algorithm == 'linbox': - raise ValueError, "linbox too broken -- currently Linbox SNF is disabled." # This fails in linbox: a = matrix(ZZ,2,[1, 1, -1, 0, 0, 0, 0, 1]) try: Index: sage/matrix/matrix_mod2_dense.pyx =================================================================== --- sage/matrix/matrix_mod2_dense.pyx (revision 4326) +++ sage/matrix/matrix_mod2_dense.pyx (revision 3931) @@ -190,7 +190,4 @@ """ cdef int i,j,e - - if entries is None: - return # scalar ? @@ -745,5 +742,5 @@ k = ((random()>>16)+random())%nc # is this safe? # 16-bit seems safe - writePackedCell(self._entries, i, k, (random()>>16) % 2) + writePackedCell(self._entries, i, j, (random()>>16) % 2) _sig_off Index: sage/matrix/matrix_modn_sparse.pyx =================================================================== --- sage/matrix/matrix_modn_sparse.pyx (revision 4329) +++ sage/matrix/matrix_modn_sparse.pyx (revision 3272) @@ -84,5 +84,5 @@ from sage.rings.integer_mod cimport IntegerMod_int, IntegerMod_abstract -from sage.misc.misc import verbose, get_verbose, graphics_filename +from sage.misc.misc import verbose, get_verbose ################ @@ -344,110 +344,5 @@ self.cache('in_echelon_form',True) - def visualize_structure(self, filename=None, maxsize=512): - """ - Write a PNG image to 'filename' which visualizes self by putting - black pixels in those positions which have nonzero entries. - - White pixels are put at positions with zero entries. If 'maxsize' - is given, then the maximal dimension in either x or y direction is - set to 'maxsize' depending on which is bigger. If the image is - scaled, the darkness of the pixel reflects how many of the - represented entries are nonzero. So if e.g. one image pixel - actually represents a 2x2 submatrix, the dot is darker the more of - the four values are nonzero. - - INPUT: - filename -- either a path or None in which case a filename in - the current directory is chosen automatically - (default:None) - maxsize -- maximal dimension in either x or y direction of the resulting - image. If None or a maxsize larger than - max(self.nrows(),self.ncols()) is given the image will have - the same pixelsize as the matrix dimensions (default: 512) - - """ - import gd - import os - - cdef Py_ssize_t i, j, k - cdef float blk,invblk - cdef int delta - cdef int x,y,r,g,b - - mr, mc = self.nrows(), self.ncols() - - if maxsize is None: - - ir = mc - ic = mr - blk = 1.0 - invblk = 1.0 - - elif max(mr,mc) > maxsize: - - maxsize = float(maxsize) - ir = int(mc * maxsize/max(mr,mc)) - ic = int(mr * maxsize/max(mr,mc)) - blk = max(mr,mc)/maxsize - invblk = maxsize/max(mr,mc) - - else: - - ir = mc - ic = mr - blk = 1.0 - invblk = 1.0 - - delta = 255.0 / blk**2 - - im = gd.image((ir,ic),1) - white = im.colorExact((255,255,255)) - im.fill((0,0),white) - - colorComponents = im.colorComponents - getPixel = im.getPixel - setPixel = im.setPixel - colorExact = im.colorExact - - for i from 0 <= i < self._nrows: - for j from 0 <= j < self.rows[i].num_nonzero: - x = (invblk * self.rows[i].positions[j]) - y = (invblk * i) - r,g,b = colorComponents( getPixel((x,y))) - setPixel( (x,y), colorExact((r-delta,g-delta,b-delta)) ) - - if filename is None: - filename = sage.misc.misc.graphics_filename() - - im.writePng(filename) - - def density(self): - """ - Return the density of self. - - By density we understand the ration of the number of nonzero - positions and the self.nrows() * self.ncols(), i.e. the number - of possible nonzero positions. - - - EXAMPLE: - - First, note that the density parameter does not ensure - the density of a matrix, it is only an upper bound. - - sage: A = random_matrix(GF(127),200,200,density=0.3, sparse=True) - sage: A.density() # somewhat random - 643/2500 - - sage: A = matrix(QQ,3,3,[0,1,2,3,0,0,6,7,8],sparse=True) - sage: A.density() - 2/3 - """ - from sage.rings.rational_field import QQ - - cdef Py_ssize_t i, j, k - k = 0 - for i from 0 <= i < self._nrows: - for j from 0 <= j < self.rows[i].num_nonzero: - k+=1 - return QQ(k)/QQ(self.nrows()*self.ncols()) + + + Index: sage/matrix/matrix_real_double_dense.pyx =================================================================== --- sage/matrix/matrix_real_double_dense.pyx (revision 4003) +++ sage/matrix/matrix_real_double_dense.pyx (revision 3616) @@ -555,5 +555,5 @@ [0.0 1.0] [2.0 3.0] - sage: RDF(log(abs(m.determinant()))) + sage: log(abs(m.determinant())) 0.69314718056 sage: m.log_determinant() Index: sage/matrix/misc.pyx =================================================================== --- sage/matrix/misc.pyx (revision 4103) +++ sage/matrix/misc.pyx (revision 3630) @@ -111,5 +111,5 @@ cdef Integer _bnd import math - _bnd = Integer(int((N//2).sqrt_approx())) + _bnd = Integer(int((N//2).sqrt())) mpz_init_set(bnd, _bnd.value) mpz_sub(other_bnd, N.value, bnd) @@ -188,5 +188,5 @@ cdef Integer _bnd import math - _bnd = Integer(int((N//2).sqrt_approx())) + _bnd = Integer(int((N//2).sqrt())) mpz_init_set(bnd, _bnd.value) mpz_sub(other_bnd, N.value, bnd) Index: sage/misc/all.py =================================================================== --- sage/misc/all.py (revision 4338) +++ sage/misc/all.py (revision 3704) @@ -7,6 +7,4 @@ DOT_SAGE, SAGE_ROOT, SAGE_URL, SAGE_DB, SAGE_TMP, is_32_bit, is_64_bit, newton_method_sizes) - -from remote_file import get_remote_file from attach import attach @@ -52,74 +50,10 @@ from func_persist import func_persist -from functional import (additive_order, - sqrt as numerical_sqrt, - arg, - base_ring, - base_field, - basis, - category, - charpoly, - coerce, - cyclotomic_polynomial, - decomposition, - denominator, - derivative, - det, - dimension, - dim, - discriminant, - disc, - eta, - exp, - factor, - fcp, - gen, - gens, - hecke_operator, - ideal, - image, - imag, - imaginary, - integral, - integral_closure, - interval, - xinterval, - is_commutative, - is_even, - is_integrally_closed, - is_field, - is_odd, - kernel, - krull_dimension, - lift, - minimal_polynomial, - multiplicative_order, - ngens, - norm, - numerator, - objgens, - objgen, - one, - order, - rank, - real, - regulator, - round, - quotient, - quo, - show, - isqrt, - square_free_part, - squarefree_part, - transpose, - zero, - log as log_b, - parent) - +from functional import * from latex import latex, view, lprint, jsmath # disabled -- nobody uses mathml -#from mathml ml +#from mathml import mathml from trace import trace Index: sage/misc/functional.py =================================================================== --- sage/misc/functional.py (revision 4120) +++ sage/misc/functional.py (revision 3713) @@ -52,7 +52,7 @@ EXAMPLES: - sage: z = CC(1,2) + sage: z = 1+2*I sage: theta = arg(z) - sage: cos(theta)*abs(z) + sage: cos(theta)*abs(z) # slightly random output on cygwin 1.00000000000000 sage: sin(theta)*abs(z) @@ -146,4 +146,9 @@ Return the arc cosine of x. + EXAMPLES: + sage: acos(0.5) + 1.04719755119660 + sage: acos(1 + I*1.0) + 0.904556894302381 - 1.06127506190504*I """ try: return x.acos() @@ -154,4 +159,9 @@ Return the arc sine of x. + EXAMPLES: + sage: asin(0.5) + 0.523598775598299 + sage: asin(1 + I*1.0) + 0.666239432492515 + 1.06127506190504*I """ try: return x.asin() @@ -162,4 +172,9 @@ Return the arc tangent of x. + EXAMPLES: + sage: atan(1/2) + 0.463647609001 + sage: atan(1 + I) + 1.01722196789785 + 0.402359478108525*I """ try: return x.atan() @@ -286,5 +301,5 @@ EXAMPLES: sage: eta(1+I) - 0.742048775837 + 0.19883137023*I + 0.742048775836565 + 0.198831370229911*I """ try: return x.eta() @@ -330,9 +345,9 @@ except AttributeError: return factor(charpoly(x, var)) -## def floor(x): -## try: -## return x.floor() -## except AttributeError: -## return sage.rings.all.floor(x) +def floor(x): + try: + return x.floor() + except AttributeError: + return sage.rings.all.floor(x) def gen(x): @@ -408,11 +423,11 @@ sage: z = 1+2*I sage: imaginary(z) - 2 + 2.00000000000000 sage: imag(z) - 2 + 2.00000000000000 """ return imag(x) -def integral(x, *args, **kwds): +def integral(x, var=None, algorithm='maxima'): """ Return an indefinite integral of an object x. @@ -426,18 +441,19 @@ sage: integral(f) 1/5*x^5 - 1/4*x^4 + 1/3*x^3 - 1/2*x^2 + x - sage: integral(sin(x),x) - -cos(x) - sage: integral(sin(x),y) - sin(x)*y - sage: integral(sin(x), x, 0, pi/2) - 1 - sage: sin(x).integral(x, 0,pi/2) - 1 - """ - if hasattr(x, 'integral'): - return x.integral(*args, **kwds) + """ + if var is None: + try: + return x.integral() + except AttributeError: + pass + import sage.interfaces.all as I + if var is None: + var = 'x' + if algorithm == 'maxima': + return I.maxima(x).integrate(var) + elif algorithm == 'mathematica': + return I.mathematica(x).Integrate(var) else: - from sage.calculus.calculus import SR - return SR(x).integral(*args, **kwds) + raise ValueError, 'no algorithm %s'%algorithm def integral_closure(x): @@ -588,4 +604,13 @@ argument.} + EXAMPLES: + sage: log(10,2) + 3.32192809489 + sage: log(8,2) + 3.0 + sage: log(10) + 2.30258509299 + sage: log(2.718) + 0.999896315728952 """ if b is None: @@ -631,8 +656,4 @@ sage: z = 1+2*I sage: norm(z) - 5 - sage: norm(CDF(z)) - 5.0 - sage: norm(CC(z)) 5.00000000000000 """ @@ -718,5 +739,5 @@ We compute the rank of an elliptic curve: - sage: E = EllipticCurve([0,0,1,-1,0]) + sage: E=EllipticCurve([0,0,1,-1,0]) sage: rank(E) 1 @@ -731,5 +752,5 @@ sage: z = 1+2*I sage: real(z) - 1 + 1.00000000000000 """ try: return x.real() @@ -788,13 +809,8 @@ except AttributeError: pass - _do_show(x) - -def _do_show(x): if sage.server.support.EMBEDDED_MODE: print '
%s
'%sage.misc.latex.latex(x) - return sage.misc.latex.LatexExpr('') # so no visible output - from latex import view - view(x) - #raise AttributeError, "object %s does not support show."%(x, ) + return sage.misc.latex.LatexExpr('') # so not visible output + raise AttributeError, "object %s does not support show."%x def sqrt(x): @@ -867,19 +883,33 @@ squarefree_part = square_free_part -## def square_root(x): -## """ -## Return a square root of x with the same parent as x, if possible, -## otherwise raise a ValueError. -## EXAMPLES: -## sage: square_root(9) -## 3 -## sage: square_root(100) -## 10 -## """ -## try: -## return x.square_root() -## except AttributeError: -## raise NotImplementedError +def square_root(x): + """ + Return a square root of x with the same parent as x, if possible, + otherwise raise a ValueError. + + EXAMPLES: + sage: square_root(9) + 3 + sage: square_root(100) + 10 + """ + try: + return x.square_root() + except AttributeError: + raise NotImplementedError +def tan(x): + """ + Return the tangent of x. + + EXAMPLES: + sage: tan(3.1415) + -0.0000926535900581913 + sage: tan(3.1415/4) + 0.999953674278156 + """ + try: return x.tan() + except AttributeError: return RDF(x).tan() + def transpose(x): """ Index: sage/misc/hg.py =================================================================== --- sage/misc/hg.py (revision 4041) +++ sage/misc/hg.py (revision 3928) @@ -564,43 +564,4 @@ self('%s --help | %s'%(cmd, pager())) - def outgoing(self, url=None, opts=''): - """ - Use this to find changsets that are in your branch, but not in the - specified destination repository. If no destination is specified, the - official repository is used. - - From the Mercurial documentation: - Show changesets not found in the specified destination repository or the - default push location. These are the changesets that would be pushed if - a push was requested. - - See pull() for valid destination format details. - - INPUT: - url: default: self.url() -- the official repository - * http://[user@]host[:port]/[path] - * https://[user@]host[:port]/[path] - * ssh://[user@]host[:port]/[path] - * local directory (starting with a /) - * name of a branch (for hg_sage); no /'s - options: (default: none) - -M --no-merges do not show merges - -f --force run even when remote repository is unrelated - -p --patch show patch - --style display using template map file - -r --rev a specific revision you would like to push - -n --newest-first show newest record first - --template display with template - -e --ssh specify ssh command to use - --remotecmd specify hg command to run on the remote side - """ - if url is None: - url = self.__url - - if not '/' in url: - url = '%s/devel/sage-%s'%(SAGE_ROOT, url) - - self('outgoing %s %s | %s' % (opts, url, pager())) - def pull(self, url=None, options='-u'): """ Index: sage/misc/interpreter.py =================================================================== --- sage/misc/interpreter.py (revision 4356) +++ sage/misc/interpreter.py (revision 3211) @@ -226,5 +226,5 @@ name = str(eval(line[6:])) except: - name = str(line[6:].strip()) + name = line[6:].strip() try: F = open(name) @@ -261,7 +261,7 @@ # e.g., including the "from sage import *" try: - name = str(eval(line[5:])) + name = eval(line[5:]) except: - name = str(line[5:].strip()) + name = line[5:].strip() if isinstance(name, str): if not os.path.exists(name): @@ -314,7 +314,7 @@ # e.g., including the "from sage import *" try: - name = str(eval(line[7:])) + name = eval(line[7:]) except: - name = str(line[7:].strip()) + name = line[7:].strip() name = os.path.abspath(name) if not os.path.exists(name): Index: sage/misc/log.py =================================================================== --- sage/misc/log.py (revision 4179) +++ sage/misc/log.py (revision 1097) @@ -64,6 +64,4 @@ for X in loggers: X._update() - -REFRESH = '' class Log: Index: sage/misc/misc.py =================================================================== --- sage/misc/misc.py (revision 4331) +++ sage/misc/misc.py (revision 3751) @@ -21,5 +21,5 @@ "assert_attribute", "LOGFILE"] -import operator, os, socket, sys, signal, time, weakref, random, resource +import operator, os, sys, signal, time, weakref, random from banner import version, banner @@ -27,6 +27,4 @@ SAGE_ROOT = os.environ["SAGE_ROOT"] SAGE_LOCAL = SAGE_ROOT + '/local/' - -HOSTNAME = socket.gethostname() if not os.path.exists(SAGE_ROOT): @@ -59,5 +57,5 @@ else: print "Your home directory has a space in it. This" - print "will probably break some functionality of SAGE. E.g.," + print "will break some functionality of SAGE. E.g.," print "the GAP interface will not work. A workaround" print "is to set the environment variable HOME to a" @@ -65,14 +63,7 @@ print "permissions to before you start sage." -SPYX_TMP = '%s/spyx/%s'%(DOT_SAGE, HOSTNAME) - -if not os.path.exists(SPYX_TMP): - try: - os.makedirs(SPYX_TMP) - except OSError, msg: - print msg - raise OSError, " ** Error trying to create the SAGE tmp directory in your home directory. A possible cause of this might be that you built or upgraded SAGE after typing 'su'. You probably need to delete the directory $HOME/.sage." - -SAGE_TMP='%s/temp/%s/%s/'%(DOT_SAGE, HOSTNAME, os.getpid()) +SPYX_TMP = '%s/spyx'%DOT_SAGE + +SAGE_TMP='%s/tmp/%s/'%(DOT_SAGE,os.getpid()) if not os.path.exists(SAGE_TMP): try: @@ -122,7 +113,4 @@ spent by subprocesses spawned by SAGE (e.g., Gap, Singular, etc.). - This is done via a call to resource.getrusage, so it avoids the - wraparound problems in time.clock() on Cygwin. - INPUT: t -- (optional) float, time in CPU seconds @@ -145,6 +133,5 @@ except TypeError: t = 0.0 - u,s = resource.getrusage(resource.RUSAGE_SELF)[:2] - return u+s - t + return time.clock() - t def walltime(t=0): @@ -465,24 +452,17 @@ -def coeff_repr(c, is_latex=False): - if not is_latex: - try: - return c._coeff_repr() - except AttributeError: - pass +def coeff_repr(c): + try: + return c._coeff_repr() + except AttributeError: + pass if isinstance(c, (int, long, float)): return str(c) - if is_latex and hasattr(c, '_latex_'): - s = c._latex_() - else: - s = str(c).replace(' ','') + s = str(c).replace(' ','') if s.find("+") != -1 or s.find("-") != -1: - if is_latex: - return "\\left(%s\\right)"%s - else: - return "(%s)"%s + return "(%s)"%s return s -def repr_lincomb(symbols, coeffs, is_latex=False): +def repr_lincomb(symbols, coeffs): """ Compute a string representation of a linear combination of some @@ -517,18 +497,9 @@ all_atomic = True for c in coeffs: - if is_latex and hasattr(symbols[i], '_latex_'): - b = symbols[i]._latex_() - else: - b = str(symbols[i]) + b = str(symbols[i]) + if len(b) > 0: + b = "*" + b if c != 0: - coeff = coeff_repr(c, is_latex) - if coeff == "1": - coeff = "" - elif coeff == "-1": - coeff = "-" - elif not is_latex and len(b) > 0: - b = "*" + b - elif len(coeff) > 0 and b == "1": - b = "" + coeff = coeff_repr(c) if not first: coeff = " + %s"%coeff @@ -546,6 +517,4 @@ s = "-" + s[3:] s = s.replace(" 1*", " ") - if s == "": - return "1" return s @@ -1062,5 +1031,5 @@ def tmp_dir(name): r""" - Create and return a temporary directory in \code{\$HOME/.sage/temp/hostname/pid/} + Create and return a temporary directory in \code{\$HOME/.sage/tmp/pid/} """ name = str(name) @@ -1078,5 +1047,5 @@ n = 0 while True: - tmp = "/temp/tmp_%s_%s"%(name, n) + tmp = "/tmp/tmp_%s_%s"%(name, n) if not os.path.exists(tmp): break @@ -1102,14 +1071,4 @@ return tmp -def graphics_filename(): - """ - Return the next available canonical filename for a plot/graphics - file. - """ - i = 0 - while os.path.exists('sage%d.png'%i): - i += 1 - filename = 'sage%d.png'%i - return filename ################################################################# Index: sage/misc/preparser.py =================================================================== --- sage/misc/preparser.py (revision 4061) +++ sage/misc/preparser.py (revision 3425) @@ -8,6 +8,4 @@ by digits of input (like mathematica). -- Joe Wetherell (2006-04-14): * added MAGMA-style constructor preparsing. - -- Bobby Moretti (2007-01-25): * added preliminary function assignment - notation EXAMPLES: @@ -51,18 +49,4 @@ SyntaxError: invalid syntax -SYMBOLIC FUNCTIONAL NOTATION: - sage: a=10; f(theta, beta) = theta + beta; b = x^2 + theta - sage: f - (theta, beta) |--> theta + beta - sage: a - 10 - sage: b - x^2 + theta - sage: f(theta,theta) - 2*theta - - sage: a = 5; f(x,y) = x*y*sqrt(a) - sage: f - (x, y) |--> sqrt(5)*x*y RAW LITERALS: @@ -130,6 +114,6 @@ # http://www.gnu.org/licenses/ ########################################################################### + import os -import pdb def isalphadigit_(s): @@ -140,18 +124,8 @@ in_double_quote = False in_triple_quote = False - def in_quote(): return in_single_quote or in_double_quote or in_triple_quote def preparse(line, reset=True, do_time=False, ignore_prompts=False): - - # find where the parens are for function assignment notation - oparen_index = -1 - cparen_index = -1 - - # for caclulus function notation: - paren_level = 0 - first_eq_index = -1 - global in_single_quote, in_double_quote, in_triple_quote line = line.rstrip() # xreadlines leaves the '\n' at end of line @@ -186,7 +160,4 @@ in_number = False is_real = False - - in_args = False - if reset: in_single_quote = False @@ -241,4 +212,5 @@ i += 1 continue + # Decide if we should wrap a particular integer or real literal @@ -269,10 +241,4 @@ continue - elif line[i] == ";" and not in_quote(): - line = line[:i+1] + preparse(line[i+1:], reset, do_time, ignore_prompts) - i = len(line) - continue - - # Support for generator construction syntax: # "obj. = objConstructor(...)" @@ -357,98 +323,4 @@ continue - # Support for calculus-like function assignment, the line - # "f(x,y,z) = sin(x^3 - 4*y) + y^x" - # gets turnd into - # "f = SR(sin(x^3 - 4*y) + y^x).function(x,y,z)" - # AUTHORS: - # - Bobby Moretti: initial version - 02/2007 - # - William Stein: make variables become defined if they aren't already defined. - - elif (line[i] == "(") and not in_quote(): - paren_level += 1 - # we need to make sure that this is the first open paren we find - if oparen_index == -1: - oparen_index = i - i += 1 - continue - - elif (line[i] == ")") and not in_quote(): - cparen_index = i - paren_level -= 1 - i += 1 - continue - - elif (line[i] == "=") and paren_level == 0 and not in_quote(): - if first_eq_index == -1: - first_eq_index = i - eq = i - - if cparen_index == -1: - i += 1 - continue - - # make sure the '=' sign is on its own, reprsenting assignment - eq_chars = ["=", "!", ">", "<", "+", "-", "*", "/", "^"] - if line[eq-1] in eq_chars or line[eq+1] in eq_chars: - i += 1 - continue - - line_before = line[:oparen_index].strip() - if line_before == "" or not line_before.isalnum(): - i += 1 - continue - - if eq != first_eq_index: - i += 1 - continue - - vars_end = cparen_index - vars_begin = oparen_index+1 - - # figure out where the line ends - line_after = line[vars_end+1:] - try: - a = line.index("#") - except ValueError: - a = len(line) - - try: - b = line.index(";") - except ValueError: - b = len(line) - - a = min(a,b) - - vars = line[vars_begin:vars_end].split(",") - vars = [v.strip() for v in vars] - b = [] - - - # construct the parsed line - k = line[:vars_begin-1].rfind(';') - if k == -1: - k = len(line) - len(line.lstrip()) - b.append(line[:k]) - b.append('_=var("%s");'%(','.join(vars))) - b.append(line[k:vars_begin-1]) - b.append('=') - b.append('symbolic_expression(') - b.append(line[eq+1:a].strip()) - b.append(').function(') - b.append(','.join(vars)) - b.append(')') - b.append(line[a:]) - - line = ''.join(b) - - # i should get set to the position of the first paren after the new - # assignment operator - n = len(line_before) - i = line.find('=', n) + 2 - - continue - # - ####### END CALCULUS ######## - # exponents can be either ^ or ** elif line[i] == "^" and not in_quote(): Index: sage/misc/remote_file.py =================================================================== --- sage/misc/remote_file.py (revision 4338) +++ sage/misc/remote_file.py (revision 2831) @@ -7,14 +7,7 @@ "http://modular.math.washington.edu/myfile.txt" verbose -- whether to display download status - OUTPUT: creates a file in the temp directory and returns the absolute path to that file. - - EXAMPLES: - sage: g = get_remote_file("http://sage.math.washington.edu/home/novoselt/K3vertices", verbose=False) # optional -- requires an internet connection - sage: L = lattice_polytope.read_all_polytopes(g, "K3 %4d") # optional - sage: L[0] # optional - K3 0: 3-dimensional, 4 vertices. """ if verbose: Index: sage/misc/reset.pyx =================================================================== --- sage/misc/reset.pyx (revision 4270) +++ sage/misc/reset.pyx (revision 3704) @@ -1,90 +1,27 @@ import sys -def reset(vars=None): +def reset(): """ Delete all user defined variables, reset all globals variables back to their default state, and reset all interfaces to other - computer algebra systems. - - If vars is specified, just restore the value of vars and leave - all other variables alone (i.e., call restore). - - INPUT: - vars -- (default: None), a list, or space or comma separated - string. + computer algebra systems. """ - if not vars is None: - restore(vars) - return G = globals() # this is the reason the code must be in SageX. T = type(sys) for k in G.keys(): if k[0] != '_' and type(k) != T: - try: - del G[k] - except KeyError: - pass + del G[k] restore() reset_interfaces() -def restore(vars=None): +def restore(): """ - Restore predefined global variables to their default values. - - INPUT: - vars -- string or list (default: None) if not None, restores - just the given variables to the default value. - - EXAMPLES: - sage: x = 10; y = 15/3; QQ='red' - sage: QQ - 'red' - sage: restore('QQ') - sage: QQ - Rational Field - sage: x - 10 - sage: restore('x y') - sage: x - x - sage: y - y - sage: x = 10; y = 15/3; QQ='red' - sage: ww = 15 - sage: restore() - sage: x,y,QQ,ww - (x, y, Rational Field, 15) - sage: restore('ww') - sage: ww - Traceback (most recent call last): - ... - NameError: name 'ww' is not defined + Restore all predefined global variables to their default values. """ G = globals() # this is the reason the code must be in SageX. import sage.all - D = sage.all.__dict__ - _restore(G, D, vars) - import sage.calculus.calculus - _restore(sage.calculus.calculus.syms_cur, sage.calculus.calculus.syms_default, vars) - -def _restore(G, D, vars): - if vars is None: - for k, v in D.iteritems(): - G[k] = v - else: - if isinstance(vars, str): - if ',' in vars: - vars = vars.split(',') - else: - vars = vars.split() - for k in vars: - if D.has_key(k): - G[k] = D[k] - else: - try: - del G[k] # the default value was "unset" - except KeyError: - pass + for k, v in sage.all.__dict__.iteritems(): + G[k] = v Index: sage/modular/cusps.py =================================================================== --- sage/modular/cusps.py (revision 4061) +++ sage/modular/cusps.py (revision 3311) @@ -102,6 +102,5 @@ Traceback (most recent call last): ... - TypeError: Unable to coerce I () to Rational - + TypeError: Unable to coerce 1.00000000000000*I () to Rational """ return Cusp(x, parent=self) @@ -169,6 +168,6 @@ Traceback (most recent call last): ... - TypeError: unable to convert I to a rational - + TypeError: Unable to coerce 1.00000000000000*I () to Rational + sage: a = Cusp(2,3) sage: loads(a.dumps()) == a Index: sage/modular/dirichlet.py =================================================================== --- sage/modular/dirichlet.py (revision 4003) +++ sage/modular/dirichlet.py (revision 3570) @@ -672,5 +672,5 @@ sage: abs(e.gauss_sum_numerical()) 1.73205080756888 - sage: sqrt(3.0) + sage: sqrt(3) 1.73205080756888 sage: e.gauss_sum_numerical(a=2) @@ -684,5 +684,5 @@ sage: abs(e.gauss_sum_numerical()) 3.60555127546399 - sage: sqrt(13.0) + sage: sqrt(13) 3.60555127546399 """ Index: sage/modular/modform/element.py =================================================================== --- sage/modular/modform/element.py (revision 4059) +++ sage/modular/modform/element.py (revision 3616) @@ -103,6 +103,6 @@ return chi - def __nonzero__(self): - return not self.element().is_zero() + def is_zero(self): + return self.element().is_zero() def level(self): Index: sage/modular/modform/numerical.py =================================================================== --- sage/modular/modform/numerical.py (revision 4249) +++ sage/modular/modform/numerical.py (revision 3870) @@ -247,5 +247,4 @@ [28.0, 28.0, -7.92820323028, 5.92820323028] sage: m = n.modular_symbols() - sage: x = polygen(QQ, 'x') sage: m.T(2).charpoly(x).factor() (x - 9)^2 * (x^2 - 2*x - 2) Index: sage/modular/modsym/space.py =================================================================== --- sage/modular/modsym/space.py (revision 4013) +++ sage/modular/modsym/space.py (revision 3887) @@ -522,5 +522,4 @@ An example that (somewhat spuriously) is over a number field: - sage: x = polygen(QQ) sage: k = NumberField(x^2+1, 'a') sage: M = ModularSymbols(11, base_ring=k, sign=1).cuspidal_submodule() Index: sage/modules/free_module.py =================================================================== --- sage/modules/free_module.py (revision 4263) +++ sage/modules/free_module.py (revision 3665) @@ -107,5 +107,4 @@ import sage.rings.principal_ideal_domain as principal_ideal_domain import sage.rings.field as field -import sage.rings.finite_field as finite_field import sage.rings.integral_domain as integral_domain import sage.rings.ring as ring @@ -2656,22 +2655,4 @@ return self.base_ring() - def __call__(self, e, coerce=True, copy=True, check=True): - """ - - EXAMPLE: - sage: k. = GF(3^4) - sage: VS = k.vector_space() - sage: VS(a) - (0, 1, 0, 0) - - """ - try: - k = e.parent() - if finite_field.is_FiniteField(k) and k.base_ring() == self.base_ring() and k.degree() == self.degree(): - return self(e.vector()) - except AttributeError: - pass - return FreeModule_generic_field.__call__(self,e) - ############################################################################### @@ -3638,9 +3619,6 @@ return Vector_rational_dense elif sage.rings.integer_mod_ring.is_IntegerModRing(R) and not is_sparse: - from vector_modn_dense import Vector_modn_dense, MAX_MODULUS - if R.order() < MAX_MODULUS: - return Vector_modn_dense - else: - return free_module_element.FreeModuleElement_generic_dense + from vector_modn_dense import Vector_modn_dense + return Vector_modn_dense else: if is_sparse: Index: sage/modules/free_module_element.pyx =================================================================== --- sage/modules/free_module_element.pyx (revision 4160) +++ sage/modules/free_module_element.pyx (revision 3887) @@ -353,5 +353,7 @@ Ambient free module of rank 4 over the principal ideal domain Integer Ring """ - return self.parent()([x % p for x in self.list()], copy=False, coerce=False, check=False) + return eval('self.parent()([x % p for x in self.list()], \ + copy=False, coerce=False, check=False)', + {'self':self, 'p':p}) def Mod(self, p): @@ -404,5 +406,5 @@ if d == 0: return "()" # compute column widths - S = [str(x) for x in self.list(copy=False)] + S = eval('[str(x) for x in self.list(copy=False)]', {'self':self}) #width = max([len(x) for x in S]) s = "(" @@ -536,53 +538,12 @@ 2 """ - if not PY_TYPE_CHECK(right, FreeModuleElement): + if not isinstance(right, FreeModuleElement): raise TypeError, "right must be a free module element" - r = right.list(copy=False) - l = self.list(copy=False) + r = right.list() + l = self.list() if len(r) != len(l): raise ArithmeticError, "degrees (%s and %s) must be the same"%(len(l),len(r)) - if len(r) == 0: - return self._parent.base_ring()(0) - sum = l[0] * r[0] - cdef Py_ssize_t i - for i from 1 <= i < len(l): - sum += l[i] * r[i] - return sum - - def pairwise_product(self, right): - """ - Return the dot product of self and right, which is a vector of - of the product of the corresponding entries. - - This is a synonym for self * right. - - INPUT: - right -- vector of the same degree as self. it need not - be in the same vector space as self, as long as - the coefficients can be multiplied. - - EXAMPLES: - sage: V = FreeModule(ZZ, 3) - sage: v = V([1,2,3]) - sage: w = V([4,5,6]) - sage: v.pairwise_product(w) - (4, 10, 18) - sage: sum(v.pairwise_product(w)) == v.dot_product(w) - True - - sage: W = VectorSpace(GF(3),3) - sage: w = W([0,1,2]) - sage: w.pairwise_product(v) - (0, 2, 0) - sage: w.pairwise_product(v).parent() - Vector space of dimension 3 over Finite Field of size 3 - - Implicit coercion is well defined (irregardless of order), so - we get 2 even if we do the dot product in the other order. - - sage: v.pairwise_product(w).parent() - Vector space of dimension 3 over Finite Field of size 3 - """ - return self * right + zero = self.parent().base_ring()(0) + return sum(eval('[l[i]*r[i] for i in xrange(len(l))]', {'l':l,'r':r}), zero) def element(self): @@ -672,6 +633,6 @@ z = self.base_ring()(0) v = self.list() - cdef Py_ssize_t i - return [i for i from 0 <= i < self.degree() if v[i] != z] + return eval('[i for i in xrange(self.degree()) if v[i] != z]', + {'self':self, 'z':z, 'v':v}) def support(self): # do not override. @@ -770,5 +731,5 @@ if coerce: try: - entries = [R(x) for x in entries] + entries = eval('[R(x) for x in entries]',{'R':R, 'entries':entries}) except TypeError: raise TypeError, "Unable to coerce entries (=%s) to %s"%(entries, R) @@ -919,5 +880,5 @@ """ x = set(v.keys()).intersection(set(w.keys())) - return sum([v[k]*w[k] for k in x]) + return eval('sum([v[k]*w[k] for k in x])', {'v':v, 'w':w, 'x':x}) def make_FreeModuleElement_generic_sparse(parent, entries, degree): Index: sage/modules/real_double_vector.pyx =================================================================== --- sage/modules/real_double_vector.pyx (revision 4021) +++ sage/modules/real_double_vector.pyx (revision 3616) @@ -333,36 +333,4 @@ memcpy(self.v.data,p,self.v.size*sizeof(double)) - ############################# - # statistics - ############################# - def mean(self): - return gsl_stats_mean(self.v.data, self.v.stride, self.v.size) - - def variance(self): - return gsl_stats_variance(self.v.data, self.v.stride, self.v.size) - - #def covariance(self): - # return gsl_stats_covariance(self.v.data, self.v.stride, self.v.size) - - def standard_deviation(self): - """ - EXAMPLES: - sage: v = vector(RDF, 5, [1,2,3,4,5]) - sage: v.standard_deviation() - 1.5811388300841898 - """ - return gsl_stats_sd(self.v.data, self.v.stride, self.v.size) - - def stats_skew(self): - return gsl_stats_skew(self.v.data, self.v.stride, self.v.size) - - def stats_kurtosis(self): - return gsl_stats_kurtosis(self.v.data, self.v.stride, self.v.size) - - def stats_lag1_autocorrelation(self): - return gsl_stats_lag1_autocorrelation(self.v.data, self.v.stride, self.v.size) - - - def unpickle_v0(parent, entries, degree): # If you think you want to change this function, don't. Index: sage/modules/vector_modn_dense.pyx =================================================================== --- sage/modules/vector_modn_dense.pyx (revision 4263) +++ sage/modules/vector_modn_dense.pyx (revision 3544) @@ -7,6 +7,4 @@ EXAMPLES: sage: v = vector(Integers(8),[1,2,3,4,5]) - sage: type(v) - sage: v (1, 2, 3, 4, 5) @@ -64,6 +62,4 @@ cimport free_module_element - -MAX_MODULUS = MOD_INT_OVERFLOW cdef class Vector_modn_dense(free_module_element.FreeModuleElement): Index: sage/monoids/free_monoid.py =================================================================== --- sage/monoids/free_monoid.py (revision 4007) +++ sage/monoids/free_monoid.py (revision 1840) @@ -154,8 +154,6 @@ """ ## There should really some careful type checking here... - if isinstance(x, FreeMonoidElement) and x.parent() is self: - return x if isinstance(x, FreeMonoidElement) and x.parent() == self: - return FreeMonoidElement(self,x._element_list,check) + return x elif isinstance(x, (int, long, Integer)) and x == 1: return FreeMonoidElement(self, x, check) Index: sage/monoids/free_monoid_element.py =================================================================== --- sage/monoids/free_monoid_element.py (revision 4007) +++ sage/monoids/free_monoid_element.py (revision 1859) @@ -67,21 +67,21 @@ raise TypeError, "Argument x (= %s) is of the wrong type."%x -## def __cmp__(left, right): -## """ -## Compare two free monoid elements with the same parents. - -## The ordering is the one on the underlying sorted list of -## (monomial,coefficients) pairs. - -## EXAMPLES: -## sage: R. = FreeMonoid(2) -## sage: x < y -## True -## sage: x * y < y * x -## True -## sage: x * y * x^2 < x * y * x^3 -## True -## """ -## return cmp(left._element_list, right._element_list) + def __cmp__(left, right): + """ + Compare two free monoid elements with the same parents. + + The ordering is the one on the underlying sorted list of + (monomial,coefficients) pairs. + + EXAMPLES: + sage: R. = FreeMonoid(2) + sage: x < y + True + sage: x * y < y * x + True + sage: x * y * x^2 < x * y * x^3 + True + """ + return cmp(left._element_list, right._element_list) def _repr_(self): @@ -254,3 +254,4 @@ return 1 return 0 # x = self and y are equal - + + Index: sage/plot/all.py =================================================================== --- sage/plot/all.py (revision 4332) +++ sage/plot/all.py (revision 3581) @@ -4,8 +4,7 @@ polar_plot, contour_plot, arrow, plot_vector_field, matrix_plot, bar_chart, - is_Graphics, - show_default) + is_Graphics) -from plot3d import (Graphics3d, point3d, line3d, plot3d) +from plot3d import (Graphics3d, point3d, line3d) from plot3dsoya_wrap import plot3dsoya Index: sage/plot/plot.py =================================================================== --- sage/plot/plot.py (revision 4330) +++ sage/plot/plot.py (revision 3712) @@ -81,8 +81,7 @@ see the first few zeros: - sage: i = CDF.0 # define i this way for maximum speed. - sage: p1 = plot(lambda t: arg(zeta(0.5+t*i)), 1,27,rgbcolor=(0.8,0,0)) - sage: p2 = plot(lambda t: abs(zeta(0.5+t*i)), 1,27,rgbcolor=hue(0.7)) - sage: p1 + p2 + sage: p1 = plot(lambda t: arg(zeta(0.5+t*I)), 1,27,rgbcolor=(0.8,0,0)) + sage: p2 = plot(lambda t: abs(zeta(0.5+t*I)), 1,27,rgbcolor=hue(0.7)) + sage: p1+p2 Graphics object consisting of 2 graphics primitives sage: (p1+p2).save() @@ -128,6 +127,4 @@ #***************************************************************************** -import pdb - from sage.structure.sage_object import SageObject @@ -141,34 +138,4 @@ EMBEDDED_MODE = False SHOW_DEFAULT = False - -def show_default(default=None): - """ - Set the default for showing plots using the following commands: - plot, parametric_plot, polar_plot, and list_plot. - - If called with no arguments, returns the current default. - - EXAMPLES: - The default starts out as False: - sage: show_default() - False - - We set it to True. - sage: show_default(True) - - We see that it is True. - sage: show_default() - True - - Now certain plot commands will display their plots by default. - - Turn of default display. - sage: show_default(False) - - """ - global SHOW_DEFAULT - if default is None: - return SHOW_DEFAULT - SHOW_DEFAULT = bool(default) do_verify = True @@ -501,6 +468,8 @@ EXAMPLES: - sage: g1 = plot(abs(sqrt(x^3 - 1)), 1, 5) - sage: g2 = plot(-abs(sqrt(x^3 - 1)), 1, 5) + sage: h1 = lambda x : abs(sqrt(x^3 - 1)) + sage: h2 = lambda x : -abs(sqrt(x^3 - 1)) + sage: g1 = plot(h1, 1, 5) + sage: g2 = plot(h2, 1, 5) sage: h = g1 + g2 sage: h.save() @@ -618,5 +587,5 @@ self._extend_y_axis(ymax) - def plot(self, *args, **kwds): + def _plot_(self, **args): return self @@ -721,5 +690,9 @@ from matplotlib.figure import Figure if filename is None: - filename = sage.misc.misc.graphics_filename() + i = 0 + while os.path.exists('sage%s.png'%i): + i += 1 + filename = 'sage%s.png'%i + try: ext = os.path.splitext(filename)[1].lower() @@ -1610,5 +1583,5 @@ def __call__(self, points, coerce=True, **kwds): try: - return points.plot(**kwds) + return points._plot_(**kwds) except AttributeError: pass @@ -1921,5 +1894,6 @@ A red, blue, and green "cool cat": - sage: G = plot(-cos(x), -2, 2, thickness=5, rgbcolor=(0.5,1,0.5)) + sage: ncos = lambda x: -cos(x) + sage: G = plot(ncos, -2, 2, thickness=5, rgbcolor=(0.5,1,0.5)) sage: P = polygon([[1,2], [5,6], [5,0]], rgbcolor=(1,0,0)) sage: Q = polygon([(-x,y) for x,y in P[0]], rgbcolor=(0,0,1)) @@ -2191,5 +2165,5 @@ where X is a SAGE object that either is callable and returns - numbers that can be coerced to floats, or has a plot method + numbers that can be coerced to floats, or has a _plot_ method that returns a GraphicPrimitive object. @@ -2277,5 +2251,5 @@ We can change the line style to one of '--' (dashed), '-.' (dash dot), '-' (solid), 'steps', ':' (dotted): - sage: g = plot(sin(x), 0, 10, linestyle='-.') + sage: g = plot(sin,0,10, linestyle='-.') sage: g.save('sage.png') """ @@ -2283,28 +2257,30 @@ o = self.options o['plot_points'] = 200 - o['plot_division'] = 1000 - o['max_bend'] = 0.1 - o['rgbcolor'] = (0,0,0) + o['plot_division'] = 1000 # is this a good value? + o['max_bend'] = 0.1 # is this good as well? def _repr_(self): return "plot; type plot? for help and examples." - def __call__(self, funcs, *args, **kwds): - do_show = False - if kwds.has_key('show') and kwds['show']: - do_show = True - del kwds['show'] - if hasattr(funcs, 'plot'): - G = funcs.plot(*args, **kwds) - else: - G = self._call(funcs, *args, **kwds) - if do_show: - G.show() - return G - - def _call(self, funcs, xmin=None, xmax=None, parametric=False, - polar=False, label='', show=None, **kwds): + def __call__(self, funcs, xmin=None, xmax=None, parametric=False, + polar=False, label='', + show=None, **kwds): if show is None: show = SHOW_DEFAULT + try: + G = funcs._plot_(xmin=xmin, xmax=xmax, **kwds) + if show: + G.show(**kwds) + return G + except AttributeError: + pass + try: + G = funcs.plot(xmin=xmin, xmax=xmax, **kwds) + if show: + G.show(**kwds) + return G + except AttributeError: + pass + if xmin is None: xmin = -1 @@ -2319,5 +2295,5 @@ G = Graphics() for i in range(0, len(funcs)): - G += plot(funcs[i], xmin=xmin, xmax=xmax, polar=polar, **kwds) + G += plot(funcs[i], xmin, xmax, polar=polar, **kwds) if show: G.show(**kwds) @@ -2337,5 +2313,4 @@ data = [] dd = delta - exceptions = 0; msg='' for i in xrange(plot_points + 1): x = xmin + i*delta @@ -2346,12 +2321,10 @@ else: x = xmax # guarantee that we get the last point. - try: y = f(x) data.append((x, float(y))) - except (ZeroDivisionError, TypeError, ValueError), msg: + except (TypeError, ValueError), msg: sage.misc.misc.verbose("%s\nUnable to compute f(%s)"%(msg, x),1) - exceptions += 1 - + pass # adaptive refinement i, j = 0, 0 @@ -2363,20 +2336,12 @@ if abs(data[i+1][1] - data[i][1]) > max_bend: x = (data[i+1][0] + data[i][0])/2 - try: - y = float(f(x)) - data.insert(i+1, (x, y)) - except (ZeroDivisionError, TypeError, ValueError), msg: - sage.misc.misc.verbose("%s\nUnable to compute f(%s)"%(msg, x),1) - exceptions += 1 - + y = float(f(x)) + data.insert(i+1, (x, y)) j += 1 if j > plot_division: + #is this wrong? break else: i += 1 - - if (len(data) == 0 and exceptions > 0) or exceptions > 10: - print "WARNING: When plotting, failed to evaluate function at %s points."%exceptions - print "Last error message: '%s'"%msg if parametric: data = [(fdata, g(x)) for x, fdata in data] @@ -2459,11 +2424,11 @@ tmin -- start value of t tmax -- end value of t - show -- bool or None - (default: use the default as set by the show_default command) - whether or not to show the plot immediately + show -- whether or not to show the plot immediately (default: True) other options -- passed to plot. EXAMPLE: - sage: G = parametric_plot( (sin(t), sin(2*t)), 0, 2*pi, rgbcolor=hue(0.6) ) + sage: f = lambda t: sin(t) + sage: g = lambda t: sin(2*t) + sage: G = parametric_plot((f,g),0,2*pi,rgbcolor=hue(0.6)) sage: G.save() """ @@ -2810,7 +2775,7 @@ Make some plots of $\sin$ functions: - sage: f(x) = sin(x) - sage: g(x) = sin(2*x) - sage: h(x) = sin(4*x) + sage: f = lambda x: sin(x) + sage: g = lambda x: sin(2*x) + sage: h = lambda x: sin(4*x) sage: p1 = plot(f,-2*pi,2*pi,rgbcolor=hue(0.5)) sage: p2 = plot(g,-2*pi,2*pi,rgbcolor=hue(0.9)) @@ -2851,8 +2816,8 @@ """ # TODO: figure out WTF from sage.rings.integer import Integer - from math import floor - rr = Integer(int(floor(r*255))).str(base=16) - gg = Integer(int(floor(g*255))).str(base=16) - bb = Integer(int(floor(b*255))).str(base=16) + from sage.rings.arith import floor + rr = Integer(floor(r*255)).str(base=16) + gg = Integer(floor(g*255)).str(base=16) + bb = Integer(floor(b*255)).str(base=16) rr = '0'*(2-len(rr)) + rr gg = '0'*(2-len(gg)) + gg @@ -2875,5 +2840,5 @@ ['#ff0000', '#ffda00', '#48ff00', '#00ff91', '#0091ff', '#4800ff', '#ff00da'] """ - from math import floor + from sage.rings.arith import floor R = [] for i in range(n): Index: sage/plot/plot3d.py =================================================================== --- sage/plot/plot3d.py (revision 4332) +++ sage/plot/plot3d.py (revision 2021) @@ -293,5 +293,9 @@ """ if filename is None: - filename = sage.misc.misc.graphics_filename() + i = 0 + while os.path.exists('sage%s.png'%i): + i += 1 + filename = 'sage%s.png'%i + try: ext = os.path.splitext(filename)[1].lower() @@ -614,15 +618,2 @@ # unique surface factory surface = SurfaceFactory() - - -def plot3d(x, *args, **kwds): - """ - Plot a given object in 3d. - - NOTE: 3d plotting in SAGE is *not* very powerful yet. - """ - if hasattr(x, 'plot3d'): - return x.plot3d(*args, **kwds) - else: - raise NotImplementedError, "3d Plotting of %s not yet implemented"%x - Index: sage/plot/tachyon.py =================================================================== --- sage/plot/tachyon.py (revision 4359) +++ sage/plot/tachyon.py (revision 3161) @@ -1,4 +1,4 @@ r""" -3D Plotting Using Tachyon +3D Ray Tracer AUTHOR: @@ -35,5 +35,5 @@ antialiasing = False, aspectratio = 1.0, - raydepth = 5, + raydepth = 12, camera_center = (-3, 0, 0), updir = (0, 0, 1), @@ -164,5 +164,5 @@ antialiasing = False, aspectratio = 1.0, - raydepth = 8, + raydepth = 12, camera_center = (-3, 0, 0), updir = (0, 0, 1), @@ -192,5 +192,4 @@ def save(self, filename='sage.png', verbose=0, block=True, extra_opts=''): """ - INPUT: filename -- (default: 'sage.png') output filename; the extension of @@ -218,5 +217,8 @@ import sage.server.support if sage.server.support.EMBEDDED_MODE: - filename = sage.misc.misc.graphics_filename() + i = 0 + while os.path.exists('sage%s.png'%i): + i += 1 + filename = 'sage%s.png'%i self.save(filename, verbose=verbose, extra_opts=extra_opts) else: @@ -224,4 +226,5 @@ self.save(filename, verbose=verbose, extra_opts=extra_opts) os.system('%s %s 2>/dev/null 1>/dev/null &'%(sage.misc.viewer.browser(), filename)) + def _res(self): Index: sage/probability/random_variable.py =================================================================== --- sage/probability/random_variable.py (revision 4022) +++ sage/probability/random_variable.py (revision 2617) @@ -2,8 +2,7 @@ Random variables and probability spaces -This introduces a class of random variables, with the focus on -discrete random variables (i.e. on a discrete probability space). -This avoids the problem of defining a measure space and measurable -functions. +This introduces a class of random variables, with the focus on discrete random +variables (i.e. on a discrete probability space). This avoids the problem of +defining a measure space and measurable functions. """ Index: sage/quadratic_forms/binary_qf.py =================================================================== --- sage/quadratic_forms/binary_qf.py (revision 4063) +++ sage/quadratic_forms/binary_qf.py (revision 1894) @@ -268,5 +268,5 @@ ## Find the range of allowed b's - bmax = (-D / ZZ(3)).sqrt_approx().ceil() + bmax = (-D / ZZ(3)).sqrt().ceil() b_range = range(-bmax, bmax+1) Index: age/rings/algebraic_real.py =================================================================== --- sage/rings/algebraic_real.py (revision 4135) +++ (revision ) @@ -1,2169 +1,0 @@ -""" -Field of Algebraic Reals - -AUTHOR: - -- Carl Witty (2007-01-27): initial version - -This is an implementation of the algebraic reals (the real numbers which -are the zero of a polynomial in ZZ[x]). All computations are exact. - -As with many other implementations of the algebraic numbers, we avoid -computing a number field and working in the number field whenever possible. - -Algebraic numbers exist in one of the following forms: - -* a rational number -* the sum, difference, product, or quotient of algebraic numbers -* a particular root of a polynomial, given as a polynomial with -algebraic coefficients, an isolating interval (given as a -RealIntervalFieldElement) which encloses exactly one root, and -the multiplicity of the root -* a polynomial in a generator, where the generator is an algebraic -number given as the root of an irreducible polynomial with integral -coefficients and the polynomial is given as a NumberFieldElement - -An algebraic number can be coerced into RealIntervalField; every algebraic -number has a cached interval of the highest precision yet calculated. - -Everything is done with intervals except for comparisons. By default, -comparisons compute the two algebraic numbers with 128-bit precision -intervals; if this does not suffice to prove that the numbers are different, -then we fall back on exact computation. - -Note that division involves an implicit comparison of the divisor against -zero, and may thus trigger exact computation. - -Also, using an algebraic number in the leading coefficient of -a polynomial also involves an implicit comparison against zero, which -again may trigger exact computation. - -Note that we work fairly hard to avoid computing new number fields; -to halp, we keep a lattice of already-computed number fields and -their inclusions. - -EXAMPLES: - sage: sqrt(AA(2)) > 0 - True - sage: (sqrt(5 + 2*sqrt(AA(6))) - sqrt(AA(3)))**2 == 2 - True - -For a monic cubic polynomial x^3 + b*x^2 + c*x + d with roots -s1,s2,s3, the discriminant is defined as (s1-s2)^2(s1-s3)^2(s2-s3)^2 -and can be computed as b^2c^2 - 4*b^3d - 4*c^3 + 18*bcd - 27*d^2. -We can test that these definitions do give the same result. - - sage: def disc1(b, c, d): - ... return b^2*c^2 - 4*b^3*d - 4*c^3 + 18*b*c*d - 27*d^2 - sage: def disc2(s1, s2, s3): - ... return ((s1-s2)*(s1-s3)*(s2-s3))^2 - sage: x = polygen(AA) - sage: p = x*(x-2)*(x-4) - sage: cp = AA.common_polynomial(p) - sage: d, c, b, _ = p.list() - sage: s1 = AA.polynomial_root(cp, RIF(-1, 1)) - sage: s2 = AA.polynomial_root(cp, RIF(1, 3)) - sage: s3 = AA.polynomial_root(cp, RIF(3, 5)) - sage: disc1(b, c, d) == disc2(s1, s2, s3) - True - sage: p = p + 1 - sage: cp = AA.common_polynomial(p) - sage: d, c, b, _ = p.list() - sage: s1 = AA.polynomial_root(cp, RIF(-1, 1)) - sage: s2 = AA.polynomial_root(cp, RIF(1, 3)) - sage: s3 = AA.polynomial_root(cp, RIF(3, 5)) - sage: disc1(b, c, d) == disc2(s1, s2, s3) - True - sage: p = (x-sqrt(AA(2)))*(x-AA(2).nth_root(3))*(x-sqrt(AA(3))) - sage: cp = AA.common_polynomial(p) - sage: d, c, b, _ = p.list() - sage: s1 = AA.polynomial_root(cp, RIF(1.4, 1.5)) - sage: s2 = AA.polynomial_root(cp, RIF(1.7, 1.8)) - sage: s3 = AA.polynomial_root(cp, RIF(1.2, 1.3)) - sage: disc1(b, c, d) == disc2(s1, s2, s3) - True - -Some computation with radicals: - - sage: phi = (1 + sqrt(AA(5))) / 2 - sage: phi^2 == phi + 1 - True - sage: tau = (1 - sqrt(AA(5))) / 2 - sage: tau^2 == tau + 1 - True - sage: phi + tau == 1 - True - sage: tau < 0 - True - - sage: rt23 = sqrt(AA(2/3)) - sage: rt35 = sqrt(AA(3/5)) - sage: rt25 = sqrt(AA(2/5)) - sage: rt23 * rt35 == rt25 - True - -Algebraic reals which are known to be rational print as rationals; otherwise -they print as intervals (with 53-bit precision). - - sage: AA(2)/3 - 2/3 - sage: AA(5/7) - 5/7 - sage: two = sqrt(AA(4)); two - [1.9999999999999997 .. 2.0000000000000005] - sage: two == 2; two - True - 2 - sage: phi - [1.6180339887498946 .. 1.6180339887498950] - -The paper _ARPREC: An Arbitrary Precision Computation Package_ discusses -this result. Evidently it is difficult to find, but we can easily -verify it. - - sage: alpha = AA.polynomial_root(x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1, RIF(1, 1.2)) - sage: lhs = alpha^630 - 1 - sage: rhs_num = (alpha^315 - 1) * (alpha^210 - 1) * (alpha^126 - 1)^2 * (alpha^90 - 1) * (alpha^3 - 1)^3 * (alpha^2 - 1)^5 * (alpha - 1)^3 - sage: rhs_den = (alpha^35 - 1) * (alpha^15 - 1)^2 * (alpha^14 - 1)^2 * (alpha^5 - 1)^6 * alpha^68 - sage: rhs = rhs_num / rhs_den - sage: lhs - [2.6420403358193507e44 .. 2.6420403358193520e44] - sage: rhs - [2.6420403358193507e44 .. 2.6420403358193520e44] - sage: lhs - rhs - [-62883485433074552000000000000 .. 63753912023197085000000000000] - sage: lhs == rhs - True - sage: lhs - rhs - [0.00000000000000000 .. 0.00000000000000000] - sage: lhs._exact_value() - -242494609856316402264822833062350847769474540*a^9 + 862295472068289472491654837785947906234680703*a^8 - 829559238431038252116584538075753012193290520*a^7 - 125882239615006638366472766103700441555126185*a^6 + 1399067970863104691667276008776398309383579345*a^5 - 1561176687069361567616835847286958553574223422*a^4 + 761706318888840943058230840550737823821027895*a^3 + 580740464974951394762758666210754821723780266*a^2 - 954587496403409756503464154898858512440951323*a + 546081123623099782018260884934770383777092602 where a^10 - 4*a^9 + 5*a^8 - a^7 - 6*a^6 + 9*a^5 - 6*a^4 - a^3 + 5*a^2 - 4*a + 1 = 0 and a in [0.44406334400909258 .. 0.44406334400909265] -""" - -import sage.rings.ring -from sage.structure.sage_object import SageObject -from sage.structure.parent_gens import ParentWithGens -from sage.rings.real_mpfr import RR -from sage.rings.real_mpfi import RealIntervalField, RIF, RealIntervalFieldElement -from sage.rings.polynomial_ring import PolynomialRing -from sage.rings.integer_ring import ZZ -from sage.rings.rational_field import QQ -from sage.rings.number_field.number_field import NumberField -from sage.rings.number_field.number_field_element import is_NumberFieldElement -from sage.rings.arith import factor -from sage.libs.pari.gen import pari - -# Singleton object implementation copied from integer_ring.py -_obj = None -class _uniq_alg(object): - def __new__(cls): - global _obj - if _obj is None: - _obj = sage.rings.ring.Field.__new__(cls) - return _obj - -class AlgebraicRealField(_uniq_alg, sage.rings.ring.Field): - r""" - The field of algebraic reals. - - """ - - def __init__(self): - ParentWithGens.__init__(self, self, ('x',), normalize=False) - self._default_interval_field = RealIntervalField(64) - - def _repr_(self): - return "Algebraic Field" - - # Is there a standard representation for this? - def _latex_(self): - return "\\mathbf{A}" - - def __call__(self, x): - """ - Coerce x into the field of algebraic numbers. - - """ - - if isinstance(x, AlgebraicRealNumber): - return x - return AlgebraicRealNumber(x) - - def _coerce_impl(self, x): - if isinstance(x, (int, long, sage.rings.integer.Integer, - sage.rings.rational.Rational)): - return self(x) - raise TypeError, 'no implicit coercion of element to the algebraic numbers' - - def _is_valid_homomorphism_(self, codomain, im_gens): - try: - return im_gens[0] == codomain._coerce_(self.gen(0)) - except TypeError: - return False - - def default_interval_field(self): - return self._default_interval_field - - def gens(self): - return (self(1), ) - - def gen(self, n=0): - if n == 0: - return self(1) - else: - raise IndexError, "n must be 0" - - def ngens(self): - return 1 - - def is_finite(self): - return False - - def is_atomic_repr(self): - return True - - def characteristic(self): - return sage.rings.integer.Integer(0) - - def order(self): - return infinity.infinity - - def zeta(self, n=2): - if n == 1: - return self(1) - elif n == 2: - return self(-1) - else: - raise ValueError, "no n-th root of unity in algebraic reals" - - def common_polynomial(self, poly): - """ - Given a polynomial with algebraic coefficients, returns a - wrapper that caches high-precision calculations and - factorizations. This wrapper can be passed to polynomial_root - in place of the polynomial. - - Using common_polynomial makes no semantic difference, but will - improve efficiency if you are dealing with multiple roots - of a single polynomial. - - EXAMPLES: - sage: x = polygen(AA) - sage: p = AA.common_polynomial(x^2 - x - 1) - sage: phi = AA.polynomial_root(p, RIF(0, 2)) - sage: tau = AA.polynomial_root(p, RIF(-2, 0)) - sage: phi + tau == 1 - True - sage: phi * tau == -1 - True - """ - return AlgebraicPolynomialTracker(poly) - - def polynomial_root(self, poly, interval, multiplicity=1): - """ - Given a polynomial with algebraic coefficients and an interval - enclosing exactly one root of the polynomial, constructs - an algebraic real representation of that root. - - The polynomial need not be irreducible, or even squarefree; but - if the given root is a multiple root, its multiplicity must be - specified. (IMPORTANT NOTE: Currently, multiplicity-k roots - are handled by taking the (k-1)th derivative of the polynomial. - This means that the interval must enclose exactly one root - of this derivative.) - - The conditions on the arguments (that the interval encloses exactly - one root, and that multiple roots match the given multiplicity) - are not checked; if they are not satisfied, an error may be - thrown (possibly later, when the algebraic number is used), - or wrong answers may result. - - Note that if you are constructing multiple roots of a single - polynomial, it is better to use AA.common_polynomial - to get a shared polynomial. - - EXAMPLES: - sage: x = polygen(AA) - sage: phi = AA.polynomial_root(x^2 - x - 1, RIF(0, 2)); phi - [1.6180339887498946 .. 1.6180339887498950] - sage: p = (x-1)^7 * (x-2) - sage: r = AA.polynomial_root(p, RIF(9/10, 11/10), multiplicity=7) - sage: r; r == 1 - [0.99999999999999988 .. 1.0000000000000003] - True - sage: p = (x-phi)*(x-sqrt(AA(2))) - sage: r = AA.polynomial_root(p, RIF(1, 3/2)) - sage: r; r == sqrt(AA(2)) - [1.4142135623730949 .. 1.4142135623730952] - True - """ - return AlgebraicRealNumber(AlgebraicRealNumberRoot(poly, interval, multiplicity)) - -def is_AlgebraicRealField(F): - return isinstance(F, AlgebraicRealField) - -AA = AlgebraicRealField() - -def rif_seq(): - # XXX Should do some testing to see where the efficiency breaks are - # in MPFR. - bits = 64 - while True: - yield RealIntervalField(bits) - bits = bits * 2 -# # XXX temporary debugging: -# if bits > 1024: -# break - -def clear_denominators(poly): - """ - Takes a monic polynomial and rescales the variable to get a monic - polynomial with "integral" coefficients. Works on any univariate - polynomial whose base ring has a denominator() method that returns - integers; for example, the base ring might be QQ or a number - field. - - Returns the scale factor and the new polynomial. - - (Inspired by Pari's primitive_pol_to_monic().) - - EXAMPLES: - sage: from sage.rings.algebraic_real import clear_denominators - - sage: _. = QQ['x'] - sage: clear_denominators(x + 3/2) - (2, x + 3) - sage: clear_denominators(x^2 + x/2 + 1/4) - (2, x^2 + x + 1) - - """ - - d = poly.denominator() - if d == 1: - return d, poly - deg = poly.degree() - factors = {} - for i in range(deg): - d = poly[i].denominator() - df = factor(d) - for f, e in df: - oe = 0 - if f in factors: - oe = factors[f] - min_e = (e + (deg-i) - 1) // (deg-i) - factors[f] = max(oe, min_e) - change = 1 - for f, e in factors.iteritems(): - change = change * f**e - poly = poly * (change ** deg) - poly = poly(poly.parent().gen() / change) - return change, poly - -def do_polred(poly): - """ - Find the polynomial of lowest discriminant that generates - the same field as poly, out of those returned by the Pari - polred routine. Returns a triple: (elt_fwd, elt_back, new_poly), - where new_poly is the new polynomial, elt_fwd is a polynomial - expression for a root of the new polynomial in terms of a root - of the original polynomial, and elt_back is a polynomial expression - for a root of the original polynomial in terms of a root of the - new polynomial. - - EXAMPLES: - sage: from sage.rings.algebraic_real import do_polred - - sage: _. = QQ['x'] - sage: do_polred(x^2-5) - (-1/2*x + 1/2, -2*x + 1, x^2 - x - 1) - sage: do_polred(x^2-x-11) - (-1/3*x + 2/3, -3*x + 2, x^2 - x - 1) - sage: do_polred(x^3 + 123456) - (-1/4*x, -4*x, x^3 - 1929) - """ - degree = poly.degree() - pari_poly = pari(poly) - - red_table = pari_poly.polred(3) - - best = None - best_discr = None - - for i in range(red_table.nrows()): - red_poly = red_table[i,1] - if red_poly.poldegree() < degree: - continue - red_discr = red_poly.poldisc().abs() - if best_discr is None or red_discr < best_discr: - best = red_poly - best_discr = red_discr - best_elt = red_table[i,0] - - assert(best is not None) - parent = poly.parent() - return parent(best_elt), parent(best_elt.Mod(pari_poly).modreverse().lift()), parent(best) - -def isolating_interval(intv_fn, pol): - """ - intv_fn is a function that takes a RealIntervalField and returns an - interval containing some particular root of pol. (It must return - better approximations as the RealIntervalField precision increases.) - pol is an irreducible polynomial with rational coefficients. - - Returns an interval containing at most one root of pol. - - EXAMPLES: - sage: from sage.rings.algebraic_real import isolating_interval - - sage: _. = QQ['x'] - sage: isolating_interval(lambda rif: sqrt(rif(2)), x^2 - 2) - [1.41421356237309504876 .. 1.41421356237309504888] - - And an example that requires more precision: - sage: delta = 10^(-70) - sage: p = (x - 1) * (x - 1 - delta) * (x - 1 + delta) - sage: isolating_interval(lambda rif: rif(1 + delta), p) - [1.00000000000000000000000000000000000000000000000000000000000000000000009999999999999999999999999999999999999999999999999999999999999999999999999999999999998 .. 1.00000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000014] - """ - for rif in rif_seq(): - intv = intv_fn(rif) - - # We need to verify that pol has exactly one root in the - # interval intv. We know (because it is a precondition of - # calling this function) that it has at least one root in the - # interval, so we only need to verify that it has at most one - # root (that the interval is sufficiently narrow). - - # Call the root we want alpha, and the bounding interval - # [bot .. top]. By Budan's theorem, if pol(x+bot) - # has at most one more sign change than pol(x+top), - # then pol has at most one root in this interval. - # This is true if all coefficients in pol(x+bot) - # except for the constant coefficient have the same - # sign as the corresponding coefficients in pol(x+top). - - # We can check this by computing pol(x+new_intv). - # This gives an "interval polynomial" which represents - # a set of polynomials, including pol(x+bot) and - # pol(x+top). If all coefficients except the constant - # coefficient of this interval polynomial are bounded away from - # zero, then pol has at most one root in intv. - - # This is a sufficient condition, but not a necessary one. - # It is not obvious that there always exists a bounding - # interval for alpha with this property. Fortunately, - # in the current case with pol irreducible, such - # a bounding interval does exist. - - # Consider the polynomial pol(x+alpha), where - # pol has degree n. The coefficient of x^k in this - # polynomial has an exact representation as a polynomial - # in QQ[alpha] of degree (n-k). If this coefficient is - # equal to zero, then that means that alpha is a zero - # of this degree (n-k) polynomial. But if k>0, then this - # is impossible, since alpha is a root of an irreducible - # degree n polynomial. - - # Thus, all coefficients of pol(x+alpha) are nonzero, - # except for the constant coefficient, which is zero. We - # are guaranteed that if we select a sufficiently tight bounding - # interval around alpha, then pol(x+[bot..top]) - # will have coefficients bounded away from zero, which proves - # that there is only one root within that bounding interval. - - zero = rif(0) - rif_poly_ring = rif['x'] - - ip = pol(rif_poly_ring.gen() + intv) - coeffs = ip.list() - assert(zero in coeffs[0]) - - ok = True - - for c in coeffs[1:]: - if zero in c: - ok = False - break - if ok: - return intv - -def find_zero_result(fn, l): - """ - l is a list of some sort. - fn is a function which maps an element of l and a RealIntervalField - into an interval, such that for a sufficiently precise RealIntervalField, - exactly one element of l results in an interval containing 0. - Returns that one element of l. - - EXAMPLES: - sage: from sage.rings.algebraic_real import find_zero_result - sage: _. = QQ['x'] - sage: delta = 10^(-70) - sage: p1 = x - 1 - sage: p2 = x - 1 - delta - sage: p3 = x - 1 + delta - sage: p2 == find_zero_result(lambda p, rif: p(rif(1 + delta)), [p1, p2, p3]) - True - """ - for rif in rif_seq(): - result = None - ambig = False - for v in l: - intv = fn(v, rif) - if 0 in intv: - if result is not None: - ambig = True - break - result = v - if ambig: - continue - if result is None: - raise ValueError, 'find_zero_result could not find any zeroes' - return result - -# Cache some commonly-used polynomial rings -QQx = QQ['x'] -QQx_x = QQx.gen() -QQy = QQ['y'] -QQy_y = QQy.gen() -QQxy = QQ['x', 'y'] -QQxy_x = QQxy.gen(0) -QQxy_y = QQxy.gen(1) - -class AlgebraicGeneratorRelation(SageObject): - """ - A simple class for maintaining relations in the lattice of algebraic - extensions. - """ - def __init__(self, child1, child1_poly, child2, child2_poly, parent): - self.child1 = child1 - self.child1_poly = child1_poly - self.child2 = child2 - self.child2_poly = child2_poly - self.parent = parent - -algebraic_generator_counter = 0 - -class AlgebraicGenerator(SageObject): - """ - An AlgebraicGenerator represents both an algebraic real alpha and - the number field QQ[alpha]. There is a single AlgebraicGenerator - representing QQ (with alpha==1). - - The AlgebraicGenerator class is private, and should not be used - directly. - """ - - def __init__(self, field, root): - """ - Construct an AlgebraicGenerator object. - - sage: from sage.rings.algebraic_real import AlgebraicRealNumberRoot, AlgebraicGenerator, unit_generator - sage: _. = QQ['y'] - sage: x = polygen(AA) - sage: nf = NumberField(y^2 - y - 1, name='a', check=False) - sage: root = AlgebraicRealNumberRoot(x^2 - x - 1, RIF(1, 2)) - sage: x = AlgebraicGenerator(nf, root) - sage: x - Number Field in a with defining polynomial y^2 - y - 1 with a in [1.6180339887498946 .. 1.6180339887498950] - sage: x.field() - Number Field in a with defining polynomial y^2 - y - 1 - sage: x.is_unit() - False - sage: x.union(unit_generator) is x - True - sage: unit_generator.union(x) is x - True - """ - self._field = field - self._unit = (field is None) - self._root = root - self._unions = {} - global algebraic_generator_counter - self._index = algebraic_generator_counter - algebraic_generator_counter += 1 - - def __hash__(self): - return self._index - - def __cmp__(self, other): - return cmp(self._index, other._index) - - def _repr_(self): - if self._unit: - return 'Unit generator' - else: - return '%s with a in %s'%(self._field, self._root.interval_fast(RIF)) - - def is_unit(self): - """ - Returns true iff this is the unit generator (alpha == 1), which - does not actually extend the rationals. - - EXAMPLES: - sage: from sage.rings.algebraic_real import unit_generator - sage: unit_generator.is_unit() - True - """ - return self._unit - - def field(self): - return self._field - - def interval_fast(self, field): - """ - Returns an interval containing this generator, to the specified - precision. - """ - return self._root.interval_fast(field) - - def union(self, other): - """ - Given generators alpha and beta, alpha.union(beta) gives a generator - for the number field QQ[alpha][beta]. - - EXAMPLES: - sage: from sage.rings.algebraic_real import AlgebraicRealNumberRoot, AlgebraicGenerator, unit_generator - sage: _. = QQ['y'] - sage: x = polygen(AA) - sage: nf2 = NumberField(y^2 - 2, name='a', check=False) - sage: root2 = AlgebraicRealNumberRoot(x^2 - 2, RIF(1, 2)) - sage: gen2 = AlgebraicGenerator(nf2, root2) - sage: gen2 - Number Field in a with defining polynomial y^2 - 2 with a in [1.4142135623730949 .. 1.4142135623730952] - sage: nf3 = NumberField(y^2 - 3, name='a', check=False) - sage: root3 = AlgebraicRealNumberRoot(x^2 - 3, RIF(1, 2)) - sage: gen3 = AlgebraicGenerator(nf3, root3) - sage: gen3 - Number Field in a with defining polynomial y^2 - 3 with a in [1.7320508075688771 .. 1.7320508075688775] - sage: gen2.union(unit_generator) is gen2 - True - sage: unit_generator.union(gen3) is gen3 - True - sage: gen2.union(gen3) - Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a in [0.51763809020504147 .. 0.51763809020504159] - """ - if self._unit: - return other - if other._unit: - return self - if self is other: - return self - if other in self._unions: - return self._unions[other].parent - if self._field.polynomial().degree() < other._field.polynomial().degree(): - self, other = other, self - sp = self._field.polynomial() - op = other._field.polynomial() - op = QQx(op) - # print sp - # print op - # print self._field.polynomial() - # print self._field.polynomial().degree() - pari_nf = self._field.pari_nf() - # print pari_nf[0] - factors = list(pari_nf.nffactor(op).lift())[0] - # print factors - x, y = QQxy.gens() - # XXX Go through strings because multivariate polynomial coercion from - # Pari is not implemented - factors_sage = [QQxy(str(p)) for p in factors] - # print factors_sage - def find_fn(p, rif): - rif_poly = rif['x', 'y'] - ip = rif_poly(p) - return ip(other._root.interval_fast(rif), self._root.interval_fast(rif)) - my_factor = find_zero_result(find_fn, factors_sage) - - if my_factor.degree(x) == 1 and my_factor.coefficient(x) == 1: - value = (-my_factor + x).univariate_polynomial(QQy) - # print value - rel = AlgebraicGeneratorRelation(self, QQy_y, - other, value, - self) - self._unions[other] = rel - other._unions[self] = rel - return rel.parent - - # XXX need more caching here - newpol, self_pol, k = pari_nf.rnfequation(my_factor, 1) - k = int(k) - # print newpol - # print self_pol - # print k - - newpol_sage = QQx(newpol) - newpol_sage_y = QQy(newpol_sage) - - red_elt, red_back, red_pol = do_polred(newpol_sage_y) - - red_back_x = QQx(red_back) - - new_nf = NumberField(red_pol, name='a', check=False) - - self_pol_sage = QQx(self_pol.lift()) - - new_nf_a = new_nf.gen() - - def intv_fn(rif): - return red_elt(self._root.interval_fast(rif) * k + other._root.interval_fast(rif)) - new_intv = isolating_interval(intv_fn, red_pol) - - new_gen = AlgebraicGenerator(new_nf, AlgebraicRealNumberRoot(QQx(red_pol), new_intv)) - rel = AlgebraicGeneratorRelation(self, self_pol_sage(red_back_x), - other, (QQx_x - k*self_pol_sage)(red_back_x), - new_gen) - self._unions[other] = rel - other._unions[self] = rel - return rel.parent - - def super_poly(self, super, checked=None): - """ - Given a generator gen and another generator super, where super - is the result of a tree of union() operations where one of the - leaves is gen, gen.super_poly(super) returns a polynomial - expressing the value of gen in terms of the value of super. - (Except that if gen is unit_generator, super_poly() always - returns None.) - - EXAMPLES: - sage: from sage.rings.algebraic_real import AlgebraicGenerator, AlgebraicRealNumberRoot, unit_generator - sage: _. = QQ['y'] - sage: x = polygen(AA) - sage: nf2 = NumberField(y^2 - 2, name='a', check=False) - sage: root2 = AlgebraicRealNumberRoot(x^2 - 2, RIF(1, 2)) - sage: gen2 = AlgebraicGenerator(nf2, root2) - sage: gen2 - Number Field in a with defining polynomial y^2 - 2 with a in [1.4142135623730949 .. 1.4142135623730952] - sage: nf3 = NumberField(y^2 - 3, name='a', check=False) - sage: root3 = AlgebraicRealNumberRoot(x^2 - 3, RIF(1, 2)) - sage: gen3 = AlgebraicGenerator(nf3, root3) - sage: gen3 - Number Field in a with defining polynomial y^2 - 3 with a in [1.7320508075688771 .. 1.7320508075688775] - sage: gen2_3 = gen2.union(gen3) - sage: gen2_3 - Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a in [0.51763809020504147 .. 0.51763809020504159] - sage: unit_generator.super_poly(gen2) is None - True - sage: gen2.super_poly(gen2_3) - -a^3 + 3*a - sage: gen3.super_poly(gen2_3) - -a^2 + 2 - - """ - if checked is None: - checked = {} - checked[self] = True - if super is self: - return self._field.gen() - for u in self._unions.values(): - # print self, u.parent - # print checked - if u.parent in checked: - continue - poly = u.parent.super_poly(super, checked) - # print poly, u.child1, u.child2 - if poly is None: - continue - if self is u.child1: - return u.child1_poly(poly) - assert(self is u.child2) - return u.child2_poly(poly) - return None - - def __call__(self, elt): - """ - Takes an AlgebraicRealNumber which is represented as either a rational - or a number field element, and which is in a subfield of the - field generated by this generator. Lifts the number into the - field of this generator, and returns either a Rational or a - NumberFieldElement depending on whether this is the unit generator. - - EXAMPLES: - sage: from sage.rings.algebraic_real import AlgebraicRealNumberRoot, AlgebraicGenerator, AlgebraicRealNumberExtensionElement, AlgebraicRealNumberRational - sage: _. = QQ['y'] - sage: x = polygen(AA) - sage: nf2 = NumberField(y^2 - 2, name='a', check=False) - sage: root2 = AlgebraicRealNumberRoot(x^2 - 2, RIF(1, 2)) - sage: gen2 = AlgebraicGenerator(nf2, root2) - sage: gen2 - Number Field in a with defining polynomial y^2 - 2 with a in [1.4142135623730949 .. 1.4142135623730952] - sage: sqrt2 = AlgebraicRealNumberExtensionElement(gen2, nf2.gen()) - sage: nf3 = NumberField(y^2 - 3, name='a', check=False) - sage: root3 = AlgebraicRealNumberRoot(x^2 - 3, RIF(1, 2)) - sage: gen3 = AlgebraicGenerator(nf3, root3) - sage: gen3 - Number Field in a with defining polynomial y^2 - 3 with a in [1.7320508075688771 .. 1.7320508075688775] - sage: sqrt3 = AlgebraicRealNumberExtensionElement(gen3, nf3.gen()) - sage: gen2_3 = gen2.union(gen3) - sage: gen2_3 - Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a in [0.51763809020504147 .. 0.51763809020504159] - sage: gen2_3(sqrt2) - -a^3 + 3*a - sage: gen2_3(AlgebraicRealNumberRational(1/7)) - 1/7 - sage: gen2_3(sqrt3) - -a^2 + 2 - """ - if self._unit: - return elt.rational_value() - if elt.is_rational(): - return self._field(elt.rational_value()) - if elt.generator() is self: - return elt.field_element_value() - gen = elt.generator() - sp = gen.super_poly(self) - # print gen - # print self - assert(not(sp is None)) - # print sp - # print self._field - return self._field(elt.field_element_value().polynomial()(sp)) - -class AlgebraicRealNumberDescr(SageObject): - """ - An AlgebraicRealNumber is a wrapper around an AlgebraicRealNumberDescr object. - AlgebraicRealNumberDescr is an abstract base class, which should never - be directly instantiated; its concrete subclasses are - AlgebraicRealNumberRational, AlgebraicRealNumberExpression, - AlgebraicRealNumberRoot, and AlgebraicRealNumberExtensionElement. - AlgebraicRealNumberDescr and all of its subclasses are private, and - should not be used directly. - """ - def is_exact(self): - """ - Returns True if self is an AlgebraicRealNumberRational or an - AlgebraicRealNumberExtensionElement. - - EXAMPLES: - sage: from sage.rings.algebraic_real import AlgebraicRealNumberRational - sage: AlgebraicRealNumberRational(1/2).is_exact() - True - """ - return False - - def is_rational(self): - """ - Returns True if self is an AlgebraicRealNumberRational or - an AlgebraicRealNumberExtensionElement which is actually rational. - - EXAMPLES: - sage: from sage.rings.algebraic_real import AlgebraicRealNumberRational - sage: AlgebraicRealNumberRational(3/7).is_rational() - True - """ - return False - - def is_field_element(self): - """ - Returns True if self is an AlgebraicRealNumberExtensionElement. - - sage: from sage.rings.algebraic_real import AlgebraicRealNumberExtensionElement, AlgebraicRealNumberRoot, AlgebraicGenerator - sage: _. = QQ['y'] - sage: x = polygen(AA) - sage: nf2 = NumberField(y^2 - 2, name='a', check=False) - sage: root2 = AlgebraicRealNumberRoot(x^2 - 2, RIF(1, 2)) - sage: gen2 = AlgebraicGenerator(nf2, root2) - sage: sqrt2 = AlgebraicRealNumberExtensionElement(gen2, nf2.gen()) - sage: sqrt2.is_field_element() - True - """ - return False - -class AlgebraicRealNumberRational(AlgebraicRealNumberDescr): - """ - The subclass of AlgebraicRealNumberDescr that represents an arbitrary - rational. This class is private, and should not be used directly. - """ - - def __init__(self, x): - if isinstance(x, (int, long, sage.rings.integer.Integer, - sage.rings.rational.Rational)): - self._value = x - else: - raise TypeError, "Illegal initializer for algebraic number rational" - - def _repr_(self): - return repr(self._value) - - def interval_fast(self, rif): - return rif(self._value) - - def is_rational(self): - return True - - def rational_value(self): - return self._value - - def exactify(self): - return self - - def is_exact(self): - return True - -class AlgebraicRealNumberExpression(AlgebraicRealNumberDescr): - """ - The subclass of AlgebraicRealNumberDescr that represents the sum, - difference, product, or quotient of two algebraic numbers. - This class is private, and should not be used directly. - """ - - def __init__(self, left, right, op): - """ - op can be '+', '-', '*', or '/'. - left and right are algebraic numbers. - If op is '-', then left can be None (in which case it is taken to - mean 0). - if op is '/', then left can be None (in which case it is taken to - mean 1). - - EXAMPLES: - sage: from sage.rings.algebraic_real import AlgebraicRealNumberExpression - sage: AlgebraicRealNumberExpression(AA(1/3), AA(1/2), '+') - [0.83333333333333325 .. 0.83333333333333338] (1/3 + 1/2) - sage: AlgebraicRealNumberExpression(AA(1/3), AA(1/2), '-') - [-0.16666666666666669 .. -0.16666666666666662] (1/3 - 1/2) - sage: AlgebraicRealNumberExpression(AA(1/3), AA(1/2), '*') - [0.16666666666666665 .. 0.16666666666666669] (1/3 * 1/2) - sage: AlgebraicRealNumberExpression(AA(1/3), AA(1/2), '/') - [0.66666666666666662 .. 0.66666666666666675] (1/3 / 1/2) - sage: AlgebraicRealNumberExpression(None, AA(1/2), '-') - [-0.50000000000000000 .. -0.50000000000000000] (None - 1/2) - sage: AlgebraicRealNumberExpression(None, AA(1/2), '/') - [2.0000000000000000 .. 2.0000000000000000] (None / 1/2) - """ - self._left = left - self._right = right - self._op = op - - def _repr_(self): - return '%s (%s %s %s)'%(self.interval_fast(RIF), self._left, self._op, self._right) - - def interval_fast(self, field): - """ - Returns an interval containing self with precision given by the - field argument (which must be a RealIntervalField). Note that - the result may not be a minimal-width interval. - - EXAMPLES: - sage: from sage.rings.algebraic_real import AlgebraicRealNumberExpression - sage: five_sixths = AlgebraicRealNumberExpression(AA(1/2), AA(1/3), '+') - sage: five_sixths.interval_fast(RealIntervalField(4)) - [0.812 .. 0.875] - sage: five_sixths.interval_fast(RealIntervalField(70)) - [0.83333333333333333333305 .. 0.83333333333333333333390] - """ - op = self._op - if op == '+': - return self._left.interval_fast(field) + self._right.interval_fast(field) - if op == '-': - if self._left is None: - return -self._right.interval_fast(field) - else: - return self._left.interval_fast(field) - self._right.interval_fast(field) - if op == '*': - return self._left.interval_fast(field) * self._right.interval_fast(field) - if op == '/': - # Note: this may trigger exact computation - if self._right.sign() == 0: - raise ZeroDivisionError, 'Division by zero in algebraic number field' - if self._left is None: - return ~self._right.interval_fast(field) - else: - return self._left.interval_fast(field) / self._right.interval_fast(field) - raise ValueError, 'Illegal operation for AlgebraicRealNumberExpression' - - def exactify(self): - """ - Return a new exact AlgebraicRealNumberDescr with the same value as self. - - EXAMPLES: - sage: from sage.rings.algebraic_real import AlgebraicRealNumberExpression - sage: five_sixths = AlgebraicRealNumberExpression(AA(1/2), AA(1/3), '+') - sage: five_sixths.exactify() - 5/6 - """ - op = self._op - left = self._left - if left is None: - if op == '-': - left = AA(0) - else: - left = AA(1) - left.exactify() - - right = self._right - right.exactify() - - gen = left._exact_field().union(right._exact_field()) - - # print gen - # print left - # print right - - left_value = gen(left._exact_value()) - right_value = gen(right._exact_value()) - - if op == '+': - value = left_value + right_value - if op == '-': - value = left_value - right_value - if op == '*': - value = left_value * right_value - if op == '/': - value = left_value / right_value - - # print gen - # print left, left_value - # print right, right_value - - if gen.is_unit(): - return AlgebraicRealNumberRational(value) - else: - return AlgebraicRealNumberExtensionElement(gen, value) - - -class AlgebraicRealNumber(sage.structure.element.FieldElement): - """ - An algebraic real (a real number which is the zero of a polynomial - in ZZ[x]). - - AlgebraicRealNumber objects can be created using AA (== AlgebraicRealNumberField()); - either by coercing a rational, or by using the AA.polynomial_root() - method to construct a particular root of a polynomial with algebraic - coefficients. Also, AlgebraicRealNumber is closed under addition, - subtraction, multiplication, division (except by 0), and rational - powers (including roots), except for negative numbers and powers - with an even denominator. - - AlgebraicRealNumber objects can be approximated to any desired precision. - They can be compared exactly; if the two numbers are very close, - this may require exact computation, which can be extremely slow. - - As long as exact computation is not triggered, computation with - algebraic reals should not be too much slower than computation with - intervals. As mentioned above, exact computation is triggered - when comparing two algebraic reals which are very close together. - This can be an explicit comparison in user code, but the following - list of actions (not necessarily complete) can also trigger exact - computation: - Dividing by an algebraic real which is very close to 0. - Using an algebraic real which is very close to 0 as the leading coefficient - in a polynomial. - Taking a root of an alebraic real which is very close to 0. - - The exact definition of "very close" is subject to change; currently, - we compute our best approximation of the two numbers using 128-bit - arithmetic, and see if that's sufficient to decide the comparison. - Note that comparing two algebraic reals which are actually equal will - always trigger exact computation. - - EXAMPLES: - sage: sqrt(AA(2)) - [1.4142135623730949 .. 1.4142135623730952] - sage: sqrt(AA(2))^2 == 2 - True - sage: x = polygen(AA) - sage: phi = AA.polynomial_root(x^2 - x - 1, RIF(0, 2)) - sage: phi - [1.6180339887498946 .. 1.6180339887498950] - sage: phi^2 == phi+1 - True - """ - - def __init__(self, x): - """ - Initialize an algebraic number. The argument must be either - a rational number or a subclass of AlgebraicRealNumberDescr. - - sage: from sage.rings.algebraic_real import AlgebraicRealNumberExpression - sage: AlgebraicRealNumber(22/7) - 22/7 - sage: AlgebraicRealNumber(AlgebraicRealNumberExpression(AA(1/2), AA(1/5), '+')) - [0.69999999999999995 .. 0.70000000000000007] - """ - sage.structure.element.FieldElement.__init__(self, AA) - if isinstance(x, (int, long, sage.rings.integer.Integer, - sage.rings.rational.Rational)): - self._descr = AlgebraicRealNumberRational(x) - elif isinstance(x, (AlgebraicRealNumberDescr)): - self._descr = x - else: - raise TypeError, "Illegal initializer for algebraic number" - - self._value = self._descr.interval_fast(AA.default_interval_field()) - - def _repr_(self): - if self._descr.is_rational(): - return repr(self._descr) - return repr(RIF(self._value)) - - def _mul_(self, other): - sd = self._descr - od = other._descr - if sd.is_rational() and od.is_rational(): - value = sd.rational_value() * od.rational_value() - return AlgebraicRealNumber(AlgebraicRealNumberRational(value)) - elif sd.is_field_element() and \ - od.is_field_element() and \ - sd.field_parent() == od.field_parent(): - value = sd.field_element_value() * od.field_element_value() - return AlgebraicRealNumber(AlgebraicRealNumberExtensionElement(sd.field_parent(), value)) - else: - value = AlgebraicRealNumberExpression(self, other, '*') - return AlgebraicRealNumber(value) - - def _div_(self, other): - sd = self._descr - od = other._descr - if sd.is_rational() and od.is_rational(): - value = sd.rational_value() / od.rational_value() - return AlgebraicRealNumber(AlgebraicRealNumberRational(value)) - elif sd.is_field_element() and \ - od.is_field_element() and \ - sd.field_parent() == od.field_parent(): - value = sd.field_element_value() / od.field_element_value() - return AlgebraicRealNumber(AlgebraicRealNumberExtensionElement(sd.field_parent(), value)) - else: - value = AlgebraicRealNumberExpression(self, other, '/') - return AlgebraicRealNumber(value) - - def __invert__(self): - sd = self._descr - if sd.is_rational: - value = ~sd.rational_value() - return AlgebraicRealNumber(AlgebraicRealNumberRational(value)) - elif sd.is_field_element(): - value = ~sd.field_element_value() - return AlgebraicRealNumber(AlgebraicRealNumberExtensionElement(sd.field_parent(), value)) - else: - value = AlgebraicRealNumberExpression(None, self, '/') - return AlgebraicRealNumber(value) - - def _add_(self, other): - sd = self._descr - od = other._descr - if sd.is_rational() and od.is_rational(): - value = sd.rational_value() + od.rational_value() - return AlgebraicRealNumber(AlgebraicRealNumberRational(value)) - elif sd.is_field_element() and \ - od.is_field_element() and \ - sd.field_parent() == od.field_parent(): - value = sd.field_element_value() + od.field_element_value() - return AlgebraicRealNumber(AlgebraicRealNumberExtensionElement(sd.field_parent(), value)) - else: - value = AlgebraicRealNumberExpression(self, other, '+') - return AlgebraicRealNumber(value) - - def _sub_(self, other): - sd = self._descr - od = other._descr - if sd.is_rational() and od.is_rational(): - value = sd.rational_value() - od.rational_value() - return AlgebraicRealNumber(AlgebraicRealNumberRational(value)) - elif sd.is_field_element() and \ - od.is_field_element() and \ - sd.field_parent() == od.field_parent(): - value = sd.field_element_value() - od.field_element_value() - return AlgebraicRealNumber(AlgebraicRealNumberExtensionElement(sd.field_parent(), value)) - else: - value = AlgebraicRealNumberExpression(self, other, '-') - return AlgebraicRealNumber(value) - - def _neg_(self): - sd = self._descr - if sd.is_rational(): - value = -sd.rational_value() - return AlgebraicRealNumber(AlgebraicRealNumberRational(value)) - elif sd.is_field_element(): - value = -sd.field_element_value() - return AlgebraicRealNumber(AlgebraicRealNumberExtensionElement(sd.field_parent(), value)) - else: - value = AlgebraicRealNumberExpression(None, self, '-') - return AlgebraicRealNumber(value) - - def __cmp__(self, other): - if other._descr.is_rational() and other._descr.rational_value() == 0: - return self.sign() - elif self._descr.is_rational() and self._descr.rational_value() == 0: - return -other.sign() - else: - return self._sub_(other).sign() - - def __pow__(self, e): - """ - self^p returns the p'th power of self (where p can be an arbitrary - rational). - - EXAMPLES: - sage: AA(2)^(1/2) - [1.4142135623730949 .. 1.4142135623730952] - sage: AA(8)^(2/3) - [3.9999999999999995 .. 4.0000000000000009] - sage: AA(8)^(2/3) == 4 - True - sage: x = polygen(AA) - sage: phi = AA.polynomial_root(x^2 - x - 1, RIF(0, 2)) - sage: tau = AA.polynomial_root(x^2 - x - 1, RIF(-2, 0)) - sage: rt5 = AA(5)^(1/2) - sage: phi^10 / rt5 - [55.003636123247410 .. 55.003636123247418] - sage: tau^10 / rt5 - [0.0036361232474132654 .. 0.0036361232474132659] - sage: (phi^10 - tau^10) / rt5 - [54.999999999999992 .. 55.000000000000008] - sage: (phi^10 - tau^10) / rt5 == fibonacci(10) - True - sage: (phi^50 - tau^50) / rt5 == fibonacci(50) - True - """ - e = QQ._coerce_(e) - n = e.numerator() - d = e.denominator() - if d == 1: - if n == 0: - # implements 0^0 == 1 - return AlgebraicRealNumber(1) - elif n < 0: - return (~self).__pow__(-n) - elif n == 1: - return self - else: - pow_n2 = self.__pow__(n//2) - if n % 2 == 1: - return pow_n2 * pow_n2 * self - else: - return pow_n2 * pow_n2 - # Without this special case, we don't know the multiplicity - # of the desired root - if self.sign() == 0: - return AlgebriacNumber(0) - if d % 2 == 0: - if self.sign() < 0: - raise ValueError, 'complex algebraics not implemented' - pow_n = self**n - poly = AAPoly.gen()**d - pow_n - range = pow_n.interval_fast(RIF) - if d % 2 == 0: - result_min = 0 - else: - result_min = min(range.lower(), -1) - result_max = max(range.upper(), 1) - return AlgebraicRealNumber(AlgebraicRealNumberRoot(poly, RIF(result_min, result_max))) - - def sqrt(self): - return self.__pow__(~ZZ(2)) - - def nth_root(self, n): - return self.__pow__(~ZZ(n)) - - def exactify(self): - """ - Compute an exact representation for this number. - - EXAMPLES: - sage: two = sqrt(AA(4)) - sage: two - [1.9999999999999997 .. 2.0000000000000005] - sage: two.exactify() - sage: two - 2 - """ - od = self._descr - self._descr = self._descr.exactify() - new_val = self._descr.interval_fast(self.parent().default_interval_field()) - self._value = self._value.intersection(new_val) -# if self._value != self._descr.interval_fast(RIF): -# v1 = self._value -# v2 = self._descr.interval_fast(RIF) -# # print (v1 != v2) -# # print v1 - v2 -# # print v1, v2 -# # print 'try1', self._descr.interval_fast(RIF), 'done' -# global od2 -# od2 = od -# # print 'a', od.interval_fast(RIF), self._value -# # print 'b', self._descr.interval_fast(RIF), v2 -# # print 'try2', self._descr.interval_fast(RIF), 'done' -# raise ValueError - - def _exact_field(self): - """ - Returns a generator for a number field that includes this number - (not necessarily the smallest such number field). - - EXAMPLES: - sage: AA(2)._exact_field() - Unit generator - sage: (sqrt(AA(2)) + sqrt(AA(19)))._exact_field() - Number Field in a with defining polynomial y^4 - 20*y^2 + 81 with a in [3.7893137826710354 .. 3.7893137826710360] - sage: (AA(7)^(3/5))._exact_field() - Number Field in a with defining polynomial y^5 - 7 with a in [1.4757731615945519 .. 1.4757731615945522] - """ - - sd = self._descr - if sd.is_rational(): - return unit_generator - if sd.is_field_element(): - return sd.field_parent() - self.exactify() - return self._exact_field() - - def _exact_value(self): - """ - Returns either an AlgebraicRealNumberRational or an - AlgebraicRealNumberExtensionElement representing this value. - - EXAMPLES: - sage: AA(2)._exact_value() - 2 - sage: (sqrt(AA(2)) + sqrt(AA(19)))._exact_value() - 1/9*a^3 + a^2 - 11/9*a - 10 where a^4 - 20*a^2 + 81 = 0 and a in [3.7893137826710354 .. 3.7893137826710360] - sage: (AA(7)^(3/5))._exact_value() - a^3 where a^5 - 7 = 0 and a in [1.4757731615945519 .. 1.4757731615945522] - """ - sd = self._descr - if sd.is_exact(): - return sd - self.exactify() - return self._exact_value() - - def _more_precision(self): - """ - Recompute the interval bounding this number with higher-precision - interval arithmetic. - - EXAMPLES: - sage: rt2 = sqrt(AA(2)) - sage: rt2._value - [1.41421356237309504876 .. 1.41421356237309504888] - sage: rt2._more_precision() - sage: rt2._value - [1.414213562373095048801688724209698078568 .. 1.414213562373095048801688724209698078575] - sage: rt2._more_precision() - sage: rt2._value - [1.414213562373095048801688724209698078569671875376948073176679737990732478462101 .. 1.414213562373095048801688724209698078569671875376948073176679737990732478462120] - """ - prec = self._value.prec() - field = RealIntervalField(prec * 2) - self._value = self._descr.interval_fast(field) - - def sign(self): - """ - Compute the sign of this algebraic number (return -1 if negative, - 0 if zero, or 1 if positive). - - Computes an interval enclosing this number using 128-bit interval - arithmetic; if this interval includes 0, then fall back to - exact computation (which can be very slow). - - EXAMPLES: - sage: AA(-5).nth_root(7).sign() - -1 - sage: (AA(2).sqrt() - AA(2).sqrt()).sign() - 0 - """ - if self._value.lower() > 0: - return 1 - elif self._value.upper() < 0: - return -1 - elif self._descr.is_rational(): - val = self._descr.rational_value() - if val > 0: - return 1 - elif val < 0: - return -1 - else: - return 0 - elif self._descr.is_field_element(): - if not self._descr.is_irrational(): - self._descr = AlgebraicRealNumberRational(self._descr.rational_value()) - return self.sign() - # An irrational number must eventually be different from 0 - self._more_precision() - return self.sign() - elif self._value.prec() < 128: - # OK, we'll try adding precision one more time - # print self._value - self._more_precision() - return self.sign() - else: - # Sigh... - self.exactify() - return self.sign() - - def interval_fast(self, field): - """ - Given a RealIntervalField, compute the value of this number - using interval arithmetic of at least the precision of the field, - and return the value in that field. (More precision may be used - in the computation.) The returned interval may be arbitrarily - imprecise, if this number is the result of a sufficiently long - computation chain. - - EXAMPLES: - sage: x = AA(2).sqrt() - sage: x.interval_fast(RIF) - [1.4142135623730949 .. 1.4142135623730952] - sage: x.interval_fast(RealIntervalField(200)) - [1.4142135623730950488016887242096980785696718753769480731766796 .. 1.4142135623730950488016887242096980785696718753769480731766809] - sage: x = AA(4).sqrt() - sage: (x-2).interval_fast(RIF) - [-1.0842021724855045e-19 .. 2.1684043449710089e-19] - """ - if field.prec() == self._value.prec(): - return self._value - elif field.prec() > self._value.prec(): - self._more_precision() - return self.interval_fast(field) - else: - return field(self._value) - - def interval_diameter(self, diam): - """ - Compute an interval representation of self with diameter() at - most diam. The precision of the returned value is unpredictable. - - EXAMPLES: - sage: AA(2).sqrt().interval_diameter(1e-10) - [1.41421356237309504876 .. 1.41421356237309504888] - sage: AA(2).sqrt().interval_diameter(1e-30) - [1.414213562373095048801688724209698078568 .. 1.414213562373095048801688724209698078575] - """ - if diam <= 0: - raise ValueError, 'diameter must be positive in interval_diameter' - - while self._value.diameter() > diam: - self._more_precision() - - return self._value - - def interval(self, field): - """ - Given a RealIntervalField of precision p, compute an interval - representation of self with diameter() at most 2^-p; then round - that representation into the given field. Here diameter() is - relative diameter for intervals not containing 0, and absolute - diameter for intervals that do contain 0; thus, if the returned - interval does not contain 0, it has at least p-1 good bits. - - EXAMPLES: - sage: RIF64 = RealIntervalField(64) - sage: x = AA(2).sqrt() - sage: y = x*x - sage: y = 1000 * y - 999 * y - sage: y.interval_fast(RIF64) - [1.99999999999999966693 .. 2.00000000000000033307] - sage: y.interval(RIF64) - [1.99999999999999999989 .. 2.00000000000000000022] - """ - target = RR(1.0) >> field.prec() - val = self.interval_diameter(target) - return field(val) - - def interval_exact(self, field): - """ - Given a RealIntervalField, compute the best possible - approximation of this number in that field. Note that if this - number is sufficiently close to some floating-point number - (and, in particular, if this number is exactly representable in - floating-point), then this will trigger exact computation, which - may be very slow. - - EXAMPLES: - sage: x = AA(2).sqrt() - sage: y = x*x - sage: x.interval(RIF) - [1.4142135623730949 .. 1.4142135623730952] - sage: x.interval_exact(RIF) - [1.4142135623730949 .. 1.4142135623730952] - sage: y.interval(RIF) - [1.9999999999999997 .. 2.0000000000000005] - sage: y.interval_exact(RIF) - [2.0000000000000000 .. 2.0000000000000000] - sage: z = 1 + AA(2).sqrt() / 2^200 - sage: z.interval(RIF) - [1.0000000000000000 .. 1.0000000000000003] - sage: z.interval_exact(RIF) - [1.0000000000000000 .. 1.0000000000000003] - """ - for extra in (0, 40): - target = RR(1.0) >> field.prec() - # p==precise; pr==precise rounded - pval = self.interval_diameter(target) - pbot = pval.lower() - ptop = pval.upper() - val = field(pval) - bot = val.lower() - top = val.upper() - prbot = pbot.parent()(bot) - prtop = ptop.parent()(top) - if bot == top or (bot.nextabove() == top and - prbot < pbot and ptop < prtop): - return val - - # Even 40 extra bits of precision aren't enough to prove that - # self is not an exactly representable float. - self.exactify() - while True: - # p==precise; pr==precise rounded - pval = self._value - pbot = pval.lower() - ptop = pval.upper() - val = field(pval) - bot = val.lower() - top = val.upper() - prbot = pbot.parent()(bot) - prtop = ptop.parent()(top) - if bot == top or (bot.nextabove() == top and - prbot < pbot and ptop < prtop): - return val - - self._more_precision() - - def real(self, field): - """ - Given a RealField, compute a good approximation to self in that field. - The approximation will be off by at most two ulp's, except for - numbers which are very close to 0, which will have an absolute - error at most 2^(-(field.prec()-1)). Also, the rounding mode of the - field is respected. - - EXAMPLES: - sage: x = AA(2).sqrt()^2 - sage: x.real(RR) - 2.00000000000000 - sage: x.real(RealField(53, rnd='RNDD')) - 1.99999999999999 - sage: x.real(RealField(53, rnd='RNDU')) - 2.00000000000001 - sage: x.real(RealField(53, rnd='RNDZ')) - 1.99999999999999 - sage: (-x).real(RR) - -2.00000000000000 - sage: (-x).real(RealField(53, rnd='RNDD')) - -2.00000000000001 - sage: (-x).real(RealField(53, rnd='RNDU')) - -1.99999999999999 - sage: (-x).real(RealField(53, rnd='RNDZ')) - -1.99999999999999 - sage: (x-2).real(RR) - 5.42101086242752e-20 - sage: (x-2).real(RealField(53, rnd='RNDD')) - -1.08420217248551e-19 - sage: (x-2).real(RealField(53, rnd='RNDU')) - 2.16840434497101e-19 - sage: (x-2).real(RealField(53, rnd='RNDZ')) - 0.000000000000000 - sage: y = AA(2).sqrt() - sage: y.real(RR) - 1.41421356237309 - sage: y.real(RealField(53, rnd='RNDD')) - 1.41421356237309 - sage: y.real(RealField(53, rnd='RNDU')) - 1.41421356237310 - sage: y.real(RealField(53, rnd='RNDZ')) - 1.41421356237309 - """ - v = self.interval(RealIntervalField(field.prec())) - - mode = field.rounding_mode() - if mode == 'RNDN': - return v.center() - if mode == 'RNDD': - return v.lower() - if mode == 'RNDU': - return v.upper() - if mode == 'RNDZ': - if v > 0: - return field(v.lower()) - elif v < 0: - return field(v.upper()) - else: - return field(0) - - def real_exact(self, field): - """ - Given a RealField, compute the best possible approximation of - this number in that field. Note that if this number is sufficiently - close to some floating-point number in the field (and, in particular, - if this number is exactly representable in the field), then - this will trigger exact computation, which may be very slow. - - The rounding mode of the field is respected. - - EXAMPLES: - sage: x = AA(2).sqrt()^2 - sage: x.real_exact(RR) - 2.00000000000000 - sage: x.real_exact(RealField(53, rnd='RNDD')) - 2.00000000000000 - sage: x.real_exact(RealField(53, rnd='RNDU')) - 2.00000000000000 - sage: x.real_exact(RealField(53, rnd='RNDZ')) - 2.00000000000000 - sage: (-x).real_exact(RR) - -2.00000000000000 - sage: (-x).real_exact(RealField(53, rnd='RNDD')) - -2.00000000000000 - sage: (-x).real_exact(RealField(53, rnd='RNDU')) - -2.00000000000000 - sage: (-x).real_exact(RealField(53, rnd='RNDZ')) - -2.00000000000000 - sage: (x-2).real_exact(RR) - 0.000000000000000 - sage: (x-2).real_exact(RealField(53, rnd='RNDD')) - 0.000000000000000 - sage: (x-2).real_exact(RealField(53, rnd='RNDU')) - 0.000000000000000 - sage: (x-2).real_exact(RealField(53, rnd='RNDZ')) - 0.000000000000000 - sage: y = AA(2).sqrt() - sage: y.real_exact(RR) - 1.41421356237310 - sage: y.real_exact(RealField(53, rnd='RNDD')) - 1.41421356237309 - sage: y.real_exact(RealField(53, rnd='RNDU')) - 1.41421356237310 - sage: y.real_exact(RealField(53, rnd='RNDZ')) - 1.41421356237309 - """ - for extra in (0, 40): - target = RR(1.0) >> field.prec() - val = self.interval_diameter(target) - fbot = field(val.lower()) - ftop = field(val.upper()) - if fbot == ftop: - return ftop - - # Even 40 extra bits of precision aren't enough to determine the - # answer. - rifp1 = RealIntervalField(field.prec() + 1) - rifp2 = RealIntervalField(field.prec() + 2) - - val = self.interval_exact(rifp1) - - # Call the largest floating-point number <= self 'x'. Then - # val may be [x .. x], [x .. x + 1/2 ulp], - # [x + 1/2 ulp .. x + 1/2 ulp], or [x + 1/2 ulp .. x + 1 ulp]; - # in the second and fourth cases, the true value is not equal - # to either of the interval endpoints. - - mid = rifp2(val).center() - - # Now mid may be x, x + 1/4 ulp, x + 1/2 ulp, or x + 3/4 ulp; in - # the first and third cases, mid is the exact, true value of self; - # in the second and fourth cases, self is close to mid, and is - # neither x, x + 1/2 ulp, nor x + 1 ulp. - - # In all of these cases, in all rounding modes, the rounded value - # of mid is the same as the rounded value of self. - - return field(mid) - -def is_AlgebraicRealNumber(x): - return isinstance(x, AlgebraicRealNumber) - -AAPoly = PolynomialRing(AA, 'x') - -class AlgebraicPolynomialTracker(SageObject): - """ - Keeps track of a polynomial used for algebraic numbers. - - If multiple algebraic numbers are created as roots of a single - polynomial, this allows the polynomial and information about - the polynomial to be shared. This reduces work if the polynomial - must be recomputed at higher precision, or if it must be made - exact. - - This class is private, and should only be constructed by - AlgebraicRealField.common_polynomial(), and should only be used as - an argument to AlgebraicRealField.polynomial_root(). - - EXAMPLES: - sage: x = polygen(AA) - sage: P = AA.common_polynomial(x^2 - x - 1) - sage: P - x^2 - x - 1 - sage: AA.polynomial_root(P, RIF(0, 2)) - [1.6180339887498946 .. 1.6180339887498950] - """ - - def __init__(self, poly): - poly = AAPoly._coerce_(poly) - self._poly = poly - self._exact = False - - def _repr_(self): - return repr(self._poly) - - def poly(self): - return self._poly - - def exactify(self): - """ - Compute a common field that holds all of the algebraic coefficients - of this polynomial, then factor the polynomial over that field. - Store the factors for later use (ignoring multiplicity). - """ - if self._exact: - return - - self._exact = True - - gen = unit_generator - - for c in self._poly.list(): - c.exactify() - gen = gen.union(c._exact_field()) - - self._gen = gen - - coeffs = [gen(c._exact_value()) for c in self._poly.list()] - - if gen.is_unit(): - qp = QQy(coeffs) - self._factors = [fac_exp[0] for fac_exp in qp.factor()] - else: - fld = gen.field() - fld_poly = fld['x'] - - fp = fld_poly(coeffs) - - self._factors = [fac_exp[0] for fac_exp in fp.factor()] - - def factors(self): - self.exactify() - return self._factors - - def generator(self): - self.exactify() - return self._gen - -class AlgebraicRealNumberRoot(AlgebraicRealNumberDescr): - """ - The subclass of AlgebraicRealNumberDescr that represents a particular - root of a polynomial with algebraic coefficients. - This class is private, and should not be used directly. - """ - def __init__(self, poly, interval, multiplicity=1): - if not isinstance(poly, AlgebraicPolynomialTracker): - poly = AlgebraicPolynomialTracker(poly) - self._poly = poly - # Rethink precision control...integer bitcounts vs. - # RealIntervalField values - self._multiplicity = multiplicity - self._interval = self.refine_interval(interval, 64) - - def _repr_(self): - return 'Root %s of %s'%(self._interval, self._poly) - - def refine_interval(self, interval, precision): - """ - Takes an interval which is assumed to enclose exactly one root - of the polynomial (or, with multiplicity=k, exactly one root - of the k-1'th derivative); and a precision, in bits. - - Tries to find a narrow interval enclosing the root using - interval arithmetic of the given precision. (No particular - number of resulting bits of precision is guaranteed.) - - Uses a combination of Newton's method (adapted for interval - arithmetic) and bisection. The algorithm will converge very - quickly if started with a sufficiently narrow interval. - - EXAMPLES: - sage: from sage.rings.algebraic_real import AlgebraicRealNumberRoot - sage: x = polygen(AA) - sage: rt2 = AlgebraicRealNumberRoot(x^2 - 2, RIF(0, 2)) - sage: rt2.refine_interval(RIF(0, 2), 75) - [1.41421356237309504880163 .. 1.41421356237309504880175] - """ - p = self._poly.poly() - dp = p.derivative() - for i in xrange(0, self._multiplicity - 1): - p = dp - dp = p.derivative() - - # Don't throw away bits in the original interval; doing so might - # invalidate it (include an extra root) - field = RealIntervalField(max(precision, interval.prec())) - zero = field(0) - interval = field(interval) - poly_ring = field['x'] - - # XXX Once this is properly integrated into SAGE, - # then the following should be replaced with: - # interval_p = poly_ring(p) - # (and similarly for interval_dp) - coeffs = [c.interval_fast(field) for c in p.list()] - interval_p = poly_ring(coeffs) - - # This special case is important: this is the only way we could - # refine "infinitely deep" (we could get an interval of diameter - # about 2^{-2^31}, and then hit floating-point underflow); avoiding - # this case here means we don't have to worry about iterating too - # many times later - if coeffs[0] == zero and zero in interval: - return zero - - dcoeffs = [c.interval_fast(field) for c in dp.list()] - interval_dp = poly_ring(dcoeffs) - - linfo = {} - uinfo = {} - - # print coeffs - # print p - - def update_info(info, x): - info['endpoint'] = x - val = interval_p(field(x)) - info['value'] = val - # sign == 1 if val is bounded greater than 0 - # sign == -1 if val is bounded less than 0 - # sign == 0 if val might be 0 - if val > zero: - info['sign'] = 1 - elif val < zero: - info['sign'] = -1 - else: - info['sign'] = 0 - - update_info(linfo, interval.lower()) - update_info(uinfo, interval.upper()) - - newton_lower = True - - while True: - if linfo['sign'] == 0 and uinfo['sign'] == 0: - # We take it on faith that there is a root in interval, - # even though we can't prove it at the current precision. - # We can't do any more refining... - return interval - - # print interval - - if linfo['sign'] == uinfo['sign']: - # Oops... - print self._poly.poly() - print interval_p - print linfo['endpoint'], linfo['value'], linfo['sign'] - print uinfo['endpoint'], uinfo['value'], uinfo['sign'] - raise ValueError, "Refining interval that does not bound unique root!" - - # Use a simple algorithm: - # Try an interval Newton-Raphson step. If this does not add at - # least one bit of information, or if it fails (because the - # slope is not bounded away from zero), then try bisection. - # If this fails because the value at the midpoint is not - # bounded away from zero, then also try the 1/4 and 3/4 points. - # If all of these fail, then return the current interval. - - slope = interval_dp(interval) - - newton_success = False - diam = interval.diameter() - - if not (zero in slope): - # OK, we try Newton-Raphson. - # I have no idea if it helps, but each time through the loop, - # we either do Newton-Raphson from the left endpoint or - # the right endpoint, alternating. - - newton_lower = not newton_lower - if newton_lower: - new_range = linfo['endpoint'] - linfo['value'] / slope - else: - new_range = uinfo['endpoint'] - uinfo['value'] / slope - - if new_range.lower() in interval: - interval = field(new_range.lower(), interval.upper()) - update_info(linfo, interval.lower()) - if new_range.upper() in interval: - interval = field(interval.lower(), new_range.upper()) - update_info(uinfo, interval.upper()) - - new_diam = interval.diameter() - - if new_diam == 0: - # Wow, we managed to find the answer exactly. - # (I think this can only happen with a linear polynomial, - # in which case we shouldn't have been in this - # function at all; but oh well.) - return interval - - if (new_diam << 1) <= diam: - # We got at least one bit. - newton_success = True - - if not newton_success: - center = interval.center() - - def try_bisection(mid): - minfo = {} - update_info(minfo, mid) - if minfo['sign'] == 0: - return interval, False - # We check to make sure the new interval is actually - # narrower; this might not be true if the interval - # is less than 4 ulp's wide - if minfo['sign'] == linfo['sign'] and mid > interval.lower(): - linfo['endpoint'] = minfo['endpoint'] - linfo['value'] = minfo['value'] - linfo['sign'] = minfo['sign'] - return field(mid, interval.upper()), True - if minfo['sign'] == uinfo['sign'] and mid < interval.upper(): - uinfo['endpoint'] = minfo['endpoint'] - uinfo['value'] = minfo['value'] - uinfo['sign'] = minfo['sign'] - return field(interval.lower(), mid), True - return interval, False - - interval, bisect_success = try_bisection(center) - - if not bisect_success: - uq = (center + interval.upper()) / 2 - interval, bisect_success = try_bisection(uq) - if not bisect_success: - lq = (center + interval.lower()) / 2 - interval, bisect_success = try_bisection(lq) - - if not bisect_success: - # OK, we've refined about as much as we can. - # (We might be able to trim a little more off the edges, - # but the interval is no more than twice as wide as the - # narrowest possible.) - return interval - - - def exactify(self): - """ - Returns either an AlgebraicRealNumberRational or an - AlgebraicRealNumberExtensionElement with the same value as this number. - - EXAMPLES: - sage: from sage.rings.algebraic_real import AlgebraicRealNumberRoot - sage: x = polygen(AA) - sage: two = AlgebraicRealNumberRoot((x-2)*(x-sqrt(AA(2))), RIF(1.5, 3)) - sage: two.exactify() - 2 where a^2 - 2 = 0 and a in [1.4142135623730949 .. 1.4142135623730952] - sage: two.exactify().rational_value() - 2 - sage: strange = AlgebraicRealNumberRoot(x^2 + sqrt(AA(3))*x - sqrt(AA(2)), RIF(-1, 3)) - sage: strange.exactify() - a where a^8 - 6*a^6 + 5*a^4 - 12*a^2 + 4 = 0 and a in [0.60510122651395104 .. 0.60510122651395116] - """ - gen = self._poly.generator() - - if gen.is_unit(): - qpf = self._poly.factors() - def find_fn(factor, rif): - return factor(self.interval_fast(rif)) - my_factor = find_zero_result(find_fn, qpf) - - # Factoring always returns monic polynomials over the rationals - assert(my_factor.is_monic()) - - if my_factor.degree() == 1: - return AlgebraicRealNumberRational(-my_factor[0]) - - den, my_factor = clear_denominators(my_factor) - - red_elt, red_back, red_pol = do_polred(my_factor) - - field = NumberField(red_pol, 'a', check=False) - - def intv_fn(rif): - return red_elt(self.interval_fast(rif) * den) - new_intv = isolating_interval(intv_fn, red_pol) - root = AlgebraicRealNumberRoot(AAPoly(red_pol), new_intv) - new_gen = AlgebraicGenerator(field, root) - - return AlgebraicRealNumberExtensionElement(new_gen, red_back(field.gen())/den) - else: - fld = gen.field() - - fpf = self._poly.factors() - # print fpf - def find_fn(factor, rif): - rif_poly = rif['x'] - gen_val = gen.interval_fast(rif) - self_val = self.interval_fast(rif) - ip = rif_poly([c.polynomial()(gen_val) for c in factor]) - return ip(self_val) - my_factor = find_zero_result(find_fn, fpf) - - # print my_factor - assert(my_factor.is_monic()) - - if my_factor.degree() == 1: - return AlgebraicRealNumberExtensionElement(gen, -my_factor[0]) - - # rnfequation needs a monic polynomial with integral coefficients. - # We achieve this with a change of variables. - - den, my_factor = clear_denominators(my_factor) - - pari_nf = gen.field().pari_nf() - # print pari_nf[0] - x, y = QQxy.gens() - my_factor = QQxy['z']([c.polynomial()(y) for c in my_factor])(x) - # print my_factor - - # XXX much duplicate code with AlgebraicGenerator.union() - - # XXX need more caching here - newpol, self_pol, k = pari_nf.rnfequation(my_factor, 1) - k = int(k) - # print newpol - # print self_pol - # print k - - newpol_sage = QQx(newpol) - newpol_sage_y = QQy(newpol_sage) - - red_elt, red_back, red_pol = do_polred(newpol_sage_y) - - new_nf = NumberField(red_pol, name='a', check=False) - - self_pol_sage = QQx(self_pol.lift()) - - new_nf_a = new_nf.gen() - - def intv_fn(rif): - return red_elt(gen.interval_fast(rif) * k + self.interval_fast(rif) * den) - new_intv = isolating_interval(intv_fn, red_pol) - - root = AlgebraicRealNumberRoot(QQx(red_pol), new_intv) - new_gen = AlgebraicGenerator(new_nf, root) - red_back_a = red_back(new_nf.gen()) - new_poly = ((QQx_x - k * self_pol_sage)(red_back_a)/den) - return AlgebraicRealNumberExtensionElement(new_gen, new_poly) - - def _more_precision(self): - """ - Recompute the interval enclosing this root at higher - precision. - """ - prec = self._interval.prec() - self._interval = self.refine_interval(self._interval, prec*2) - - def interval_fast(self, field): - """ - Given a RealIntervalField, compute the value of this number - using interval arithmetic of at least the precision of the field, - and return the value in that field. (More precision may be used - in the computation.) - """ - if field.prec() == self._interval.prec(): - return self._interval - if field.prec() < self._interval.prec(): - return field(self._interval) - self._more_precision() - return self.interval_fast(field) - -unit_generator = AlgebraicGenerator(None, AlgebraicRealNumberRoot(AAPoly.gen() - 1, RIF(1))) - -class AlgebraicRealNumberExtensionElement(AlgebraicRealNumberDescr): - """ - The subclass of AlgebraicRealNumberDescr that represents a number field - element in terms of a specific generator. Consists of a polynomial - with rational coefficients in terms of the generator, and the - generator itself, an AlgebraicGenerator. - """ - - # XXX Should override __new__, and return an AlgebraicRealNumberRational - # if the value is rational. - - def __init__(self, generator, value): - self._generator = generator - self._value = value - - def _repr_(self): - return '%s where %s = 0 and a in %s'%(self._value, - self._generator.field().polynomial()._repr(name='a'), - self._generator.interval_fast(RIF)) - - def generator(self): - return self._generator - - def is_exact(self): - return True - - def is_field_element(self): - return True - - def field_parent(self): - return self._generator - - def exactify(self): - return self - - def rational_value(self): - poly = self._value.polynomial() - assert(poly.is_constant()) - return poly[0] - - def is_irrational(self): - return self._value.polynomial().degree() >= 1 - - def field_element_value(self): - return self._value - - def interval_fast(self, field): - gen_val = self._generator.interval_fast(field) - # XXX Coercion to field() below is necessary in case this is - # a constant polynomial (that is, this is a rational number). - # If we maintain the invariant that AlgebraicRealNumberExtensionElement - # values are never rational, then the coercion is redundant. - return field(self._value.polynomial()(gen_val)) - -ax = AAPoly.gen() -# def heptadecagon(): -# # Compute the exact (x,y) coordinates of the vertices of a 34-gon. -# # (Take every other coordinate to get the vertices of a -# # heptadecagon.) -# # Formulas from: -# # Weisstein, Eric W. "Trigonometry Angles--Pi/17." From -# # MathWorld--A Wolfram Web Resource. -# # http://mathworld.wolfram.com/TrigonometryAnglesPi17.html - -# rt17 = AA(17).sqrt() -# rt2 = AA(2).sqrt() -# eps = (17 + rt17).sqrt() -# epss = (17 - rt17).sqrt() -# delta = rt17 - 1 -# alpha = (34 + 6*rt17 + rt2*delta*epss - 8*rt2*eps).sqrt() -# beta = 2*(17 + 3*rt17 - 2*rt2*eps - rt2*epss).sqrt() -# x = rt2*(15 + rt17 + rt2*(alpha + epss)).sqrt()/8 -# y = rt2*(epss**2 - rt2*(alpha + epss)).sqrt()/8 - -# cx, cy = 1, 0 -# for i in range(34): -# cx, cy = x*cx-y*cy, x*cy+y*cx -# print cx, cy -# print cx.sign(), cy.sign() -# print "Yo!" -# print (cx-1).sign() -# print "OK" -# # heptadecagon() - -# def heptadecagon2(): -# # Compute the exact (x,y) coordinates of the vertices of a 34-gon. -# # (Take every other coordinate to get the vertices of a -# # heptadecagon.) -# # Formulas from: -# # Weisstein, Eric W. "Heptadecagon." From MathWorld--A Wolfram -# # Web Resource. http://mathworld.wolfram.com/Heptadecagon.html - -# x = AA.polynomial_root(256*ax**8 - 128*ax**7 - 448*ax**6 + 192*ax**5 + 240*ax**4 - 80*ax**3 - 40*ax**2 + 8*ax + 1, RIF(0.9829, 0.983)) -# y = (1-x**2).sqrt() - -# cx, cy = 1, 0 -# for i in range(34): -# cx, cy = x*cx-y*cy, x*cy+y*cx -# print cx, cy -# print cx.sign(), cy.sign() -# print (cx-1).sign() -# return x, y -# # heptadecagon2() Index: sage/rings/all.py =================================================================== --- sage/rings/all.py (revision 4255) +++ sage/rings/all.py (revision 3543) @@ -88,10 +88,5 @@ # Quad double -from real_rqdf import RealQuadDoubleField, RQDF, QuadDoubleElement - -# Algebraic reals (the intersection of the algebraic closure of the rationals -# with the reals) -from algebraic_real import (AlgebraicRealField, is_AlgebraicRealField, AA, - AlgebraicRealNumber, is_AlgebraicRealNumber) +#from real_qdrf import RealQuadDoubleField, RQDF # Intervals @@ -121,5 +116,5 @@ # Laurent series ring in one variable from laurent_series_ring import LaurentSeriesRing, is_LaurentSeriesRing -from laurent_series_ring_element import LaurentSeries, is_LaurentSeries +from laurent_series_ring_element import LaurentSeries # Float interval arithmetic Index: sage/rings/arith.py =================================================================== --- sage/rings/arith.py (revision 4254) +++ sage/rings/arith.py (revision 3926) @@ -62,5 +62,5 @@ This example involves a complex number. - sage: z = (1/2)*(1 + RDF(sqrt(3)) *CC.0); z + sage: z = (1/2)*(1 + sqrt(3) *CC.0); z 0.500000000000000 + 0.866025403784439*I sage: p = algdep(z, 6); p @@ -422,8 +422,7 @@ def is_prime_power(n, flag=0): r""" - Returns True if $x$ is a prime power, and False otherwise. - The result is proven correct -- {\em this is NOT a - pseudo-primality test!}. - + Returns True if $x$ is a prime power, and False otherwise. The result + is proven correct -- {\em this is NOT a pseudo-primality test!}. + INPUT: n -- an integer @@ -433,4 +432,8 @@ 2: certify primality using the APRCL test. + OUTPUT: + bool -- True or False + + IMPLEMENTATION: Calls the PARI isprime and ispower functions. EXAMPLES:: @@ -451,5 +454,15 @@ """ Z = integer_ring.ZZ - return Z(n).is_prime_power(flag=flag) + n = Z(n) + if n == 1: + return True + if n < 1: + return False + if is_prime(n, flag): + return True + k, g = pari(n).ispower() + if not k: + return False + return g.isprime(flag) def valuation(m, p): @@ -542,7 +555,7 @@ """ if stop is None: - start, stop = 2, int(start) + start, stop = 2, start try: - v = pari.primes_up_to_n(int(stop-1)) + v = pari.primes_up_to_n(stop-1) except OverflowError: return list(primes(start, stop)) # lame but works. @@ -554,5 +567,5 @@ if start <= 2: return [Z(p) for p in v] # this dominates runtime! - start = pari(int(start)) + start = pari(start) return [Z(p) for p in v if p >= start] # this dominates runtime! @@ -1211,5 +1224,5 @@ 1/8 """ - if bool(a == one): + if a == one: return a if m < 0: @@ -1470,5 +1483,5 @@ # primes at most a given limit. -def factor(n, proof=True, int_=False, algorithm='pari', verbose=0, **kwds): +def factor(n, proof=True, int_=False, algorithm='pari', verbose=0): """ Returns the factorization of the integer n as a sorted list of @@ -1523,5 +1536,5 @@ if not isinstance(n, (int,long, integer.Integer)): try: - return n.factor(**kwds) + return n.factor() except AttributeError: raise TypeError, "unable to factor n" @@ -2247,5 +2260,5 @@ [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1] sage: continued_fraction_list(sqrt(4/19)) - [0, 2, 5, 1, 1, 2, 1, 16, 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 18] + [0, 2, 5, 1, 1, 2, 1, 16, 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 15, 2] sage: continued_fraction_list(RR(pi), partial_convergents=True) ([3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 3], @@ -2271,21 +2284,6 @@ [2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, 1, 1, 16, 1, 1, 18, 1, 1, 20, 1, 1, 22, 1, 1, 24, 1, 1, 26, 1, 1, 28, 1, 1, 30, 1, 1, 32, 1, 1, 34, 1, 1, 36, 1, 1, 38, 1, 1] """ - import sage.calculus.calculus - import sage.functions.constants - # if x is a SymbolicExpression, try coercing it to a real number if not bits is None: - try: - x = sage.rings.real_mpfr.RealField(bits)(x) - except TypeError: - raise TypeError, "can only find the continued fraction of a number" - elif isinstance(x, float): - from real_double import RDF - x = RDF(x) - elif isinstance(x, (sage.calculus.calculus.SymbolicExpression, - sage.functions.constants.Constant)): - try: - x = sage.rings.real_mpfr.RealField(53)(x) - except TypeError: - raise TypeError, "can only find the continued fraction of a number" + x = sage.rings.real_mpfr.RealField(bits)(x) elif not partial_convergents and \ isinstance(x, (integer.Integer, sage.rings.rational.Rational, @@ -2617,9 +2615,7 @@ 0.652965496420167 + 0.343065839816545*I sage: falling_factorial(1+I, 4) - (I - 2)*(I - 1)*I*(I + 1) - sage: expand(falling_factorial(1+I, 4)) - 4*I + 2 + 2.00000000000000 + 4.00000000000000*I sage: falling_factorial(I, 4) - (I - 3)*(I - 2)*(I - 1)*I + -10.0000000000000 sage: M = MatrixSpace(ZZ, 4, 4) @@ -2680,12 +2676,10 @@ 0.266816390637832 + 0.122783354006372*I - sage: a = rising_factorial(I, 4); a - I*(I + 1)*(I + 2)*(I + 3) - sage: expand(a) - -10 + sage: rising_factorial(I, 4) + -10.0000000000000 See falling_factorial(I, 4). - sage: x = polygen(ZZ) + sage: R = ZZ['x'] sage: rising_factorial(x, 4) x^4 + 6*x^3 + 11*x^2 + 6*x @@ -2702,11 +2696,7 @@ -def integer_ceil(x): +def ceil(x): """ Return the ceiling of x. - - EXAMPLES: - sage: integer_ceil(5.4) - 6 """ try: @@ -2717,7 +2707,9 @@ except TypeError: pass - raise NotImplementedError, "computation of floor of %s not implemented"%repr(x) - -def integer_floor(x): + raise NotImplementedError, "computation of floor of %s not implemented"%x + +ceiling = ceil + +def floor(x): r""" Return the largest integer $\leq x$. @@ -2730,11 +2722,11 @@ EXAMPLES: - sage: integer_floor(5.4) + sage: floor(5.4) 5 - sage: integer_floor(float(5.4)) + sage: floor(float(5.4)) 5 - sage: integer_floor(-5/2) + sage: floor(-5/2) -3 - sage: integer_floor(RDF(-5/2)) + sage: floor(RDF(-5/2)) -3 """ Index: sage/rings/complex_double.pyx =================================================================== --- sage/rings/complex_double.pyx (revision 4258) +++ sage/rings/complex_double.pyx (revision 3710) @@ -82,8 +82,4 @@ CC = complex_field.ComplexField() -import real_mpfr -RR = real_mpfr.RealField() - -from real_double import RealDoubleElement # PREC is the precision (in decimal digits) that all PARI computations with doubles @@ -100,7 +96,4 @@ from sage.structure.parent_gens import ParentWithGens - -def is_ComplexDoubleField(x): - return bool(PY_TYPE_CHECK(x, ComplexDoubleField_class)) cdef class ComplexDoubleField_class(sage.rings.ring.Field): @@ -236,6 +229,4 @@ else: return t - elif hasattr(x, '_complex_double_'): - return x._complex_double_(self) else: return ComplexDoubleElement(x, 0) @@ -261,12 +252,26 @@ 3.4 - Thus the sum of a CDF and a symbolic object is symbolic: + Symbolic constants canonically coerce into the complex double field, + but CDF does not canonically coerce to symbolic constants: + sage: CDF._coerce_(pi) + 3.14159265359 + sage: R = parent(pi) + sage: R(CDF.0) + 1.0*I + sage: R._coerce_(CDF.0) + Traceback (most recent call last): + ... + TypeError: no canonical coercion of element into self. + + Thus the sum of a CDF and a symbolic constant is in CDF: sage: a = pi + CDF.0; a - 1.00000000000000*I + pi + 3.14159265359 + 1.0*I sage: parent(a) - Symbolic Ring - """ + Complex Double Field + """ + import sage.functions.constants return self._coerce_try(x, [self.real_double_field(), - CC, RR]) + sage.functions.constants.ConstantRing, + CC]) @@ -656,11 +661,11 @@ 0.0 sage: CDF(0,1).arg() - 1.57079632679 + 1.5707963267948966 sage: CDF(0,-1).arg() - -1.57079632679 + -1.5707963267948966 sage: CDF(-1,0).arg() - 3.14159265359 - """ - return RealDoubleElement(gsl_complex_arg(self._complex)) + 3.1415926535897931 + """ + return gsl_complex_arg(self._complex) def __abs__(self): @@ -670,5 +675,5 @@ EXAMPLES: sage: abs(CDF(1,2)) - 2.2360679775 + 2.2360679774997898 sage: abs(CDF(1,0)) 1.0 @@ -676,5 +681,5 @@ 3.6055512754639891 """ - return RealDoubleElement(gsl_complex_abs(self._complex)) + return gsl_complex_abs(self._complex) def abs(self): @@ -686,5 +691,5 @@ 3.6055512754639891 """ - return RealDoubleElement(gsl_complex_abs(self._complex)) + return gsl_complex_abs(self._complex) def abs2(self): @@ -697,5 +702,5 @@ 13.0 """ - return RealDoubleElement(gsl_complex_abs2(self._complex)) + return gsl_complex_abs2(self._complex) def norm(self): @@ -708,5 +713,5 @@ 13.0 """ - return RealDoubleElement(gsl_complex_abs2(self._complex)) + return gsl_complex_abs2(self._complex) def logabs(self): @@ -720,14 +725,19 @@ EXAMPLES: + We try it out. sage: CDF(1.1,0.1).logabs() - 0.0994254293726 + 0.099425429372582669 sage: log(abs(CDF(1.1,0.1))) 0.0994254293726 + Which is better? sage: log(abs(ComplexField(200)(1.1,0.1))) 0.099425429372582675602989386713555936556752871164033127857198 - """ - return RealDoubleElement(gsl_complex_logabs(self._complex)) - + + Indeed, the logabs, wins. + """ + return gsl_complex_logabs(self._complex) + + # TODO: real and imag should be elements of RealDoubleField, when that exists. def real(self): """ @@ -739,5 +749,5 @@ 3.0 """ - return RealDoubleElement(self._complex.dat[0]) + return self._complex.dat[0] def imag(self): @@ -750,5 +760,5 @@ -2.0 """ - return RealDoubleElement(self._complex.dat[1]) + return self._complex.dat[1] def parent(self): @@ -769,14 +779,9 @@ # Elementary Complex Functions ####################################################################### - def sqrt(self, all=False, **kwds): - r""" - The square root function. - - INPUT: - all -- bool (default: False); if True, return a list - of all square roots. - - If all is False, the branch cut is the negative real axis. The - result always lies in the right half of the complex plane. + def sqrt(self): + r""" + This function returns the square root of the complex number $z$, + $\sqrt{z}$. The branch cut is the negative real axis. The result + always lies in the right half of the complex plane. EXAMPLES: @@ -793,25 +798,22 @@ sage: a = CDF(-1) sage: a.sqrt() - 1.0*I - - We compute all square roots: - sage: CDF(-2).sqrt(all=True) - [1.41421356237*I, -1.41421356237*I] - sage: CDF(0).sqrt(all=True) - [0] - """ - z = self._new_c(gsl_complex_sqrt(self._complex)) - if all: - if z.is_zero(): - return [z] - else: - return [z, -z] - return z - - def is_square(self): - """ - This function always returns true as $\C$ is algebraically closed. - """ - return True + 1.0*I + """ + return self._new_c(gsl_complex_sqrt(self._complex)) + + def square_root(self): + r""" + This function returns the square root of the complex number $z$, + $\sqrt{z}$. The branch cut is the negative real axis. The result + always lies in the right half of the complex plane. + + EXAMPLES: + sage: a = CDF(2,3) + sage: b = a.square_root(); b + 1.67414922804 + 0.89597747613*I + sage: b^2 + 2.0 + 3.0*I + """ + return self._new_c(gsl_complex_sqrt(self._complex)) def _pow_(self, ComplexDoubleElement a): @@ -852,11 +854,5 @@ 0.5 + 0.866025403784*I sage: a^3 # slightly random-ish arch dependent output - -1.0 + 1.22460635382e-16*I - - We raise to symbolic powers: - sage: CDF(1.2)^x - 1.20000000000000^x - sage: CDF(1.2)^(x^n + n^x) - 1.20000000000000^(x^n + n^x) + -1.0 + 1.22460635382e-16*I """ try: @@ -867,12 +863,5 @@ except TypeError: # a is not a complex number - try: - return z._pow_(CDF(a)) - except TypeError: - try: - return a.parent()(z)**a - except AttributeError: - raise TypeError - + return z._pow_(CDF(a)) def exp(self): @@ -1267,5 +1256,5 @@ sage: z = CDF(1,1); z.eta() 0.742048775837 + 0.19883137023*I - sage: i = CDF(0,1); pi = CDF(pi) + sage: i = CDF(0,1) sage: exp(pi * i * z / 12) * prod([1-exp(2*pi*i*n*z) for n in range(1,10)]) 0.742048775837 + 0.19883137023*I @@ -1275,5 +1264,4 @@ sage: z.eta(omit_frac=True) 0.998129069926 - sage: pi = CDF(pi) sage: prod([1-exp(2*pi*i*n*z) for n in range(1,10)]) # slightly random-ish arch dependent output 0.998129069926 + 4.5908467128e-19*I @@ -1357,7 +1345,6 @@ EXAMPLES: - sage: i = CDF(I) - sage: (1+i).agm(2-i) - 1.62780548487 + 0.136827548397*I + sage: (1+I).agm(2-I) + 1.62780548487271 + 0.136827548397369*I """ cdef pari_sp sp @@ -1447,5 +1434,5 @@ EXAMPLE: - sage: z = (1/2)*(1 + RDF(sqrt(3)) *CDF.0); z + sage: z = (1/2)*(1 + sqrt(3) *CDF.0); z 0.5 + 0.866025403784*I sage: p = z.algdep(5); p Index: sage/rings/complex_field.py =================================================================== --- sage/rings/complex_field.py (revision 4057) +++ sage/rings/complex_field.py (revision 3290) @@ -30,5 +30,5 @@ cache = {} -def ComplexField(prec=53, names=None): +def ComplexField(prec=53): """ Return the complex field with real and imaginary parts having prec @@ -42,7 +42,4 @@ sage: ComplexField(100).base_ring() Real Field with 100 bits of precision - sage: i = ComplexField(200).gen() - sage: i^2 - -1.0000000000000000000000000000000000000000000000000000000000 """ global cache @@ -102,7 +99,4 @@ sage: loads(CC.dumps()) == CC True - sage: k = ComplexField(100) - sage: loads(dumps(k)) == k - True This illustrates basic properties of a complex field. @@ -127,7 +121,4 @@ self._prec = int(prec) ParentWithGens.__init__(self, self._real_field(), ('I',), False) - - def __reduce__(self): - return ComplexField, (self._prec, ) def is_exact(self): Index: sage/rings/complex_number.pyx =================================================================== --- sage/rings/complex_number.pyx (revision 4258) +++ sage/rings/complex_number.pyx (revision 3484) @@ -38,5 +38,5 @@ def is_ComplexNumber(x): - return bool(isinstance(x, ComplexNumber)) + return isinstance(x, ComplexNumber) cdef class ComplexNumber(sage.structure.element.FieldElement): @@ -45,5 +45,6 @@ EXAMPLES: - sage: I = CC.0 + sage: C = ComplexField() + sage: I = C.0 sage: b = 1.5 + 2.5*I sage: loads(b.dumps()) == b @@ -116,9 +117,9 @@ EXAMPLES: - sage: a = CC(1 + I) + sage: a = 1+I sage: loads(dumps(a)) == a True """ - # TODO: This is potentially slow -- make a 1 version that + # TODO: This is potentially very slow -- make a 1 version that # is native and much faster -- doesn't use .real()/.imag() return (make_ComplexNumber0, (self._parent, self._multiplicative_order, self.real(), self.imag())) @@ -260,5 +261,5 @@ """ EXAMPLES: - sage: C. = ComplexField(20) + sage: C, i = ComplexField(20).objgen() sage: a = i^2; a -1.0000 @@ -278,17 +279,9 @@ if isinstance(right, (int, long, integer.Integer)): return sage.rings.ring_element.RingElement.__pow__(self, right) - try: - P = self.parent() - right = P(right) - z = self._pari_() - w = P(right)._pari_() - m = z**w - return P(m) - except TypeError: - try: - self = right.parent()(self) - return self**right - except AttributeError: - raise TypeError + z = self._pari_() + P = (self)._parent + w = P(right)._pari_() + m = z**w + return P(m) def prec(self): @@ -355,5 +348,4 @@ EXAMPLES: - sage: I = CC.0 sage: a = ~(5+I) sage: a * (5+I) @@ -424,5 +416,5 @@ EXAMPLES: - sage: C. = ComplexField() + sage: C, i = ComplexField().objgen() sage: i.multiplicative_order() 4 @@ -437,5 +429,5 @@ sage: C(2).multiplicative_order() +Infinity - sage: w = (1+sqrt(-3.0))/2; w + sage: w = (1+sqrt(-3))/2; w 0.500000000000000 + 0.866025403784439*I sage: abs(w) @@ -470,5 +462,5 @@ """ EXAMPLES: - sage: (1+CC(I)).acos() + sage: (1+I).acos() 0.904556894302381 - 1.06127506190504*I """ @@ -478,5 +470,5 @@ """ EXAMPLES: - sage: (1+CC(I)).acosh() + sage: (1+I).acosh() 1.06127506190504 + 0.904556894302381*I """ @@ -486,5 +478,5 @@ """ EXAMPLES: - sage: (1+CC(I)).asin() + sage: (1+I).asin() 0.666239432492515 + 1.06127506190504*I """ @@ -494,5 +486,5 @@ """ EXAMPLES: - sage: (1+CC(I)).asinh() + sage: (1+I).asinh() 1.06127506190504 + 0.666239432492515*I """ @@ -502,5 +494,5 @@ """ EXAMPLES: - sage: (1+CC(I)).atan() + sage: (1+I).atan() 1.01722196789785 + 0.402359478108525*I """ @@ -510,38 +502,13 @@ """ EXAMPLES: - sage: (1+CC(I)).atanh() + sage: (1+I).atanh() 0.402359478108525 + 1.01722196789785*I """ return self._parent(self._pari_().atanh()) - def coth(self): - """ - EXAMPLES: - sage: ComplexField(100)(1,1).coth() - 0.86801414289592494863584920892 - 0.21762156185440268136513424361*I - """ - return 1/self.tanh() - - def csch(self): - """ - EXAMPLES: - sage: ComplexField(100)(1,1).csch() - 0.30393100162842645033448560451 - 0.62151801717042842123490780586*I - """ - return 1/self.sinh() - - def sech(self): - """ - EXAMPLES: - sage: ComplexField(100)(1,1).sech() - 0.49833703055518678521380589177 - 0.59108384172104504805039169297*I - """ - return 1/self.cosh() - - def cotan(self): """ EXAMPLES: - sage: (1+CC(I)).cotan() + sage: (1+I).cotan() 0.217621561854403 - 0.868014142895925*I sage: i = ComplexField(200).0 @@ -557,5 +524,5 @@ """ EXAMPLES: - sage: (1+CC(I)).cos() + sage: (1+I).cos() 0.833730025131149 - 0.988897705762865*I """ @@ -565,5 +532,5 @@ """ EXAMPLES: - sage: (1+CC(I)).cosh() + sage: (1+I).cosh() 0.833730025131149 + 0.988897705762865*I """ @@ -601,5 +568,4 @@ sage: z = 1 + i; z.eta() 0.742048775836565 + 0.198831370229911*I - sage: pi = CC(pi) # otherwise we will get a symbolic result. sage: exp(pi * i * z / 12) * prod([1-exp(2*pi*i*n*z) for n in range(1,10)]) 0.742048775836565 + 0.198831370229911*I @@ -624,5 +590,5 @@ You can also use functional notation. - sage: eta(1+CC(I)) + sage: eta(1+I) 0.742048775836565 + 0.198831370229911*I """ @@ -636,5 +602,5 @@ """ EXAMPLES: - sage: (1+CC(I)).sin() + sage: (1+I).sin() 1.29845758141598 + 0.634963914784736*I """ @@ -644,5 +610,5 @@ """ EXAMPLES: - sage: (1+CC(I)).sinh() + sage: (1+I).sinh() 0.634963914784736 + 1.29845758141598*I """ @@ -652,5 +618,5 @@ """ EXAMPLES: - sage: (1+CC(I)).tan() + sage: (1+I).tan() 0.271752585319512 + 1.08392332733869*I """ @@ -660,5 +626,5 @@ """ EXAMPLES: - sage: (1+CC(I)).tanh() + sage: (1+I).tanh() 1.08392332733869 + 0.271752585319512*I """ @@ -669,5 +635,5 @@ """ EXAMPLES: - sage: (1+CC(I)).agm(2-I) + sage: (1+I).agm(2-I) 1.62780548487271 + 0.136827548397369*I """ @@ -798,12 +764,6 @@ return infinity.infinity - def sqrt(self, all=False, **kwds): - """ - The square root function. - - INPUT: - all -- bool (default: False); if True, return a list - of all square roots. - + def sqrt(self): + """ EXAMPLES: sage: C, i = ComplexField(30).objgen() @@ -818,17 +778,16 @@ 0.70710678118654752440084436210484903928483593768847403658834 + 0.70710678118654752440084436210484903928483593768847403658834*I """ - z = self._parent(self._pari_().sqrt()) - if all: - if z.is_zero(): - return [z] - else: - return [z, -z] - return z - - def is_square(self): - """ - This function always returns true as $\C$ is algebraically closed. - """ - return True + return self._parent(self._pari_().sqrt()) + + def square_root(self): + """ + Return square root, which is a complex number. + + EXAMPLES: + sage: i = ComplexField(100).0 + sage: (-i).sqrt() + 0.70710678118654752440084436210 - 0.70710678118654752440084436210*I + """ + return self.sqrt() def zeta(self): @@ -857,5 +816,5 @@ EXAMPLE: sage: C = ComplexField() - sage: z = (1/2)*(1 + sqrt(3.0) *C.0); z + sage: z = (1/2)*(1 + sqrt(3) *C.0); z 0.500000000000000 + 0.866025403784439*I sage: p = z.algdep(5); p Index: sage/rings/contfrac.py =================================================================== --- sage/rings/contfrac.py (revision 4257) +++ sage/rings/contfrac.py (revision 3538) @@ -626,14 +626,23 @@ return s - def sqrt(self, prec=53, all=False): - """ - Return continued fraction approximation to square root of the - value of this continued fraction. - - INPUT: - prec -- integer (default: 53) precision of square root - that is approximated - all -- bool (default: False); if True, return all square - roots of self, instead of just one. + def square_root(self): + """ + Return exact square root or raise an error if self + is not a perfect square. + + EXAMPLES: + sage: a = CFF(4/25).square_root(); a + [0, 2, 2] + sage: a.value() + 2/5 + """ + return ContinuedFraction(self.parent(), + self._rational_().square_root()) + + def sqrt(self): + """ + Return continued fraction approximation to the + square root of the value of this continued fraction, if + it is positive. If self is negative raise a ValueError. EXAMPLES: @@ -646,24 +655,9 @@ sage: float(b.value()^2 - a) 4.0935373134057017e-17 - sage: b = a.sqrt(prec=100); b - [0, 2, 5, 1, 1, 2, 1, 16, 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 16, 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 16, 1, 2, 1, 1, 5, 5] - sage: b^2 - [0, 4, 1, 3, 49545773063556658177372134479, 1, 3, 4] - sage: a.sqrt(all=True) - [[0, 2, 5, 1, 1, 2, 1, 16, 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 15, 2], - [-1, 1, 1, 5, 1, 1, 2, 1, 16, 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 15, 2]] - sage: a = CFF(4/25).sqrt(); a - [0, 2, 2] - sage: a.value() - 2/5 """ r = self._rational_() if r < 0: raise ValueError, "self must be positive" - X = r.sqrt(all=all, prec=prec) - if not all: - return ContinuedFraction(self.parent(), X) - else: - return [ContinuedFraction(self.parent(), x) for x in X] + return ContinuedFraction(self.parent(), r.sqrt()) def list(self): @@ -770,7 +764,7 @@ return self._rational_().is_one() - def __nonzero__(self): - """ - Return False if self is zero. + def is_zero(self): + """ + Return True if self is zero. EXAMPLES: @@ -780,5 +774,5 @@ False """ - return not self._rational_().is_zero() + return self._rational_().is_zero() def _pari_(self): Index: age/rings/cygwinfix.h =================================================================== --- sage/rings/cygwinfix.h (revision 4357) +++ (revision ) @@ -1,7 +1,0 @@ -// cygwinfix.h -// defines NAN and INFINITY on cygwin - -#if defined(__CYGWIN__) -#define NAN (0.0/0.0) -#define INFINITY HUGE_VALF -#endif Index: sage/rings/finite_field.py =================================================================== --- sage/rings/finite_field.py (revision 3991) +++ sage/rings/finite_field.py (revision 3125) @@ -500,5 +500,4 @@ Nonconstant polynomials do not coerce: - sage: x = polygen(QQ) sage: k(x) Traceback (most recent call last): Index: sage/rings/finite_field_givaro.pyx =================================================================== --- sage/rings/finite_field_givaro.pyx (revision 4074) +++ sage/rings/finite_field_givaro.pyx (revision 3125) @@ -762,38 +762,4 @@ return ret - def fetch_int(FiniteField_givaro self, int n): - r""" - Given an integer $n$ return a finite field element in self - which equals $n$ under the condition that self.gen() is set to - self.characteristic(). - - EXAMPLE: - sage: k. = GF(2^8) - sage: k.fetch_int(8) - a^3 - sage: e = k.fetch_int(151); e - a^7 + a^4 + a^2 + a + 1 - sage: 2^7 + 2^4 + 2^2 + 2 + 1 - 151 - """ - cdef GivaroGfq *k = self.objectptr - cdef int ret = k.zero - cdef int a = k.sage_generator() - cdef int at = k.one - cdef unsigned int ch = k.characteristic() - cdef int _n, t, i - - if n<0 or n>k.cardinality(): - raise TypeError, "n must be between 0 and self.order()" - - _n = n - - for i from 0 <= i < k.exponent(): - t = k.initi(t, _n%ch) - ret = k.axpy(ret, t, at, ret) - at = k.mul(at,at,a) - _n = _n/ch - return make_FiniteField_givaroElement(self, ret) - def polynomial(self): """ @@ -1225,7 +1191,7 @@ return (self._parent) - def __nonzero__(FiniteField_givaroElement self): + def is_zero(FiniteField_givaroElement self): r""" - Return True if \code{self != k(0)}. + Return True if \code{self == k(0)}. EXAMPLES: @@ -1237,5 +1203,5 @@ True """ - return not bool((self._parent).objectptr.isZero(self.element)) + return bool((self._parent).objectptr.isZero(self.element)) def is_one(FiniteField_givaroElement self): @@ -1904,38 +1870,4 @@ raise ValueError, "must be a perfect square." - def vector(FiniteField_givaroElement self): - """ - Return a vector in self.parent().vector_space() matching self. - - EXAMPLES: - sage: k. = GF(2^4) - sage: e = a^2 + 1 - sage: v = e.vector() - sage: v - (1, 0, 1, 0) - sage: k(v) - a^2 + 1 - - sage: k. = GF(3^4) - sage: e = 2*a^2 + 1 - sage: v = e.vector() - sage: v - (1, 0, 2, 0) - sage: k(v) - 2*a^2 + 1 - - """ - cdef FiniteField_givaro k = self._parent - - quo = k.log_to_int(self.element) - b = int(k.characteristic()) - - ret = [] - for i in range(k.degree()): - coeff = quo%b - ret.append(coeff) - quo = quo/b - return k.vector_space()(ret) - def __reduce__(FiniteField_givaroElement self): """ Index: sage/rings/fraction_field.py =================================================================== --- sage/rings/fraction_field.py (revision 4057) +++ sage/rings/fraction_field.py (revision 2399) @@ -4,40 +4,4 @@ AUTHOR: William Stein (with input from David Joyner, David Kohel, and Joe Wetherell) - -EXAMPLES: -Quotienting is a constructor for an element of the fraction field: - sage: R. = QQ[] - sage: (x^2-1)/(x+1) - x - 1 - sage: parent((x^2-1)/(x+1)) - Fraction Field of Univariate Polynomial Ring in x over Rational Field - - -The GCD is not taken (since it doesn't converge sometimes) in the inexact case. - sage: Z. = CC[] - sage: I = CC.gen() - sage: (1+I+z)/(z+0.1*I) - (1.00000000000000*z + 1.00000000000000 + 1.00000000000000*I)/(1.00000000000000*z + 0.100000000000000*I) - sage: (1+I*z)/(z+1.1) - (1.00000000000000*I*z + 1.00000000000000)/(1.00000000000000*z + 1.10000000000000) - - -TESTS: - sage: F = FractionField(IntegerRing()) - sage: F == loads(dumps(F)) - True - - sage: F = FractionField(PolynomialRing(RationalField(),'x')) - sage: F == loads(dumps(F)) - True - - sage: F = FractionField(PolynomialRing(IntegerRing(),'x')) - sage: F == loads(dumps(F)) - True - - sage: F = FractionField(MPolynomialRing(RationalField(),2,'x')) - sage: F == loads(dumps(F)) - True - """ @@ -195,20 +159,5 @@ R = self.ring() x = R(x); y = R(y) - return fraction_field_element.FractionFieldElement(self, x, y, - coerce=False, reduce = self.is_exact()) - - def is_exact(self): - """ - EXAMPLES: - sage: Z.=CC[] - sage: Z.is_exact() - False - """ - try: - return self.__is_exact - except AttributeError: - r = self.ring().is_exact() - self.__is_exact = r - return r + return fraction_field_element.FractionFieldElement(self, x, y, coerce=False) def _coerce_impl(self, x): Index: sage/rings/fraction_field_element.py =================================================================== --- sage/rings/fraction_field_element.py (revision 4181) +++ sage/rings/fraction_field_element.py (revision 2725) @@ -57,5 +57,5 @@ self.__numerator = numerator self.__denominator = denominator - if reduce and parent.is_exact(): + if reduce: try: self.reduce() Index: sage/rings/groebner_fan.py =================================================================== --- sage/rings/groebner_fan.py (revision 4077) +++ sage/rings/groebner_fan.py (revision 2487) @@ -42,12 +42,4 @@ sage: g.reduced_groebner_bases() [[1 - y^2 + x^2], [-1 + y^2 - x^2]] - -TESTS: - sage: x,y = QQ['x,y'].gens() - sage: i = ideal(x^2 - y^2 + 1) - sage: g = i.groebner_fan() - sage: g == loads(dumps(g)) - True - """ @@ -105,5 +97,5 @@ sage: G = I.groebner_fan(); G Groebner fan of the ideal: - Ideal (-1*z + x^2*y, y^2*z - x, -1*y + x*z^2) of Polynomial Ring in x, y, z over Rational Field + Ideal (y^2*z - x, -1*y + x*z^2, -1*z + x^2*y) of Polynomial Ring in x, y, z over Rational Field """ self.__is_groebner_basis = is_groebner_basis @@ -128,7 +120,4 @@ self.__ring = S - def __eq__(self,right): - return type(self) == type(right) and self.ideal() == right.ideal() - def ideal(self): """ @@ -206,5 +195,5 @@ sage: G = R.ideal([x^2*y - z, y^2*z - x, z^2*x - y]).groebner_fan() sage: G._gfan_ideal() - '{-1*c + a^2*b, b^2*c - a, -1*b + a*c^2}' + '{-1*b + a*c^2, b^2*c - a, -1*c + a^2*b}' """ try: Index: sage/rings/homset.py =================================================================== --- sage/rings/homset.py (revision 4057) +++ sage/rings/homset.py (revision 2369) @@ -36,24 +36,4 @@ return "Set of Homomorphisms from %s to %s"%(self.domain(), self.codomain()) - def has_coerce_map_from(self, x): - """ - The default for coercion maps between ring homomorphism - spaces is very restrictive (until more implementation work - is done). - """ - return (x.domain() == self.domain() and x.codomain() == self.codomain()) - - def _coerce_impl(self, x): - if not isinstance(x, morphism.RingHomomorphism): - raise TypeError - if x.parent() is self: - return x - if x.parent() == self: - if isinstance(x, morphism.RingHomomorphism_im_gens): - return morphism.RingHomomorphism_im_gens(self, x.im_gens()) - elif isinstance(x, morphism.RingHomomorphism_cover): - return morphism.RingHomomorphism_cover(self) - raise TypeError - def __call__(self, im_gens, check=True): """ @@ -64,20 +44,9 @@ ... TypeError: images do not define a valid homomorphism - - TESTS: - sage: H = Hom(ZZ, QQ) - sage: H == loads(dumps(H)) - True """ - if isinstance(im_gens, (morphism.RingHomomorphism_im_gens, morphism.RingHomomorphism_cover) ): - return self._coerce_impl(im_gens) try: return morphism.RingHomomorphism_im_gens(self, im_gens, check=check) except (NotImplementedError, ValueError), err: - try: - return self._coerce_impl(im_gens) - except TypeError: - raise TypeError, "images do not define a valid homomorphism" - + raise TypeError, "images do not define a valid homomorphism" def natural_map(self): @@ -100,21 +69,6 @@ sage: phi(b) a - - TESTS: - We test pickling of a homset from a quotient. - sage: R. = PolynomialRing(QQ, 2) - sage: S. = R.quotient(x^2 + y^2) - sage: H = S.Hom(R) - sage: H == loads(dumps(H)) - True - - We test pickling of actual homomorphisms in a quotient: - sage: phi = S.hom([b,a]) - sage: phi == loads(dumps(phi)) - True """ def __call__(self, im_gens, check=True): - if isinstance(im_gens, morphism.RingHomomorphism_from_quotient): - return morphism.RingHomomorphism_from_quotient(self, im_gens._phi()) try: pi = self.domain().cover() @@ -122,15 +76,5 @@ return morphism.RingHomomorphism_from_quotient(self, phi) except (NotImplementedError, ValueError), err: - try: - return self._coerce_impl(im_gens) - except TypeError: - raise TypeError, "images do not define a valid homomorphism" - - def _coerce_impl(self, x): - if not isinstance(x, morphism.RingHomomorphism_from_quotient): - raise TypeError - if x.parent() is self: - return x - if x.parent() == self: - return morphism.RingHomomorphism_from_quotient(self, x._phi()) - raise TypeError + raise TypeError, "images do not define a valid homomorphism" + + Index: sage/rings/ideal.py =================================================================== --- sage/rings/ideal.py (revision 4077) +++ sage/rings/ideal.py (revision 3311) @@ -60,15 +60,18 @@ sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2]) sage: I - Ideal (x^2 + 3*x + 4, x^2 + 1) of Univariate Polynomial Ring in x over Integer Ring + Ideal (x^2 + 1, x^2 + 3*x + 4) of Univariate Polynomial Ring in x over Integer Ring sage: Ideal(R, [4 + 3*x + x^2, 1 + x^2]) - Ideal (x^2 + 3*x + 4, x^2 + 1) of Univariate Polynomial Ring in x over Integer Ring + Ideal (x^2 + 1, x^2 + 3*x + 4) of Univariate Polynomial Ring in x over Integer Ring sage: Ideal((4 + 3*x + x^2, 1 + x^2)) - Ideal (x^2 + 3*x + 4, x^2 + 1) of Univariate Polynomial Ring in x over Integer Ring + Ideal (x^2 + 1, x^2 + 3*x + 4) of Univariate Polynomial Ring in x over Integer Ring sage: ideal(x^2-2*x+1, x^2-1) - Ideal (x^2 - 1, x^2 - 2*x + 1) of Univariate Polynomial Ring in x over Integer Ring + Ideal (x^2 - 2*x + 1, x^2 - 1) of Univariate Polynomial Ring in x over Integer Ring sage: ideal([x^2-2*x+1, x^2-1]) - Ideal (x^2 - 1, x^2 - 2*x + 1) of Univariate Polynomial Ring in x over Integer Ring - + Ideal (x^2 - 2*x + 1, x^2 - 1) of Univariate Polynomial Ring in x over Integer Ring + sage: ideal(x^2-2*x+1, x^2-1) + Ideal (x^2 - 2*x + 1, x^2 - 1) of Univariate Polynomial Ring in x over Integer Ring + sage: ideal([x^2-2*x+1, x^2-1]) + Ideal (x^2 - 2*x + 1, x^2 - 1) of Univariate Polynomial Ring in x over Integer Ring This example illustrates how SAGE finds a common ambient ring for the ideal, even though @@ -77,24 +80,9 @@ sage: i = ideal(1,t,t^2) sage: i - Ideal (t, 1, t^2) of Univariate Polynomial Ring in t over Integer Ring + Ideal (1, t, t^2) of Univariate Polynomial Ring in t over Integer Ring sage: i = ideal(1/2,t,t^2) Traceback (most recent call last): ... TypeError: unable to find common ring into which all ideal generators map - - TESTS: - sage: R, x = PolynomialRing(ZZ, 'x').objgen() - sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2]) - sage: I == loads(dumps(I)) - True - - sage: I = Ideal(R, [4 + 3*x + x^2, 1 + x^2]) - sage: I == loads(dumps(I)) - True - - sage: I = Ideal((4 + 3*x + x^2, 1 + x^2)) - sage: I == loads(dumps(I)) - True - """ if isinstance(R, Ideal_generic): @@ -135,4 +123,5 @@ gens = list(set(gens)) + if isinstance(R, sage.rings.principal_ideal_domain.PrincipalIdealDomain): # Use GCD algorithm to obtain a principal ideal @@ -141,5 +130,5 @@ g = R.gcd(g, h) return Ideal_pid(R, g) - + if len(gens) == 1: return Ideal_principal(R, gens[0]) @@ -161,5 +150,18 @@ if coerce: gens = [ring(x) for x in gens] - + gens = list(set(gens)) + + # Regarding the "important" comment below: Otherwise the + # generators will be in a completely random order, given the + # code that comes before that line. A basic design choice in + # SAGE is that as much as possible lists of objects (e.g., + # list(R), where R is finite), should not be in a random + # order. Feel free to add this as a comment. It would be + # fine to replace the sort by something else, if it yields the + # same answer. However, I don't think randomizing the orders + # of things, e.g., lists of generators, for no reason, is a + # good idea in SAGE. This is another "rule of thumb" for the + # programmer's guide. + gens.sort() # important! self.__gens = tuple(gens) MonoidElement.__init__(self, ring.ideal_monoid()) @@ -168,13 +170,9 @@ return '(%s)'%(', '.join([str(x) for x in self.gens()])) - def __repr__(self): + def _repr_(self): return "Ideal %s of %s"%(self._repr_short(), self.ring()) def __cmp__(self, other): - S = set(self.gens()) - T = set(other.gens()) - if S == T: - return 0 - return cmp(self.gens(), other.gens()) + return cmp(set(self.gens()), set(other.gens())) def __contains__(self, x): @@ -188,6 +186,6 @@ raise NotImplementedError - def __nonzero__(self): - return self.gens() != [self.ring()(0)] + def is_zero(self): + return self.gens() == [self.ring()(0)] def base_ring(self): @@ -264,5 +262,5 @@ Ideal_generic.__init__(self, ring, [gen]) - def __repr__(self): + def _repr_(self): return "Principal ideal (%s) of %s"%(self.gen(), self.ring()) Index: sage/rings/ideal_monoid.py =================================================================== --- sage/rings/ideal_monoid.py (revision 4019) +++ sage/rings/ideal_monoid.py (revision 1851) @@ -7,5 +7,4 @@ from sage.structure.parent_gens import ParentWithGens import sage.rings.integer_ring -import ideal def IdealMonoid(R): @@ -26,13 +25,7 @@ def __call__(self, x): - if isinstance(x, ideal.Ideal_generic): - x = x.gens() return self.__R.ideal(x) def _coerce_impl(self, x): R = self.__R - if isinstance(x, ideal.Ideal_generic): - x = [R._coerce_impl(y) for y in x.gens()] - return R.ideal(x) - else: - return R.ideal(R._coerce_(x)) + return R.ideal(R._coerce_(x)) Index: sage/rings/infinity.py =================================================================== --- sage/rings/infinity.py (revision 4022) +++ sage/rings/infinity.py (revision 3335) @@ -116,14 +116,4 @@ ... SignError: cannot add positive finite value to negative finite value - -TESTS: - sage: P = InfinityRing - sage: P == loads(dumps(P)) - True - - sage: P(2) == loads(dumps(P(2))) - True - - """ @@ -271,7 +261,4 @@ def _repr_(self): return "Infinity" - - def _maxima_init_(self): - return "inf" def lcm(self, x): @@ -481,12 +468,4 @@ return "-Infinity" - def _maxima_init_(self): - """ - EXAMPLES: - sage: maxima(-oo) - minf - """ - return "minf" - def _latex_(self): return "-\\infty" @@ -558,12 +537,4 @@ return "+Infinity" - def _maxima_init_(self): - """ - EXAMPLES: - sage: maxima(oo) - inf - """ - return "inf" - def _latex_(self): return "+\\infty" @@ -626,19 +597,18 @@ infinity = InfinityRing.gen(0) Infinity = infinity -minus_infinity = InfinityRing.gen(1) - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + Index: sage/rings/integer.pyx =================================================================== --- sage/rings/integer.pyx (revision 4260) +++ sage/rings/integer.pyx (revision 3927) @@ -13,5 +13,4 @@ -- Rishikesh (2007-02-25): changed quo_rem so that the rem is positive -- David Harvey, Martin Albrecht, Robert Bradshaw (2007-03-01): optimized Integer constructor and pool - -- Pablo De Napoli (2007-04-01): multiplicative_order should return +infinity for non zero numbers -- Robert Bradshaw (2007-04-12): is_perfect_power, Jacobi symbol (with Kronecker extension) Convert some methods to use GMP directly rather than pari, Integer() -> PY_NEW(Integer) @@ -394,4 +393,5 @@ base. + EXAMPLES: sage: Integer(2^10).str(2) @@ -655,5 +655,8 @@ 1 sage: (-1)^(1/3) - -1 + Traceback (most recent call last): + ... + TypeError: exponent (=1/3) must be an integer. + Coerce your numbers to real or complex numbers first. The base need not be an integer (it can be a builtin @@ -673,22 +676,4 @@ RuntimeError: exponent must be at most 4294967294 # 32-bit RuntimeError: exponent must be at most 18446744073709551614 # 64-bit - - We raise 2 to various interesting exponents: - sage: 2^x # symbolic x - 2^x - sage: 2^1.5 # real number - 2.82842712474619 - sage: 2^I # complex number - 2^I - sage: f = 2^(sin(x)-cos(x)); f - 2^(sin(x) - cos(x)) - sage: f(3) - 2^(sin(3) - cos(3)) - sage: 2^(x+y+z) - 2^(z + y + x) - sage: 2^(1/2) - sqrt(2) - sage: 2^(-1/2) - 1/sqrt(2) """ cdef Integer _n @@ -705,9 +690,5 @@ _n = Integer(n) except TypeError: - try: - s = n.parent()(self) - return s**n - except AttributeError: - raise TypeError, "exponent (=%s) must be an integer.\nCoerce your numbers to real or complex numbers first."%n + raise TypeError, "exponent (=%s) must be an integer.\nCoerce your numbers to real or complex numbers first."%n if _n < 0: @@ -778,5 +759,5 @@ if n < 1: raise ValueError, "n (=%s) must be positive" % n - if (mpz_sgn(self.value) < 0) and not (n & 1): + if (self < 0) and not (n & 1): raise ValueError, "cannot take even root of negative number" cdef Integer x @@ -825,7 +806,7 @@ sage: x = 3^100000 - sage: RR(log(RR(x), 3)) + sage: log(RR(x), 3) 100000.000000000 - sage: RR(log(RR(x + 100000), 3)) + sage: log(RR(x + 100000), 3) 100000.000000000 @@ -846,5 +827,5 @@ raise ValueError, "base of log must be an integer" m = _m - if mpz_sgn(self.value) <= 0: + if self <= 0: raise ValueError, "self must be positive" if m < 2: @@ -1086,4 +1067,7 @@ return mpz_get_pylong(self.value) + def __nonzero__(self): + return not self.is_zero() + def __float__(self): return mpz_get_d(self.value) @@ -1277,5 +1261,5 @@ 6 720 """ - if mpz_sgn(self.value) < 0: + if self < 0: raise ValueError, "factorial -- self = (%s) must be nonnegative"%self @@ -1334,7 +1318,7 @@ return bool(mpz_cmp_si(self.value, 1) == 0) - def __nonzero__(self): + def is_zero(self): r""" - Returns \code{True} if the integers is not $0$, otherwise \code{False}. + Returns \code{True} if the integers is $0$, otherwise \code{False}. EXAMPLES: @@ -1344,5 +1328,5 @@ True """ - return bool(mpz_cmp_si(self.value, 0) != 0) + return bool(mpz_cmp_si(self.value, 0) == 0) def is_unit(self): @@ -1371,44 +1355,5 @@ False """ - if mpz_sgn(self.value) < 0: - return False return bool(mpz_perfect_square_p(self.value)) - - def is_prime_power(self, flag=0): - r""" - Returns True if $x$ is a prime power, and False otherwise. - - INPUT: - flag (for primality testing) -- int - 0 (default): use a combination of algorithms. - 1: certify primality using the Pocklington-Lehmer Test. - 2: certify primality using the APRCL test. - - EXAMPLES: - sage: (-10).is_prime_power() - False - sage: (10).is_prime_power() - False - sage: (64).is_prime_power() - True - sage: (3^10000).is_prime_power() - True - sage: (10000).is_prime_power(flag=1) - False - """ - if self.is_zero(): - return False - elif self.is_one(): - return True - elif mpz_sgn(self.value) < 0: - return False - if self.is_prime(): - return True - if not self.is_perfect_power(): - return False - k, g = self._pari_().ispower() - if not k: - raise RuntimeError, "inconsistent results between GMP and pari" - return g.isprime(flag=flag) def is_prime(self): @@ -1458,6 +1403,6 @@ def jacobi(self, b): - r""" - Calculate the Jacobi symbol $\left(\frac{self}{b}\right)$. + """ + Calculate the Jacobi symbol $\left(\frac{self,b}\right)$. EXAMPLES: @@ -1494,7 +1439,6 @@ def kronecker(self, b): - r""" - Calculate the Kronecker symbol - $\left(\frac{self}{b}\right)$ with the Kronecker extension + """ + Calculate the Jacobi symbol $\left(\frac{self,b}\right)$ with the Kronecker extension $(self/2)=(2/self)$ when self odd, or $(self/2)=0$ when $self$ even. @@ -1584,5 +1528,6 @@ def multiplicative_order(self): r""" - Return the multiplicative order of self. + Return the multiplicative order of self, if self is a unit, or raise + \code{ArithmeticError} otherwise. EXAMPLES: @@ -1592,15 +1537,18 @@ 2 sage: ZZ(0).multiplicative_order() - +Infinity + Traceback (most recent call last): + ... + ArithmeticError: no power of 0 is a unit sage: ZZ(2).multiplicative_order() - +Infinity - """ - import sage.rings.infinity - if mpz_cmp_si(self.value, 1) == 0: - return Integer(1) + Traceback (most recent call last): + ... + ArithmeticError: no power of 2 is a unit + """ + if mpz_cmp_si(self.value, 1) == 0: + return Integer(1) elif mpz_cmp_si(self.value, -1) == 0: - return Integer(2) + return Integer(2) else: - return sage.rings.infinity.infinity + raise ArithmeticError, "no power of %s is a unit"%self def is_squarefree(self): @@ -1667,8 +1615,8 @@ Traceback (most recent call last): ... - ValueError: square root of negative integer not defined. - """ - if mpz_sgn(self.value) < 0: - raise ValueError, "square root of negative integer not defined." + ValueError: square root of negative number not defined. + """ + if self < 0: + raise ValueError, "square root of negative number not defined." cdef Integer x x = PY_NEW(Integer) @@ -1680,97 +1628,81 @@ return x - def sqrt_approx(self, prec=None, all=False): - """ - EXAMPLES: - sage: 5.sqrt_approx(prec=200) - 2.2360679774997896964091736687312762354406183596115257242709 - sage: 5.sqrt_approx() - 2.23606797749979 - sage: 4.sqrt_approx() + + def sqrt(self, bits=None): + r""" + Returns the positive square root of self, possibly as a + \emph{a real or complex number} if self is not a perfect + integer square. + + INPUT: + bits -- number of bits of precision. + If bits is not specified, the number of + bits of precision is at least twice the + number of bits of self (the precision + is always at least 53 bits if not specified). + OUTPUT: + integer, real number, or complex number. + + For the guaranteed integer square root of a perfect square + (with error checking), use \code{self.square_root()}. + + EXAMPLE: + sage: Z = IntegerRing() + sage: Z(4).sqrt() 2 - """ - try: - return self.sqrt(extend=False,all=all) - except ValueError: - pass - if prec is None: - prec = max(53, 2*(mpz_sizeinbase(self.value, 2)+2)) - return self.sqrt(prec=prec, all=all) - - def sqrt(self, prec=None, extend=True, all=False): - """ - The square root function. - - INPUT: - prec -- integer (default: None): if None, returns an exact - square root; otherwise returns a numerical square - root if necessary, to the given bits of precision. - extend -- bool (default: True); if True, return a square - root in an extension ring, if necessary. Otherwise, - raise a ValueError if the square is not in the base - ring. - all -- bool (default: False); if True, return all square - roots of self, instead of just one. - - EXAMPLES: - sage: Integer(144).sqrt() + sage: Z(4).sqrt(53) + 2.00000000000000 + sage: Z(2).sqrt(53) + 1.41421356237310 + sage: Z(2).sqrt(100) + 1.4142135623730950488016887242 + sage: n = 39188072418583779289; n.square_root() + 6260037733 + sage: (100^100).sqrt() + 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sage: (-1).sqrt() + 1.00000000000000*I + sage: sqrt(-2) + 1.41421356237310*I + sage: sqrt(97) + 9.84885780179610 + sage: n = 97; n.sqrt(200) + 9.8488578017961047217462114149176244816961362874427641717232 + """ + if bits is None: + try: + return self.square_root() + except ValueError: + pass + bits = max(53, 2*(mpz_sizeinbase(self.value, 2)+2)) + + if self < 0: + import sage.rings.complex_field + x = sage.rings.complex_field.ComplexField(bits)(self) + return x.sqrt() + else: + import real_mpfr + R = real_mpfr.RealField(bits) + return R(self).sqrt() + + def square_root(self): + """ + Return the positive integer square root of self, or raises a ValueError + if self is not a perfect square. + + EXAMPLES: + sage: Integer(144).square_root() 12 - sage: Integer(102).sqrt() - sqrt(102) - - sage: n = 2 - sage: n.sqrt(all=True) - [sqrt(2), -sqrt(2)] - sage: n.sqrt(prec=10) - 1.4 - sage: n.sqrt(prec=100) - 1.4142135623730950488016887242 - sage: n.sqrt(prec=100,all=True) - [1.4142135623730950488016887242, -1.4142135623730950488016887242] - sage: n.sqrt(extend=False) + sage: Integer(102).square_root() Traceback (most recent call last): ... - ValueError: square root of 2 not an integer - sage: Integer(0).sqrt(all=True) - [0] - """ - if mpz_sgn(self.value) < 0: - if not extend: - raise ValueError, "square root of negative number not an integer" - if prec: - from sage.rings.complex_field import ComplexField - K = ComplexField(prec) - return K(self).sqrt(all=all) - from sage.calculus.calculus import sqrt - return sqrt(self, all=all) - - - cdef int non_square - cdef Integer z = PY_NEW(Integer) - cdef mpz_t tmp - _sig_on - mpz_init(tmp) - mpz_sqrtrem(z.value, tmp, self.value) - non_square = mpz_sgn(tmp) != 0 - mpz_clear(tmp) - _sig_off - - if non_square: - if not extend: - raise ValueError, "square root of %s not an integer"%self - if prec: - from sage.rings.real_mpfr import RealField - K = RealField(prec) - return K(self).sqrt(all=all) - from sage.calculus.calculus import sqrt - return sqrt(self, all=all) - - if all: - if z.is_zero(): - return [z] - else: - [z, -z] - return z - + ValueError: self (=102) is not a perfect square + """ + n = self.isqrt() + if n * n == self: + return n + raise ValueError, "self (=%s) is not a perfect square"%self + + def _xgcd(self, Integer n): r""" Index: sage/rings/integer_mod.pyx =================================================================== --- sage/rings/integer_mod.pyx (revision 4291) +++ sage/rings/integer_mod.pyx (revision 3926) @@ -22,5 +22,4 @@ -- William Stein (editing and polishing; new arith architecture) -- Robert Bradshaw (implement native is_square and square_root) - -- William Stein (sqrt) TESTS: @@ -111,9 +110,5 @@ if res < 0: res = res + modulus.int64 - a = modulus.lookup(res) - if (a)._parent is not parent: - (a)._parent = parent -# print (a)._parent, " is not ", parent - return a + return modulus.lookup(res) if modulus.int32 != -1: return IntegerMod_int(parent, value) @@ -477,5 +472,5 @@ def is_square(self): - r""" + """ EXAMPLES: sage: Mod(3,17).is_square() @@ -499,5 +494,5 @@ ALGORITHM: - Calculate the Jacobi symbol $(self/p)$ at each prime $p$ dividing $n$ + Calculate the Jacobi symbol $(self/p)$ at each prime $p dividing $n$ It must be 1 or 0 for each prime, and if it is 0 mod $p$, where $p^k || n$, then $ord_p(self)$ must be even or greater than $k$. @@ -540,156 +535,65 @@ return 1 - def sqrt(self, extend=True, all=False): - r""" - Returns square root or square roots of self modulo n. - - INPUT: - extend -- bool (default: True); if True, return a square - root in an extension ring, if necessary. Otherwise, - raise a ValueError if the square is not in the base ring. - all -- bool (default: False); if True, return *all* square - roots of self, instead of just one. - - ALGORITHM: Calculates the square roots mod $p$ for each of the - primes $p$ dividing the order of the ring, then lifts them - p-adically and uses the CRT to find a square root mod $n$. - - See also \code{square_root_mod_prime_power} and - \code{square_root_mod_prime} (in this module) for more - algorithmic details. + + def sqrt(self): + """ + Same as self.square_root(). + """ + return self.square_root() + + + def square_root(self): + """ + Calculates the square roots mod $p$ for each of the primes $p$ dividing + the order of the ring, then lifts p-adically and uses crt to + find a square root mod $n$. + + See also \code{square_root_mod_prime_power} and \code{square_root_mod_prime} + (in this module) algorithms details. EXAMPLES: - sage: mod(-1, 17).sqrt() + sage: mod(-1, 17).square_root() 4 - sage: mod(5, 389).sqrt() + sage: mod(5, 389).square_root() 86 - sage: mod(7, 18).sqrt() + sage: mod(7, 18).square_root() 5 - sage: a = mod(14, 5^60).sqrt() + sage: a = mod(14, 5^60).square_root() sage: a*a 14 - sage: mod(15, 389).sqrt(extend=False) + sage: mod(15, 389).square_root() Traceback (most recent call last): ... - ValueError: self must be a square - sage: Mod(1/9, next_prime(2^40)).sqrt()^(-2) + ValueError: Self must be a square. + sage: Mod(1/9, next_prime(2^40)).square_root()^(-2) 9 - sage: Mod(1/25, next_prime(2^90)).sqrt()^(-2) + sage: Mod(1/25, next_prime(2^90)).square_root()^(-2) 25 - sage: a = Mod(3,5); a - 3 - sage: x = Mod(-1, 360) - sage: x.sqrt(extend=False) - Traceback (most recent call last): - ... - ValueError: self must be a square - sage: y = x.sqrt(); y - sqrt359 - sage: y.parent() - Univariate Quotient Polynomial Ring in sqrt359 over Ring of integers modulo 360 with modulus x^2 + 1 - sage: y^2 - 359 - - We compute all square roots in several cases: - sage: R = Integers(5*2^3*3^2); R - Ring of integers modulo 360 - sage: R(40).sqrt(all=True) - [20, 160, 200, 340] - sage: [x for x in R if x^2 == 40] # Brute force verification - [20, 160, 200, 340] - sage: R(1).sqrt(all=True) - [1, 19, 71, 89, 91, 109, 161, 179, 181, 199, 251, 269, 271, 289, 341, 359] - - sage: R = Integers(5*13^2*37); R - Ring of integers modulo 31265 - sage: v = R(-1).sqrt(all=True); v - [3817, 5507, 13252, 14942, 16323, 18013, 25758, 27448] - sage: [x^2 for x in v] - [31264, 31264, 31264, 31264, 31264, 31264, 31264, 31264] - - Modulo a power of 2: - sage: R = Integers(2^7); R - Ring of integers modulo 128 - sage: a = R(17) - sage: a.sqrt() - 23 - sage: a.sqrt(all=True) - [23, 41, 87, 105] - sage: [x for x in R if x^2==17] - [23, 41, 87, 105] - """ - if self.is_one(): - if all: - return list(self.parent().square_roots_of_one()) - else: - return self + """ + if self.is_zero() or self.is_one(): + return self if not self.is_square_c(): - if extend: - R = self.parent()['x'] - y = 'sqrt%s'%self - Q = R.quotient(R.gen()**2 - R(self), names=(y,)) - z = Q.gen() - if all: - # TODO - raise NotImplementedError - return z - raise ValueError, "self must be a square" - - F = self._parent.factored_order() - if len(F) == 1: - p, e = F[0] - - if all and e > 1 and not self.is_unit(): - # TODO -- this is the *one* and only remaining STUPID SLOW CASE. - v = [x for x in self.parent() if x*x == self] - return v - + raise ValueError, "Self must be a square." + + moduli = self._parent.factored_order() + if len(moduli) == 1: + p, e = moduli[0] if e > 1: x = square_root_mod_prime_power(mod(self, p**e), p, e) else: x = square_root_mod_prime(self, p) - x = x._balanced_abs() - - if not all: - return x - - v = list(set([x*a for a in self.parent().square_roots_of_one()])) - v.sort() - return v - else: - if not all: - # Use CRT to combine together a square root modulo each prime power - sqrts = [square_root_mod_prime(mod(self, p), p) for p, e in F if e == 1] + \ - [square_root_mod_prime_power(mod(self, p**e), p, e) for p, e in F if e != 1] - - x = sqrts.pop() - for y in sqrts: - x = x.crt(y) - return x._balanced_abs() - else: - # Use CRT to combine together all square roots modulo each prime power - vmod = [] - moduli = [] - P = self.parent() - from integer_mod_ring import IntegerModRing - for p, e in F: - k = p**e - R = IntegerModRing(p**e) - w = [P(x) for x in R(self).sqrt(all=True)] - vmod.append(w) - moduli.append(k) - # Now combine in all possible ways using the CRT - from arith import CRT_basis - basis = CRT_basis(moduli) - from sage.misc.mrange import cartesian_product_iterator - v = [] - for x in cartesian_product_iterator(vmod): - # x is a specific choice of roots modulo each prime power divisor - a = sum([basis[i]*x[i] for i in range(len(x))]) - v.append(a) - v.sort() - return v + + else: + # return product of square roots mod each prime power + sqrts = [square_root_mod_prime(mod(self, p), p) for p, e in moduli if e == 1] + \ + [square_root_mod_prime_power(mod(self, p**e), p, e) for p, e in moduli if e != 1] + + x = sqrts.pop() + for y in sqrts: + x = x.crt(y) + + return x._balanced_abs() def _balanced_abs(self): @@ -931,7 +835,7 @@ return bool(mpz_cmp_si(self.value, 1) == 0) - def __nonzero__(IntegerMod_gmp self): - """ - Returns \code{True} if this is not $0$, otherwise \code{False}. + def is_zero(IntegerMod_gmp self): + """ + Returns \code{True} if this is $0$, otherwise \code{False}. EXAMPLES: @@ -941,5 +845,5 @@ True """ - return bool(mpz_cmp_si(self.value, 0) != 0) + return bool(mpz_cmp_si(self.value, 0) == 0) def is_unit(self): @@ -1192,6 +1096,6 @@ return self.__modulus.lookup(value) cdef IntegerMod_int x = PY_NEW(IntegerMod_int) + x.__modulus = self.__modulus x._parent = self._parent - x.__modulus = self.__modulus x.ivalue = value return x @@ -1256,7 +1160,7 @@ return bool(self.ivalue == 1) - def __nonzero__(IntegerMod_int self): - """ - Returns \code{True} if this is not $0$, otherwise \code{False}. + def is_zero(IntegerMod_int self): + """ + Returns \code{True} if this is $0$, otherwise \code{False}. EXAMPLES: @@ -1266,5 +1170,5 @@ True """ - return bool(self.ivalue != 0) + return bool(self.ivalue == 0) def is_unit(IntegerMod_int self): @@ -1767,7 +1671,7 @@ return bool(self.ivalue == 1) - def __nonzero__(IntegerMod_int64 self): - """ - Returns \code{True} if this is not $0$, otherwise \code{False}. + def is_zero(IntegerMod_int64 self): + """ + Returns \code{True} if this is $0$, otherwise \code{False}. EXAMPLES: @@ -1777,5 +1681,5 @@ True """ - return bool(self.ivalue != 0) + return bool(self.ivalue == 0) def is_unit(IntegerMod_int64 self): @@ -2136,10 +2040,10 @@ def square_root_mod_prime_power(IntegerMod_abstract a, p, e): - r""" - Calculates the square root of $a$, where $a$ is an integer mod $p^e$. + """ + Calculates the square root of a, where a is an integer mod p^e. ALGORITHM: Perform p-adically by stripping off even powers of $p$ to get - a unit and lifting $\sqrt{unit} mod p$ via Newton's method. + a unit and lifting $\sqrt{unit} mod p$ via newton's method. AUTHOR: @@ -2150,10 +2054,5 @@ if p == 2: - if e == 1: - return a - # TODO: implement something that isn't totally idiotic. - for x in a.parent(): - if x**2 == a: - return x + return a if e == 1 else self.parent()(self.pari().sqrt()) # TODO: implement directly # strip off even powers of p @@ -2169,6 +2068,6 @@ x = unit.parent()(square_root_mod_prime(mod(unit, p), p)) - # lift p-adically using Newton iteration - # this is done to higher precision than neccesary except at the last step + # lift p-adically using newton iteration + # this is done to higher precision than neccisary except at the last step one_half = ~(a._new_c_from_long(2)) for i from 0 <= i < ceil(log(e)/log(2)) - val/2: @@ -2179,19 +2078,17 @@ x *= p**(val // 2) return x - + def square_root_mod_prime(IntegerMod_abstract a, p=None): - r""" + """ Calculates the square root of a, where a is an integer mod p. ALGORITHM: Several cases based on residue class of p mod 16. - - \begin{verbatim} - $p$ mod 2 = 0 $\Rightarrow$ p = 2 so \sqrt{a} = a$. - $p$ mod 4 = 3 $\Rightarrow \sqrt{a} = a^{(p+1)/4}$. - $p$ mod 8 = 5 $\Rightarrow \sqrt{a} = \zeta i a$ where $\zeta = (2a)^{(p-5)/8}$, $i=\sqrt{-1}$. - $p$ mod 16 = 9$ Similar, work in a bi-quadratic extension of $\FF_p$. - $p$ mod 16 = 1$ Variant of Cipolla-Lehmer, using Lucas functions. - \end{verbatim} + + $p$ mod 2 = 0 \Rightarrow p = 2 so \sqrt{a} = a$. + $p$ mod 4 = 3 \Rightarrow \sqrt{a} = a^{(p+1)/4}$. + $p$ mod 8 = 5 \Rightarrow \sqrt{a} = \zeta i a$ where $\zeta = (2a)^{(p-5)/8}, i=\sqrt{-1}$. + $p$ mod 16 = 9$ Similar, work in a bi-quadratic extension of $\F_p$. + $p$ mod 16 = 1$ Variant of Cipolla\202\304\354Lehmer, using Lucas functions. REFERENCES: @@ -2205,5 +2102,5 @@ H. Postl. 'Fast evaluation of Dickson Polynomials' - Contrib. to General Algebra, Vol. 6 (1988) pp. 223--225 + Contrib. to General Algebra, Vol. 6 (1988) pp. 223\202\304\354225 AUTHOR: @@ -2212,5 +2109,5 @@ TESTS: Every case appears in the first hundred primes. - sage: all([(a*a).sqrt()^2 == a*a for p in prime_range(100) for a in Integers(p)]) + sage: all([(a*a).square_root()^2 == a*a for p in prime_range(100) for a in Integers(p)]) True """ @@ -2229,22 +2126,22 @@ elif p_mod_16 % 4 == 3: - return a ** ((p+1)//4) + return a ** ((p+1)/4) elif p_mod_16 % 8 == 5: two_a = a+a - zeta = two_a ** ((p-5)//8) - i = two_a ** ((p-1)//4) + zeta = two_a ** ((p-5)/8) + i = two_a ** ((p-1)/4) return zeta*a*(i-1) elif p_mod_16 == 9: - s = (a+a) ** ((p-1)//4) + s = (a+a) ** ((p-1)/4) if s.is_one(): d = a._parent.quadratic_nonresidue() d2 = d*d - z = (2 * d2 * a) ** ((p-9)//16) + z = (2 * d2 * a) ** ((p-9)/16) i = 2 * d2 * z*z * a return z*d*a*(i-1) else: - z = (a+a) ** ((p-9)//16) + z = (a+a) ** ((p-9)/16) i = 2 * z*z * a return z*a*(i-1) Index: sage/rings/integer_mod_ring.py =================================================================== --- sage/rings/integer_mod_ring.py (revision 4278) +++ sage/rings/integer_mod_ring.py (revision 3925) @@ -32,6 +32,5 @@ -- William Stein (initial code) -- David Joyner (2005-12-22): most examples - -- Robert Bradshaw (2006-08-24): convert to SageX - -- William Stein (2007-04-29): square_roots_of_one + -- Robert Bradshaw (2006-08-24) Convert to SageX """ @@ -400,14 +399,4 @@ def quadratic_nonresidue(self): - """ - Return a quadratic non-residue in self. - - EXAMPLES: - sage: R = Integers(17) - sage: R.quadratic_nonresidue() - 3 - sage: R(3).is_square() - False - """ try: return self._nonresidue @@ -417,73 +406,4 @@ self._nonresidue = a return a - - def square_roots_of_one(self): - """ - Return all square roots of 1 in self, i.e., all solutions - to $x^2 - 1$. - - OUTPUT: - tuple -- the square roots of 1 in self. - - EXAMPLES: - sage: R = Integers(2^10) - sage: [x for x in R if x^2 == 1] - [1, 511, 513, 1023] - sage: R.square_roots_of_one() - (1, 511, 513, 1023) - - sage: v = Integers(9*5).square_roots_of_one(); v - (1, 19, 26, 44) - sage: [x^2 for x in v] - [1, 1, 1, 1] - sage: v = Integers(9*5*8).square_roots_of_one(); v - (1, 19, 71, 89, 91, 109, 161, 179, 181, 199, 251, 269, 271, 289, 341, 359) - sage: [x^2 for x in v] - [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] - """ - try: - return self.__square_roots_of_one - except AttributeError: - pass - n = self.__order - if n.is_prime_power(): - if n % 2 == 0: - # power of 2 - if n == 2: - v = [self(1)] - elif n == 4: - v = [self(1), self(3)] - else: # n >= 8 - half_ord = n//2 - v = [self(1), self(-1), self(half_ord-1), self(half_ord+1)] - else: - v = [self(1), self(-1)] - else: - # Reduce to the prime power case. - F = self.factored_order() - vmod = [] - moduli = [] - for p, e in F: - k = p**e - R = IntegerModRing(p**e) - w = [self(x) for x in R.square_roots_of_one()] - vmod.append(w) - moduli.append(k) - # Now combine in all possible ways using the CRT - from arith import CRT_basis - basis = CRT_basis(moduli) - from sage.misc.mrange import cartesian_product_iterator - v = [] - for x in cartesian_product_iterator(vmod): - # x is a specific choice of roots modulo each prime power divisor - a = sum([basis[i]*x[i] for i in range(len(x))]) - v.append(a) - #end for - #end if - - v.sort() - v = tuple(v) - self.__square_roots_of_one = v - return v def factored_order(self): Index: sage/rings/integer_ring.pxd =================================================================== --- sage/rings/integer_ring.pxd (revision 4057) +++ sage/rings/integer_ring.pxd (revision 3665) @@ -8,3 +8,2 @@ cdef Integer _coerce_ZZ(self, ntl_c_ZZ *z) cdef int _randomize_mpz(self, mpz_t value, x, y, distribution) except -1 - cdef object _zero Index: sage/rings/integer_ring.pyx =================================================================== --- sage/rings/integer_ring.pyx (revision 4010) +++ sage/rings/integer_ring.pyx (revision 3665) @@ -28,10 +28,4 @@ sage: Z('94803849083985934859834583945394') 94803849083985934859834583945394 - -TESTS: - sage: Z = IntegerRing() - sage: Z == loads(dumps(Z)) - True - """ Index: sage/rings/laurent_series_ring.py =================================================================== --- sage/rings/laurent_series_ring.py (revision 4279) +++ sage/rings/laurent_series_ring.py (revision 3665) @@ -100,7 +100,4 @@ ParentWithGens.__init__(self, base_ring, name) self.__sparse = sparse - - def change_ring(self, R): - return LaurentSeriesRing(R, self.variable_name(), sparse=self.__sparse) def is_sparse(self): Index: age/rings/laurent_series_ring_element.pxd =================================================================== --- sage/rings/laurent_series_ring_element.pxd (revision 4158) +++ (revision ) @@ -1,5 +1,0 @@ -from sage.structure.element cimport AlgebraElement - -cdef class LaurentSeries(AlgebraElement): - cdef object __u - cdef long __n Index: sage/rings/laurent_series_ring_element.py =================================================================== --- sage/rings/laurent_series_ring_element.py (revision 3919) +++ sage/rings/laurent_series_ring_element.py (revision 3919) @@ -0,0 +1,768 @@ +""" +Laurent Series + +EXAMPLES: + + sage: R. = LaurentSeriesRing(GF(7), 't'); R + Laurent Series Ring in t over Finite Field of size 7 + sage: f = 1/(1-t+O(t^10)); f + 1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + t^9 + O(t^10) + +Laurent series are immutable: + sage: f[2] + 1 + sage: f[2] = 5 + Traceback (most recent call last): + ... + IndexError: Laurent series are immutable + +We compute with a Laurent series over the complex mpfr numbers. + sage: K. = Frac(CC[['q']]) + sage: K + Laurent Series Ring in q over Complex Field with 53 bits of precision + sage: q + 1.00000000000000*q + +Saving and loading. + sage: loads(q.dumps()) == q + True + sage: loads(K.dumps()) == K + True + +IMPLEMENTATION: Laurent series in SAGE are represented internally as a + power of the variable times the unit part (which need not be a + unit -- it's a polynomial with nonzero constant term). The zero + Laurent series has unit part 0. + + +AUTHORS: + -- William Stein: original version + -- David Joyner: added examples 2006-01-22 +""" + +import operator + +from infinity import infinity + +import laurent_series_ring +import power_series_ring_element +import power_series_ring +import polynomial_element as polynomial +import sage.misc.latex as latex +import sage.rings.ring_element as ring_element + +class LaurentSeries(ring_element.RingElement): + """ + A Laurent Series. + """ + def __init__(self, parent, f, n=0): + r""" + Create the Laurent series $t^n \cdot f$. The default is n=0. + + INPUT: + parent -- a Laurent series ring + f -- a power series (or something can be coerced to one); note that + f does *not* have to be a unit. + n -- integer (default 0) + + OUTPUT: + a Laurent series + """ + ring_element.RingElement.__init__(self, parent) + if isinstance(f, LaurentSeries): + n += f.__n + f = parent.power_series_ring()(f.__u) + elif not isinstance(f, power_series_ring_element.PowerSeries): + f = parent.power_series_ring()(f) + + # self is that t^n * u: + if f.is_zero(): + self.__n = n + self.__u = f + else: + self.__n = n + f.valuation() # power of the variable + self.__u = f.valuation_zero_part() # unit part + + def is_unit(self): + """ + Returns True if this is Laurent series is a unit in this ring. + + EXAMPLES: + sage: R. = LaurentSeriesRing(QQ) + sage: (2+t).is_unit() + True + sage: f = 2+t^2+O(t^10); f.is_unit() + True + sage: 1/f + 1/2 - 1/4*t^2 + 1/8*t^4 - 1/16*t^6 + 1/32*t^8 + O(t^10) + sage: R(0).is_unit() + False + sage: R. = LaurentSeriesRing(ZZ) + sage: f = 2 + s^2 + O(s^10) + sage: f.is_unit() + False + sage: 1/f + Traceback (most recent call last): + ... + ArithmeticError: division not defined + + ALGORITHM: A Laurent series is a unit if and only if + its "unit part" is a unit. + """ + return self.__u.is_unit() + + def is_zero(self): + """ + EXAMPLES: + sage: x = Frac(QQ[['x']]).0 + sage: f = 1/x + x + x^2 + 3*x^4 + O(x^7) + sage: f.is_zero() + 0 + sage: z = 0*f + sage: z.is_zero() + 1 + """ + return self.__u.is_zero() + + def _im_gens_(self, codomain, im_gens): + return codomain(self(im_gens[0])) + + def __normalize(self): + r""" + A Laurent series is a pair (u(t), n), where either u=0 (to + some precision) or u is a unit. This pair corresponds to + $t^n\cdot u(t)$. + """ + if self.is_zero(): + return + v = self.__u.valuation() + if v == 0: + return + self.__n += v + self.__u = self.__u.valuation_zero_part() + + def _repr_(self): + """ + EXAMPLES: + sage: R. = LaurentSeriesRing(QQ) + sage: (2 + (2/3)*t^3).__repr__() + '2 + 2/3*t^3' + """ + if self.is_zero(): + if self.prec() == infinity: + return "0" + else: + return "O(%s^%s)"%(self.parent().variable_name(),self.prec()) + s = " " + v = self.__u.list() + valuation = self.__n + m = len(v) + X = self.parent().variable_name() + atomic_repr = self.parent().base_ring().is_atomic_repr() + first = True + for n in xrange(m): + x = v[n] + e = n + valuation + if x != 0: + if not first: + s += " + " + x = str(x) + if not atomic_repr and (x[1:].find("+") != -1 or x[1:].find("-") != -1): + x = "(%s)"%x + if e == 1: + var = "*%s"%X + elif e == 0: + var = "" + else: + var = "*%s^%s"%(X,e) + s += "%s%s"%(x,var) + first = False + if atomic_repr: + s = s.replace(" + -", " - ") + s = s.replace(" 1*"," ") + s = s.replace(" -1*", " -") + if self.prec() == 0: + bigoh = "O(1)" + elif self.prec() == 1: + bigoh = "O(%s)"%self.parent().variable_name() + else: + bigoh = "O(%s^%s)"%(self.parent().variable_name(),self.prec()) + if self.prec() != infinity: + if s == " ": + return bigoh + s += " + %s"%bigoh + return s[1:] + + def _latex_(self): + r""" + EXAMPLES: + sage: x = Frac(QQ[['x']]).0 + sage: f = (17/2)*x^-2 + x + x^2 + 3*x^4 + O(x^7) + sage: latex(f) + \frac{\frac{17}{2}}{x^{2}} + x + x^{2} + 3x^{4} + O(\text{x}^{7}) + """ + if self.is_zero(): + if self.prec() == infinity: + return "0" + else: + return "0 + \\cdots" + s = " " + v = self.__u.list() + valuation = self.__n + m = len(v) + X = self.parent().variable_name() + atomic_repr = self.parent().base_ring().is_atomic_repr() + first = True + for n in xrange(m): + x = v[n] + e = n + valuation + if x != 0: + if not first: + s += " + " + x = latex.latex(x) + if not atomic_repr and n > 0 and (x[1:].find("+") != -1 or x[1:].find("-") != -1): + x = "\\left(%s\\right)"%x + if e == 1: + var = "|%s"%X + elif e == 0: + var = "" + elif e > 0: + var = "|%s^{%s}"%(X,e) + if e >= 0: + s += "%s%s"%(x,var) + else: # negative e + if e == -1: + s += "\\frac{%s}{%s}"%(x, X) + else: + s += "\\frac{%s}{%s^{%s}}"%(x, X,-e) + first = False + if atomic_repr: + s = s.replace(" + -", " - ") + s = s.replace(" 1|"," ") + s = s.replace(" -1|", " -") + s = s.replace("|","") + pr = self.prec() + if pr != infinity: + if pr == 0: + bigoh = "O(1)" + elif pr == 1: + bigoh = "O(%s)"%latex.latex(self.parent().variable_name()) + else: + bigoh = "O(%s^{%s})"%(latex.latex(self.parent().variable_name()),pr) + if s == " ": + return bigoh + s += " + %s"%bigoh + return s[1:] + + def __getitem__(self, i): + """ + EXAMPLES: + sage: R. = LaurentSeriesRing(QQ) + sage: f = -5/t^(10) + t + t^2 - 10/3*t^3; f + -5*t^-10 + t + t^2 - 10/3*t^3 + sage: f[-10] + -5 + sage: f[1] + 1 + sage: f[3] + -10/3 + sage: f[-9] + 0 + """ + return self.__u[i-self.__n] + + def __getslice__(self, i, j): + """ + EXAMPLES: + sage: R. = LaurentSeriesRing(QQ) + sage: f = -5/t^(10) + 1/3 + t + t^2 - 10/3*t^3 + O(t^5); f + -5*t^-10 + 1/3 + t + t^2 - 10/3*t^3 + O(t^5) + sage: f[-10:2] + -5*t^-10 + 1/3 + t + O(t^5) + sage: f[0:] + 1/3 + t + t^2 - 10/3*t^3 + O(t^5) + """ + f = self.__u[i-self.__n:j-self.__n] + return LaurentSeries(self.parent(), f, self.__n) + + def __iter__(self): + """ + Iterate through the coefficients from the first nonzero one to + the last nonzero one. + + EXAMPLES: + sage: R. = LaurentSeriesRing(QQ) + sage: f = -5/t^(2) + t + t^2 - 10/3*t^3; f + -5*t^-2 + t + t^2 - 10/3*t^3 + sage: for a in f: print a + -5 + 0 + 0 + 1 + 1 + -10/3 + """ + for i in range(self.valuation(), self.degree()+1): + yield self[i] + + def __setitem__(self, n, value): + """ + EXAMPLES: + sage: R. = LaurentSeriesRing(QQ) + sage: f = t^2 + t^3 + O(t^10) + sage: f[2] = 5 + Traceback (most recent call last): + ... + IndexError: Laurent series are immutable + """ + raise IndexError, "Laurent series are immutable" + + def _unsafe_mutate(self, i, value): + """ + SAGE assumes throughout that commutative ring elements are immutable. + This is relevant for caching, etc. But sometimes you need to change + a Laurent series and you really know what you're doing. That's + when this function is for you. + + EXAMPLES: + + """ + j = i - self.__n + if j >= 0: + self.__u._unsafe_mutate(j, value) + else: # off to the left + if value != 0: + self.__n = self.__n + j + R = self.parent().base_ring() + coeffs = [value] + [R(0) for _ in range(1,-j)] + self.__u.list() + self.__u = self.__u.parent()(coeffs) + self.__normalize() + + def _add_(self, right): + """ + Add two power series with the same parent. + + EXAMPLES: + sage: R. = LaurentSeriesRing(QQ) + sage: t + t + 2*t + sage: f = 1/t + t^2 + t^3 - 17/3 * t^4 + O(t^5) + sage: g = 1/(1-t + O(t^7)); g + 1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + O(t^7) + sage: f + g + t^-1 + 1 + t + 2*t^2 + 2*t^3 - 14/3*t^4 + O(t^5) + sage: f + 0 + t^-1 + t^2 + t^3 - 17/3*t^4 + O(t^5) + sage: 0 + f + t^-1 + t^2 + t^3 - 17/3*t^4 + O(t^5) + sage: R(0) + R(0) + 0 + sage: (t^3 + O(t^10)) + (t^-3 +O(t^9)) + t^-3 + t^3 + O(t^9) + + ALGORITHM: + Multiply both Laurent series by a power of the variable to make them + power series, add those power series, then divide. + """ + # Add together two Laurent series over the same base ring. + + # 1. Special case when one or the other is 0. + if right.is_zero(): + return self.add_bigoh(right.prec()) + if self.is_zero(): + return right.add_bigoh(self.prec()) + + # 2. Find power of t that we can multiply both by to get power series. + m = - min(self.valuation(), right.valuation()) + # Now t^m times each one is a polynomial. + # Get the polynomial indeterminate t + t = self.__u.parent().gen() + # Compute t^m times self + f1 = t**(self.__n + m) * self.__u + # Compute t^m times right + f2 = t**(right.__n + m) * right.__u + # Now add f1 and f2 as power series + g = f1 + f2 + # Finally construct the Laurent series associated to the sum time t^(-m). + return LaurentSeries(self.parent(), g, -m) + + def add_bigoh(self, prec): + """ + EXAMPLES: + sage: R. = LaurentSeriesRing(QQ) + sage: f = t^2 + t^3 + O(t^10); f + t^2 + t^3 + O(t^10) + sage: f.add_bigoh(5) + t^2 + t^3 + O(t^5) + """ + if prec == infinity or prec >= self.prec(): + return self + u = self.__u.add_bigoh(prec - self.__n) + return LaurentSeries(self.parent(), u, self.__n) + + def degree(self): + """ + Return the degree of a polynomial equivalent to this power + series modulo big oh of the precision. + + EXAMPLES: + sage: x = Frac(QQ[['x']]).0 + sage: g = x^2 - x^4 + O(x^8) + sage: g.degree() + 4 + sage: g = -10/x^5 + x^2 - x^4 + O(x^8) + sage: g.degree() + 4 + """ + return self.__u.degree() + self.valuation() + +# todo: +# I commented out the following __sub__ because it doesn't fit the new +# arithmetic architecture rules. Perhaps a native _sub_ implementation would +# be reasonable to have, but it seems pretty well covered by the default +# RingElement._sub_c_impl() implementation anyway, so why bother. +# -- David Harvey +# +# BTW should mention that David Roe is working on reimplementing laurent +# series from scratch. (hopefully!) +# +# def __sub__(self, right): +# """ +# EXAMPLES: +# sage: R = LaurentSeriesRing(ZZ, 't') +# sage: f = t^2 + t^3 + O(t^10) +# sage: g = 3/t^4 + t^3 + O(t^5) +# sage: f - g +# -3*t^-4 + t^2 + O(t^5) +# sage: g - f +# 3*t^-4 - t^2 + O(t^5) +# """ +# return self + right.__neg__() + +# ditto with __neg__: +# +# def __neg__(self): +# """ +# EXAMPLES: +# sage: R. = LaurentSeriesRing(ZZ) +# sage: f = 3/t^2 + t^2 + t^3 + O(t^10) +# sage: f.__neg__() +# -3*t^-2 - t^2 - t^3 + O(t^10) +# """ +# return (-1)*self + + def _mul_(self, right): + """ + EXAMPLES: + sage: x = Frac(QQ[['x']]).0 + sage: f = 1/x^3 + x + x^2 + 3*x^4 + O(x^7) + sage: g = 1 - x + x^2 - x^4 + O(x^8) + sage: f*g + x^-3 - x^-2 + x^-1 + 4*x^4 + O(x^5) + """ + return LaurentSeries(self.parent(), + self.__u * right.__u, + self.__n + right.__n) + def __pow__(self, r): + """ + EXAMPLES: + sage: x = Frac(QQ[['x']]).0 + sage: f = x + x^2 + 3*x^4 + O(x^7) + sage: g = 1/x^10 - x + x^2 - x^4 + O(x^8) + sage: f^7 + x^7 + 7*x^8 + 21*x^9 + 56*x^10 + 161*x^11 + 336*x^12 + O(x^13) + sage: g^7 + x^-70 - 7*x^-59 + 7*x^-58 - 7*x^-56 + O(x^-52) + """ + right=int(r) + if right != r: + raise ValueError, "exponent must be an integer" + return LaurentSeries(self.parent(), self.__u**right, self.__n*right) + + + def _div_(self, right): + """ + EXAMPLES: + sage: x = Frac(QQ[['x']]).0 + sage: f = x + x^2 + 3*x^4 + O(x^7) + sage: g = 1/x^7 - x + x^2 - x^4 + O(x^8) + sage: f/x + 1 + x + 3*x^3 + O(x^6) + sage: f/g + x^8 + x^9 + 3*x^11 + O(x^14) + """ + if right.__u.is_zero(): + raise ZeroDivisionError + try: + return LaurentSeries(self.parent(), + self.__u / right.__u, + self.__n - right.__n) + except TypeError, msg: + # todo: this could also make something in the formal fraction field. + raise ArithmeticError, "division not defined" + + + def common_prec(self, f): + r""" + Returns minimum precision of $f$ and self. + + EXAMPLES: + sage: R. = LaurentSeriesRing(QQ) + + sage: f = t^(-1) + t + t^2 + O(t^3) + sage: g = t + t^3 + t^4 + O(t^4) + sage: f.common_prec(g) + 3 + sage: g.common_prec(f) + 3 + + sage: f = t + t^2 + O(t^3) + sage: g = t^(-3) + t^2 + sage: f.common_prec(g) + 3 + sage: g.common_prec(f) + 3 + + sage: f = t + t^2 + sage: f = t^2 + sage: f.common_prec(g) + +Infinity + + sage: f = t^(-3) + O(t^(-2)) + sage: g = t^(-5) + O(t^(-1)) + sage: f.common_prec(g) + -2 + """ + if self.prec() is infinity: + return f.prec() + elif f.prec() is infinity: + return self.prec() + return min(self.prec(), f.prec()) + + def __cmp__(self, right): + r""" + Comparison of self and right. + + We say two approximate laurent series are equal, if they agree + for all coefficients up to the *minimum* of the precisions of + each. Comparison is done in dictionary order from lowest degree to + highest degree coefficients (this is different than + polynomials). + + See power_series_ring_element.__cmp__() for more information. + + EXAMPLES: + sage: R. = LaurentSeriesRing(QQ) + sage: f = x^(-1) + 1 + x + O(x^2) + sage: g = x^(-1) + 1 + O(x) + sage: f == g + True + + sage: f = x^(-1) + 1 + x + O(x^2) + sage: g = x^(-1) + 2 + O(x) + sage: f == g + False + sage: f < g + True + sage: f > g + False + + sage: f = x^(-2) + 1 + x + O(x^2) + sage: g = x^(-1) + 2 + O(x) + sage: f == g + False + sage: f < g + False + sage: f > g + True + """ + prec = self.common_prec(right) + n = min(self.__n, right.__n) + + # zero pad coefficients on the left, to line them up for comparison + zero = self.base_ring()(0) + x = [zero] * (self.__n - n) + self.__u.list() + y = [zero] * (right.__n - n) + right.__u.list() + + # zero pad on right to make the lists the same length + # (this is necessary since the power series list() function just + # returns the coefficients of the underlying polynomial, which may + # have zeroes in the high coefficients) + if len(x) < len(y): + x.extend([zero] * (len(y) - len(x))) + elif len(y) < len(x): + y.extend([zero] * (len(x) - len(y))) + + if not (prec is infinity): + x = x[:(prec-n)] + y = y[:(prec-n)] + + return cmp(x,y) + + def valuation_zero_part(self): + """ + EXAMPLES: + sage: x = Frac(QQ[['x']]).0 + sage: f = x + x^2 + 3*x^4 + O(x^7) + sage: f/x + 1 + x + 3*x^3 + O(x^6) + sage: f.valuation_zero_part() + 1 + x + 3*x^3 + O(x^6) + sage: g = 1/x^7 - x + x^2 - x^4 + O(x^8) + sage: g.valuation_zero_part() + 1 - x^8 + x^9 - x^11 + O(x^15) + """ + return self.__u + + def valuation(self): + """ + EXAMPLES: + sage: x = Frac(QQ[['x']]).0 + sage: f = 1/x + x^2 + 3*x^4 + O(x^7) + sage: g = 1 - x + x^2 - x^4 + O(x^8) + sage: f.valuation() + -1 + sage: g.valuation() + 0 + """ + return self.__n + + def variable(self): + """ + EXAMPLES: + sage: x = Frac(QQ[['x']]).0 + sage: f = 1/x + x^2 + 3*x^4 + O(x^7) + sage: f.variable() + 'x' + """ + return self.parent().variable_name() + + def prec(self): + """ + This function returns the n so that the Laurent series is + of the form (stuff) + $O(t^n)$. It doesn't matter how many + negative powers appear in the expansion. In particular, + prec could be negative. + + EXAMPLES: + sage: x = Frac(QQ[['x']]).0 + sage: f = x^2 + 3*x^4 + O(x^7) + sage: f.prec() + 7 + sage: g = 1/x^10 - x + x^2 - x^4 + O(x^8) + sage: g.prec() + 8 + """ + return self.__u.prec() + self.valuation() + + def copy(self): + return LaurentSeries(self.parent(), self.__u.copy(), self.__n) + + def derivative(self): + """ + The formal derivative of this Laurent series. + + EXAMPLES: + sage: x = Frac(QQ[['x']]).0 + sage: f = x^2 + 3*x^4 + O(x^7) + sage: f.derivative() + 2*x + 12*x^3 + O(x^6) + sage: g = 1/x^10 - x + x^2 - x^4 + O(x^8) + sage: g.derivative() + -10*x^-11 - 1 + 2*x - 4*x^3 + O(x^7) + """ + if self.is_zero(): + return LaurentSeries(self.parent(), 0, self.__u.prec() - 1) + n = self.__n + a = self.__u.list() + v = [(n+m)*a[m] for m in range(len(a))] + u = self.parent().power_series_ring()(v, self.__u.prec()) + return LaurentSeries(self.parent(), u, n-1) + + def integral(self): + r""" + The formal integral of this Laurent series with 0 constant term. + + EXAMPLES: + The integral may or may not be defined if the base ring + is not a field. + sage: t = LaurentSeriesRing(ZZ, 't').0 + sage: f = 2*t^-3 + 3*t^2 + O(t^4) + sage: f.integral() + -t^-2 + t^3 + O(t^5) + + sage: f = t^3 + sage: f.integral() + Traceback (most recent call last): + ... + ArithmeticError: Coefficients of integral cannot be coerced into the base ring + + + The integral of 1/t is $\log(t)$, which is not given by a Laurent series: + + sage: t = Frac(QQ[['t']]).0 + sage: f = -1/t^3 - 31/t + O(t^3) + sage: f.integral() + Traceback (most recent call last): + ... + ArithmeticError: The integral of is not a Laurent series, since t^-1 has nonzero coefficient. + """ + n = self.__n + a = self.__u.list() + if self[-1] != 0: + raise ArithmeticError, \ + "The integral of is not a Laurent series, since t^-1 has nonzero coefficient." + + if n < 0: + v = [a[i]/(n+i+1) for i in range(-1-n)] + [0] + else: + v = [] + v += [a[i]/(n+i+1) for i in range(max(-n,0), len(a))] + try: + u = self.parent().power_series_ring()(v, self.__u.prec()) + except TypeError: + raise ArithmeticError, "Coefficients of integral cannot be coerced into the base ring" + return LaurentSeries(self.parent(), u, n+1) + + + def power_series(self): + """ + EXAMPLES: + sage: R. = LaurentSeriesRing(ZZ) + sage: f = 1/(1-t+O(t^10)); f.parent() + Laurent Series Ring in t over Integer Ring + sage: g = f.power_series(); g + 1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + t^9 + O(t^10) + sage: parent(g) + Power Series Ring in t over Integer Ring + sage: f = 3/t^2 + t^2 + t^3 + O(t^10) + sage: f.power_series() + Traceback (most recent call last): + ... + ArithmeticError: self is a not a power series + """ + if self.__n < 0: + raise ArithmeticError, "self is a not a power series" + u = self.__u + t = u.parent().gen() + return t**(self.__n) * u + + def __call__(self, *x): + """ + Compute value of this Laurent series at x. + + EXAMPLES: + sage: R. = LaurentSeriesRing(ZZ) + sage: f = t^(-2) + t^2 + O(t^8) + sage: f(2) + 17/4 + sage: f(-1) + 2 + sage: f(1/3) + 82/9 + """ + if isinstance(x[0], tuple): + x = x[0] + return self.__u(x) * (x[0]**self.__n) + + Index: age/rings/laurent_series_ring_element.pyx =================================================================== --- sage/rings/laurent_series_ring_element.pyx (revision 4279) +++ (revision ) @@ -1,850 +1,0 @@ -""" -Laurent Series - -EXAMPLES: - - sage: R. = LaurentSeriesRing(GF(7), 't'); R - Laurent Series Ring in t over Finite Field of size 7 - sage: f = 1/(1-t+O(t^10)); f - 1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + t^9 + O(t^10) - -Laurent series are immutable: - sage: f[2] - 1 - sage: f[2] = 5 - Traceback (most recent call last): - ... - IndexError: Laurent series are immutable - -We compute with a Laurent series over the complex mpfr numbers. - sage: K. = Frac(CC[['q']]) - sage: K - Laurent Series Ring in q over Complex Field with 53 bits of precision - sage: q - 1.00000000000000*q - -Saving and loading. - sage: loads(q.dumps()) == q - True - sage: loads(K.dumps()) == K - True - -IMPLEMENTATION: Laurent series in SAGE are represented internally as a - power of the variable times the unit part (which need not be a - unit -- it's a polynomial with nonzero constant term). The zero - Laurent series has unit part 0. - - -AUTHORS: - -- William Stein: original version - -- David Joyner: added examples 2006-01-22 - -- Robert Bradshaw: optimizations, shifting 2007-04 - -- Robert Bradshaw: SageX version -""" - -import operator - -from infinity import infinity - -import laurent_series_ring -import power_series_ring_element -import power_series_ring -import polynomial_element as polynomial -import sage.misc.latex -import sage.rings.ring_element as ring_element -from sage.rings.integer import Integer - -from sage.structure.element cimport Element, ModuleElement, RingElement, AlgebraElement - -def is_LaurentSeries(x): - return isinstance(x, LaurentSeries) - -cdef class LaurentSeries(AlgebraElement): - """ - A Laurent Series. - """ - - def __init__(self, parent, f, n=0): - r""" - Create the Laurent series $t^n \cdot f$. The default is n=0. - - INPUT: - parent -- a Laurent series ring - f -- a power series (or something can be coerced to one); note that - f does *not* have to be a unit. - n -- integer (default 0) - - OUTPUT: - a Laurent series - """ - AlgebraElement.__init__(self, parent) - if isinstance(f, LaurentSeries): - n += (f).__n - f = parent.power_series_ring()((f).__u) - elif not isinstance(f, power_series_ring_element.PowerSeries): - f = parent.power_series_ring()(f) - - # self is that t^n * u: - if f.is_zero(): - if n == infinity: - self.__n = 0 - self.__u = parent.power_series_ring()(0) - else: - self.__n = n - self.__u = f - else: - self.__n = n + f.valuation() # power of the variable - self.__u = f.valuation_zero_part() # unit part - - def __reduce__(self): - return make_element_from_parent, (self._parent, self.__u, self.__n) - - def change_ring(self, R): - return self.parent().change_ring(R)(self) - - def is_unit(self): - """ - Returns True if this is Laurent series is a unit in this ring. - - EXAMPLES: - sage: R. = LaurentSeriesRing(QQ) - sage: (2+t).is_unit() - True - sage: f = 2+t^2+O(t^10); f.is_unit() - True - sage: 1/f - 1/2 - 1/4*t^2 + 1/8*t^4 - 1/16*t^6 + 1/32*t^8 + O(t^10) - sage: R(0).is_unit() - False - sage: R. = LaurentSeriesRing(ZZ) - sage: f = 2 + s^2 + O(s^10) - sage: f.is_unit() - False - sage: 1/f - Traceback (most recent call last): - ... - ArithmeticError: division not defined - - ALGORITHM: A Laurent series is a unit if and only if - its "unit part" is a unit. - """ - return self.__u.is_unit() - - def is_zero(self): - """ - EXAMPLES: - sage: x = Frac(QQ[['x']]).0 - sage: f = 1/x + x + x^2 + 3*x^4 + O(x^7) - sage: f.is_zero() - 0 - sage: z = 0*f - sage: z.is_zero() - 1 - """ - return self.__u.is_zero() - - def _im_gens_(self, codomain, im_gens): - return codomain(self(im_gens[0])) - - def __normalize(self): - r""" - A Laurent series is a pair (u(t), n), where either u=0 (to - some precision) or u is a unit. This pair corresponds to - $t^n\cdot u(t)$. - """ - if self.is_zero(): - return - v = self.__u.valuation() - if v == 0: - return - self.__n += v - self.__u = self.__u.valuation_zero_part() - - def _repr_(self): - """ - EXAMPLES: - sage: R. = LaurentSeriesRing(QQ) - sage: (2 + (2/3)*t^3).__repr__() - '2 + 2/3*t^3' - """ - if self.is_zero(): - if self.prec() == infinity: - return "0" - else: - return "O(%s^%s)"%(self._parent.variable_name(),self.prec()) - s = " " - v = self.__u.list() - valuation = self.__n - m = len(v) - X = self._parent.variable_name() - atomic_repr = self._parent.base_ring().is_atomic_repr() - first = True - for n in xrange(m): - x = v[n] - e = n + valuation - if x != 0: - if not first: - s += " + " - x = str(x) - if not atomic_repr and (x[1:].find("+") != -1 or x[1:].find("-") != -1): - x = "(%s)"%x - if e == 1: - var = "*%s"%X - elif e == 0: - var = "" - else: - var = "*%s^%s"%(X,e) - s += "%s%s"%(x,var) - first = False - if atomic_repr: - s = s.replace(" + -", " - ") - s = s.replace(" 1*"," ") - s = s.replace(" -1*", " -") - if self.prec() == 0: - bigoh = "O(1)" - elif self.prec() == 1: - bigoh = "O(%s)"%self._parent.variable_name() - else: - bigoh = "O(%s^%s)"%(self._parent.variable_name(),self.prec()) - if self.prec() != infinity: - if s == " ": - return bigoh - s += " + %s"%bigoh - return s[1:] - - def _latex_(self): - r""" - EXAMPLES: - sage: x = Frac(QQ[['x']]).0 - sage: f = (17/2)*x^-2 + x + x^2 + 3*x^4 + O(x^7) - sage: latex(f) - \frac{\frac{17}{2}}{x^{2}} + x + x^{2} + 3x^{4} + O(\text{x}^{7}) - """ - if self.is_zero(): - if self.prec() == infinity: - return "0" - else: - return "0 + \\cdots" - s = " " - v = self.__u.list() - valuation = self.__n - m = len(v) - X = self._parent.variable_name() - atomic_repr = self._parent.base_ring().is_atomic_repr() - first = True - for n in xrange(m): - x = v[n] - e = n + valuation - if x != 0: - if not first: - s += " + " - x = sage.misc.latex.latex(x) - if not atomic_repr and n > 0 and (x[1:].find("+") != -1 or x[1:].find("-") != -1): - x = "\\left(%s\\right)"%x - if e == 1: - var = "|%s"%X - elif e == 0: - var = "" - elif e > 0: - var = "|%s^{%s}"%(X,e) - if e >= 0: - s += "%s%s"%(x,var) - else: # negative e - if e == -1: - s += "\\frac{%s}{%s}"%(x, X) - else: - s += "\\frac{%s}{%s^{%s}}"%(x, X,-e) - first = False - if atomic_repr: - s = s.replace(" + -", " - ") - s = s.replace(" 1|"," ") - s = s.replace(" -1|", " -") - s = s.replace("|","") - pr = self.prec() - if pr != infinity: - if pr == 0: - bigoh = "O(1)" - elif pr == 1: - bigoh = "O(%s)"%sage.misc.latex.latex(self._parent.variable_name()) - else: - bigoh = "O(%s^{%s})"%(sage.misc.latex.latex(self._parent.variable_name()),pr) - if s == " ": - return bigoh - s += " + %s"%bigoh - return s[1:] - - def __getitem__(self, i): - """ - EXAMPLES: - sage: R. = LaurentSeriesRing(QQ) - sage: f = -5/t^(10) + t + t^2 - 10/3*t^3; f - -5*t^-10 + t + t^2 - 10/3*t^3 - sage: f[-10] - -5 - sage: f[1] - 1 - sage: f[3] - -10/3 - sage: f[-9] - 0 - """ - return self.__u[i-self.__n] - - def __getslice__(self, i, j): - """ - EXAMPLES: - sage: R. = LaurentSeriesRing(QQ) - sage: f = -5/t^(10) + 1/3 + t + t^2 - 10/3*t^3 + O(t^5); f - -5*t^-10 + 1/3 + t + t^2 - 10/3*t^3 + O(t^5) - sage: f[-10:2] - -5*t^-10 + 1/3 + t + O(t^5) - sage: f[0:] - 1/3 + t + t^2 - 10/3*t^3 + O(t^5) - """ - if j > self.__u.degree(): - j = self.__u.degree() - f = self.__u[i-self.__n:j-self.__n] - return LaurentSeries(self._parent, f, self.__n) - - def __iter__(self): - """ - Iterate through the coefficients from the first nonzero one to - the last nonzero one. - - EXAMPLES: - sage: R. = LaurentSeriesRing(QQ) - sage: f = -5/t^(2) + t + t^2 - 10/3*t^3; f - -5*t^-2 + t + t^2 - 10/3*t^3 - sage: for a in f: print a - -5 - 0 - 0 - 1 - 1 - -10/3 - """ - return iter(self.__u) - - def list(self): - return self.__u.list() - - def __setitem__(self, n, value): - """ - EXAMPLES: - sage: R. = LaurentSeriesRing(QQ) - sage: f = t^2 + t^3 + O(t^10) - sage: f[2] = 5 - Traceback (most recent call last): - ... - IndexError: Laurent series are immutable - """ - raise IndexError, "Laurent series are immutable" - - def _unsafe_mutate(self, i, value): - """ - SAGE assumes throughout that commutative ring elements are immutable. - This is relevant for caching, etc. But sometimes you need to change - a Laurent series and you really know what you're doing. That's - when this function is for you. - - EXAMPLES: - - """ - j = i - self.__n - if j >= 0: - self.__u._unsafe_mutate(j, value) - else: # off to the left - if value != 0: - self.__n = self.__n + j - R = self._parent.base_ring() - coeffs = [value] + [R(0) for _ in range(1,-j)] + self.__u.list() - self.__u = self.__u._parent(coeffs) - self.__normalize() - - cdef ModuleElement _add_c_impl(self, ModuleElement right_m): - """ - Add two power series with the same parent. - - EXAMPLES: - sage: R. = LaurentSeriesRing(QQ) - sage: t + t - 2*t - sage: f = 1/t + t^2 + t^3 - 17/3 * t^4 + O(t^5) - sage: g = 1/(1-t + O(t^7)); g - 1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + O(t^7) - sage: f + g - t^-1 + 1 + t + 2*t^2 + 2*t^3 - 14/3*t^4 + O(t^5) - sage: f + 0 - t^-1 + t^2 + t^3 - 17/3*t^4 + O(t^5) - sage: 0 + f - t^-1 + t^2 + t^3 - 17/3*t^4 + O(t^5) - sage: R(0) + R(0) - 0 - sage: (t^3 + O(t^10)) + (t^-3 +O(t^9)) - t^-3 + t^3 + O(t^9) - - ALGORITHM: - Shift the unit parts to align them, then add. - """ - cdef LaurentSeries right = right_m - cdef long m - - # 1. Special case when one or the other is 0. - if right.is_zero(): - return self.add_bigoh(right.prec()) - if self.is_zero(): - return right.add_bigoh(self.prec()) - - # 2. Align the unit parts. - if self.__n < right.__n: - m = self.__n - f1 = self.__u - f2 = right.__u << right.__n - m - else: - m = right.__n - f1 = self.__u << self.__n - m - f2 = right.__u - # 3. Add - return LaurentSeries(self._parent, f1 + f2, m) - - cdef ModuleElement _sub_c_impl(self, ModuleElement right_m): - """ - Subtract two power series with the same parent. - - EXAMPLES: - sage: R. = LaurentSeriesRing(QQ) - sage: t - t - 0 - sage: t^5 + 2 * t^-5 - 2*t^-5 + t^5 - - ALGORITHM: - Shift the unit parts to align them, then subtract. - """ - cdef LaurentSeries right = right_m - cdef long m - - # 1. Special case when one or the other is 0. - if right.is_zero(): - return self.add_bigoh(right.prec()) - if self.is_zero(): - return -right.add_bigoh(self.prec()) - - # 2. Align the unit parts. - if self.__n < right.__n: - m = self.__n - f1 = self.__u - f2 = right.__u << right.__n - m - else: - m = right.__n - f1 = self.__u << self.__n - m - f2 = right.__u - # 3. Subtract - return LaurentSeries(self._parent, f1 - f2, m) - - - def add_bigoh(self, prec): - """ - EXAMPLES: - sage: R. = LaurentSeriesRing(QQ) - sage: f = t^2 + t^3 + O(t^10); f - t^2 + t^3 + O(t^10) - sage: f.add_bigoh(5) - t^2 + t^3 + O(t^5) - """ - if prec == infinity or prec >= self.prec(): - return self - u = self.__u.add_bigoh(prec - self.__n) - return LaurentSeries(self._parent, u, self.__n) - - def degree(self): - """ - Return the degree of a polynomial equivalent to this power - series modulo big oh of the precision. - - EXAMPLES: - sage: x = Frac(QQ[['x']]).0 - sage: g = x^2 - x^4 + O(x^8) - sage: g.degree() - 4 - sage: g = -10/x^5 + x^2 - x^4 + O(x^8) - sage: g.degree() - 4 - """ - return self.__u.degree() + self.__n - - - def __neg__(self): - """ - sage: R. = LaurentSeriesRing(QQ) - sage: -(1+t^5) - -1 - t^5 - sage: -(1/(1+t+O(t^5))) - -1 + t - t^2 + t^3 - t^4 + O(t^5) - """ - return LaurentSeries(self._parent, -self.__u, self.__n) - - cdef RingElement _mul_c_impl(self, RingElement right_r): - """ - EXAMPLES: - sage: x = Frac(QQ[['x']]).0 - sage: f = 1/x^3 + x + x^2 + 3*x^4 + O(x^7) - sage: g = 1 - x + x^2 - x^4 + O(x^8) - sage: f*g - x^-3 - x^-2 + x^-1 + 4*x^4 + O(x^5) - """ - cdef LaurentSeries right = right_r - return LaurentSeries(self._parent, - self.__u * right.__u, - self.__n + right.__n) - - cdef ModuleElement _rmul_c_impl(self, RingElement c): - return LaurentSeries(self._parent, c * self.__u, self.__n) - - cdef ModuleElement _lmul_c_impl(self, RingElement c): - return LaurentSeries(self._parent, self.__u * c, self.__n) - - def __pow__(_self, r, dummy): - """ - EXAMPLES: - sage: x = Frac(QQ[['x']]).0 - sage: f = x + x^2 + 3*x^4 + O(x^7) - sage: g = 1/x^10 - x + x^2 - x^4 + O(x^8) - sage: f^7 - x^7 + 7*x^8 + 21*x^9 + 56*x^10 + 161*x^11 + 336*x^12 + O(x^13) - sage: g^7 - x^-70 - 7*x^-59 + 7*x^-58 - 7*x^-56 + O(x^-52) - """ - cdef LaurentSeries self = _self - right=int(r) - if right != r: - raise ValueError, "exponent must be an integer" - return LaurentSeries(self._parent, self.__u**right, self.__n*right) - - def shift(self, k): - r""" - Returns this laurent series multiplied by the power $t^n$. Does not - change this series. - - NOTE: - Despite the fact that higher order terms are printed to the - right in a power series, right shifting decreases the powers - of $t$, while left shifting increases them. This is to be - consistant with polynomials, integers, etc. - - EXAMPLES: - sage: R. = LaurentSeriesRing(QQ['y']) - sage: f = (t+t^-1)^4; f - t^-4 + 4*t^-2 + 6 + 4*t^2 + t^4 - sage: f.shift(10) - t^6 + 4*t^8 + 6*t^10 + 4*t^12 + t^14 - sage: f >> 10 - t^-14 + 4*t^-12 + 6*t^-10 + 4*t^-8 + t^-6 - sage: t << 4 - t^5 - sage: t + O(t^3) >> 4 - t^-3 + O(t^-1) - - AUTHOR: - -- Robert Bradshaw (2007-04-18) - """ - return LaurentSeries(self._parent, self.__u, self.__n + k) - - def __lshift__(LaurentSeries self, k): - return LaurentSeries(self._parent, self.__u, self.__n + k) - - def __rshift__(LaurentSeries self, k): - return LaurentSeries(self._parent, self.__u, self.__n - k) - - - cdef RingElement _div_c_impl(self, RingElement right_r): - """ - EXAMPLES: - sage: x = Frac(QQ[['x']]).0 - sage: f = x + x^2 + 3*x^4 + O(x^7) - sage: g = 1/x^7 - x + x^2 - x^4 + O(x^8) - sage: f/x - 1 + x + 3*x^3 + O(x^6) - sage: f/g - x^8 + x^9 + 3*x^11 + O(x^14) - """ - cdef LaurentSeries right = right_r - if right.__u.is_zero(): - raise ZeroDivisionError - try: - return LaurentSeries(self._parent, - self.__u / right.__u, - self.__n - right.__n) - except TypeError, msg: - # todo: this could also make something in the formal fraction field. - raise ArithmeticError, "division not defined" - - - def common_prec(self, f): - r""" - Returns minimum precision of $f$ and self. - - EXAMPLES: - sage: R. = LaurentSeriesRing(QQ) - - sage: f = t^(-1) + t + t^2 + O(t^3) - sage: g = t + t^3 + t^4 + O(t^4) - sage: f.common_prec(g) - 3 - sage: g.common_prec(f) - 3 - - sage: f = t + t^2 + O(t^3) - sage: g = t^(-3) + t^2 - sage: f.common_prec(g) - 3 - sage: g.common_prec(f) - 3 - - sage: f = t + t^2 - sage: f = t^2 - sage: f.common_prec(g) - +Infinity - - sage: f = t^(-3) + O(t^(-2)) - sage: g = t^(-5) + O(t^(-1)) - sage: f.common_prec(g) - -2 - """ - if self.prec() is infinity: - return f.prec() - elif f.prec() is infinity: - return self.prec() - return min(self.prec(), f.prec()) - - cdef int _cmp_c_impl(self, Element right_r) except -2: - r""" - Comparison of self and right. - - We say two approximate laurent series are equal, if they agree - for all coefficients up to the *minimum* of the precisions of - each. Comparison is done in dictionary order from lowest degree to - highest degree coefficients (this is different than - polynomials). - - See power_series_ring_element.__cmp__() for more information. - - EXAMPLES: - sage: R. = LaurentSeriesRing(QQ) - sage: f = x^(-1) + 1 + x + O(x^2) - sage: g = x^(-1) + 1 + O(x) - sage: f == g - True - - sage: f = x^(-1) + 1 + x + O(x^2) - sage: g = x^(-1) + 2 + O(x) - sage: f == g - False - sage: f < g - True - sage: f > g - False - - sage: f = x^(-2) + 1 + x + O(x^2) - sage: g = x^(-1) + 2 + O(x) - sage: f == g - False - sage: f < g - False - sage: f > g - True - """ - cdef LaurentSeries right = right_r - - prec = self.common_prec(right) - n = min(self.__n, right.__n) - - # zero pad coefficients on the left, to line them up for comparison - zero = self.base_ring()(0) - x = [zero] * (self.__n - n) + self.__u.list() - y = [zero] * (right.__n - n) + right.__u.list() - - # zero pad on right to make the lists the same length - # (this is necessary since the power series list() function just - # returns the coefficients of the underlying polynomial, which may - # have zeroes in the high coefficients) - if len(x) < len(y): - x.extend([zero] * (len(y) - len(x))) - elif len(y) < len(x): - y.extend([zero] * (len(x) - len(y))) - - if not (prec is infinity): - x = x[:(prec-n)] - y = y[:(prec-n)] - - return cmp(x,y) - - def valuation_zero_part(self): - """ - EXAMPLES: - sage: x = Frac(QQ[['x']]).0 - sage: f = x + x^2 + 3*x^4 + O(x^7) - sage: f/x - 1 + x + 3*x^3 + O(x^6) - sage: f.valuation_zero_part() - 1 + x + 3*x^3 + O(x^6) - sage: g = 1/x^7 - x + x^2 - x^4 + O(x^8) - sage: g.valuation_zero_part() - 1 - x^8 + x^9 - x^11 + O(x^15) - """ - return self.__u - - def valuation(self): - """ - EXAMPLES: - sage: x = Frac(QQ[['x']]).0 - sage: f = 1/x + x^2 + 3*x^4 + O(x^7) - sage: g = 1 - x + x^2 - x^4 + O(x^8) - sage: f.valuation() - -1 - sage: g.valuation() - 0 - """ - return self.__n - - def variable(self): - """ - EXAMPLES: - sage: x = Frac(QQ[['x']]).0 - sage: f = 1/x + x^2 + 3*x^4 + O(x^7) - sage: f.variable() - 'x' - """ - return self._parent.variable_name() - - def prec(self): - """ - This function returns the n so that the Laurent series is - of the form (stuff) + $O(t^n)$. It doesn't matter how many - negative powers appear in the expansion. In particular, - prec could be negative. - - EXAMPLES: - sage: x = Frac(QQ[['x']]).0 - sage: f = x^2 + 3*x^4 + O(x^7) - sage: f.prec() - 7 - sage: g = 1/x^10 - x + x^2 - x^4 + O(x^8) - sage: g.prec() - 8 - """ - return self.__u.prec() + self.__n - - def copy(self): - return LaurentSeries(self._parent, self.__u.copy(), self.__n) - - def derivative(self): - """ - The formal derivative of this Laurent series. - - EXAMPLES: - sage: x = Frac(QQ[['x']]).0 - sage: f = x^2 + 3*x^4 + O(x^7) - sage: f.derivative() - 2*x + 12*x^3 + O(x^6) - sage: g = 1/x^10 - x + x^2 - x^4 + O(x^8) - sage: g.derivative() - -10*x^-11 - 1 + 2*x - 4*x^3 + O(x^7) - """ - if self.is_zero(): - return LaurentSeries(self._parent, 0, self.__u.prec() - 1) - cdef long m, n = self.__n - a = self.__u.list() - v = [(n+m)*a[m] for m from 0 <= m < len(a)] - u = self._parent.power_series_ring()(v, self.__u.prec()) - return LaurentSeries(self._parent, u, n-1) - - def integral(self): - r""" - The formal integral of this Laurent series with 0 constant term. - - EXAMPLES: - The integral may or may not be defined if the base ring - is not a field. - sage: t = LaurentSeriesRing(ZZ, 't').0 - sage: f = 2*t^-3 + 3*t^2 + O(t^4) - sage: f.integral() - -t^-2 + t^3 + O(t^5) - - sage: f = t^3 - sage: f.integral() - Traceback (most recent call last): - ... - ArithmeticError: Coefficients of integral cannot be coerced into the base ring - - - The integral of 1/t is $\log(t)$, which is not given by a Laurent series: - - sage: t = Frac(QQ[['t']]).0 - sage: f = -1/t^3 - 31/t + O(t^3) - sage: f.integral() - Traceback (most recent call last): - ... - ArithmeticError: The integral of is not a Laurent series, since t^-1 has nonzero coefficient. - """ - cdef long i, n = self.__n - a = self.__u.list() - if self[-1] != 0: - raise ArithmeticError, \ - "The integral of is not a Laurent series, since t^-1 has nonzero coefficient." - - if n < 0: - v = [a[i]/(n+i+1) for i in range(-1-n)] + [0] - else: - v = [] - v += [a[i]/(n+i+1) for i in range(max(-n,0), len(a))] - try: - u = self._parent.power_series_ring()(v, self.__u.prec()) - except TypeError: - raise ArithmeticError, "Coefficients of integral cannot be coerced into the base ring" - return LaurentSeries(self._parent, u, n+1) - - - def power_series(self): - """ - EXAMPLES: - sage: R. = LaurentSeriesRing(ZZ) - sage: f = 1/(1-t+O(t^10)); f.parent() - Laurent Series Ring in t over Integer Ring - sage: g = f.power_series(); g - 1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + t^9 + O(t^10) - sage: parent(g) - Power Series Ring in t over Integer Ring - sage: f = 3/t^2 + t^2 + t^3 + O(t^10) - sage: f.power_series() - Traceback (most recent call last): - ... - ArithmeticError: self is a not a power series - """ - if self.__n < 0: - raise ArithmeticError, "self is a not a power series" - u = self.__u - t = u.parent().gen() - return t**(self.__n) * u - - def __call__(self, *x): - """ - Compute value of this Laurent series at x. - - EXAMPLES: - sage: R. = LaurentSeriesRing(ZZ) - sage: f = t^(-2) + t^2 + O(t^8) - sage: f(2) - 17/4 - sage: f(-1) - 2 - sage: f(1/3) - 82/9 - """ - if isinstance(x[0], tuple): - x = x[0] - return self.__u(x) * (x[0]**self.__n) - - -def make_element_from_parent(parent, *args): - return parent(*args) Index: sage/rings/morphism.py =================================================================== --- sage/rings/morphism.py (revision 4077) +++ sage/rings/morphism.py (revision 3354) @@ -282,20 +282,4 @@ sage: i(beta^2 + 1) 2.58740105196820 - -TESTS: - sage: H = Hom(ZZ, QQ) - sage: H == loads(dumps(H)) - True - - sage: K. = CyclotomicField(7) - sage: c = K.hom([1/zeta7]) - sage: c == loads(dumps(c)) - True - - sage: R. = PowerSeriesRing(GF(5)) - sage: f = R.hom([t^5]) - sage: f == loads(dumps(f)) - True - """ @@ -345,9 +329,7 @@ sage: S.lift() Set-theoretic ring morphism: - From: Quotient of Polynomial Ring in x, y over Rational Field by the ideal (y^2 + x^2, y) + From: Quotient of Polynomial Ring in x, y over Rational Field by the ideal (y, y^2 + x^2) To: Polynomial Ring in x, y over Rational Field Defn: Choice of lifting map - sage: S.lift() == 0 - False """ def __init__(self, R, S): @@ -360,9 +342,4 @@ raise TypeError, "No natural lift map" - def __cmp__(self, other): - if not isinstance(other, RingMap_lift): - return cmp(type(self), type(other)) - return 0 - def _repr_defn(self): return "Choice of lifting map" @@ -402,8 +379,5 @@ A *ring* homomorphism is considered to be 0 if and only if it sends the 1 element of the domain to the 0 element of the - codomain. Since rings in SAGE all have a 1 element, the - zero homomorphism is only to a ring of order 1, where 1==0, - e.g., the ring \code{Integers(1)}. - + codomain. EXAMPLES: @@ -479,48 +453,4 @@ return self.__im_gens - def __cmp__(self, other): - """ - EXAMPLES: - A single variate quotient over QQ. - sage: R. = QQ[] - sage: Q. = R.quotient(x^2 + x + 1) - sage: f1 = R.hom([a]) - sage: f2 = R.hom([a + a^2 + a + 1]) - sage: f1 == f2 - True - sage: f1 == R.hom([a^2]) - False - sage: f1(x^3 + x) - a + 1 - sage: f2(x^3 + x) - a + 1 - - TESTS: - sage: loads(dumps(f2)) == f2 - True - - EXAMPLES: - A multivariate quotient over a finite field. - sage: R. = GF(7)[] - sage: Q. = R.quotient([x^2 + x + 1, y^2 + y + 1]) - sage: f1 = R.hom([a, b]) - sage: f2 = R.hom([a + a^2 + a + 1, b + b^2 + b + 1]) - sage: f1 == f2 - True - sage: f1 == R.hom([b,a]) - False - sage: f1(x^3 + x + y^2) - 6*b + a - sage: f2(x^3 + x + y^2) - 6*b + a - - TEST: - sage: loads(dumps(f2)) == f2 - True - """ - if not isinstance(other, RingHomomorphism_im_gens): - return cmp(type(self), type(other)) - return cmp(self.__im_gens, other.__im_gens) - def _repr_defn(self): D = self.domain() @@ -547,6 +477,6 @@ b + a """ - def __init__(self, parent): - RingHomomorphism.__init__(self, parent) + def __init__(self, ring, quotient_ring): + RingHomomorphism.__init__(self, ring.Hom(quotient_ring)) def _call_(self, x): @@ -559,18 +489,4 @@ return self.codomain().defining_ideal() - def __cmp__(self, other): - """ - EXAMPLES: - sage: R. = PolynomialRing(QQ, 2) - sage: S. = R.quo(x^2 + y^2) - sage: phi = S.cover() - sage: phi == loads(dumps(phi)) - True - sage: phi == R.quo(x^2 + y^3).cover() - False - """ - if not isinstance(other, RingHomomorphism_cover): - return cmp(type(self), type(other)) - return 0 # since parents are the same, i.e., both cover maps with same domain and codomain class RingHomomorphism_from_quotient(RingHomomorphism): @@ -601,6 +517,5 @@ sage: phi(a+b+c) c + b + a - sage: loads(dumps(phi)) == phi - True + Validity of the homomorphism is determined, when possible, and a @@ -624,30 +539,6 @@ self.__phi = phi - def _phi(self): - """ - Underlying morphism used to define this quotient map (i.e., - morphism from the cover of the domain). - """ - return self.__phi - def morphism_from_cover(self): return self.__phi - - def __cmp__(self, other): - """ - EXAMPLES: - sage: R. = PolynomialRing(GF(19), 3) - sage: S. = R.quo(x^3 + y^3 + z^3) - sage: phi = S.hom([b, c, a]) - sage: psi = S.hom([c, b, a]) - sage: f = S.hom([b, c, a + a^3 + b^3 + c^3]) - sage: phi == psi - False - sage: phi == f - True - """ - if not isinstance(other, RingHomomorphism_from_quotient): - return cmp(type(self), type(other)) - return cmp(self.__phi, other.__phi) def _repr_defn(self): Index: sage/rings/mpfr.pxi =================================================================== --- sage/rings/mpfr.pxi (revision 4151) +++ sage/rings/mpfr.pxi (revision 3927) @@ -37,6 +37,4 @@ void mpfr_clear (mpfr_t x) - mp_prec_t mpfr_get_prec (mpfr_t x) - int mpfr_set (mpfr_t rop, mpfr_t op, mp_rnd_t rnd) int mpfr_set_si (mpfr_t rop, long int op, mp_rnd_t rnd) Index: age/rings/multi_polynomial.pxd =================================================================== --- sage/rings/multi_polynomial.pxd (revision 4091) +++ (revision ) @@ -1,4 +1,0 @@ -from sage.structure.element cimport CommutativeRingElement - -cdef class MPolynomial(CommutativeRingElement): - pass Index: age/rings/multi_polynomial.pyx =================================================================== --- sage/rings/multi_polynomial.pyx (revision 4091) +++ (revision ) @@ -1,5 +1,0 @@ - - -def is_MPolynomial(x): - return isinstance(x, MPolynomial) - Index: sage/rings/multi_polynomial_element.py =================================================================== --- sage/rings/multi_polynomial_element.py (revision 4171) +++ sage/rings/multi_polynomial_element.py (revision 3134) @@ -58,10 +58,8 @@ from integer_ring import ZZ -from multi_polynomial import MPolynomial - def is_MPolynomial(x): return isinstance(x, MPolynomial) -class MPolynomial_element(MPolynomial): +class MPolynomial(CommutativeRingElement): def __init__(self, parent, x): CommutativeRingElement.__init__(self, parent) @@ -70,66 +68,4 @@ def _repr_(self): return "%s"%self.__element - - #################### - # Some standard conversions - #################### - def __int__(self): - if self.degree() == 0: - return int(self.constant_coefficient()) - else: - raise TypeError - - def __long__(self): - if self.degree() == 0: - return long(self.constant_coefficient()) - else: - raise TypeError - - def __float__(self): - if self.degree() == 0: - return float(self.constant_coefficient()) - else: - raise TypeError - - def _mpfr_(self, R): - if self.degree() == 0: - return R(self.constant_coefficient()) - else: - raise TypeError - - def _complex_mpfr_field_(self, R): - if self.degree() == 0: - return R(self.constant_coefficient()) - else: - raise TypeError - - def _complex_double_(self, R): - if self.degree() == 0: - return R(self.constant_coefficient()) - else: - raise TypeError - - def _real_double_(self, R): - if self.degree() == 0: - return R(self.constant_coefficient()) - else: - raise TypeError - - def _rational_(self): - if self.degree() == 0: - from rational import Rational - return Rational(repr(self)) - else: - raise TypeError - - def _integer_(self): - if self.degree() == 0: - from integer import Integer - return Integer(repr(self)) - else: - raise TypeError - - - #################### def __call__(self, *x): @@ -215,5 +151,5 @@ try: return self.__element.compare(right.__element, - self.parent().term_order().compare_tuples) + self.parent()._MPolynomialRing_generic__term_order.compare_tuples) except AttributeError: return self.__element.compare(right.__element) @@ -289,5 +225,5 @@ def __pow__(self, n): if not isinstance(n, (int, long, integer.Integer)): - n = integer.Integer(n) + raise TypeError, "The exponent must be an integer." if n < 0: return 1/(self**(-n)) @@ -301,7 +237,4 @@ def element(self): return self.__element - - def change_ring(self, R): - return self.parent().change_ring(R)(self) @@ -336,5 +269,5 @@ -class MPolynomial_polydict(Polynomial_singular_repr, MPolynomial_macaulay2_repr, MPolynomial_element): +class MPolynomial_polydict(Polynomial_singular_repr, MPolynomial_macaulay2_repr, MPolynomial): def __init__(self, parent, x): """ @@ -348,5 +281,5 @@ if not isinstance(x, polydict.PolyDict): x = polydict.PolyDict(x, parent.base_ring()(0), remove_zero=True) - MPolynomial_element.__init__(self, parent, x) + MPolynomial.__init__(self, parent, x) def __neg__(self): @@ -372,5 +305,5 @@ INPUT: - x -- multivariate polynomial (a generator of the parent of self) + x -- multivariate polynmial (a generator of the parent of self) If x is not specified (or is None), return the total degree, which is the maximum degree of any monomial. @@ -554,6 +487,6 @@ sage: c = f.coefficient(x*y); c 2 - sage: c.parent() - Polynomial Ring in x, y over Rational Field + sage: c in QQ + False sage: c in MPolynomialRing(RationalField(), 2, names = ['x','y']) True @@ -741,5 +674,5 @@ one = self.parent().base_ring()(1) self.__monomials = [ MPolynomial_polydict(ring, polydict.PolyDict( {m:one}, force_int_exponents=False, force_etuples=False ) ) \ - for m in self._MPolynomial_element__element.dict().keys() ] + for m in self._MPolynomial__element.dict().keys() ] return self.__monomials @@ -832,5 +765,5 @@ R = polynomial_ring.PolynomialRing(self.base_ring(),'x') - monomial_coefficients = self._MPolynomial_element__element.dict() + monomial_coefficients = self._MPolynomial__element.dict() if( not self.is_constant() ): @@ -856,5 +789,5 @@ def _variable_indices_(self): - ETuples = self._MPolynomial_element__element.dict().keys() + ETuples = self._MPolynomial__element.dict().keys() idx = set() @@ -930,5 +863,5 @@ def __hash__(self): #requires base field elements are hashable! - return hash(tuple(self._MPolynomial_element__element.dict().items())) + return hash(tuple(self._MPolynomial__element.dict().items())) def lm(self): @@ -976,5 +909,5 @@ return self R = self.parent() - f = self._MPolynomial_element__element.lcmt( R.term_order().greater_tuple ) + f = self._MPolynomial__element.lcmt( R._MPolynomialRing_generic__term_order.greater_tuple ) one = R.base_ring()(1) self.__lm = MPolynomial_polydict(R,polydict.PolyDict({f:one},force_int_exponents=False, force_etuples=False)) @@ -992,6 +925,6 @@ return self R = self.parent() - f = self._MPolynomial_element__element.dict() - self.__lc = f[self._MPolynomial_element__element.lcmt( R.term_order().greater_tuple )] + f = self._MPolynomial__element.dict() + self.__lc = f[self._MPolynomial__element.lcmt( R._MPolynomialRing_generic__term_order.greater_tuple )] return self.__lc @@ -1006,6 +939,6 @@ return self R = self.parent() - f = self._MPolynomial_element__element.dict() - res = self._MPolynomial_element__element.lcmt( R.term_order().greater_tuple ) + f = self._MPolynomial__element.dict() + res = self._MPolynomial__element.lcmt( R._MPolynomialRing_generic__term_order.greater_tuple ) self.__lt = MPolynomial_polydict(R,polydict.PolyDict({res:f[res]},force_int_exponents=False, force_etuples=False)) return self.__lt @@ -1017,10 +950,10 @@ # we want comparison with zero to be fast if right == 0: - if self._MPolynomial_element__element.dict()=={}: + if self._MPolynomial__element.dict()=={}: return True else: return False return self._richcmp_(right,2) - return self._MPolynomial_element__element == right._MPolynomial_element__element + return self._MPolynomial__element == right._MPolynomial__element def __ne__(self,right): @@ -1030,5 +963,5 @@ # we want comparison with zero to be fast if right == 0: - if self._MPolynomial_element__element.dict()=={}: + if self._MPolynomial__element.dict()=={}: return False else: @@ -1036,13 +969,13 @@ # maybe add constant elements as well return self._richcmp_(right,3) - return self._MPolynomial_element__element != right._MPolynomial_element__element - - def __nonzero__(self): - """ - Returns True if self != 0 + return self._MPolynomial__element != right._MPolynomial__element + + def is_zero(self): + """ + Returns True if self == 0 \note{This is much faster than actually writing self == 0} """ - return self._MPolynomial_element__element.dict()!={} + return self._MPolynomial__element.dict()=={} ############################################################################ @@ -1116,5 +1049,5 @@ sage: M = f.lift(I) sage: M - [y^7, y^4 + x*y^5 + x^2*y^3 + x^3*y^4 + x^4*y^2 + x^5*y^3 + x^6*y + x^7*y^2 + x^8] + [y^4 + x*y^5 + x^2*y^3 + x^3*y^4 + x^4*y^2 + x^5*y^3 + x^6*y + x^7*y^2 + x^8, y^7] sage: sum( map( mul , zip( M, I.gens() ) ) ) == f True Index: sage/rings/multi_polynomial_ideal.py =================================================================== --- sage/rings/multi_polynomial_ideal.py (revision 4261) +++ sage/rings/multi_polynomial_ideal.py (revision 3645) @@ -61,11 +61,4 @@ sage: 0 in j # optional True - -TESTS: - sage: x,y,z = QQ['x,y,z'].gens() - sage: I = ideal(x^5 + y^4 + z^3 - 1, x^3 + y^3 + z^2 - 1) - sage: I == loads(dumps(I)) - True - """ @@ -177,6 +170,6 @@ sage: S = I._singular_() sage: S - x^3+y, - y + y, + x^3+y """ if singular is None: singular = singular_default @@ -337,8 +330,8 @@ return self.__dimension - def _groebner_basis_using_singular(self, algorithm="groebner"): + def _singular_groebner_basis(self, algorithm="groebner"): """ Return a Groebner basis of this ideal. If a groebner basis for - this ideal has been calculated before the cached Groebner + this ideal has been calculated before the cached groebner basis is returned regardless of the requested algorithm. @@ -364,6 +357,12 @@ sage: I = sage.rings.ideal.Cyclic(R,4); I Ideal (d + c + b + a, c*d + b*c + a*d + a*b, b*c*d + a*c*d + a*b*d + a*b*c, -1 + a*b*c*d) of Polynomial Ring in a, b, c, d over Rational Field - sage: I._groebner_basis_using_singular() + sage: I.groebner_basis() [1 - d^4 - c^2*d^2 + c^2*d^6, -1*d - c + c^2*d^3 + c^3*d^2, -1*d + d^5 - b + b*d^4, -1*d^2 - d^6 + c*d + c^2*d^4 - b*d^5 + b*c, d^2 + 2*b*d + b^2, d + c + b + a] + + \note{Some Groebner basis calculations crash on 64-bit + opterons with \SAGE's singular build, but work fine with an + official binary. If you download and install a Singular + binary from the Singular website it will not have this problem + (you can use it with \SAGE by putting it in local/bin/).} """ try: @@ -388,25 +387,4 @@ return self.__groebner_basis - def _singular_groebner_basis(self): - try: - S = self.__singular_groebner_basis - except AttributeError: - G = self.groebner_basis() - try: - return self.__singular_groebner_basis - except AttributeError: - pass - - try: - S._check_valid() - return S - except ValueError: - G = self.groebner_basis() - S = singular(G) - self.__singular_groebner_basis = S - return S - - - def genus(self): """ @@ -439,5 +417,5 @@ Ideal (y^2, x*y, x^2) of Polynomial Ring in x, y over Rational Field sage: IJ = I*J; IJ - Ideal (x^2*y^2, x^3, y^3, x*y) of Polynomial Ring in x, y over Rational Field + Ideal (y^3, x*y, x^2*y^2, x^3) of Polynomial Ring in x, y over Rational Field sage: IJ == K False @@ -463,5 +441,5 @@ sage: I = (p*q^2, y-z^2)*R sage: I.minimal_associated_primes () - [Ideal (2 + z^3, -1*z^2 + y) of Polynomial Ring in x, y, z over Rational Field, Ideal (1 + z^2, -1*z^2 + y) of Polynomial Ring in x, y, z over Rational Field] + [Ideal (-1*z^2 + y, 2 + z^3) of Polynomial Ring in x, y, z over Rational Field, Ideal (-1*z^2 + y, 1 + z^2) of Polynomial Ring in x, y, z over Rational Field] ALGORITHM: Uses Singular. @@ -531,5 +509,5 @@ sage: I = ideal([x^2,x*y^4,y^5]) sage: I.integral_closure() - [x^2, y^5, -1*x*y^3] + [x^2, y^5, x*y^3] ALGORITHM: Use Singular @@ -569,31 +547,20 @@ if self.base_ring() == sage.rings.integer_ring.ZZ: return self._reduce_using_macaulay2(f) - - try: - singular = self._singular_groebner_basis().parent() - except (AttributeError, RuntimeError): - singular = self._singular_groebner_basis().parent() + + try: + singular = self.__singular_groebner_basis.parent() + except AttributeError: + self.groebner_basis() + singular = self.__singular_groebner_basis.parent() f = self.ring()(f) g = singular(f) try: - self.ring()._singular_(singular).set_ring() - h = g.reduce(self._singular_groebner_basis()) + h = g.reduce(self.__singular_groebner_basis) except TypeError: # This is OK, since f is in the right ring -- type error # just means it's a rational return f - try: - return self.ring()(h) - except (TypeError, RuntimeError): - return self.ring()(h[1]) - # Why the above? - # For mysterious reasons, sometimes Singular returns a length - # one vector with the reduced polynomial in it. - # This occurs in the following example: - #sage: R. = PolynomialRing(QQ, 2) - #sage: S. = R.quotient(x^2 + y^2) - #sage: phi = S.hom([b,a]) - #sage: loads(dumps(phi)) + return self.ring()(h) @@ -842,5 +809,5 @@ sage: R. = PolynomialRing(IntegerRing(), 2, order='lex') sage: R.ideal([x, y]) - Ideal (x, y) of Polynomial Ring in x, y over Integer Ring + Ideal (y, x) of Polynomial Ring in x, y over Integer Ring sage: R. = GF(3)[] sage: R.ideal([x0^2, x1^3]) @@ -889,7 +856,7 @@ return self._macaulay2_groebner_basis() else: - return self._groebner_basis_using_singular("groebner") + return self._singular_groebner_basis("groebner") elif algorithm.startswith('singular:'): - return self._groebner_basis_using_singular(algorithm[9:]) + return self._singular_groebner_basis(algorithm[9:]) elif algorithm == 'macaulay2:gb': return self._macaulay2_groebner_basis() Index: age/rings/multi_polynomial_libsingular.pxd =================================================================== --- sage/rings/multi_polynomial_libsingular.pxd (revision 4095) +++ (revision ) @@ -1,20 +1,0 @@ -include "../libs/singular/singular-cdefs.pxi" - -cimport sage.rings.multi_polynomial -from sage.rings.multi_polynomial_ring_generic cimport MPolynomialRing_generic -from sage.structure.parent cimport Parent - -cdef class MPolynomial_libsingular(sage.rings.multi_polynomial.MPolynomial): - cdef object __singular - cdef poly *_poly - cdef _repr_short_c(self) - cdef _singular_init_c(self,singular, have_ring) - cdef int is_constant_c(self) - -cdef class MPolynomialRing_libsingular(MPolynomialRing_generic): - cdef ring *_ring - cdef object __singular - cdef object __macaulay2 - cdef MPolynomial_libsingular _zero - cdef int _cmp_c_impl(left, Parent right) except -2 - Index: age/rings/multi_polynomial_libsingular.pyx =================================================================== --- sage/rings/multi_polynomial_libsingular.pyx (revision 4369) +++ (revision ) @@ -1,2678 +1,0 @@ -"""nodoctest -Multivariate polynomials over QQ and GF(p) implemented using SINGULAR as backend. - -AUTHORS: - Martin Albrecht - - - -TODO: - * implement every method from multi_polynomial_ring and multi_polynomial_element - * check SINGULAR code base for 'interesting' methods to add - * add required methods for F4, Buchberger etc. - * implement GF(p^n) - * implement block orderings - * implement Real, Complex - * test under CYGWIN (link error) - -""" -# We do this as we get a link error for init_csage(). However, on -# obscure plattforms (Windows) we might need to link to csage anyway. - -cdef extern from "stdsage.h": - ctypedef void PyObject - object PY_NEW(object t) - int PY_TYPE_CHECK(object o, object t) - PyObject** FAST_SEQ_UNSAFE(object o) - void init_csage() - - void sage_free(void *p) - void* sage_realloc(void *p, size_t n) - void* sage_malloc(size_t) - -import os -import sage.rings.memory - -from sage.libs.singular.singular import Conversion -from sage.libs.singular.singular cimport Conversion - -cdef Conversion co -co = Conversion() - -from sage.rings.multi_polynomial_ring import singular_name_mapping, TermOrder -from sage.rings.multi_polynomial_ideal import MPolynomialIdeal -from sage.rings.polydict import ETuple - -from sage.rings.rational_field import RationalField -from sage.rings.finite_field import FiniteField_prime_modn - -from sage.rings.rational cimport Rational - -from sage.interfaces.singular import singular as singular_default, is_SingularElement, SingularElement -from sage.interfaces.macaulay2 import macaulay2 as macaulay2_default, is_Macaulay2Element -from sage.structure.factorization import Factorization - -from complex_field import is_ComplexField -from real_mpfr import is_RealField - - -from sage.rings.integer_ring import IntegerRing -from sage.structure.element cimport EuclideanDomainElement, \ - RingElement, \ - ModuleElement, \ - Element, \ - CommutativeRingElement - -from sage.rings.integer cimport Integer - -from sage.structure.parent cimport Parent -from sage.structure.parent_base cimport ParentWithBase -from sage.structure.parent_gens cimport ParentWithGens - -from sage.misc.sage_eval import sage_eval -from sage.misc.latex import latex - - -# shared library loading -cdef extern from "dlfcn.h": - void *dlopen(char *, long) - char *dlerror() - -cdef extern from "string.h": - char *strdup(char *s) - - -cdef init_singular(): - """ - This initializes the Singular library. Right now, this is a hack. - - SINGULAR has a concept of compiled extension modules similar to - SAGE. For this, the compiled modules need to see the symbols from - the main programm. However, SINGULAR is a shared library in this - context these symbols are not known globally. The work around so - far is to load the library again and to specifiy RTLD_GLOBAL. - """ - - cdef void *handle - - - for extension in ["so", "dylib", "dll"]: - lib = os.environ['SAGE_LOCAL']+"/lib/libsingular."+extension - if os.path.exists(lib): - handle = dlopen(lib, 256+1) - break - - if handle == NULL: - print dlerror() - raise ImportError, "cannot load libSINGULAR library" - - # Load Singular - siInit(lib) - - # Steal Memory Manager back or weird things may happen - sage.rings.memory.pmem_malloc() - -# call it -init_singular() - -cdef class MPolynomialRing_libsingular(MPolynomialRing_generic): - """ - A multivariate polynomial ring over QQ or GF(p) implemented using SINGULAR. - - """ - def __init__(self, base_ring, n, names, order='degrevlex'): - """ - - Construct a multivariate polynomial ring subject to the following conditions: - - INPUT: - base_ring -- base ring (must be either GF(p) (p prime) or QQ) - n -- number of variables (must be at least 1) - names -- names of ring variables, may be string of list/tuple - order -- term order (default: degrevlex) - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: P - Polynomial Ring in x, y, z over Rational Field - - sage: f = 27/113 * x^2 + y*z + 1/2; f - 27/113*x^2 + y*z + 1/2 - - sage: P.term_order() - Degree reverse lexicographic term order - - sage: P = MPolynomialRing_libsingular(GF(127),3,names='abc', order='lex') - sage: P - Polynomial Ring in a, b, c over Finite Field of size 127 - - sage: a,b,c = P.gens() - sage: f = 57 * a^2*b + 43 * c + 1; f - 57*a^2*b + 43*c + 1 - - sage: P.term_order() - Lexicographic term order - - """ - cdef char **_names - cdef char *_name - cdef int i - cdef int characteristic - - n = int(n) - if n<1: - raise ArithmeticError, "number of variables must be at least 1" - - self.__ngens = n - - MPolynomialRing_generic.__init__(self, base_ring, n, names, TermOrder(order)) - - self._has_singular = True - - _names = sage_malloc(sizeof(char*)*len(self._names)) - - for i from 0 <= i < n: - _name = self._names[i] - _names[i] = strdup(_name) - - if PY_TYPE_CHECK(base_ring, FiniteField_prime_modn): - characteristic = base_ring.characteristic() - - elif PY_TYPE_CHECK(base_ring, RationalField): - characteristic = 0 - - else: - raise NotImplementedError, "Only GF(p) and QQ are supported right now, sorry" - - try: - order = singular_name_mapping[order] - except KeyError: - pass - - self._ring = rDefault(characteristic, n, _names) - if(self._ring != currRing): rChangeCurrRing(self._ring) - - rUnComplete(self._ring) - - omFree(self._ring.wvhdl) - omFree(self._ring.order) - omFree(self._ring.block0) - omFree(self._ring.block1) - - self._ring.wvhdl = omAlloc0(3 * sizeof(int*)) - self._ring.order = omAlloc0(3* sizeof(int *)) - self._ring.block0 = omAlloc0(3 * sizeof(int *)) - self._ring.block1 = omAlloc0(3 * sizeof(int *)) - - if order == "dp": - self._ring.order[0] = ringorder_dp - elif order == "Dp": - self._ring.order[0] = ringorder_Dp - elif order == "lp": - self._ring.order[0] = ringorder_lp - elif order == "rp": - self._ring.order[0] = ringorder_rp - else: - self._ring.order[0] = ringorder_lp - - self._ring.order[1] = ringorder_C - - self._ring.block0[0] = 1 - self._ring.block1[0] = n - - rComplete(self._ring, 1) - self._ring.ShortOut = 0 - - self._zero = new_MP(self,NULL) - - for i from 0 <= i < n: - free(_names[i]) # strdup() --> free() - sage_free(_names) - - def __dealloc__(self): - """ - """ - rDelete(self._ring) - - cdef _coerce_c_impl(self, element): - """ - - Coerces elements to self. - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - - We can coerce elements of self to self - - sage: P._coerce_(x*y + 1/2) - x*y + 1/2 - - We can coerce elements for a ring with the same algebraic properties - - sage: R. = MPolynomialRing_libsingular(QQ,3) - sage: P == R - True - - sage: P is R - False - - sage: P._coerce_(x*y + 1) - x*y + 1 - - We can coerce base ring elements - - sage: P._coerce_(3/2) - 3/2 - - sage: P._coerce_(ZZ(1)) - 1 - - sage: P._coerce_(int(1)) - 1 - - """ - cdef poly *_p - cdef ring *_ring - cdef number *_n - cdef poly *rm - cdef int ok - cdef int i - cdef ideal *from_id, *to_id, *res_id - - _ring = self._ring - - if(_ring != currRing): rChangeCurrRing(_ring) - - if PY_TYPE_CHECK(element, MPolynomial_libsingular): - if element.parent() is self: - return element - elif element.parent() == self: - # is this safe? - _p = p_Copy((element)._poly, _ring) - else: - raise TypeError, "parents do not match" - - elif PY_TYPE_CHECK(element, CommutativeRingElement): - # Accepting ZZ - if element.parent() is IntegerRing(): - _p = p_ISet(int(element), _ring) - - elif element.parent() is self._base: - # Accepting GF(p) - if PY_TYPE_CHECK(self._base, FiniteField_prime_modn): - _p = p_ISet(int(element), _ring) - - # Accepting QQ - elif PY_TYPE_CHECK(self._base, RationalField): - _n = co.sa2si_QQ(element,_ring) - _p = p_NSet(_n, _ring) - else: - raise NotImplementedError - else: - raise TypeError, "base rings must be identical" - - # Accepting int - elif PY_TYPE_CHECK(element, int): - _p = p_ISet(int(element), _ring) - else: - raise TypeError, "Cannot coerce element" - - return new_MP(self,_p) - - def __call__(self, element): - """ - Construct a new element in self. - - INPUT: - element -- several types are supported, see below - - EXAMPLE: - Call supports all conversions _coerce_ supports, plus: - - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: P('x+y + 1/4') - x + y + 1/4 - - sage: P._singular_() - // characteristic : 0 - // number of vars : 3 - // block 1 : ordering dp - // : names x y z - // block 2 : ordering C - - sage: P(singular('x + 3/4')) - x + 3/4 - """ - cdef poly *_m, *_p, *_tmp - cdef ring *_r - - if PY_TYPE_CHECK(element, SingularElement): - element = str(element) - - if PY_TYPE_CHECK(element,str): - # let python do the the parsing - return sage_eval(element,self.gens_dict()) - - # this almost does what I want, besides variables with 0-9 in their names -## _r = self._ring -## if(_r != currRing): rChangeCurrRing(_r) -## # improve this -## element = element.replace('^','').replace('**','').replace(' ','').strip() -## monomials = element.split('+') -## _p = p_ISet(0, _r) -## # wrong for e.g. x0,..,x7 -## for m in monomials: -## _m = p_ISet(1,_r) -## for var in m.split('*'): -## p_Read(var, _tmp, _r) -## _m = p_Mult_q(_m,_tmp, _r) - -## _p = p_Add_q(_p,_m,_r) - -## return new_MP(self,_p) - - return self._coerce_c_impl(element) - - def _repr_(self): - """ - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ, 2) - sage: P - Polynomial Ring in x, y over Rational Field - - """ - varstr = ", ".join([ rRingVar(i,self._ring) for i in range(self.__ngens) ]) - return "Polynomial Ring in %s over %s"%(varstr,self._base) - - def ngens(self): - """ - Returns the number of variables in self. - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ, 2) - sage: P.ngens() - 2 - - sage: P = MPolynomialRing_libsingular(GF(127),1000,'x') - sage: P.ngens() - 1000 - - """ - return int(self.__ngens) - - def gens(self): - """ - Return the tuple of variables in self. - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: P.gens() - (x, y, z) - - sage: P = MPolynomialRing_libsingular(QQ,10,'x') - sage: P.gens() - (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9) - - sage: P. = MPolynomialRing_libsingular(QQ,2) # weird names - sage: P.gens() - (SAGE, SINGULAR) - - """ - return tuple([self.gen(i) for i in range(self.__ngens) ]) - - def gen(self, int n=0): - """ - Returns the n-th generator of self. - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: P.gen(),P.gen(1) - (x, y) - - sage: P = MPolynomialRing_libsingular(GF(127),1000,'x') - sage: P.gen(500) - x500 - - sage: P. = MPolynomialRing_libsingular(QQ,2) # weird names - sage: P.gen(1) - SINGULAR - - """ - cdef poly *_p - cdef ring *_ring - - if n < 0 or n >= self.__ngens: - raise ValueError, "Generator not defined." - - _ring = self._ring - _p = p_ISet(1,_ring) - - # oddly enough, Singular starts counting a 1!!! - p_SetExp(_p, n+1, 1, _ring) - p_Setm(_p, _ring); - - return new_MP(self,_p) - - def ideal(self, gens, coerce=True): - """ - Create an ideal in this polynomial ring. - - INPUT: - gens -- generators of the ideal - coerce -- shall the generators be coerced first (default:True) - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: sage.rings.ideal.Katsura(P) - Ideal (x + 2*y + 2*z - 1, x^2 + 2*y^2 + 2*z^2 - x, 2*x*y + 2*y*z - y) of Polynomial Ring in x, y, z over Rational Field - - sage: P.ideal([x + 2*y + 2*z-1, 2*x*y + 2*y*z-y, x^2 + 2*y^2 + 2*z^2-x]) - Ideal (x + 2*y + 2*z - 1, 2*x*y + 2*y*z - y, x^2 + 2*y^2 + 2*z^2 - x) of Polynomial Ring in x, y, z over Rational Field - - """ - if is_SingularElement(gens): - gens = list(gens) - coerce = True - if is_Macaulay2Element(gens): - gens = list(gens) - coerce = True - elif not isinstance(gens, (list, tuple)): - gens = [gens] - if coerce: - gens = [self(x) for x in gens] # this will even coerce from singular ideals correctly! - return MPolynomialIdeal(self, gens, coerce=False) - - def _macaulay2_(self, macaulay2=macaulay2_default): - """ - Create a M2 representation of self if Macaulay2 is installed. - - INPUT: - macaulay2 -- M2 interpreter (default: macaulay2_default) - """ - try: - R = self.__macaulay2 - if R is None or not (R.parent() is macaulay2): - raise ValueError - R._check_valid() - return R - except (AttributeError, ValueError): - if self.base_ring().is_prime_field(): - if self.characteristic() == 0: - base_str = "QQ" - else: - base_str = "ZZ/" + str(self.characteristic()) - elif is_IntegerRing(self.base_ring()): - base_str = "ZZ" - else: - raise TypeError, "no conversion of to a Macaulay2 ring defined" - self.__macaulay2 = macaulay2.ring(base_str, str(self.gens()), \ - self.term_order().macaulay2_str()) - return self.__macaulay2 - - def _singular_(self, singular=singular_default): - """ - Create a SINGULAR (as in the CAS) representation of self. The - result is cached. - - INPUT: - singular -- SINGULAR interpreter (default: singular_default) - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: P._singular_() - // characteristic : 0 - // number of vars : 3 - // block 1 : ordering dp - // : names x y z - // block 2 : ordering C - - sage: P._singular_() is P._singular_() - True - - sage: P._singular_().name() == P._singular_().name() - True - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,1) - sage: P._singular_() - // characteristic : 0 - // number of vars : 1 - // block 1 : ordering lp - // : names x - // block 2 : ordering C - - """ - try: - R = self.__singular - if R is None or not (R.parent() is singular): - raise ValueError - R._check_valid() - if self.base_ring().is_prime_field(): - return R - if self.base_ring().is_finite(): - R.set_ring() #sorry for that, but needed for minpoly - if singular.eval('minpoly') != self.__minpoly: - singular.eval("minpoly=%s"%(self.__minpoly)) - return R - except (AttributeError, ValueError): - return self._singular_init_(singular) - - def _singular_init_(self, singular=singular_default): - """ - Create a SINGULAR (as in the CAS) representation of self. The - result is NOT cached. - - INPUT: - singular -- SINGULAR interpreter (default: singular_default) - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: P._singular_init_() - // characteristic : 0 - // number of vars : 3 - // block 1 : ordering dp - // : names x y z - // block 2 : ordering C - sage: P._singular_init_() is P._singular_init_() - False - - sage: P._singular_init_().name() == P._singular_init_().name() - False - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,1) - sage: P._singular_init_() - // characteristic : 0 - // number of vars : 1 - // block 1 : ordering lp - // : names x - // block 2 : ordering C - - """ - if self.ngens()==1: - _vars = str(self.gen()) - if "*" in _vars: # 1.000...000*x - _vars = _vars.split("*")[1] - order = 'lp' - else: - _vars = str(self.gens()) - order = self.term_order().singular_str() - - if is_RealField(self.base_ring()): - # singular converts to bits from base_10 in mpr_complex.cc by: - # size_t bits = 1 + (size_t) ((float)digits * 3.5); - precision = self.base_ring().precision() - digits = sage.rings.arith.ceil((2*precision - 2)/7.0) - self.__singular = singular.ring("(real,%d,0)"%digits, _vars, order=order) - - elif is_ComplexField(self.base_ring()): - # singular converts to bits from base_10 in mpr_complex.cc by: - # size_t bits = 1 + (size_t) ((float)digits * 3.5); - precision = self.base_ring().precision() - digits = sage.rings.arith.ceil((2*precision - 2)/7.0) - self.__singular = singular.ring("(complex,%d,0,I)"%digits, _vars, order=order) - - elif self.base_ring().is_prime_field(): - self.__singular = singular.ring(self.characteristic(), _vars, order=order) - - elif self.base_ring().is_finite(): #must be extension field - gen = str(self.base_ring().gen()) - r = singular.ring( "(%s,%s)"%(self.characteristic(),gen), _vars, order=order) - self.__minpoly = "("+(str(self.base_ring().modulus()).replace("x",gen)).replace(" ","")+")" - singular.eval("minpoly=%s"%(self.__minpoly) ) - - self.__singular = r - else: - raise TypeError, "no conversion to a Singular ring defined" - - return self.__singular - - def __hash__(self): - """ - Return a hash for self, that is, a hash of the string representation of self - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: hash(P) - -6257278808099690586 # 64-bit - -1767675994 # 32-bit - """ - return hash(self.__repr__()) - - def __richcmp__(left, right, int op): - return (left)._richcmp(right, op) - - cdef int _cmp_c_impl(left, Parent right) except -2: - """ - Multivariate polynomial rings are said to be equal if: - * their base rings match - * their generator names match - * their term orderings match - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: R. = MPolynomialRing_libsingular(QQ,3) - sage: P == R - True - - sage: R. = MPolynomialRing_libsingular(GF(127),3) - sage: P == R - False - - sage: R. = MPolynomialRing_libsingular(QQ,2) - sage: P == R - False - - sage: R. = MPolynomialRing_libsingular(QQ,3,order='revlex') - sage: P == R - False - - - """ - if PY_TYPE_CHECK(right, MPolynomialRing_libsingular): - return cmp( (left.base_ring(), map(str, left.gens()), left.term_order()), - (right.base_ring(), map(str, right.gens()), right.term_order()) - ) - else: - return cmp(type(left),type(right)) - - def __reduce__(self): - """ - Serializes self. - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3, order='degrevlex') - sage: P == loads(dumps(P)) - True - - sage: P = MPolynomialRing_libsingular(GF(127),3,names='abc') - sage: P == loads(dumps(P)) - True - - """ - return sage.rings.multi_polynomial_libsingular.unpickle_MPolynomialRing_libsingular, ( self.base_ring(), - map(str, self.gens()), - self.term_order() ) - - ### The following methods are handy for implementing e.g. F4. They - ### do only superficial type/sanity checks and should be called - ### carefully. - - def monomial_m_div_n(self, MPolynomial_libsingular f, MPolynomial_libsingular g, coeff=False): - """ - Return f/g, where both f and g are treated as - monomials. Coefficients are ignored by default. - - INPUT: - f -- monomial - g -- monomial - coeff -- divide coefficents as well (default: False) - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: P.monomial_m_div_n(3/2*x*y,x) - y - - sage: P.monomial_m_div_n(3/2*x*y,x,coeff=True) - 3/2*y - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R.=MPolynomialRing_libsingular(QQ,3) - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: P.monomial_m_div_n(x*y,x) - y - - sage: P.monomial_m_div_n(x*y,R.gen()) - y - - sage: P.monomial_m_div_n(P(0),P(1)) - 0 - - sage: P.monomial_m_div_n(P(1),P(0)) - Traceback (most recent call last): - ... - ZeroDivisionError - - sage: P.monomial_m_div_n(P(3/2),P(2/3), coeff=True) - 9/4 - - sage: P.monomial_m_div_n(x,y) # Note the wrong result - x*y^1048575*z^1048575 # 64-bit - x*y^65535*z^65535 # 32-bit - - sage: P.monomial_m_div_n(x,P(1)) - x - - NOTE: Assumes that the head term of f is a multiple of the - head term of g and return the multiplicant m. If this rule is - violated, funny things may happen. - - - - """ - cdef poly *res - cdef ring *r = self._ring - - if not self is f._parent: - f = self._coerce_c(f) - if not self is g._parent: - g = self._coerce_c(g) - - if(r != currRing): rChangeCurrRing(r) - - if not f._poly: - return self._zero - if not g._poly: - raise ZeroDivisionError - - res = pDivide(f._poly,g._poly) - if coeff: - p_SetCoeff(res, n_Div( p_GetCoeff(f._poly, r) , p_GetCoeff(g._poly, r), r), r) - else: - p_SetCoeff(res, n_Init(1, r), r) - return new_MP(self, res) - - def monomial_lcm(self, MPolynomial_libsingular f, MPolynomial_libsingular g): - """ - LCM for monomials. Coefficients are ignored. - - INPUT: - f -- monomial - g -- monomial - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: P.monomial_lcm(3/2*x*y,x) - x*y - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R.=MPolynomialRing_libsingular(QQ,3) - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: P.monomial_lcm(x*y,R.gen()) - x*y - - sage: P.monomial_lcm(P(3/2),P(2/3)) - 1 - - sage: P.monomial_lcm(x,P(1)) - x - - """ - cdef poly *m = p_ISet(1,self._ring) - - if not self is f._parent: - f = self._coerce_c(f) - if not self is g._parent: - g = self._coerce_c(g) - - if f._poly == NULL: - if g._poly == NULL: - return self._zero - else: - raise ArithmeticError, "cannot compute lcm of zero and nonzero element" - if g._poly == NULL: - raise ArithmeticError, "cannot compute lcm of zero and nonzero element" - - if(self._ring != currRing): rChangeCurrRing(self._ring) - - pLcm(f._poly, g._poly, m) - return new_MP(self,m) - - def monomial_reduce(self, MPolynomial_libsingular f, G): - """ - Try to find a g in G where g.lm() divides f. If found (g,flt) - is returned, (0,0) otherwise, where flt is f/g.lm(). - - It is assumed that G is iterable and contains ONLY elements in - self. - - INPUT: - f -- monomial - G -- list/set of mpolynomials - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: f = x*y^2 - sage: G = [ 3/2*x^3 + y^2 + 1/2, 1/4*x*y + 2/7, 1/2 ] - sage: P.monomial_reduce(f,G) - (1/4*x*y + 2/7, y) - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: f = x*y^2 - sage: G = [ 3/2*x^3 + y^2 + 1/2, 1/4*x*y + 2/7, 1/2 ] - - sage: P.monomial_reduce(P(0),G) - (0, 0) - - sage: P.monomial_reduce(f,[P(0)]) - (0, 0) - - """ - cdef poly *m = f._poly - cdef ring *r = self._ring - cdef poly *flt - - if not m: - return f,f - - for g in G: - if PY_TYPE_CHECK(g, MPolynomial_libsingular) \ - and (g) \ - and p_LmDivisibleBy((g)._poly, m, r): - flt = pDivide(f._poly, (g)._poly) - #p_SetCoeff(flt, n_Div( p_GetCoeff(f._poly, r) , p_GetCoeff((g)._poly, r), r), r) - p_SetCoeff(flt, n_Init(1, r), r) - return g, new_MP(self,flt) - return self._zero,self._zero - - def monomial_pairwise_prime(self, MPolynomial_libsingular g, MPolynomial_libsingular h): - """ - Return True if h and g are pairwise prime. Both are treated as monomials. - - INPUT: - h -- monomial - g -- monomial - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: P.monomial_pairwise_prime(x^2*z^3, y^4) - True - - sage: P.monomial_pairwise_prime(1/2*x^3*y^2, 3/4*y^3) - False - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: Q. = MPolynomialRing_libsingular(QQ,3) - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: P.monomial_pairwise_prime(x^2*z^3, Q('y^4')) - True - - sage: P.monomial_pairwise_prime(1/2*x^3*y^2, Q(0)) - True - - sage: P.monomial_pairwise_prime(P(1/2),x) - False - - - """ - cdef int i - cdef ring *r - cdef poly *p, *q - - if h._parent is not g._parent: - g = (h._parent)._coerce_c(g) - - r = (h._parent)._ring - p = g._poly - q = h._poly - - if p == NULL: - if q == NULL: - return False #GCD(0,0) = 0 - else: - return True #GCD(x,0) = 1 - - elif q == NULL: - return True # GCD(0,x) = 1 - - elif p_IsConstant(p,r) or p_IsConstant(q,r): # assuming a base field - return False - - for i from 1 <= i <= r.N: - if p_GetExp(p,i,r) and p_GetExp(q,i,r): - return False - return True - - def monomial_is_divisible_by(self, MPolynomial_libsingular a, MPolynomial_libsingular b): - """ - Return 0 if b does not divide a and the factor - otherwise. Coefficients are ignored. - - INPUT: - a -- monomial - b -- monomial - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: P.monomial_is_divisible_by(x^3*y^2*z^4, x*y*z) - x^2*y*z^3 - sage: P.monomial_is_divisible_by(x*y*z, x^3*y^2*z^4) - 0 - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: P.monomial_is_divisible_by(P(0),P(1)) - 0 - sage: P.monomial_is_divisible_by(x,P(1)) - x - - """ - cdef poly *_a - cdef poly *_b - cdef ring *_r - cdef poly *ret - if a._parent is not b._parent: - b = (a._parent)._coerce_c(b) - - _a = a._poly - _b = b._poly - _r = (a._parent)._ring - if(_r != currRing): rChangeCurrRing(_r) - - if _b == NULL: - raise ZeroDivisionError - if _a == NULL: - return self._zero - - if not p_DivisibleBy(_b, _a, _r): - return self._zero - else: - ret = pDivide(_a,_b) - p_SetCoeff(ret, n_Init(1, _r), _r) - return new_MP(self,ret) - -def unpickle_MPolynomialRing_libsingular(base_ring, names, term_order): - """ - inverse function for MPolynomialRing_libsingular.__reduce__ - - """ - return MPolynomialRing_libsingular(base_ring, len(names), names, term_order) - -cdef MPolynomial_libsingular new_MP(MPolynomialRing_libsingular parent, poly *juice): - """ - Construct a new MPolynomial_libsingular element - """ - cdef MPolynomial_libsingular p - p = PY_NEW(MPolynomial_libsingular) - p._parent = parent - p._poly = juice - return p - -cdef class MPolynomial_libsingular(sage.rings.multi_polynomial.MPolynomial): - """ - A multivariate polynomial implemented using libSINGULAR. - """ - def __init__(self, MPolynomialRing_libsingular parent): - """ - Construct a zero element in parent. - """ - self._parent = parent - - def __dealloc__(self): - if self._poly: - p_Delete(&self._poly, (self._parent)._ring) - - def __call__(self, *x): - """ - Evaluate a polynomial at the given point x - - INPUT: - x -- a list of elements in self.parent() - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: f = 3/2*x^2*y + 1/7 * y^2 + 13/27 - sage: f(0,0,0) - 13/27 - - sage: f(1,1,1) - 803/378 - sage: 3/2 + 1/7 + 13/27 - 803/378 - - sage: f(45/2,19/3,1) - 7281167/1512 - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: P(0)(1,2,3) - 0 - sage: P(3/2)(1,2,3) - 3/2 - """ - cdef int l = len(x) - cdef MPolynomialRing_libsingular parent = (self._parent) - cdef ring *_ring = parent._ring - - cdef poly *_p - - if l != parent._ring.N: - raise TypeError, "number of arguments does not match number of variables in parent" - - return self.fix(dict(zip(parent.gens(), x))) - - ### the following is going to be faster at some size, but slower in general - ### TODO: find the crossover point -## cdef ideal *to_id = idInit(l,1) - -## try: -## for i from 0 <= i < l: -## e = x[i] # TODO: optimize this line -## to_id.m[i]= p_Copy( ((parent._coerce_c(x[i])))._poly, _ring) - -## except TypeError: -## id_Delete(&to_id, _ring) -## raise TypeError, "cannot coerce in arguments" - -## cdef ideal *from_id=idInit(1,1) -## from_id.m[0] = p_Copy(self._poly, _ring) - -## cdef ideal *res_id = fast_map(from_id, _ring, to_id, _ring) -## cdef poly *res = p_Copy(res_id.m[0], _ring) - -## id_Delete(&to_id, _ring) -## id_Delete(&from_id, _ring) -## id_Delete(&res_id, _ring) -## return new_MP(parent, res) - - def __richcmp__(left, right, int op): - return (left)._richcmp(right, op) - - cdef int _cmp_c_impl(left, Element right) except -2: - """ - Compare left and right and return -1, 0, and 1 for <,==, and > respectively. - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3, order='degrevlex') - sage: x == x - True - - sage: x > y - True - sage: y^2 > x - True - - sage: (2/3*x^2 + 1/2*y + 3) > (2/3*x^2 + 1/4*y + 10) - True - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3, order='degrevlex') - sage: x > P(0) - True - - sage: P(0) == P(0) - True - - sage: P(0) < P(1) - True - - sage: x > P(1) - True - - sage: 1/2*x < 3/4*x - True - - sage: (x+1) > x - True - - sage: f = 3/4*x^2*y + 1/2*x + 2/7 - sage: f > f - False - sage: f < f - False - sage: f == f - True - - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(GF(127),3, order='degrevlex') - sage: (66*x^2 + 23) > (66*x^2 + 2) - True - - - """ - cdef ring *r - cdef poly *p, *q - cdef number *h - cdef int ret = 0 - - r = (left._parent)._ring - if(r != currRing): rChangeCurrRing(r) - p = (left)._poly - q = (right)._poly - - # handle special cases first (slight slowdown, as special - # cases are - well - special - if p==NULL: - if q==NULL: - # compare 0, 0 - return 0 - elif p_IsConstant(q,r): - # compare 0, const - return 1-2*n_GreaterZero(p_GetCoeff(q,r), r) # -1: <, 1: > # - elif q==NULL: - if p_IsConstant(p,r): - # compare const, 0 - return -1+2*n_GreaterZero(p_GetCoeff(p,r), r) # -1: <, 1: > - #else - - while ret==0 and p!=NULL and q!=NULL: - ret = p_Cmp( p, q, r) - - if ret==0: - h = n_Sub(p_GetCoeff(p, r),p_GetCoeff(q, r), r) - # compare coeffs - ret = -1+n_IsZero(h, r)+2*n_GreaterZero(h, r) # -1: <, 0:==, 1: > - n_Delete(&h, r) - p = pNext(p) - q = pNext(q) - - if ret==0: - if p==NULL and q != NULL: - ret = -1 - elif p!=NULL and q==NULL: - ret = 1 - - return ret - - cdef ModuleElement _add_c_impl( left, ModuleElement right): - """ - Add left and right. - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: 3/2*x + 1/2*y + 1 - 3/2*x + 1/2*y + 1 - - """ - cdef MPolynomial_libsingular res - - cdef poly *_l, *_r, *_p - cdef ring *_ring - - _ring = (left._parent)._ring - - _l = p_Copy(left._poly, _ring) - _r = p_Copy((right)._poly, _ring) - - if(_ring != currRing): rChangeCurrRing(_ring) - _p= p_Add_q(_l, _r, _ring) - - return new_MP((left._parent),_p) - - cdef ModuleElement _sub_c_impl( left, ModuleElement right): - """ - Subtract left and right. - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: 3/2*x - 1/2*y - 1 - 3/2*x - 1/2*y - 1 - - """ - cdef MPolynomial_libsingular res - - cdef poly *_l, *_r, *_p - cdef ring *_ring - - _ring = (left._parent)._ring - - _l = p_Copy(left._poly, _ring) - _r = p_Copy((right)._poly, _ring) - - if(_ring != currRing): rChangeCurrRing(_ring) - _p= p_Add_q(_l, p_Neg(_r, _ring), _ring) - - return new_MP((left._parent),_p) - - - cdef ModuleElement _rmul_c_impl(self, RingElement left): - """ - Multiply self with a base ring element. - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: 3/2*x - 3/2*x - """ - - cdef number *_n - cdef ring *_ring - cdef poly *_p - - _ring = (self._parent)._ring - - if(_ring != currRing): rChangeCurrRing(_ring) - - if PY_TYPE_CHECK((self._parent)._base, FiniteField_prime_modn): - _n = n_Init(int(left),_ring) - - elif PY_TYPE_CHECK((self._parent)._base, RationalField): - _n = co.sa2si_QQ(left,_ring) - - _p = pp_Mult_nn(self._poly,_n,_ring) - n_Delete(&_n, _ring) - return new_MP((self._parent),_p) - - cdef RingElement _mul_c_impl(left, RingElement right): - """ - Multiply left and right. - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: (3/2*x - 1/2*y - 1) * (3/2*x + 1/2*y + 1) - 9/4*x^2 - 1/4*y^2 - y - 1 - """ - cdef poly *_l, *_r, *_p - cdef ring *_ring - - _ring = (left._parent)._ring - - if(_ring != currRing): rChangeCurrRing(_ring) - _p = pp_Mult_qq(left._poly, (right)._poly, _ring) - - return new_MP(left._parent,_p) - - cdef RingElement _div_c_impl(left, RingElement right): - """ - Divide left by right - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R.=MPolynomialRing_libsingular(QQ,2) - sage: f = (x + y)/3 - sage: f.parent() - Polynomial Ring in x, y over Rational Field - - Note that / is still a constructor for elements of the - fraction field in all cases as long as both arguments have the - same parent. - - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R.=MPolynomialRing_libsingular(QQ,2) - sage: f = x^3 + y - sage: g = x - sage: h = f/g; h - (x^3 + y)/x - sage: h.parent() - Fraction Field of Polynomial Ring in x, y over Rational Field - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R.=MPolynomialRing_libsingular(QQ,2) - sage: x/0 - Traceback (most recent call last): - ... - ZeroDivisionError - - """ - cdef poly *p - cdef ring *r - cdef number *n - if (right).is_constant_c(): - - p = (right)._poly - if p == NULL: - raise ZeroDivisionError - r = ((left)._parent)._ring - n = p_GetCoeff(p, r) - n = nInvers(n) - p = pp_Mult_nn(left._poly, n, r) - n_Delete(&n,r) - return new_MP(left._parent, p) - else: - return (left._parent).fraction_field()(left,right) - - def __pow__(MPolynomial_libsingular self,int exp,ignored): - """ - Return self^(exp). - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R.=MPolynomialRing_libsingular(QQ,2) - sage: f = x^3 + y - sage: f^2 - x^6 + 2*x^3*y + y^2 - - """ - cdef ring *_ring - _ring = (self._parent)._ring - - cdef poly *_p - - - if exp < 0: - raise ArithmeticError, "Cannot comput negative powers of polynomials" - - if(_ring != currRing): rChangeCurrRing(_ring) - _p = pPower( p_Copy(self._poly,(self._parent)._ring),exp) - return new_MP((self._parent),_p) - - - def __neg__(self): - """ - Return -self. - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R.=MPolynomialRing_libsingular(QQ,2) - sage: f = x^3 + y - sage: -f - -x^3 - y - """ - cdef ring *_ring - _ring = (self._parent)._ring - if(_ring != currRing): rChangeCurrRing(_ring) - - return new_MP((self._parent),\ - p_Neg(p_Copy(self._poly,_ring),_ring)) - - def _repr_(self): - s = self._repr_short_c() - s = s.replace("+"," + ").replace("-"," - ") - if s.startswith(" - "): - return "-" + s[3:] - else: - return s - - def _repr_short(self): - """ - This is a faster but less pretty way to print polynomials. If available - it uses the short SINGULAR notation. - """ - cdef ring *_ring = (self._parent)._ring - if _ring.CanShortOut: - _ring.ShortOut = 1 - s = self._repr_short_c() - _ring.ShortOut = 0 - else: - s = self._repr_short_c() - return s - - cdef _repr_short_c(self): - """ - Raw SINGULAR printing. - """ - s = p_String(self._poly, (self._parent)._ring, (self._parent)._ring) - return s - - def _latex_(self): - """ - Return a polynomial latex representation of self. - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: f = - 1*x^2*y - 25/27 * y^3 - z^2 - sage: latex(f) - - x^{2}y - \frac{25}{27} y^{3} - z^{2} - - """ - cdef ring *_ring = (self._parent)._ring - cdef int n = _ring.N - cdef int j, e - cdef poly *p = self._poly - poly = "" - gens = self.parent().gens() - base = self.parent().base() - - while p: - sign_switch = False - - # First determine the multinomial: - multi = "" - for j from 1 <= j <= n: - e = p_GetExp(p, j, _ring) - if e > 0: - multi += str(gens[j-1]) - if e > 1: - multi += "^{%d}"%e - - # Next determine coefficient of multinomial - c = co.si2sa( p_GetCoeff(p, _ring), _ring, base) - if len(multi) == 0: - multi = latex(c) - elif c != 1: - if c == -1: - if len(poly) > 0: - sign_switch = True - else: - multi = "- %s"%(multi) - else: - multi = "%s %s"%(latex(c),multi) - - # Now add on coefficiented multinomials - if len(poly) > 0: - if sign_switch: - poly = poly + " - " - else: - poly = poly + " + " - poly = poly + multi - - p = pNext(p) - - poly = poly.lstrip().rstrip() - poly = poly.replace("+ -","- ") - - if len(poly) == 0: - return "0" - return poly - - def _repr_with_changed_varnames(self, varnames): - """ - Return string representing self but change the variable names - to varnames. - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: f = - 1*x^2*y - 25/27 * y^3 - z^2 - sage: print f._repr_with_changed_varnames(['FOO', 'BAR', 'FOOBAR']) - -FOO^2*BAR - 25/27*BAR^3 - FOOBAR^2 - - """ - cdef ring *_ring = (self._parent)._ring - cdef char **_names - cdef char **_orig_names - cdef char *_name - cdef int i - - if len(varnames) != _ring.N: - raise TypeError, "len(varnames) doesn't equal self.parent().ngens()" - - - _names = sage_malloc(sizeof(char*)*_ring.N) - for i from 0 <= i < _ring.N: - _name = varnames[i] - _names[i] = strdup(_name) - - _orig_names = _ring.names - _ring.names = _names - s = str(self) - _ring.names = _orig_names - - for i from 0 <= i < _ring.N: - free(_names[i]) # strdup() --> free() - sage_free(_names) - - return s - - def degree(self, MPolynomial_libsingular x=None): - """ - Return the maximal degree of self in x, where x must be one of the - generators for the parent of self. - - INPUT: - x -- multivariate polynmial (a generator of the parent of self) - If x is not specified (or is None), return the total degree, - which is the maximum degree of any monomial. - - OUTPUT: - integer - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R. = MPolynomialRing_libsingular(QQ, 2) - sage: f = y^2 - x^9 - x - sage: f.degree(x) - 9 - sage: f.degree(y) - 2 - sage: (y^10*x - 7*x^2*y^5 + 5*x^3).degree(x) - 3 - sage: (y^10*x - 7*x^2*y^5 + 5*x^3).degree(y) - 10 - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ, 2) - sage: P(0).degree(x) - 0 - sage: P(1).degree(x) - 0 - - """ - cdef ring *r = (self._parent)._ring - cdef poly *p = self._poly - cdef int deg, _deg - - deg = 0 - - if not x: - return self.total_degree() - - # TODO: we can do this faster - if not x in self._parent.gens(): - raise TypeError, "x must be one of the generators of the parent." - for i from 1 <= i <= r.N: - if p_GetExp(x._poly, i, r): - break - while p: - _deg = p_GetExp(p,i,r) - if _deg > deg: - deg = _deg - p = pNext(p) - - return deg - - def newton_polytope(self): - """ - Return the Newton polytope of this polynomial. - - You should have the optional polymake package installed. - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R. = MPolynomialRing_libsingular(QQ,2) - sage: f = 1 + x*y + x^3 + y^3 - sage: P = f.newton_polytope() - sage: P - Convex hull of points [[1, 0, 0], [1, 0, 3], [1, 1, 1], [1, 3, 0]] - sage: P.facets() - [(0, 1, 0), (3, -1, -1), (0, 0, 1)] - sage: P.is_simple() - True - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R. = MPolynomialRing_libsingular(QQ,2) - sage: R(0).newton_polytope() - Convex hull of points [] - sage: R(1).newton_polytope() - Convex hull of points [[1, 0, 0]] - - """ - from sage.geometry.all import polymake - e = self.exponents() - a = [[1] + list(v) for v in e] - P = polymake.convex_hull(a) - return P - - def total_degree(self): - """ - Return the total degree of self, which is the maximum degree - of all monomials in self. - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R. = MPolynomialRing_libsingular(QQ, 3) - sage: f=2*x*y^3*z^2 - sage: f.total_degree() - 6 - sage: f=4*x^2*y^2*z^3 - sage: f.total_degree() - 7 - sage: f=99*x^6*y^3*z^9 - sage: f.total_degree() - 18 - sage: f=x*y^3*z^6+3*x^2 - sage: f.total_degree() - 10 - sage: f=z^3+8*x^4*y^5*z - sage: f.total_degree() - 10 - sage: f=z^9+10*x^4+y^8*x^2 - sage: f.total_degree() - 10 - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R. = MPolynomialRing_libsingular(QQ, 3) - sage: R(0).total_degree() - 0 - sage: R(1).total_degree() - 0 - """ - cdef poly *p = self._poly - cdef ring *r = (self._parent)._ring - cdef int l - if self._poly == NULL: - return 0 - if(r != currRing): rChangeCurrRing(r) - return pLDeg(p,&l,r) - - def monomial_coefficient(self, MPolynomial_libsingular mon): - """ - Return the coefficient of the monomial mon in self, where mon - must have the same parent as self. - - INPUT: - mon -- a monomial - - OUTPUT: - ring element - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ, 2) - - The coefficient returned is an element of the base ring of self; in - this case, QQ. - - sage: f = 2 * x * y - sage: c = f.monomial_coefficient(x*y); c - 2 - sage: c in QQ - True - - sage: f = y^2 - x^9 - 7*x + 5*x*y - sage: f.monomial_coefficient(y^2) - 1 - sage: f.monomial_coefficient(x*y) - 5 - sage: f.monomial_coefficient(x^9) - -1 - sage: f.monomial_coefficient(x^10) - 0 - """ - cdef poly *p = self._poly - cdef poly *m = mon._poly - cdef ring *r = (self._parent)._ring - - if not mon._parent is self._parent: - raise TypeError, "mon must have same parent as self" - - while(p): - if p_ExpVectorEqual(p, m, r) == 1: - return co.si2sa(p_GetCoeff(p, r), r, (self._parent)._base) - p = pNext(p) - - return (self._parent)._base(0) - - def dict(self): - """ - Return a dictionary representing self. This dictionary is in - the same format as the generic MPolynomial: The dictionary - consists of ETuple:coefficient pairs. - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R. = MPolynomialRing_libsingular(QQ, 3) - sage: f=2*x*y^3*z^2 + 1/7*x^2 + 2/3 - sage: f.dict() - {(2, 0, 0): 1/7, (0, 0, 0): 2/3, (1, 3, 2): 2} - """ - cdef poly *p - cdef ring *r - cdef int n - cdef int v - r = (self._parent)._ring - base = (self._parent)._base - p = self._poly - pd = dict() - while p: - d = dict() - for v from 1 <= v <= r.N: - n = p_GetExp(p,v,r) - if n!=0: - d[v-1] = n - - pd[ETuple(d,r.N)] = co.si2sa(p_GetCoeff(p, r), r, base) - - p = pNext(p) - return pd - - def __getitem__(self,x): - """ - same as self.monomial_coefficent but for exponent vectors. - - INPUT: - x -- a tuple or, in case of a single-variable MPolynomial - ring x can also be an integer. - - EXAMPLES: - sage: R. = PolynomialRing(QQ, 2) - sage: f = -10*x^3*y + 17*x*y - sage: f[3,1] - -10 - sage: f[1,1] - 17 - sage: f[0,1] - 0 - - sage: R. = PolynomialRing(GF(7),1); R - Polynomial Ring in x over Finite Field of size 7 - sage: f = 5*x^2 + 3; f - 3 + 5*x^2 - sage: f[2] - 5 - """ - - cdef poly *m - cdef poly *p = self._poly - cdef ring *r = (self._parent)._ring - cdef int i - - if PY_TYPE_CHECK(x, MPolynomial_libsingular): - return self.monomial_coefficient(x) - if not PY_TYPE_CHECK(x, tuple): - try: - x = tuple(x) - except TypeError: - x = (x,) - - if len(x) != (self._parent).__ngens: - raise TypeError, "x must have length self.ngens()" - - m = p_ISet(1,r) - i = 1 - for e in x: - p_SetExp(m, i, int(e), r) - i += 1 - p_Setm(m, r) - - while(p): - if p_ExpVectorEqual(p, m, r) == 1: - p_Delete(&m,r) - return co.si2sa(p_GetCoeff(p, r), r, (self._parent)._base) - p = pNext(p) - - p_Delete(&m,r) - return (self._parent)._base(0) - - def coefficient(self, mon): - raise NotImplementedError - - def exponents(self): - """ - Return the exponents of the monomials appearing in self. - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R. = MPolynomialRing_libsingular(QQ, 3) - sage: f = a^3 + b + 2*b^2 - sage: f.exponents() - [(3, 0, 0), (0, 2, 0), (0, 1, 0)] - """ - cdef poly *p - cdef ring *r - cdef int v - r = (self._parent)._ring - - p = self._poly - - pl = list() - while p: - ml = list() - for v from 1 <= v <= r.N: - ml.append(p_GetExp(p,v,r)) - pl.append(tuple(ml)) - - p = pNext(p) - return pl - - def is_unit(self): - """ - Return True if self is a unit. - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R. = MPolynomialRing_libsingular(QQ, 2) - sage: (x+y).is_unit() - False - sage: R(0).is_unit() - False - sage: R(-1).is_unit() - True - sage: R(-1 + x).is_unit() - False - sage: R(2).is_unit() - True - """ - return bool(p_IsUnit(self._poly, (self._parent)._ring)) - - def inverse_of_unit(self): - """ - Return the inverse of self if self is a unit. - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R. = MPolynomialRing_libsingular(QQ, 2) - """ - cdef ring *_ring = (self._parent)._ring - if(_ring != currRing): rChangeCurrRing(_ring) - - if not p_IsUnit(self._poly, _ring): - raise ArithmeticError, "is not a unit" - else: - return new_MP(self._parent,pInvers(0,self._poly,NULL)) - - def is_homogeneous(self): - """ - Return True if self is a homogeneous polynomial. - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(RationalField(), 2) - sage: (x+y).is_homogeneous() - True - sage: (x.parent()(0)).is_homogeneous() - True - sage: (x+y^2).is_homogeneous() - False - sage: (x^2 + y^2).is_homogeneous() - True - sage: (x^2 + y^2*x).is_homogeneous() - False - sage: (x^2*y + y^2*x).is_homogeneous() - True - - """ - cdef ring *_ring = (self._parent)._ring - if(_ring != currRing): rChangeCurrRing(_ring) - return bool(pIsHomogeneous(self._poly)) - - def homogenize(self, var='h'): - """ - Return self is self is homogeneous. Otherwise return a - homogeneous polynomial. If a string is given, return a - polynomial in one more variable such that setting that - variable equal to 1 yields self. This variable is added to the - end of the variables. If either a variable in self.parent() or - an index is given, this variable is used to homogenize the - polynomial. - - INPUT: - var -- either a string (default: 'h'); a variable name for the new variable - to be added in when homogenizing or a variable/index to specify the existing - variable to be used. - - OUTPUT: - a multivariate polynomial - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,2) - sage: f = x^2 + y + 1 + 5*x*y^10 - sage: g = f.homogenize('z'); g - 5*x*y^10 + x^2*z^9 + y*z^10 + z^11 - sage: g.parent() - Polynomial Ring in x, y, z over Rational Field - sage: f.homogenize(x) - 2*x^11 + x^10*y + 5*x*y^10 - - """ - cdef MPolynomialRing_libsingular parent = self._parent - cdef MPolynomial_libsingular f - - if self.is_homogeneous(): - return self - - if PY_TYPE_CHECK(var, MPolynomial_libsingular): - if (var)._parent is self._parent: - var = var._variable_indices_() - if len(var) == 1: - var = var[0] - else: - raise TypeError, "parameter var must be single variable" - - if PY_TYPE_CHECK(var,str): - names = [str(e) for e in parent.gens()] + [var] - P = MPolynomialRing_libsingular(parent.base(),parent.ngens()+1, names, order=parent.term_order()) - f = P(str(self)) - return new_MP(P, pHomogen(f._poly,len(names))) - elif PY_TYPE_CHECK(var,int) or PY_TYPE_CHECK(var,Integer): - if var < parent._ring.N: - return new_MP(parent, pHomogen(p_Copy(self._poly, parent._ring), var+1)) - else: - raise TypeError, "var must be < self.parent().ngens()" - else: - raise TypeError, "parameter var not understood" - - def is_monomial(self): - return not self._poly.next - - def fix(self, fixed): - """ - Fixes some given variables in a given multivariate polynomial and - returns the changed multivariate polynomials. The polynomial - itself is not affected. The variable,value pairs for fixing are - to be provided as dictionary of the form {variable:value}. - - This is a special case of evaluating the polynomial with some of - the variables constants and the others the original variables, but - should be much faster if only few variables are to be fixed. - - INPUT: - fixed -- dict with variable:value pairs - - OUTPUT: - new MPolynomial - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: x, y = MPolynomialRing_libsingular(QQ,2,'xy').gens() - sage: f = x^2 + y + x^2*y^2 + 5 - sage: f(5,y) - 25*y^2 + y + 30 - sage: f.fix({x:5}) - 25*y^2 + y + 30 - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: f = y - sage: f.fix({y:x}).fix({x:z}) - z - - """ - cdef int mi, i - - cdef MPolynomialRing_libsingular parent = self._parent - cdef ring *_ring = parent._ring - - if(_ring != currRing): rChangeCurrRing(_ring) - - cdef poly *_p = p_Copy(self._poly, _ring) - - for m,v in fixed.iteritems(): - if PY_TYPE_CHECK(m,int) or PY_TYPE_CHECK(m,Integer): - mi = m+1 - elif PY_TYPE_CHECK(m,MPolynomial_libsingular) and m.parent() is parent: - for i from 0 < i <= _ring.N: - if p_GetExp((m)._poly, i, _ring) != 0: - mi = i - break - if i > _ring.N: - raise TypeError, "key does not match" - else: - raise TypeError, "keys do not match self's parent" - _p = pSubst(_p, mi, (parent._coerce_c(v))._poly) - - return new_MP(parent,_p) - - def monomials(self): - """ - Return the list of monomials in self. The returned list is - ordered by the term ordering of self.parent(). - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: f = x + 3/2*y*z^2 + 2/3 - sage: f.monomials() - [y*z^2, x, 1] - sage: f = P(3/2) - sage: f.monomials() - [1] - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: f = x - sage: f.monomials() - [x] - sage: f = P(0) - sage: f.monomials() - [0] - - - """ - l = list() - cdef MPolynomialRing_libsingular parent = self._parent - cdef ring *_ring = parent._ring - cdef poly *p = p_Copy(self._poly, _ring) - cdef poly *t - - if p == NULL: - return [parent._zero] - - while p: - t = pNext(p) - p.next = NULL - p_SetCoeff(p, n_Init(int(1),_ring), _ring) - p_Setm(p, _ring) - l.append( new_MP(parent,p) ) - p = t - - return l - - def constant_coefficent(self): - raise NotImplementedError - - def is_univariate(self): - """ - Return True if self is a univariate polynomial, that is if - self contains only one variable. - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(GF(2),3) - sage: f = x^2 + 1 - sage: f.is_univariate() - True - sage: f = y*x^2 + 1 - sage: f.is_univariate() - False - sage: f = P(0) - sage: f.is_univariate() - True - """ - return bool(len(self._variable_indices_(sort=False))<2) - - def univariate_polynomial(self, R=None): - raise NotImplementedError - - def _variable_indices_(self, sort=True): - """ - Return the indices of all variables occuring in self. - This index is the index as SAGE uses them (starting at zero), not - as SINGULAR uses them (starting at one). - - INPUT: - sort -- specifies whether the indices shall be sorted - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(GF(2),3) - sage: f = x*z^2 + z + 1 - sage: f._variable_indices_() - [0, 2] - - """ - cdef poly *p - cdef ring *r = (self._parent)._ring - cdef int i - s = set() - p = self._poly - while p: - for i from 1 <= i <= r.N: - if p_GetExp(p,i,r): - s.add(i-1) - p = pNext(p) - if sort: - return sorted(s) - else: - return list(s) - - def variables(self, sort=True): - """ - Return a list of all variables occuring in self. - - INPUT: - sort -- specifies whether the indices shall be sorted - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(GF(2),3) - sage: f = x*z^2 + z + 1 - sage: f.variables() - [z, x] - sage: f.variables(sort=False) - [x, z] - - """ - cdef poly *p, *v - cdef ring *r = (self._parent)._ring - cdef int i - l = list() - si = set() - p = self._poly - while p: - for i from 1 <= i <= r.N: - if i not in si and p_GetExp(p,i,r): - v = p_ISet(1,r) - p_SetExp(v, i, 1, r) - p_Setm(v, r) - l.append(new_MP(self._parent, v)) - si.add(i) - p = pNext(p) - if sort: - return sorted(l) - else: - return l - - def variable(self, i=0): - """ - Return the i-th variable occuring in self. The index i is the - index in self.variables(). - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(GF(2),3) - sage: f = x*z^2 + z + 1 - sage: f.variables() - [z, x] - sage: f.variable(1) - x - """ - return self.variables()[i] - - def nvariables(self): - """ - """ - return self._variable_indices_(sort=False) - - def is_constant(self): - """ - """ - return bool(p_IsConstant(self._poly, (self._parent)._ring)) - - cdef int is_constant_c(self): - return p_IsConstant(self._poly, (self._parent)._ring) - - def __hash__(self): - """ - """ - s = p_String(self._poly, (self._parent)._ring, (self._parent)._ring) - return hash(s) - - def lm(MPolynomial_libsingular self): - """ - Returns the lead monomial of self with respect to the term - order of self.parent(). In SAGE a monomial is a product of - variables in some power without a coefficient. - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - - sage: R.=MPolynomialRing_libsingular(GF(7),3,order='lex') - sage: f = x^1*y^2 + y^3*z^4 - sage: f.lm() - x*y^2 - sage: f = x^3*y^2*z^4 + x^3*y^2*z^1 - sage: f.lm() - x^3*y^2*z^4 - - sage: R.=MPolynomialRing_libsingular(QQ,3,order='deglex') - sage: f = x^1*y^2*z^3 + x^3*y^2*z^0 - sage: f.lm() - x*y^2*z^3 - sage: f = x^1*y^2*z^4 + x^1*y^1*z^5 - sage: f.lm() - x*y^2*z^4 - - sage: R.=PolynomialRing(GF(127),3,order='degrevlex') - sage: f = x^1*y^5*z^2 + x^4*y^1*z^3 - sage: f.lm() - x*y^5*z^2 - sage: f = x^4*y^7*z^1 + x^4*y^2*z^3 - sage: f.lm() - x^4*y^7*z - - """ - cdef poly *_p - cdef ring *_ring - _ring = (self._parent)._ring - _p = p_Head(self._poly, _ring) - p_SetCoeff(_p, n_Init(int(1),_ring), _ring) - p_Setm(_p,_ring) - return new_MP((self._parent), _p) - - - def lc(MPolynomial_libsingular self): - """ - Leading coefficient of self. See self.lm() for details. - """ - - cdef poly *_p - cdef ring *_ring - cdef number *_n - _ring = (self._parent)._ring - - if(_ring != currRing): rChangeCurrRing(_ring) - - _p = p_Head(self._poly, _ring) - _n = p_GetCoeff(_p, _ring) - - return co.si2sa(_n, _ring, (self._parent)._base) - - def lt(MPolynomial_libsingular self): - """ - Leading term of self. In SAGE a term is a product of variables - in some power AND a coefficient. - - See self.lm() for details - """ - return new_MP((self._parent), - p_Head(self._poly,(self._parent)._ring)) - - def is_zero(self): - if self._poly is NULL: - return True - else: - return False - - def __nonzero__(self): - if self._poly: - return True - else: - return False - - def __floordiv__(self,right): - raise NotImplementedError - - def factor(self, param=0): - """ - - Return the factorization of self. - - INPUT: - param -- 0: returns factors and multiplicities, first factor is a constant. - 1: returns non-constant factors (no multiplicities). - 2: returns non-constant factors and multiplicities. - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R. = MPolynomialRing_libsingular(GF(32003),3) - sage: f = 9*(x-1)^2*(y+z) - sage: f.factor(0) - 9 * (y + z) * (x - 1)^2 - sage: f.factor(1) - (y + z) * (x - 1) - sage: f.factor(2) - (y + z) * (x - 1)^2 - - """ - cdef ring *_ring - cdef intvec *iv - cdef int *ivv - cdef ideal *I - cdef MPolynomialRing_libsingular parent - cdef int i - - parent = self._parent - _ring = parent._ring - - if(_ring != currRing): rChangeCurrRing(_ring) - - iv = NULL - I = singclap_factorize ( self._poly, &iv , int(param)) #delete iv at some point - - if param!=1: - ivv = iv.ivGetVec() - v = [(new_MP(parent, p_Copy(I.m[i],_ring)) , ivv[i]) for i in range(I.ncols)] - else: - v = [(new_MP(parent, p_Copy(I.m[i],_ring)) , 1) for i in range(I.ncols)] - - # TODO: peel of 1 - - F = Factorization(v) - F.sort() - - omFree(iv) - id_Delete(&I,_ring) - - return F - - def lift(self, I): - #m = idLift(Ideal(self), I, NULL, FALSE, TRUE ); - #matrix_to_list(m) - raise NotImplementedError - - def reduce(self,I): - """ - Return the normal form of self w.r.t. I, i.e. return the - remainder of self with respect to the polynomials in I. If the - polynomial set/list I is not a Groebner basis the result is - not canonical. - - INPUT: - I -- a list/set of polynomials in self.parent(). If I is an ideal, - the generators are used. - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: f1 = -2 * x^2 + x^3 - sage: f2 = -2 * y + x* y - sage: f3 = -x^2 + y^2 - sage: F = Ideal([f1,f2,f3]) - sage: g = x*y - 3*x*y^2 - sage: g.reduce(F) - -6*y^2 + 2*y - sage: g.reduce(F.gens()) - -6*y^2 + 2*y - - """ - cdef ideal *_I - cdef MPolynomialRing_libsingular parent = self._parent - cdef int i = 0 - cdef ring *r = (self._parent)._ring - cdef poly *res - - if(r != currRing): rChangeCurrRing(r) - - if PY_TYPE_CHECK(I, MPolynomialIdeal): - I = I.gens() - - _I = idInit(len(I),1) - for f in I: - if not (PY_TYPE_CHECK(f,MPolynomial_libsingular) \ - and (f)._parent is parent): - try: - f = parent._coerce_c(f) - except TypeError, msg: - id_Delete(&_I,r) - raise TypeError, msg - - _I.m[i] = p_Copy((f)._poly, r) - i+=1 - - #the second parameter would be qring! - res = kNF(_I, NULL, p_Copy(self._poly, r)) - return new_MP(parent,res) - - def gcd(self, right): - """ - Return the greates common divisor of self and right. - - INPUT: - right -- polynomial - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: f = (x*y*z)^6 - 1 - sage: g = (x*y*z)^4 - 1 - sage: f.gcd(g) - x^2*y^2*z^2 - 1 - sage: GCD([x^3 - 3*x + 2, x^4 - 1, x^6 -1]) - x - 1 - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: Q. = MPolynomialRing_libsingular(QQ,3) - sage: P. = MPolynomialRing_libsingular(QQ,3) - sage: P(0).gcd(Q(0)) - 0 - sage: x.gcd(1) - 1 - - """ - cdef MPolynomial_libsingular _right - cdef poly *_res - cdef ring *_ring - - _ring = (self._parent)._ring - - if(_ring != currRing): rChangeCurrRing(_ring) - - if not PY_TYPE_CHECK(right, MPolynomial_libsingular): - _right = (self._parent)._coerce_c(right) - else: - _right = (right) - - _res = singclap_gcd(p_Copy(self._poly, _ring), p_Copy(_right._poly, _ring)) - - return new_MP((self._parent), _res) - -## def lcm(self, MPolynomial_libsingular g): -## cdef ring *_ring = (self._parent)._ring -## cdef poly *ret -## if(_ring != currRing): rChangeCurrRing(_ring) - -## if self._parent is not g._parent: -## g = (self._parent)._coerce_c(g) - -## # This guy calculates on napoly not poly -## ret = singclap_alglcm(self._poly, (g)._poly) -## return new_MP(self._parent, ret) - - def is_square_free(self): - """ - """ - cdef ring *_ring = (self._parent)._ring - if(_ring != currRing): rChangeCurrRing(_ring) - return bool(singclap_isSqrFree(self._poly)) - - def quo_rem(self, MPolynomial_libsingular right): - """ - """ - if self._parent is not right._parent: - right = self._parent._coerce_c(right) - raise NotImplementedError - - def _magma_(self, magma=None): - """ - Returns the MAGMA representation of self. - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R. = MPolynomialRing_libsingular(GF(2),2) - sage: f = y*x^2 + x +1 - sage: f._magma_() #optional - x^2*y + x + 1 - """ - if magma is None: - # TODO: import this globally - import sage.interfaces.magma - magma = sage.interfaces.magma.magma - - magma_gens = [e.name() for e in self.parent()._magma_().gens()] - f = self._repr_with_changed_varnames(magma_gens) - return magma(f) - - def _singular_(self, singular=singular_default, have_ring=False): - """ - Return a SINGULAR (as in the CAS) element for this - element. The result is cached. - - INPUT: - singular -- interpreter (default: singular_default) - have_ring -- should the correct ring not be set in SINGULAR first (default:False) - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(GF(127),3) - sage: x._singular_() - x - sage: f =(x^2 + 35*y + 128); f - x^2 + 35*y + 1 - sage: x._singular_().name() == x._singular_().name() - True - - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(GF(127),3) - sage: P(0)._singular_() - 0 - - """ - if not have_ring: - self.parent()._singular_(singular).set_ring() #this is expensive - - try: - if self.__singular is None: - return self._singular_init_c(singular, True) - - self.__singular._check_valid() - - if self.__singular.parent() is singular: - return self.__singular - - except (AttributeError, ValueError): - pass - - return self._singular_init_c(singular, True) - - def _singular_init_(self,singular=singular_default, have_ring=False): - """ - Return a new SINGULAR (as in the CAS) element for this element. - - INPUT: - singular -- interpreter (default: singular_default) - have_ring -- should the correct ring not be set in SINGULAR first (default:False) - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(GF(127),3) - sage: x._singular_init_() - x - sage: (x^2+37*y+128)._singular_init_() - x^2+37*y+1 - sage: x._singular_init_().name() == x._singular_init_().name() - False - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P(0)._singular_init_() - 0 - """ - return self._singular_init_c(singular, have_ring) - - cdef _singular_init_c(self,singular, have_ring): - """ - See MPolynomial_libsingular._singular_init_ - - """ - if not have_ring: - self.parent()._singular_(singular).set_ring() #this is expensive - - self.__singular = singular(str(self)) - return self.__singular - - def sub_m_mul_q(self, MPolynomial_libsingular m, MPolynomial_libsingular q): - """ - Return self - m*q, where m must be a monomial and q a - polynomial. - - INPUT: - m -- a monomial - q -- a polynomial - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: x.sub_m_mul_q(y,z) - -y*z + x - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: Q.=MPolynomialRing_libsingular(QQ,3) - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: P(0).sub_m_mul_q(P(0),P(1)) - 0 - sage: x.sub_m_mul_q(Q.gen(1),Q.gen(2)) - -y*z + x - - """ - cdef ring *r = (self._parent)._ring - - if not self._parent is m._parent: - m = self._parent._coerce_c(m) - if not self._parent is q._parent: - q = self._parent._coerce_c(q) - - if m._poly and m._poly.next: - raise ArithmeticError, "m must be a monomial" - elif not m._poly: - return self - - return new_MP(self._parent, p_Minus_mm_Mult_qq(p_Copy(self._poly, r), m._poly, q._poly, r)) - - def add_m_mul_q(self, MPolynomial_libsingular m, MPolynomial_libsingular q): - """ - Return self + m*q, where m must be a monomial and q a - polynomial. - - INPUT: - m -- a monomial - q -- a polynomial - - EXAMPLE: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: x.add_m_mul_q(y,z) - y*z + x - - TESTS: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: R.=MPolynomialRing_libsingular(QQ,3) - sage: P.=MPolynomialRing_libsingular(QQ,3) - sage: P(0).add_m_mul_q(P(0),P(1)) - 0 - sage: x.add_m_mul_q(R.gen(),R.gen(1)) - x*y + x - """ - - cdef ring *r = (self._parent)._ring - - if not self._parent is m._parent: - m = self._parent._coerce_c(m) - if not self._parent is q._parent: - q = self._parent._coerce_c(q) - - if m._poly and m._poly.next: - raise ArithmeticError, "m must be a monomial" - elif not m._poly: - return self - - return new_MP(self._parent, p_Plus_mm_Mult_qq(p_Copy(self._poly, r), m._poly, q._poly, r)) - - - def __reduce__(self): - """ - - Serialize self. - - EXAMPLES: - sage: from sage.rings.multi_polynomial_libsingular import MPolynomialRing_libsingular - sage: P. = MPolynomialRing_libsingular(QQ,3, order='degrevlex') - sage: f = 27/113 * x^2 + y*z + 1/2 - sage: f == loads(dumps(f)) - True - - sage: P = MPolynomialRing_libsingular(GF(127),3,names='abc') - sage: a,b,c = P.gens() - sage: f = 57 * a^2*b + 43 * c + 1 - sage: f == loads(dumps(f)) - True - - """ - return sage.rings.multi_polynomial_libsingular.unpickle_MPolynomial_libsingular, ( self._parent, self.dict() ) - -def unpickle_MPolynomial_libsingular(MPolynomialRing_libsingular R, d): - """ - Deserialize a MPolynomial_libsingular object - - INPUT: - R -- the base ring - d -- a Python dictionary as returned by MPolynomial_libsingular.dict - - """ - cdef ring *r = R._ring - cdef poly *m, *p - cdef int _i, _e - p = p_ISet(0,r) - for mon,c in d.iteritems(): - m = p_Init(r) - for i,e in mon.sparse_iter(): - _i = i - if _i >= r.N: - p_Delete(&p,r) - p_Delete(&m,r) - raise TypeError, "variable index too big" - _e = e - if _e <= 0: - p_Delete(&p,r) - p_Delete(&m,r) - raise TypeError, "exponent too small" - p_SetExp(m, _i+1,_e, r) - p_SetCoeff(m, co.sa2si(c, r), r) - p_Setm(m,r) - p = p_Add_q(p,m,r) - return new_MP(R,p) Index: sage/rings/multi_polynomial_ring.py =================================================================== --- sage/rings/multi_polynomial_ring.py (revision 4259) +++ sage/rings/multi_polynomial_ring.py (revision 2665) @@ -84,6 +84,7 @@ from sage.structure.parent_gens import ParentWithGens -from multi_polynomial_ring_generic import MPolynomialRing_generic, is_MPolynomialRing - + +def is_MPolynomialRing(x): + return isinstance(x, MPolynomialRing_generic) class MPolynomialRing_macaulay2_repr: @@ -113,10 +114,219 @@ return self.__macaulay2 - def is_exact(self): - return self.base_ring().is_exact() +class MPolynomialRing_generic(commutative_ring.CommutativeRing): + def __init__(self, base_ring, n, names, order): + if not isinstance(base_ring, commutative_ring.CommutativeRing): + raise TypeError, "Base ring must be a commutative ring." + n = int(n) + if n < 0: + raise ValueError, "Multivariate Polynomial Rings must " + \ + "have more than 0 variables." + self.__ngens = n + self.__term_order = order + self._has_singular = False #cannot convert to Singular by default + ParentWithGens.__init__(self, base_ring, names) + + def is_integral_domain(self): + return self.base_ring().is_integral_domain() + + def _coerce_impl(self, x): + """ + Return the canonical coercion of x to this multivariate + polynomial ring, if one is defined, or raise a TypeError. + + The rings that canonically coerce to this polynomial ring are: + * this ring itself + * polynomial rings in the same variables over any base ring that + canonically coerces to the base ring of this ring + * any ring that canonically coerces to the base ring of this + polynomial ring. + + """ + try: + P = x.parent() + # polynomial rings in the same variable over the any base that coerces in: + if is_MPolynomialRing(P): + if P.variable_names() == self.variable_names(): + if self.has_coerce_map_from(P.base_ring()): + return self(x) + else: + raise TypeError, "no natural map between bases of polynomial rings" + else: + raise TypeError, "polynomial ring has a different indeterminate names" + + except AttributeError: + pass + + # any ring that coerces to the base ring of this polynomial ring. + return self._coerce_try(x, [self.base_ring()]) + + def __cmp__(self, right): + if not is_MPolynomialRing(right): + return cmp(type(self),type(right)) + return cmp((self.base_ring(), self.__ngens, self.variable_names(), self.__term_order), + (right.base_ring(), right.__ngens, right.variable_names(), right.__term_order)) + + def __contains__(self, x): + """ + This definition of containment does not involve a natural + inclusion from rings with less variables into rings with more. + """ + try: + return x.parent() == self + except AttributeError: + return False + + def _repr_(self): + return "Polynomial Ring in %s over %s"%(", ".join(self.variable_names()), self.base_ring()) + + def _latex_(self): + vars = str(self.latex_variable_names()).replace('\n','').replace("'",'') + return "%s[%s]"%(latex.latex(self.base_ring()), vars[1:-1]) + + + def _ideal_class_(self): + return multi_polynomial_ideal.MPolynomialIdeal + + def _is_valid_homomorphism_(self, codomain, im_gens): + try: + # all that is needed is that elements of the base ring + # of the polynomial ring canonically coerce into codomain. + # Since poly rings are free, any image of the gen + # determines a homomorphism + codomain._coerce_(self.base_ring()(1)) + except TypeError: + return False + return True + + def _magma_(self, magma=None): + """ + Used in converting this ring to the corresponding ring in MAGMA. + + EXAMPLES: + sage: R. = PolynomialRing(QQ,3) + sage: magma(R) # optional + Polynomial ring of rank 3 over Rational Field + Graded Reverse Lexicographical Order + Variables: y, z, w + + sage: magma(PolynomialRing(GF(7),4, 'x')) #optional + Polynomial ring of rank 4 over GF(7) + Graded Reverse Lexicographical Order + Variables: x0, x1, x2, x3 + + sage: magma(PolynomialRing(GF(49,'a'),10, 'x')) #optional + Polynomial ring of rank 10 over GF(7^2) + Graded Reverse Lexicographical Order + Variables: x0, x1, x2, x3, x4, x5, x6, x7, x8, x9 + + sage: magma(PolynomialRing(ZZ['a,b,c'],3, 'x')) #optional + Polynomial ring of rank 3 over Polynomial ring of rank 3 over Integer Ring + Graded Reverse Lexicographical Order + Variables: x0, x1, x2 + """ + if magma == None: + import sage.interfaces.magma + magma = sage.interfaces.magma.magma - def change_ring(self, R): - from polynomial_ring_constructor import PolynomialRing - return PolynomialRing(R, self.variable_names(), order=self.term_order()) + try: + m = self.__magma + m._check_valid() + if not m.parent() is magma: + raise ValueError + return m + except (AttributeError,ValueError): + B = magma(self.base_ring()) + R = magma('PolynomialRing(%s, %s, %s)'%(B.name(), self.ngens(),self.term_order().magma_str())) + R.assign_names(self.variable_names()) + self.__magma = R + return R + + def var_dict(self): + """ + Return dictionary of paris varname:var of the variables + of this multivariate polynomial ring. + """ + return dict([(str(g),g) for g in self.gens()]) + + + def is_finite(self): + if self.ngens() == 0: + return self.base_ring().is_finite() + return False + + def is_field(self): + """ + Return True if this multivariate polynomial ring is a field, i.e., + it is a ring in 0 generators over a field. + """ + if self.ngens() == 0: + return self.base_ring().is_field() + return False + + def term_order(self): + return self.__term_order + + def characteristic(self): + """ + Return the characteristic of this polynomial ring. + + EXAMPLES: + sage: R = MPolynomialRing(QQ, 'x', 3) + sage: R.characteristic() + 0 + sage: R = MPolynomialRing(GF(7),'x', 20) + sage: R.characteristic() + 7 + """ + return self.base_ring().characteristic() + + def gen(self, n=0): + if n < 0 or n >= self.__ngens: + raise ValueError, "Generator not defined." + return self._gens[int(n)] + + def gens(self): + return self._gens + + def krull_dimension(self): + return self.base_ring().krull_dimension() + self.ngens() + + def ngens(self): + return self.__ngens + + def _monomial_order_function(self): + raise NotImplementedError + + def latex_variable_names(self): + """ + Returns the list of variable names suitable for latex output. + + All '_SOMETHING' substrings are replaced by '_{SOMETHING}' recursively + so that subscripts of subscripts work. + + EXAMPLES: + sage: R, x = PolynomialRing(QQ,'x',12).objgens() + sage: x + (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) + sage: print R.latex_variable_names () + ['x_{0}', 'x_{1}', 'x_{2}', 'x_{3}', 'x_{4}', 'x_{5}', 'x_{6}', 'x_{7}', 'x_{8}', 'x_{9}', 'x_{10}', 'x_{11}'] + sage: f = x[0]^3 + 15/3 * x[1]^10 + sage: print latex(f) + 5 x_{1}^{10} + x_{0}^{3} + """ + try: + return self.__latex_variable_names + except AttributeError: + pass + names = [] + for g in self.variable_names(): + i = len(g)-1 + while i >= 0 and g[i].isdigit(): + i -= 1 + if i < len(g)-1: + g = '%s_{%s}'%(g[:i+1], g[i+1:]) + names.append(g) + self.__latex_variable_names = names + return names class MPolynomialRing_polydict( MPolynomialRing_macaulay2_repr, MPolynomialRing_generic): @@ -150,11 +360,4 @@ return multi_polynomial_element.MPolynomial_polydict - def __cmp__(left, right): - if not is_MPolynomialRing(right): - return cmp(type(left),type(right)) - else: - return cmp((left.base_ring(), left.ngens(), left.variable_names(), left.term_order()), - (right.base_ring(), right.ngens(), right.variable_names(), right.term_order())) - def __call__(self, x, check=True): """ @@ -195,6 +398,9 @@ Polynomial Ring in x, y over Rational Field - We dump and load a the polynomial ring S: + We create an equal but not identical copy of the integer ring + by dumping and loading: sage: S2 = loads(dumps(S)) + sage: S2 is S + False sage: S2 == S True @@ -209,30 +415,5 @@ sage: f = x^2 + 2/3*y^3 sage: S(f) - 3*b^3 + a^2 - - Coercion from symbolic variables: - sage: x,y,z = var('x,y,z') - sage: R = QQ[x,y,z] - sage: type(x) - - sage: type(R(x)) - - sage: f = R(x^3 + y^3 - z^3); f - -1*z^3 + y^3 + x^3 - sage: type(f) - - sage: parent(f) - Polynomial Ring in x, y, z over Rational Field - - A more complicated symbolic and computational mix. Behind the scenes - Singular and Maxima are doing the real work. - sage: R = QQ[x,y,z] - sage: f = (x^3 + y^3 - z^3)^10; f - (-z^3 + y^3 + x^3)^10 - sage: g = R(f); parent(g) - Polynomial Ring in x, y, z over Rational Field - sage: (f - g).expand() - 0 - + 3*b^3 + a^2 """ if isinstance(x, multi_polynomial_element.MPolynomial_polydict): @@ -257,5 +438,4 @@ elif isinstance(x, polydict.PolyDict): return multi_polynomial_element.MPolynomial_polydict(self, x) - elif isinstance(x, fraction_field_element.FractionFieldElement) and x.parent().ring() == self: if x.denominator() == 1: @@ -263,22 +443,16 @@ else: raise TypeError, "unable to coerce since the denominator is not 1" - elif is_SingularElement(x) and self._has_singular: self._singular_().set_ring() try: return x.sage_poly(self) - except TypeError: - raise TypeError, "unable to coerce singular object" - - elif hasattr(x, '_polynomial_'): - return x._polynomial_(self) - + except: + raise TypeError, "Unable to coerce singular object" elif isinstance(x , str) and self._has_singular: self._singular_().set_ring() try: return self._singular_().parent(x).sage_poly(self) - except TypeError: - raise TypeError,"unable to coerce string" - + except: + raise TypeError,"Unable to coerce string" elif is_Macaulay2Element(x): try: @@ -294,10 +468,6 @@ raise TypeError, "Unable to coerce macaulay2 object" return multi_polynomial_element.MPolynomial_polydict(self, x) - - if isinstance(x, dict): - return multi_polynomial_element.MPolynomial_polydict(self, x) - else: - c = self.base_ring()(x) - return multi_polynomial_element.MPolynomial_polydict(self, {self._zero_tuple:c}) + c = self.base_ring()(x) + return multi_polynomial_element.MPolynomial_polydict(self, {self._zero_tuple:c}) @@ -504,5 +674,5 @@ else: return cmp(type(self), type(other)) - return cmp(self.__singular_str, other.__singular_str) + return cmp(self.__name, other.__name) Index: age/rings/multi_polynomial_ring_generic.pxd =================================================================== --- sage/rings/multi_polynomial_ring_generic.pxd (revision 4091) +++ (revision ) @@ -1,13 +1,0 @@ -import ring -cimport ring - -from sage.structure.parent cimport Parent - -cdef class MPolynomialRing_generic(ring.CommutativeRing): - cdef object __ngens - cdef object __term_order - cdef object _has_singular - cdef object __magma - - cdef _coerce_c_impl(self, x) - cdef int _cmp_c_impl(left, Parent right) except -2 Index: age/rings/multi_polynomial_ring_generic.pyx =================================================================== --- sage/rings/multi_polynomial_ring_generic.pyx (revision 4109) +++ (revision ) @@ -1,255 +1,0 @@ -include '../ext/stdsage.pxi' - -from sage.structure.parent_gens cimport ParentWithGens -import sage.misc.latex - - -def is_MPolynomialRing(x): - return bool(PY_TYPE_CHECK(x, MPolynomialRing_generic)) - -cdef class MPolynomialRing_generic(ring.CommutativeRing): - def __init__(self, base_ring, n, names, order): - if not isinstance(base_ring, ring.CommutativeRing): - raise TypeError, "Base ring must be a commutative ring." - n = int(n) - if n < 0: - raise ValueError, "Multivariate Polynomial Rings must " + \ - "have more than 0 variables." - self.__ngens = n - self.__term_order = order - self._has_singular = False #cannot convert to Singular by default - ParentWithGens.__init__(self, base_ring, names) - - def is_integral_domain(self): - return self.base_ring().is_integral_domain() - - cdef _coerce_c_impl(self, x): - """ - Return the canonical coercion of x to this multivariate - polynomial ring, if one is defined, or raise a TypeError. - - The rings that canonically coerce to this polynomial ring are: - * this ring itself - * polynomial rings in the same variables over any base ring that - canonically coerces to the base ring of this ring - * any ring that canonically coerces to the base ring of this - polynomial ring. - - """ - try: - P = x.parent() - # polynomial rings in the same variable over the any base that coerces in: - if is_MPolynomialRing(P): - if P.variable_names() == self.variable_names(): - if self.has_coerce_map_from(P.base_ring()): - return self(x) - else: - raise TypeError, "no natural map between bases of polynomial rings" - else: - raise TypeError, "polynomial ring has a different indeterminate names" - - except AttributeError: - pass - - # any ring that coerces to the base ring of this polynomial ring. - return self._coerce_try(x, [self.base_ring()]) - - def __richcmp__(left, right, int op): - return (left)._richcmp(right, op) - - cdef int _cmp_c_impl(left, Parent right) except -2: - if not is_MPolynomialRing(right): - return cmp(type(left),type(right)) - else: - return cmp((left.base_ring(), left.__ngens, left.variable_names(), left.__term_order), - (right.base_ring(), (right).__ngens, right.variable_names(), (right).__term_order)) - - def __contains__(self, x): - """ - This definition of containment does not involve a natural - inclusion from rings with less variables into rings with more. - """ - try: - return x.parent() == self - except AttributeError: - return False - - def _repr_(self): - return "Polynomial Ring in %s over %s"%(", ".join(self.variable_names()), self.base_ring()) - - def _latex_(self): - vars = str(self.latex_variable_names()).replace('\n','').replace("'",'') - return "%s[%s]"%(sage.misc.latex.latex(self.base_ring()), vars[1:-1]) - - - def _ideal_class_(self): - return multi_polynomial_ideal.MPolynomialIdeal - - def _is_valid_homomorphism_(self, codomain, im_gens): - try: - # all that is needed is that elements of the base ring - # of the polynomial ring canonically coerce into codomain. - # Since poly rings are free, any image of the gen - # determines a homomorphism - codomain._coerce_(self.base_ring()(1)) - except TypeError: - return False - return True - - def _magma_(self, magma=None): - """ - Used in converting this ring to the corresponding ring in MAGMA. - - EXAMPLES: - sage: R. = PolynomialRing(QQ,3) - sage: magma(R) # optional - Polynomial ring of rank 3 over Rational Field - Graded Reverse Lexicographical Order - Variables: y, z, w - - sage: magma(PolynomialRing(GF(7),4, 'x')) #optional - Polynomial ring of rank 4 over GF(7) - Graded Reverse Lexicographical Order - Variables: x0, x1, x2, x3 - - sage: magma(PolynomialRing(GF(49,'a'),10, 'x')) #optional - Polynomial ring of rank 10 over GF(7^2) - Graded Reverse Lexicographical Order - Variables: x0, x1, x2, x3, x4, x5, x6, x7, x8, x9 - - sage: magma(PolynomialRing(ZZ['a,b,c'],3, 'x')) #optional - Polynomial ring of rank 3 over Polynomial ring of rank 3 over Integer Ring - Graded Reverse Lexicographical Order - Variables: x0, x1, x2 - """ - if magma == None: - import sage.interfaces.magma - magma = sage.interfaces.magma.magma - - try: - if self.__magma is None: - raise AttributeError - m = self.__magma - m._check_valid() - if not m.parent() is magma: - raise ValueError - return m - except (AttributeError,ValueError): - B = magma(self.base_ring()) - R = magma('PolynomialRing(%s, %s, %s)'%(B.name(), self.ngens(),self.term_order().magma_str())) - R.assign_names(self.variable_names()) - self.__magma = R - return R - - def var_dict(self): - """ - Return dictionary of paris varname:var of the variables - of this multivariate polynomial ring. - """ - return dict([(str(g),g) for g in self.gens()]) - - - def is_finite(self): - if self.ngens() == 0: - return self.base_ring().is_finite() - return False - - def is_field(self): - """ - Return True if this multivariate polynomial ring is a field, i.e., - it is a ring in 0 generators over a field. - """ - if self.ngens() == 0: - return self.base_ring().is_field() - return False - - def term_order(self): - return self.__term_order - - def characteristic(self): - """ - Return the characteristic of this polynomial ring. - - EXAMPLES: - sage: R = MPolynomialRing(QQ, 'x', 3) - sage: R.characteristic() - 0 - sage: R = MPolynomialRing(GF(7),'x', 20) - sage: R.characteristic() - 7 - """ - return self.base_ring().characteristic() - - def gen(self, n=0): - if n < 0 or n >= self.__ngens: - raise ValueError, "Generator not defined." - return self._gens[int(n)] - - def gens(self): - return self._gens - - def krull_dimension(self): - return self.base_ring().krull_dimension() + self.ngens() - - def ngens(self): - return self.__ngens - - def _monomial_order_function(self): - raise NotImplementedError - - def latex_variable_names(self): - """ - Returns the list of variable names suitable for latex output. - - All '_SOMETHING' substrings are replaced by '_{SOMETHING}' recursively - so that subscripts of subscripts work. - - EXAMPLES: - sage: R, x = PolynomialRing(QQ,'x',12).objgens() - sage: x - (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) - sage: print R.latex_variable_names () - ['x_{0}', 'x_{1}', 'x_{2}', 'x_{3}', 'x_{4}', 'x_{5}', 'x_{6}', 'x_{7}', 'x_{8}', 'x_{9}', 'x_{10}', 'x_{11}'] - sage: f = x[0]^3 + 15/3 * x[1]^10 - sage: print latex(f) - 5 x_{1}^{10} + x_{0}^{3} - """ - try: - return self.__latex_variable_names - except AttributeError: - pass - names = [] - for g in self.variable_names(): - i = len(g)-1 - while i >= 0 and g[i].isdigit(): - i -= 1 - if i < len(g)-1: - g = '%s_{%s}'%(g[:i+1], g[i+1:]) - names.append(g) - self.__latex_variable_names = names - return names - - def __reduce__(self): - """ - """ - - base_ring = self.base_ring() - n = self.ngens() - names = self.variable_names() - order = self.term_order() - - return unpickle_MPolynomialRing_generic_v1,(base_ring, n, names, order) - - -#################### -# Leave *all* old versions! - -def unpickle_MPolynomialRing_generic_v1(base_ring, n, names, order): - from sage.rings.multi_polynomial_ring import MPolynomialRing - return MPolynomialRing(base_ring, n, names=names, order=order) - - -def unpickle_MPolynomialRing_generic(base_ring, n, names, order): - from sage.rings.multi_polynomial_ring import MPolynomialRing - - return MPolynomialRing(base_ring, n, names=names, order=order) Index: sage/rings/number_field/number_field.py =================================================================== --- sage/rings/number_field/number_field.py (revision 4058) +++ sage/rings/number_field/number_field.py (revision 3650) @@ -98,7 +98,8 @@ EXAMPLES: Constructing a relative number field + sage: R. = PolynomialRing(QQ) sage: K. = NumberField(x^2 - 2) - sage: R. = K[] - sage: L = K.extension(t^3+t+a, b); L + sage: R. = K['t'] + sage: L = K.extension(t^3+t+a, 'b'); L Extension by t^3 + t + a of the Number Field in a with defining polynomial x^2 - 2 sage: L.absolute_field() @@ -124,11 +125,4 @@ name = sage.structure.parent_gens.normalize_names(1, name) - if not isinstance(polynomial, polynomial_element.Polynomial): - try: - polynomial = polynomial.polynomial(QQ) - except (AttributeError, TypeError): - raise TypeError, "polynomial (=%s) must be a polynomial."%repr(polynomial) - - key = (polynomial, name) if _nf_cache.has_key(key): @@ -185,4 +179,5 @@ """ EXAMPLES: + sage: R. = PolynomialRing(QQ) sage: K. = NumberField(x^3 - 2); K Number Field in a with defining polynomial x^3 - 2 @@ -194,5 +189,5 @@ ParentWithGens.__init__(self, QQ, name) if not isinstance(polynomial, polynomial_element.Polynomial): - raise TypeError, "polynomial (=%s) must be a polynomial"%repr(polynomial) + raise TypeError, "polynomial (=%s) must be a polynomial"%polynomial if check: @@ -214,14 +209,5 @@ def __reduce__(self): - """ - TESTS: - sage: K. = NumberField(Z^3 + Z + 1) - sage: L = loads(dumps(K)) - sage: print L - Number Field in w with defining polynomial Z^3 + Z + 1 - sage: print L == K - True - """ - return NumberField_generic_v1, (self.__polynomial, self.variable_name(), self.__latex_variable_name) + return NumberField_generic, (self.__polynomial, self.variable_name(), self.__latex_variable_name) def complex_embeddings(self, prec=53): @@ -231,5 +217,4 @@ EXAMPLES: - sage: x = polygen(QQ) sage: f = x^5 + x + 17 sage: k. = NumberField(f) @@ -257,5 +242,5 @@ self.__latex_variable_name = name - def _repr_(self): + def __repr__(self): return "Number Field in %s with defining polynomial %s"%( self.variable_name(), self.polynomial()) @@ -290,8 +275,4 @@ list)): return number_field_element.NumberFieldElement(self, x) - try: - return number_field_element.NumberFieldElement(self, x._rational_()) - except (TypeError, AttributeError): - pass raise TypeError, "Cannot coerce %s into %s"%(x,self) @@ -299,7 +280,10 @@ if isinstance(x, (rational.Rational, integer.Integer, int, long)): return number_field_element.NumberFieldElement(self, x) - elif isinstance(x, number_field_element.NumberFieldElement) and x.parent() == self: - return number_field_element.NumberFieldElement(self, x.list()) raise TypeError + + def category(self): + from sage.categories.all import NumberFields + return NumberFields() + def category(self): @@ -325,4 +309,5 @@ EXAMPLES: + sage: R. = PolynomialRing(QQ) sage: K. = NumberField(x^3-2) sage: K.ideal([a]) @@ -466,4 +451,5 @@ EXAMPLES: + sage: R. = PolynomialRing(QQ) sage: K. = NumberField(x^2+1) sage: K.elements_of_norm(3) @@ -482,30 +468,14 @@ EXAMPLES: + sage: R. = PolynomialRing(QQ) sage: K. = NumberField(x^3 - 2) - sage: R. = K[] + sage: t = K['x'].gen() sage: L. = K.extension(t^2 + a); L - Extension by t^2 + a of the Number Field in a with defining polynomial x^3 - 2 - - We create another extension. - sage: k. = NumberField(x^2 + 1); k - Number Field in a with defining polynomial x^2 + 1 - sage: m. = k.extension(y^2 + 1); m - Extension by y^2 + 1 of the Number Field in a with defining polynomial x^2 + 1 - sage: b.minpoly() - x^4 + 10*x^2 + 9 - - """ - if not isinstance(poly, polynomial_element.Polynomial): - try: - poly = poly.polynomial(self) - except (AttributeError, TypeError): - raise TypeError, "polynomial (=%s) must be a polynomial."%repr(poly) - if not names is None: - name = names - if isinstance(name, tuple): - name = name[0] + Extension by x^2 + a of the Number Field in a with defining polynomial x^3 - 2 + """ + if not names is None: name = names if name is None: raise TypeError, "the variable name must be specified." - return NumberField_extension(self, poly, repr(name)) + return NumberField_extension(self, poly, name) def factor_integer(self, n): @@ -518,4 +488,5 @@ First we form a number field defined by $x^2 + 1$: + sage: R. = PolynomialRing(QQ) sage: K. = NumberField(x^2 + 1); K Number Field in I with defining polynomial x^2 + 1 @@ -572,4 +543,6 @@ EXAMPLES: + sage: R. = PolynomialRing(QQ) + sage: NumberField(x^3-2, 'a').galois_group(pari_group=True) PARI group [6, -1, 2, "S3"] of degree 3 @@ -591,5 +564,6 @@ EXAMPLES: - sage: K. = NumberField(x^5 + 10*x + 1) + sage: x = polygen(QQ,'x') + sage: K. = NumberField(x^5+10*x+1) sage: K.integral_basis() [1, a, a^2, a^3, a^4] @@ -616,4 +590,5 @@ EXAMPLES: + sage: R. = PolynomialRing(QQ) sage: NumberField(x^3+x+9, 'a').narrow_class_group() Multiplicative Abelian Group isomorphic to C2 @@ -662,4 +637,5 @@ EXAMPLES: + sage: R. = PolynomialRing(QQ) sage: K = NumberField(x^3 + 2*x - 5, 'alpha') sage: K.polynomial_quotient_ring() @@ -676,4 +652,5 @@ EXAMPLES: + sage: R. = PolynomialRing(QQ) sage: NumberField(x^2-2, 'a').regulator() 0.88137358701954305 @@ -695,4 +672,5 @@ EXAMPLES: + sage: R. = PolynomialRing(QQ) sage: NumberField(x^2+1, 'a').signature() (0, 1) @@ -711,4 +689,5 @@ EXAMPLES: + sage: R. = PolynomialRing(QQ) sage: K. = NumberField(x^2 + 3) sage: K.trace_pairing([1,zeta3]) @@ -836,4 +815,5 @@ """ EXAMPLES: + sage: R. = PolynomialRing(QQ) sage: K. = NumberField(x^3 - 2) sage: t = K['x'].gen() @@ -850,8 +830,5 @@ raise TypeError, "base (=%s) must be a number field"%base if not isinstance(polynomial, polynomial_element.Polynomial): - try: - polynomial = polynomial.polynomial(base) - except (AttributeError, TypeError), msg: - raise TypeError, "polynomial (=%s) must be a polynomial."%repr(polynomial) + raise TypeError, "polynomial (=%s) must be a polynomial"%polynomial if name == base.variable_name(): raise ValueError, "Base field and extension cannot have the same name" @@ -897,19 +874,5 @@ self.__pari_bnf_certified = False - def __reduce__(self): - """ - TESTS: - sage: K. = NumberField(Z^3 + Z + 1) - sage: L. = K.extension(Z^3 + 2) - sage: L = loads(dumps(K)) - sage: print L - Number Field in w with defining polynomial Z^3 + Z + 1 - sage: print L == K - True - """ - return NumberField_extension_v1, (self.__base_field, self.polynomial(), self.variable_name(), - self.latex_variable_name()) - - def _repr_(self): + def __repr__(self): return "Extension by %s of the Number Field in %s with defining polynomial %s"%( self.polynomial(), self.base_field().variable_name(), @@ -1060,5 +1023,5 @@ pbn = self._pari_base_nf() prp = self.pari_relative_polynomial() - pari_poly = pbn.rnfequation(prp) + pari_poly = str(pbn.rnfequation(prp)).replace('^', '**') R = self.base_field().polynomial().parent() self.__absolute_polynomial = R(pari_poly) @@ -1222,17 +1185,5 @@ self.__zeta_order = n - def __reduce__(self): - """ - TESTS: - sage: K. = CyclotomicField(7) - sage: L = loads(dumps(K)) - sage: print L - Cyclotomic Field of order 7 and degree 6 - sage: print L == K - True - """ - return NumberField_cyclotomic_v1, (self.__zeta_order, self.variable_name()) - - def _repr_(self): + def __repr__(self): return "Cyclotomic Field of order %s and degree %s"%( self.zeta_order(), self.degree()) @@ -1540,15 +1491,4 @@ NumberField_generic.__init__(self, polynomial, name=name, check=check) - def __reduce__(self): - """ - TESTS: - sage: K. = QuadraticField(7) - sage: L = loads(dumps(K)) - sage: print L - Number Field in z7 with defining polynomial x^2 - 7 - sage: print L == K - True - """ - return NumberField_quadratic_v1, (self.polynomial(), self.variable_name()) def class_number(self, proof = True): @@ -1625,19 +1565,2 @@ (arith.is_squarefree(D) or \ (d == 0 and (D//4)%4 in [2,3] and arith.is_squarefree(D//4))) - - -################### -# For pickling -################### - -def NumberField_generic_v1(poly, name, latex_name): - return NumberField_generic(poly, name, latex_name, check=False) - -def NumberField_extension_v1(base_field, poly, name, latex_name): - return NumberField_extension(base_field, poly, name, latex_name, check=False) - -def NumberField_cyclotomic_v1(zeta_order, name): - return NumberField_cyclotomic(zeta_order, name) - -def NumberField_quadratic_v1(poly, name): - return NumberField_quadratic(poly, name, check=False) Index: sage/rings/number_field/number_field_element.pyx =================================================================== --- sage/rings/number_field/number_field_element.pyx (revision 4127) +++ sage/rings/number_field/number_field_element.pyx (revision 3650) @@ -233,4 +233,9 @@ Return PARI C-library object corresponding to self. + NOTE: This is not the actual underlying object that represents + this element, since that is a polynomial in x (as PARI + polynomials are rather constrained in their possible variable + names, e.g., I cannot be the name of a variable). + EXAMPLES: sage: k. = QuadraticField(-1) @@ -239,9 +244,4 @@ sage: pari(j) Mod(j, j^2 + 1) - - sage: y = QQ['y'].gen() - sage: k. = NumberField(y^3 - 2) - sage: pari(j) - Mod(j, j^3 - 2) If you try do coerce a generator called I to PARI, hell may @@ -276,11 +276,8 @@ else: f = self.polynomial()._pari_() - gp = self.parent().polynomial() - g = gp._pari_() - gv = str(gp.parent().gen()) + g = self.parent().polynomial()._pari_() if var != 'x': f = f.subst("x",var) - if var != gv: - g = g.subst(gv, var) + g = g.subst("x",var) h = f.Mod(g) self.__pari[var] = h Index: sage/rings/number_field/number_field_ideal.py =================================================================== --- sage/rings/number_field/number_field_ideal.py (revision 4077) +++ sage/rings/number_field/number_field_ideal.py (revision 2725) @@ -4,12 +4,4 @@ AUTHOR: -- Steven Sivek (2005-05-16) - -TESTS: - sage: K. = NumberField(x^2 - 5) - sage: I = K.ideal(2/(5+a)) - sage: I == loads(dumps(I)) - True - - """ @@ -71,5 +63,7 @@ def __cmp__(self, other): - return cmp(self.pari_hnf(), other.pari_hnf()) + if self.pari_hnf() == other.pari_hnf(): + return 0 + return -1 def _coerce_impl(self, x): @@ -235,5 +229,5 @@ sage: J = K.ideal([i+1, 2]) sage: J.gens() - (i + 1, 2) + (2, i + 1) sage: J.gens_reduced() (i + 1,) Index: sage/rings/padics/local_generic_element.py =================================================================== --- sage/rings/padics/local_generic_element.py (revision 4282) +++ sage/rings/padics/local_generic_element.py (revision 3674) @@ -204,5 +204,5 @@ raise NotImplementedError - def sqrt(self, prec=None, extend=True, all=False): + def sqrt(self): r""" Returns the square root of this local ring element @@ -213,15 +213,5 @@ local ring element -- the square root of self """ - try: - s = self.square_root() - if all: - return [s, -s] - else: - return s - except ValueError: - if extend: - raise NotImplementedError - else: - raise ValueError, "self must be a square" + return self.square_root() def square_root(self): Index: sage/rings/padics/padic_generic_element.py =================================================================== --- sage/rings/padics/padic_generic_element.py (revision 4257) +++ sage/rings/padics/padic_generic_element.py (revision 3804) @@ -6,5 +6,4 @@ -- Genya Zaytman: documentation -- David Harvey: doctests - -- William Stein: fixes to p-adic log in case input is not in Zp. """ @@ -498,13 +497,8 @@ return (self.valuation() % 2 == 0) and (self.unit_part().residue(3) == 1) - def log(self, branch=0): - r""" - Compute the p-adic logarithm of any nonzero element of $\\Q_p$. + def log(self, branch = None): + r""" + Compute the p-adic logarithm of any unit in $\Z_p$. (See below for normalization.) - - INPUT: - branch -- (default: 0). If self is not a $p$-adic unit, return - the p-adic log of the unit part plus branch times - the valuation of self. The usual power series for log with values in the additive @@ -573,24 +567,4 @@ ... assert ll == full_log ... assert ll.precision_absolute() == prec - - We compute some logs of non-units: - sage: K = Qp(5,8); K - 5-adic Field with capped relative precision 8 - sage: a = K(-5^2*17); a - 3*5^2 + 5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + O(5^10) - sage: u = a.unit_part(); u - 3 + 5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + O(5^8) - sage: b = K(1235/5); b - 2 + 4*5 + 4*5^2 + 5^3 + O(5^8) - sage: log(a) - 5 + 3*5^2 + 3*5^3 + 4*5^4 + 4*5^5 + 5^6 + O(5^8) - sage: log(a*b) - log(a) - log(b) - O(5^8) - sage: c = a^b; c - 2*5^494 + 4*5^496 + 2*5^497 + 5^499 + 3*5^500 + 5^501 + O(5^502) - - Note that we can recover b: - sage: log(c)/log(a) - 2 + 4*5 + 4*5^2 + 5^3 + O(5^7) @@ -639,5 +613,5 @@ z = self.unit_part() return (z**Integer(p-1)).log() // Integer(p-1) - elif not branch is None: + elif not branch is None and self.parent().__contains__(branch): branch = self.parent()(branch) return self.unit_part().log() + branch*self.valuation() Index: age/rings/padics/tests.py =================================================================== --- sage/rings/padics/tests.py (revision 4004) +++ (revision ) @@ -1,36 +1,0 @@ -""" -TESTS: - -sage: R = Zp(5, prec=5, type='fixed-mod') -sage: a = random_matrix(R,5) -sage: a.determinant() # random output -5 + 3*5^2 + 5^3 + 4*5^4 + O(5^5) -sage: K = Qp(3, 10,'capped-rel'); K.krull_dimension() -0 - -Computation of logs: - - sage: Qp(5)(1).log() - O(5^20) - sage: Qp(5)(-1).log() - O(5^20) - sage: Qp(5,prec=5)(7).log() - 4*5^2 + 4*5^3 + 3*5^4 + O(5^5) - sage: Qp(5,prec=10)(25*8).log(branch = 0) - 5 + 3*5^3 + 3*5^4 + 2*5^5 + 5^6 + 2*5^7 + 3*5^8 + 3*5^9 + O(5^10) - sage: Zp(5,prec=10)(8).log() - 5 + 3*5^3 + 3*5^4 + 2*5^5 + 5^6 + 2*5^7 + 3*5^8 + 3*5^9 + O(5^10) - -Loading and saving elements of various types: - - sage: a = ZpCA(5)(-3); loads(dumps(a)) == a - True - sage: a = ZpCR(5)(-3); loads(dumps(a)) == a - True - sage: a = Zp(5)(-3); loads(dumps(a)) == a - True - sage: a = ZpFM(5)(-3); loads(dumps(a)) == a - True - sage: a = ZpL(5)(-3); loads(dumps(a)) == a - True -""" Index: sage/rings/polydict.pyx =================================================================== --- sage/rings/polydict.pyx (revision 4006) +++ sage/rings/polydict.pyx (revision 2725) @@ -570,11 +570,4 @@ def __reduce__(PolyDict self): - """ - sage: from sage.rings.polydict import PolyDict - sage: f = PolyDict({(2,3):2, (1,2):3, (2,1):4}) - sage: loads(dumps(f)) == f - True - - """ return make_PolyDict,(self.__repn,) @@ -839,12 +832,4 @@ def __reduce__(ETuple self): - """ - EXAMPLES: - sage: from sage.rings.polydict import ETuple - sage: e = ETuple([1,1,0]) - sage: bool(e == loads(dumps(e))) - True - """ - return make_ETuple,(self._data,self._length) Index: sage/rings/polynomial_element.pxd =================================================================== --- sage/rings/polynomial_element.pxd (revision 4149) +++ sage/rings/polynomial_element.pxd (revision 3650) @@ -5,13 +5,11 @@ from sage.structure.element import Element, CommutativeAlgebraElement -from sage.structure.element cimport Element, CommutativeAlgebraElement, ModuleElement +from sage.structure.element cimport Element, CommutativeAlgebraElement cdef class Polynomial(CommutativeAlgebraElement): - cdef ModuleElement _neg_c_impl(self) cdef char _is_gen -cdef class Polynomial_generic_dense(Polynomial): - cdef object __coeffs # a python list - cdef void __normalize(self) +#cdef class Polynomial_generic_dense(Polynomial): +# cdef object __coeffs # a python list #cdef class Polynomial_generic_sparse(Polynomial): Index: sage/rings/polynomial_element.pyx =================================================================== --- sage/rings/polynomial_element.pyx (revision 4277) +++ sage/rings/polynomial_element.pyx (revision 3925) @@ -5,12 +5,4 @@ -- William Stein: first version -- Martin Albrecht: Added singular coercion. - -- Robert Bradshaw: Move Polynomial_generic_dense to SageX - -TESTS: - sage: R. = ZZ[] - sage: f = x^5 + 2*x^2 + (-1) - sage: f == loads(dumps(f)) - True - """ @@ -38,5 +30,4 @@ import rational_field import complex_field -import fraction_field_element #import padic_field from infinity import infinity @@ -53,5 +44,5 @@ from sage.structure.element import RingElement -from sage.structure.element cimport Element, RingElement, ModuleElement, MonoidElement +from sage.structure.element cimport Element from rational_field import QQ @@ -75,5 +66,5 @@ y*x sage: type(f) - + """ def __init__(self, parent, is_gen = False, construct=False): @@ -104,56 +95,17 @@ self._is_gen = is_gen - cdef ModuleElement _add_c_impl(self, ModuleElement right): - cdef Py_ssize_t i, min - x = self.list() - cdef ModuleElement _neg_c_impl(self): - return self.polynomial([-x for x in self.list()]) - - y = right.list() - - if len(x) > len(y): - min = len(y) - high = x[min:] - elif len(x) < len(y): - min = len(x) - high = y[min:] + def _add_(self, right): + if self.degree() >= right.degree(): + x = list(self.list()) + y = right.list() else: - min = len(x) - high = [] + x = list(right.list()) + y = self.list() - low = [x[i] + y[i] for i from 0 <= i < min] - return self.polynomial(low + high) + for i in xrange(len(y)): + x[i] += y[i] + + return self.polynomial(x) - def plot(self, xmin=0, xmax=1, *args, **kwds): - """ - Return a plot of this polynomial. - - INPUT: - xmin -- float - xmax -- float - *args, **kwds -- passed to either point or point - - OUTPUT: - returns a graphic object. - - EXAMPLES: - sage: x = polygen(GF(389)) - sage: plot(x^2 + 1, rgbcolor=(0,0,1)).save() - sage: x = polygen(QQ) - sage: plot(x^2 + 1, rgbcolor=(1,0,0)).save() - """ - R = self.base_ring() - from sage.plot.plot import plot, point, line - if R.characteristic() == 0: - return plot(self.__call__, xmin=xmin, xmax=xmax, *args, **kwds) - else: - if R.is_finite(): - v = list(R) - v.sort() - w = dict([(v[i],i) for i in range(len(v))]) - z = [(i, w[self(v[i])]) for i in range(len(v))] - return point(z, *args, **kwds) - raise NotImplementedError, "plotting of polynomials over %s not implemented"%R - def _lmul_(self, left): """ @@ -170,7 +122,5 @@ # todo -- should multiply individual coefficients?? # that could be in derived class. - # Note that we are guaranteed that right is in the base ring, so this could be fast. - if left == 0: - return self.parent()(0) + # Note that we are guaranteed that right is in the base ring, so this could be fast. return self.parent()(left) * self @@ -190,6 +140,4 @@ # that could be in derived class. # Note that we are guaranteed that right is in the base ring, so this could be fast. - if right == 0: - return self.parent()(0) return self * self.parent()(right) @@ -440,5 +388,5 @@ return long(self[0]) - cdef RingElement _mul_c_impl(self, RingElement right): + def _mul_(self, right): """ EXAMPLES: @@ -517,5 +465,5 @@ - def __pow__(self, right, dummy): + def _pow(self, right): """ EXAMPLES: @@ -531,16 +479,5 @@ if right < 0: return (~self)**(-right) - if (self)._is_gen: # special case x**n should be faster! - v = [0]*right + [1] - return self.parent()(v, check=True) - return arith.generic_power(self, right, self.parent()(1)) - - def _pow(self, right): - # TODO: fit __pow__ into the arithmatic structure - if self.degree() <= 0: - return self.parent()(self[0]**right) - if right < 0: - return (~self)**(-right) - if (self)._is_gen: # special case x**n should be faster! + if self._is_gen: # special case x**n should be faster! v = [0]*right + [1] return self.parent()(v, check=True) @@ -560,5 +497,5 @@ if n != m-1: s += " + " - x = repr(x) + x = str(x) if not atomic_repr and n > 0 and (x.find("+") != -1 or x.find("-") != -1): x = "(%s)"%x @@ -581,5 +518,4 @@ r""" EXAMPLES: - sage: x = polygen(QQ) sage: f = x^3+2/3*x^2 - 5/3 sage: f._repr_() @@ -594,5 +530,4 @@ r""" EXAMPLES: - sage: x = polygen(QQ) sage: latex(x^3+2/3*x^2 - 5/3) x^{3} + \frac{2}{3}x^{2} - \frac{5}{3} @@ -665,43 +600,12 @@ def _mul_generic(self, right): - if self is right: - return self._square_generic() - x = self.list() - y = right.list() - cdef Py_ssize_t i, k, start, end - cdef Py_ssize_t d1 = len(x)-1, d2 = len(y)-1 - if d1 == -1: - return self - elif d2 == -1: - return right - elif d1 == 0: - c = x[0] - return self._parent([c*a for a in y]) - elif d2 == 0: - c = y[0] - return self._parent([a*c for a in x]) - coeffs = [] - for k from 0 <= k <= d1+d2: - start = 0 if k <= d2 else k-d2 # max(0, k-d2) - end = k if k <= d1 else d1 # min(k, d1) - sum = x[start] * y[k-start] - for i from start < i <= end: - sum += x[i] * y[k-i] - coeffs.append(sum) - return self._parent(coeffs) - - def _square_generic(self): - x = self.list() - cdef Py_ssize_t i, j - cdef Py_ssize_t d = len(x)-1 - zero = self._parent.base_ring()(0) - two = self._parent.base_ring()(2) - coeffs = [zero] * (2 * d + 1) - for i from 0 <= i <= d: - coeffs[2*i] = x[i] * x[i] - for j from 0 <= j < i: - coeffs[i+j] += two * x[i] * x[j] - return self._parent(coeffs) - + d1 = self.degree() + d2 = right.degree() + d = d1 + d2 + w = [sum([self[i]*right[k-i] for i in range(0,min(d1,k)+1) if \ + i <= d1 and k-i <= d2 and self[i]!=0 and right[k-i]!=0]) \ + for k in range(d+1)] + return self.parent()(w) + def _mul_fateman(self, right): r""" @@ -835,5 +739,5 @@ polynomial multiplication is implemented. """ - return self._parent(do_karatsuba(self.list(), right.list())) + return self.parent()(do_karatsuba(self.list(), right.list())) def base_ring(self): @@ -975,11 +879,5 @@ def derivative(self): - if self.is_zero(): - return self - cdef Py_ssize_t n, degree = self.degree() - if degree == 0: - return self.parent()(0) - coeffs = self.list() - return self.polynomial([n*coeffs[n] for n from 1 <= n <= degree]) + return self.polynomial([self[n]*n for n in xrange(1,self.degree()+1)]) def integral(self): @@ -1041,5 +939,5 @@ Notice that the unit factor is included when we multiply $F$ back out. - sage: expand(F) + sage: F.mul() 2*x^10 + 2*x + 2*a @@ -1076,5 +974,5 @@ sage: F = factor(x^2-3); F (1.0000000000000000000000000000*x + 1.7320508075688772935274463415) * (1.0000000000000000000000000000*x - 1.7320508075688772935274463415) - sage: expand(F) + sage: F.mul() 1.0000000000000000000000000000*x^2 - 3.0000000000000000000000000000 sage: factor(x^2 + 1) @@ -1084,5 +982,5 @@ sage: F = factor(x^2+3); F (1.0000000000000000000000000000*x + -1.7320508075688772935274463415*I) * (1.0000000000000000000000000000*x + 1.7320508075688772935274463415*I) - sage: expand(F) + sage: F.mul() 1.0000000000000000000000000000*x^2 + 3.0000000000000000000000000000 sage: factor(x^2+1) @@ -1090,7 +988,7 @@ sage: f = C.0 * (x^2 + 1) ; f 1.0000000000000000000000000000*I*x^2 + 1.0000000000000000000000000000*I - sage: F = factor(f); F + sage: F=factor(f); F (1.0000000000000000000000000000*I) * (1.0000000000000000000000000000*x + -1.0000000000000000000000000000*I) * (1.0000000000000000000000000000*x + 1.0000000000000000000000000000*I) - sage: expand(F) + sage: F.mul() 1.0000000000000000000000000000*I*x^2 + 1.0000000000000000000000000000*I """ @@ -1346,4 +1244,7 @@ return bool(self._is_gen) + def is_zero(self): + return bool(self.degree() == -1) + def leading_coefficient(self): return self[self.degree()] @@ -1454,5 +1355,5 @@ def polynomial(self, *args, **kwds): - return self._parent(*args, **kwds) + return self.parent()(*args, **kwds) def newton_slopes(self, p): @@ -1510,5 +1411,5 @@ def _pari_init_(self): - return repr(self._pari_()) + return str(self._pari_()) def _magma_init_(self): @@ -1555,5 +1456,5 @@ def _gap_init_(self): - return repr(self) + return str(self) def _gap_(self, G): @@ -1664,14 +1565,7 @@ sage: x = CC['x'].0 - sage: i = CC.0 - sage: f = (x - 1)*(x - i) + sage: f = (x-1)*(x-I) sage: f.roots() [1.00000000000000*I, 1.00000000000000] - - A purely symbolic roots example: - sage: f = expand((X-1)*(X-I)^3*(X^2 - sqrt(2))); f - X^6 - 3*I*X^5 - X^5 + 3*I*X^4 - sqrt(2)*X^4 - 3*X^4 + 3*sqrt(2)*I*X^3 + I*X^3 + sqrt(2)*X^3 + 3*X^3 - 3*sqrt(2)*I*X^2 - I*X^2 + 3*sqrt(2)*X^2 - sqrt(2)*I*X - 3*sqrt(2)*X + sqrt(2)*I - sage: print f.roots() - [(I, 3), (-2^(1/4), 1), (2^(1/4), 1), (1, 1)] """ seq = [] @@ -1851,15 +1745,7 @@ sage: p.shift(2) x^4 + 2*x^3 + 4*x^2 - - One can also use the infix shift operator: - sage: f = x^3 + x - sage: f >> 2 - x - sage: f << 2 - x^5 + x^3 AUTHOR: -- David Harvey (2006-08-06) - -- Robert Bradshaw (2007-04-18) Added support for infix operator. """ if n == 0: @@ -1874,11 +1760,4 @@ else: return self.polynomial(self.coeffs()[-int(n):], check=False) - - def __lshift__(self, k): - return self.shift(k) - - def __rshift__(self, k): - return self.shift(-k) - def truncate(self, n): @@ -1902,5 +1781,5 @@ 3*x^3 - 4*x^2 + 4*x - 1 """ - return self // self.gcd(self.derivative()) + return self // self.gcd(self.derivative()) @@ -1909,5 +1788,5 @@ -cdef _karatsuba_sum(v,w): +def _karatsuba_sum(v,w): if len(v)>=len(w): x = list(v) @@ -1920,5 +1799,5 @@ return x -cdef _karatsuba_dif(v,w): +def _karatsuba_dif(v,w): if len(v)>=len(w): x = list(v) @@ -1931,13 +1810,11 @@ return x -cdef do_karatsuba(left, right): +def do_karatsuba(left, right): if len(left) == 0 or len(right) == 0: return [] if len(left) == 1: - c = left[0] - return [c*a for a in right] + return [left[0]*a for a in right] if len(right) == 1: - c = right[0] - return [c*a for a in left] + return [right[0]*a for a in left] if len(left) == 2 and len(right) == 2: b = left[0] @@ -1959,299 +1836,2 @@ t0 = bd return _karatsuba_sum(t0,_karatsuba_sum(t1,t2)) - - - -cdef class Polynomial_generic_dense(Polynomial): - """ - A generic dense polynomial. - - EXAMPLES: - sage: R. = PolynomialRing(PolynomialRing(QQ,'y')) - sage: f = x^3 - x + 17 - sage: type(f) - - sage: loads(f.dumps()) == f - True - """ - def __init__(self, parent, x=None, int check=1, is_gen=False, int construct=0, absprec=None): - Polynomial.__init__(self, parent, is_gen=is_gen) - - if x is None: - self.__coeffs = [] - return - R = parent.base_ring() - - if PY_TYPE_CHECK(x, Polynomial): - if (x)._parent is self._parent: - x = list(x.list()) - elif (x)._parent is R or (x)._parent == R: - x = [x] - elif absprec is None: - x = [R(a) for a in x.list()] - check = 0 - else: - x = [R(a, absprec = absprec) for a in x.list()] - check = 0 - - elif PY_TYPE_CHECK(x, list): - pass - - elif PY_TYPE_CHECK(x, int) and x == 0: - self.__coeffs = [] - return - - elif fraction_field_element.is_FractionFieldElement(x): - if x.denominator() != 1: - raise TypeError, "denominator must be 1" - else: - x = x.numerator() - - elif isinstance(x, dict): - zero = R(0) - n = max(x.keys()) - v = [zero for _ in xrange(n+1)] - for i, z in x.iteritems(): - v[i] = z - x = v - elif isinstance(x, pari_gen): - if absprec is None: - x = [R(w) for w in x.Vecrev()] - else: - x = [R(w, absprec = absprec) for w in x.Vecrev()] - check = True - elif not isinstance(x, list): - x = [x] # constant polynomials - if check: - if absprec is None: - self.__coeffs = [R(z) for z in x] - else: - self.__coeffs = [R(z, absprec=absprec) for z in x] - else: - self.__coeffs = x - if check: - self.__normalize() - - def __reduce__(self): - return make_generic_polynomial, (self._parent, self.__coeffs) - - def __hash__(self): - if self.degree() >= 1: - return hash(tuple(self.__coeffs)) - else: - return hash(self[0]) - - cdef void __normalize(self): - x = self.__coeffs - cdef Py_ssize_t n = len(x) - 1 - while n >= 0 and x[n].is_zero(): -# while n > 0 and x[n] == 0: - del x[n] - n -= 1 - - def __richcmp__(left, right, int op): - return (left)._richcmp(right, op) - - def __getitem__(self, Py_ssize_t n): - """ - EXAMPLES: - sage: R. = ZZ[] - sage: f = (1+2*x)^5; f - 32*x^5 + 80*x^4 + 80*x^3 + 40*x^2 + 10*x + 1 - sage: f[-1] - 0 - sage: f[2] - 40 - sage: f[6] - 0 - """ - if n < 0 or n >= len(self.__coeffs): - return self.base_ring()(0) - return self.__coeffs[n] - - def __getslice__(self, Py_ssize_t i, j): - """ - EXAMPLES: - sage: R. = RDF[] - sage: f = (1+2*x)^5; f - 32.0*x^5 + 80.0*x^4 + 80.0*x^3 + 40.0*x^2 + 10.0*x + 1.0 - sage: f[:3] - 40.0*x^2 + 10.0*x + 1.0 - sage: f[2:5] - 80.0*x^4 + 80.0*x^3 + 40.0*x^2 - sage: f[2:] - 32.0*x^5 + 80.0*x^4 + 80.0*x^3 + 40.0*x^2 - """ - if i <= 0: - i = 0 - zeros = [] - elif i > 0: - zeros = [self._parent.base_ring()(0)] * i - return self._parent(zeros + self.__coeffs[i:j]) - - def _unsafe_mutate(self, n, value): - """ - Never use this unless you really know what you are doing. - - WARNING: This could easily introduce subtle bugs, since SAGE - assumes everywhere that polynomials are immutable. It's OK to - use this if you really know what you're doing. - - EXAMPLES: - sage: R. = ZZ[] - sage: f = (1+2*x)^2; f - 4*x^2 + 4*x + 1 - sage: f._unsafe_mutate(1, -5) - sage: f - 4*x^2 - 5*x + 1 - """ - n = int(n) - value = self.base_ring()(value) - if n >= 0 and n < len(self.__coeffs): - self.__coeffs[n] = value - if n == len(self.__coeffs) and value == 0: - self.__normalize() - elif n < 0: - raise IndexError, "polynomial coefficient index must be nonnegative" - elif value != 0: - zero = self.base_ring()(0) - for _ in xrange(len(self.__coeffs), n): - self.__coeffs.append(zero) - self.__coeffs.append(value) - - def __floordiv__(self, right): - """ - Return the quotient upon division (no remainder). - - EXAMPLES: - sage: R. = QQ[] - sage: f = (1+2*x)^3 + 3*x; f - 8*x^3 + 12*x^2 + 9*x + 1 - sage: g = f // (1+2*x); g - 4*x^2 + 4*x + 5/2 - sage: f - g * (1+2*x) - -3/2 - sage: f.quo_rem(1+2*x) - (4*x^2 + 4*x + 5/2, -3/2) - """ - if right.parent() == self.parent(): - return Polynomial.__floordiv__(self, right) - d = self.parent().base_ring()(right) - return self.polynomial([c // d for c in self.__coeffs], check=False) - - cdef ModuleElement _add_c_impl(self, ModuleElement right): - cdef Py_ssize_t check=0, i, min - x = (self).__coeffs - y = (right).__coeffs - if len(x) > len(y): - min = len(y) - high = x[min:] - elif len(x) < len(y): - min = len(x) - high = y[min:] - else: - min = len(x) - low = [x[i] + y[i] for i from 0 <= i < min] - if len(x) == len(y): - res = self._parent(low, check=0) - (res).__normalize() - return res - else: - return self._parent(low + high, check=0) - - cdef ModuleElement _sub_c_impl(self, ModuleElement right): - cdef Py_ssize_t check=0, i, min - x = (self).__coeffs - y = (right).__coeffs - if len(x) > len(y): - min = len(y) - high = x[min:] - elif len(x) < len(y): - min = len(x) - high = [-y[i] for i from min <= i < len(y)] - else: - min = len(x) - low = [x[i] - y[i] for i from 0 <= i < min] - if len(x) == len(y): - res = self._parent(low, check=0) - (res).__normalize() - return res - else: - return self._parent(low + high, check=0) - - def _rmul_(self, c): - if len(self.__coeffs) == 0: - return self - v = [c * a for a in self.__coeffs] - res = self._parent(v, check=0) - if v[len(v)-1].is_zero(): - (res).__normalize() - return res - - def _lmul_(self, c): - if len(self.__coeffs) == 0: - return self - v = [a * c for a in self.__coeffs] - res = self._parent(v, check=0) - if v[len(v)-1].is_zero(): - (res).__normalize() - return res - - def list(self): - """ - Return a new copy of the list of the underlying - elements of self. - - EXAMPLES: - sage: R. = GF(17)[] - sage: f = (1+2*x)^3 + 3*x; f - 8*x^3 + 12*x^2 + 9*x + 1 - sage: f.list() - [1, 9, 12, 8] - """ - return list(self.__coeffs) - - def degree(self): - """ - EXAMPLES: - sage: R. = RDF[] - sage: f = (1+2*x^7)^5 - sage: f.degree() - 35 - """ - return len(self.__coeffs) - 1 - - def shift(self, Py_ssize_t n): - r""" - Returns this polynomial multiplied by the power $x^n$. If $n$ - is negative, terms below $x^n$ will be discarded. Does not - change this polynomial. - - EXAMPLES: - sage: R. = PolynomialRing(PolynomialRing(QQ,'y'), 'x') - sage: p = x^2 + 2*x + 4 - sage: type(p) - - sage: p.shift(0) - x^2 + 2*x + 4 - sage: p.shift(-1) - x + 2 - sage: p.shift(2) - x^4 + 2*x^3 + 4*x^2 - - AUTHOR: - -- David Harvey (2006-08-06) - """ - if n == 0: - return self - if n > 0: - output = [self.base_ring()(0)] * n - output.extend(self.__coeffs) - return self.polynomial(output, check=False) - if n < 0: - if n > len(self.__coeffs) - 1: - return self.polynomial([]) - else: - return self.polynomial(self.__coeffs[-int(n):], check=False) - -def make_generic_polynomial(parent, coeffs): - return parent(coeffs) Index: sage/rings/polynomial_element_generic.py =================================================================== --- sage/rings/polynomial_element_generic.py (revision 4276) +++ sage/rings/polynomial_element_generic.py (revision 3681) @@ -28,5 +28,5 @@ import copy -from polynomial_element import Polynomial, is_Polynomial, Polynomial_generic_dense +from polynomial_element import Polynomial, is_Polynomial from sage.structure.element import (IntegralDomainElement, EuclideanDomainElement, PrincipalIdealDomainElement) @@ -49,5 +49,216 @@ import sage.rings.polynomial_ring + +class Polynomial_generic_dense(Polynomial): + """ + A generic dense polynomial. + + EXAMPLES: + sage: R. = PolynomialRing(PolynomialRing(QQ,'y')) + sage: f = x^3 - x + 17 + sage: type(f) + + sage: loads(f.dumps()) == f + True + """ + def __init__(self, parent, x=None, check=True, is_gen=False, construct=False, absprec=None): + Polynomial.__init__(self, parent, is_gen=is_gen) + if x is None: + self.__coeffs = [] + return + R = parent.base_ring() + + if fraction_field_element.is_FractionFieldElement(x): + if x.denominator() != 1: + raise TypeError, "denominator must be 1" + else: + x = x.numerator() + if isinstance(x, Polynomial): + if x.parent() == self.parent(): + x = list(x.list()) + elif x.parent() == R: + x = [x] + elif absprec is None: + x = [R(a) for a in x.list()] + check = False + else: + x = [R(a, absprec = absprec) for a in x.list()] + check = False + elif isinstance(x, dict): + zero = R(0) + n = max(x.keys()) + v = [zero for _ in xrange(n+1)] + for i, z in x.iteritems(): + v[i] = z + x = v + elif isinstance(x, pari_gen): + if absprec is None: + x = [R(w) for w in x.Vecrev()] + else: + x = [R(w, absprec = absprec) for w in x.Vecrev()] + check = True + elif not isinstance(x, list): + x = [x] # constant polynomials + if check: + if absprec is None: + self.__coeffs = [R(z) for z in x] + else: + self.__coeffs = [R(z, absprec=absprec) for z in x] + else: + self.__coeffs = x + if check: + self.__normalize() + + def __normalize(self): + x = self.__coeffs + n = len(x)-1 + while n>=0 and x[n] == 0: + del x[n] + n -= 1 + + def __getitem__(self,n): + """ + EXAMPLES: + sage: R. = ZZ[] + sage: f = (1+2*x)^5; f + 32*x^5 + 80*x^4 + 80*x^3 + 40*x^2 + 10*x + 1 + sage: f[-1] + 0 + sage: f[2] + 40 + sage: f[6] + 0 + """ + if n < 0 or n >= len(self.__coeffs): + return self.base_ring()(0) + return self.__coeffs[n] + + def __getslice__(self, i, j): + """ + EXAMPLES: + sage: R. = RDF[] + sage: f = (1+2*x)^5; f + 32.0*x^5 + 80.0*x^4 + 80.0*x^3 + 40.0*x^2 + 10.0*x + 1.0 + sage: f[:3] + 40.0*x^2 + 10.0*x + 1.0 + sage: f[2:5] + 80.0*x^4 + 80.0*x^3 + 40.0*x^2 + sage: f[2:] + 32.0*x^5 + 80.0*x^4 + 80.0*x^3 + 40.0*x^2 + """ + if i < 0: + i = 0 + P = self.parent() + return P([0]*int(i) + self.__coeffs[i:j]) + + def _unsafe_mutate(self, n, value): + """ + Never use this unless you really know what you are doing. + + WARNING: This could easily introduce subtle bugs, since SAGE + assumes everywhere that polynomials are immutable. It's OK to + use this if you really know what you're doing. + + EXAMPLES: + sage: R. = ZZ[] + sage: f = (1+2*x)^2; f + 4*x^2 + 4*x + 1 + sage: f._unsafe_mutate(1, -5) + sage: f + 4*x^2 - 5*x + 1 + """ + n = int(n) + value = self.base_ring()(value) + if n >= 0 and n < len(self.__coeffs): + self.__coeffs[n] = value + if n == len(self.__coeffs) and value == 0: + self.__normalize() + elif n < 0: + raise IndexError, "polynomial coefficient index must be nonnegative" + elif value != 0: + zero = self.base_ring()(0) + for _ in xrange(len(self.__coeffs), n): + self.__coeffs.append(zero) + self.__coeffs.append(value) + + def __floordiv__(self, right): + """ + Return the quotient upon division (no remainder). + + EXAMPLES: + sage: R. = QQ[] + sage: f = (1+2*x)^3 + 3*x; f + 8*x^3 + 12*x^2 + 9*x + 1 + sage: g = f // (1+2*x); g + 4*x^2 + 4*x + 5/2 + sage: f - g * (1+2*x) + -3/2 + sage: f.quo_rem(1+2*x) + (4*x^2 + 4*x + 5/2, -3/2) + """ + if right.parent() == self.parent(): + return Polynomial.__floordiv__(self, right) + d = self.parent().base_ring()(right) + return self.polynomial([c // d for c in self.__coeffs], check=False) + + def list(self): + """ + Return a new copy of the list of the underlying + elements of self. + + EXAMPLES: + sage: R. = GF(17)[] + sage: f = (1+2*x)^3 + 3*x; f + 8*x^3 + 12*x^2 + 9*x + 1 + sage: f.list() + [1, 9, 12, 8] + """ + return list(self.__coeffs) + + def degree(self): + """ + EXAMPLES: + sage: R. = RDF[] + sage: f = (1+2*x^7)^5 + sage: f.degree() + 35 + """ + return len(self.__coeffs) - 1 + + def shift(self, n): + r""" + Returns this polynomial multiplied by the power $x^n$. If $n$ + is negative, terms below $x^n$ will be discarded. Does not + change this polynomial. + + EXAMPLES: + sage: R. = PolynomialRing(PolynomialRing(QQ,'y'), 'x') + sage: p = x^2 + 2*x + 4 + sage: type(p) + + sage: p.shift(0) + x^2 + 2*x + 4 + sage: p.shift(-1) + x + 2 + sage: p.shift(2) + x^4 + 2*x^3 + 4*x^2 + + AUTHOR: + -- David Harvey (2006-08-06) + """ + if n == 0: + return self + if n > 0: + output = [self.base_ring()(0)] * n + output.extend(self.__coeffs) + return self.polynomial(output, check=False) + if n < 0: + if n > self.degree(): + return self.polynomial([]) + else: + return self.polynomial(self.__coeffs[-int(n):], check=False) + + class Polynomial_generic_sparse(Polynomial): """ @@ -169,5 +380,5 @@ EXAMPLES: sage: R. = PolynomialRing(CDF, sparse=True) - sage: f = CDF(1,2) + w^5 - CDF(pi)*w + CDF(e) + sage: f = CDF(1,2) + w^5 - pi*w + e sage: f._repr() '1.0*w^5 + (-3.14159265359)*w + 3.71828182846 + 2.0*I' @@ -189,5 +400,5 @@ if n != m-1: s += " + " - x = repr(x) + x = str(x) if not atomic_repr and n > 0 and (x.find("+") != -1 or x.find("-") != -1): x = "(%s)"%x @@ -223,5 +434,4 @@ EXAMPLES: sage: R. = PolynomialRing(RDF, sparse=True) - sage: e = RDF(e) sage: f = sum(e^n*w^n for n in range(4)); f 20.0855369232*w^3 + 7.38905609893*w^2 + 2.71828182846*w + 1.0 @@ -450,5 +660,4 @@ EuclideanDomainElement, Polynomial_singular_repr): - def quo_rem(self, other): """ @@ -529,7 +738,6 @@ if fraction_field_element.is_FractionFieldElement(x): - if x.denominator().degree() >= 1: - raise TypeError, "denominator must be constant" - x = x.numerator() * (~x.denominator()[0]) + if x.denominator() != 1: + raise TypeError, "denominator must be 1" else: x = x.numerator() @@ -1406,5 +1614,5 @@ if n != m: s += " + " - x = repr(x) + x = str(x) if not atomic_repr and n > 0 and (x.find("+") != -1 or x.find("-") != -1): x = "(%s)"%x @@ -1659,15 +1867,5 @@ """ self._ntl_set_modulus() - return self.parent()(self.__poly * right.__poly, construct=True) - - def _rmul_(self, c): - self._ntl_set_modulus() - return self.parent()(ZZ_pX([c]) * self.__poly, construct=True) - - def _lmul_(self, c): - self._ntl_set_modulus() - return self.parent()(ZZ_pX([c]) * self.__poly, construct=True) - - + return self.parent()(self.__poly * right.__poly, construct=True) def quo_rem(self, right): Index: sage/rings/polynomial_pyx.pyx =================================================================== --- sage/rings/polynomial_pyx.pyx (revision 4059) +++ sage/rings/polynomial_pyx.pyx (revision 2725) @@ -1144,6 +1144,6 @@ return self.pq.degree - def __nonzero__(self): - return self.pq.degree != -1 + def is_zero(self): + return self.pq.degree == -1 def list(self): Index: sage/rings/polynomial_quotient_ring.py =================================================================== --- sage/rings/polynomial_quotient_ring.py (revision 4003) +++ sage/rings/polynomial_quotient_ring.py (revision 3480) @@ -667,5 +667,4 @@ EXAMPLES: - sage: x = polygen(QQ) sage: f = x^5 + x + 17 sage: k = QQ['x'].quotient(f) Index: sage/rings/polynomial_ring.py =================================================================== --- sage/rings/polynomial_ring.py (revision 4149) +++ sage/rings/polynomial_ring.py (revision 3679) @@ -71,5 +71,4 @@ import principal_ideal_domain import polynomial_element_generic -import polynomial_element import multi_polynomial_element import rational_field @@ -169,6 +168,6 @@ except: raise TypeError,"Unable to coerce string" - elif hasattr(x, '_polynomial_'): - return x._polynomial_(self) + # elif isinstance(x, multi_polynomial_element.MPolynomial_polydict): + # return x.univariate_polynomial(self) elif is_MagmaElement(x): x = list(x.Eltseq()) @@ -310,5 +309,5 @@ self.__polynomial_class = polynomial_element_generic.Polynomial_generic_sparse else: - self.__polynomial_class = polynomial_element.Polynomial_generic_dense + self.__polynomial_class = polynomial_element_generic.Polynomial_generic_dense def base_extend(self, R): @@ -436,7 +435,4 @@ return True return False - - def is_exact(self): - return self.base_ring().is_exact() def is_field(self): @@ -832,6 +828,4 @@ except: raise TypeError,"Unable to coerce string" - elif hasattr(x, '_polynomial_'): - return x._polynomial_(self) return polynomial_element_generic.Polynomial_dense_mod_p(self, x, check, is_gen,construct=construct) Index: sage/rings/polynomial_ring_constructor.py =================================================================== --- sage/rings/polynomial_ring_constructor.py (revision 4259) +++ sage/rings/polynomial_ring_constructor.py (revision 3828) @@ -4,5 +4,4 @@ from sage.structure.parent_gens import normalize_names -from sage.structure.element import is_Element import ring import weakref @@ -116,10 +115,4 @@ y^2 + y - Often the quotes are not needed, because of predefined symbolic variables: - sage: R = GF(9,a)[x] - sage: R - Univariate Polynomial Ring in x over Finite Field in a of size 3^2 - - In fact, since the diamond brackets on the left determine the variable name, you can omit the variable from the square brackets: @@ -198,10 +191,4 @@ """ import polynomial_ring as m - - if is_Element(arg1) and not isinstance(arg1, (int, long, m.integer.Integer)): - arg1 = repr(arg1) - if is_Element(arg2) and not isinstance(arg2, (int, long, m.integer.Integer)): - arg2 = repr(arg2) - if isinstance(arg1, (int, long, m.integer.Integer)): arg1, arg2 = arg2, arg1 @@ -219,8 +206,4 @@ R = None - if isinstance(arg1, (list, tuple)): - arg1 = [str(x) for x in arg1] - if isinstance(arg2, (list, tuple)): - arg2 = [str(x) for x in arg2] if isinstance(arg2, (int, long, m.integer.Integer)): # 3. PolynomialRing(base_ring, names, n, order='degrevlex'): Index: sage/rings/polynomial_singular_interface.py =================================================================== --- sage/rings/polynomial_singular_interface.py (revision 4269) +++ sage/rings/polynomial_singular_interface.py (revision 2641) @@ -4,9 +4,4 @@ AUTHORS: -- Martin Albrecht (2006-04-21) - -TESTS: - sage: R = MPolynomialRing(GF(2**8,'a'),10,'x', order='revlex') - sage: R == loads(dumps(R)) - True """ @@ -35,6 +30,4 @@ from complex_field import is_ComplexField from real_mpfr import is_RealField -from complex_double import is_ComplexDoubleField -from real_double import is_RealDoubleField from integer_ring import ZZ import sage.rings.arith @@ -171,6 +164,6 @@ # size_t bits = 1 + (size_t) ((float)digits * 3.5); precision = self.base_ring().precision() - digits = sage.rings.arith.integer_ceil((2*precision - 2)/7.0) - self.__singular = singular.ring("(real,%d,0)"%digits, _vars, order=order, check=False) + digits = sage.rings.arith.ceil((2*precision - 2)/7.0) + self.__singular = singular.ring("(real,%d,0)"%digits, _vars, order=order) elif is_ComplexField(self.base_ring()): @@ -178,23 +171,13 @@ # size_t bits = 1 + (size_t) ((float)digits * 3.5); precision = self.base_ring().precision() - digits = sage.rings.arith.integer_ceil((2*precision - 2)/7.0) - self.__singular = singular.ring("(complex,%d,0,I)"%digits, _vars, order=order, check=False) - - elif is_RealDoubleField(self.base_ring()): - # singular converts to bits from base_10 in mpr_complex.cc by: - # size_t bits = 1 + (size_t) ((float)digits * 3.5); - self.__singular = singular.ring("(real,15,0)", _vars, order=order, check=False) - - elif is_ComplexDoubleField(self.base_ring()): - # singular converts to bits from base_10 in mpr_complex.cc by: - # size_t bits = 1 + (size_t) ((float)digits * 3.5); - self.__singular = singular.ring("(complex,15,0,I)", _vars, order=order, check=False) + digits = sage.rings.arith.ceil((2*precision - 2)/7.0) + self.__singular = singular.ring("(complex,%d,0,I)"%digits, _vars, order=order) elif self.base_ring().is_prime_field() or (self.base_ring() is ZZ and force): - self.__singular = singular.ring(self.characteristic(), _vars, order=order, check=False) + self.__singular = singular.ring(self.characteristic(), _vars, order=order) elif self.base_ring().is_finite(): #must be extension field gen = str(self.base_ring().gen()) - r = singular.ring( "(%s,%s)"%(self.characteristic(),gen), _vars, order=order, check=False) + r = singular.ring( "(%s,%s)"%(self.characteristic(),gen), _vars, order=order) self.__minpoly = "("+(str(self.base_ring().modulus()).replace("x",gen)).replace(" ","")+")" singular.eval("minpoly=%s"%(self.__minpoly) ) @@ -219,6 +202,4 @@ or is_RealField(base_ring) or is_ComplexField(base_ring) - or is_RealDoubleField(base_ring) - or is_ComplexDoubleField(base_ring) or base_ring is ZZ ) Index: sage/rings/power_series_ring.py =================================================================== --- sage/rings/power_series_ring.py (revision 4259) +++ sage/rings/power_series_ring.py (revision 3663) @@ -30,14 +30,5 @@ Power Series Ring in t2 over Power Series Ring in t over Integer Ring sage: S.base_ring() - Power Series Ring in t over Integer Ring - -We compute with power series over the symbolic ring. - sage: K. = PowerSeriesRing(SR, 5) - sage: f = a + b*t + c*t^2 + O(t^3) - sage: f*f - a^2 + ((b + a)^2 - b^2 - a^2)*t + ((c + b + a)^2 - (c + b)^2 - (b + a)^2 + 2*b^2)*t^2 + O(t^3) - sage: f = sqrt(2) + sqrt(3)*t + O(t^3) - sage: f^2 - 2 + ((sqrt(3) + sqrt(2))^2 - 5)*t + 3*t^2 + O(t^3) + Power Series Ring in t over Integer Ring Elements are first coerced to constants in base_ring, then coerced into the @@ -65,14 +56,4 @@ -- William Stein: the code -- Jeremy Cho (2006-05-17): some examples (above) - -TESTS: - sage: R. = PowerSeriesRing(QQ) - sage: R == loads(dumps(R)) - True - - sage: R. = PowerSeriesRing(QQ, sparse=True) - sage: R == loads(dumps(R)) - True - """ @@ -123,5 +104,5 @@ Traceback (most recent call last): ... - ValueError: first letter of variable name must be a letter + TypeError: illegal variable name sage: S = PowerSeriesRing(QQ, 'x', default_prec = 15); S @@ -130,6 +111,4 @@ 15 """ - if isinstance(name, (int,long,integer.Integer)): - default_prec = name if not names is None: name = names @@ -199,22 +178,4 @@ self.__generator = self.__power_series_class(self, [0,1], check=True, is_gen=True) self.__is_sparse = sparse - self.__params = (base_ring, name, default_prec, sparse) - - def __reduce__(self): - """ - TESTS: - sage: R. = PowerSeriesRing(ZZ) - sage: S = loads(dumps(R)); S - Power Series Ring in t over Integer Ring - sage: type(S) - - sage: R. = PowerSeriesRing(QQ, default_prec=10, sparse=True); R - Sparse Power Series Ring in t over Rational Field - sage: S = loads(dumps(R)); S - Sparse Power Series Ring in t over Rational Field - sage: type(S) - - """ - return unpickle_power_series_ring_v0, self.__params def _repr_(self): @@ -637,6 +598,2 @@ return self.laurent_series_ring() - -def unpickle_power_series_ring_v0(base_ring, name, default_prec, sparse): - return PowerSeriesRing(base_ring, name=name, default_prec = default_prec, sparse=sparse) - Index: age/rings/power_series_ring_element.pxd =================================================================== --- sage/rings/power_series_ring_element.pxd (revision 4159) +++ (revision ) @@ -1,9 +1,0 @@ -from sage.structure.element cimport AlgebraElement - -cdef class PowerSeries(AlgebraElement): - cdef char __is_gen - cdef _prec - cdef common_prec_c(self, PowerSeries other) - -cdef class PowerSeries_poly(PowerSeries): - cdef __f Index: sage/rings/power_series_ring_element.py =================================================================== --- sage/rings/power_series_ring_element.py (revision 3927) +++ sage/rings/power_series_ring_element.py (revision 3927) @@ -0,0 +1,1605 @@ +""" +Power Series + +SAGE provides an implementation of dense and sparse power series over +any SAGE base ring. + +AUTHOR: + -- William Stein + -- David Harvey (2006-09-11): added solve_linear_de() method + +EXAMPLE: + sage: R. = PowerSeriesRing(ZZ) + sage: R([1,2,3]) + 1 + 2*x + 3*x^2 + sage: R([1,2,3], 10) + 1 + 2*x + 3*x^2 + O(x^10) + sage: f = 1 + 2*x - 3*x^3 + O(x^4); f + 1 + 2*x - 3*x^3 + O(x^4) + sage: f^10 + 1 + 20*x + 180*x^2 + 930*x^3 + O(x^4) + sage: g = 1/f; g + 1 - 2*x + 4*x^2 - 5*x^3 + O(x^4) + sage: g * f + 1 + O(x^4) + +In Python (as opposted to SAGE) create the power series ring and its +generator as follows: + + sage: R, x = objgen(PowerSeriesRing(ZZ, 'x')) + sage: parent(x) + Power Series Ring in x over Integer Ring + +EXAM