Ticket #14334: trac_14334_integration_multipoly-fc.patch

sage/misc/functional.py

@@ -658 +658 @@
         from sage.symbolic.ring import SR
         return SR(expression).sum(*args, **kwds)
 
+
 def integral(x, *args, **kwds):
     """
     Returns an indefinite or definite integral of an object x.
     
-    First call x.integrate() and if that fails make an object and
+    First call x.integral() and if that fails make an object and
     integrate it using Maxima, maple, etc, as specified by algorithm.
 
     For symbolic expression calls
-    ``sage.calculus.calculus.integral`` - see this function for
+    :func:`sage.calculus.calculus.integral` - see this function for
     available options.
     
     EXAMPLES::
@@ -751 +752 @@
 
 integrate = integral
 
+
 def integral_closure(x):
     """
     Returns the integral closure of x.

sage/rings/polynomial/multi_polynomial_element.py

@@ -1423 +1423 @@
         Q, _ = self.quo_rem(right)
         return Q
 
-
     def _derivative(self, var=None):
         r"""
-        Differentiates self with respect to variable var.
-        
-        If var is not one of the generators of this ring, _derivative(var)
+        Differentiates ``self`` with respect to variable ``var``.
+
+        If ``var`` is not one of the generators of this ring, _derivative(var)
         is called recursively on each coefficient of this polynomial.
-        
-        .. seealso::
 
-           :meth:`derivative`
-        
+        .. SEEALSO::
+
+            :meth:`derivative`
+
         EXAMPLES::
-        
+
             sage: R.<t> = PowerSeriesRing(QQbar)
             sage: S.<x, y> = PolynomialRing(R)
             sage: f = (t^2 + O(t^3))*x^2*y^3 + (37*t^4 + O(t^5))*x^3
-            sage: type(f)
-            <class 'sage.rings.polynomial.multi_polynomial_element.MPolynomial_polydict'>
+            sage: f.parent()
+            Multivariate Polynomial Ring in x, y over Power Series Ring in t over Algebraic Field
             sage: f._derivative(x)   # with respect to x
             (2*t^2 + O(t^3))*x*y^3 + (111*t^4 + O(t^5))*x^2
+            sage: f._derivative(x).parent()
+            Multivariate Polynomial Ring in x, y over Power Series Ring in t over Algebraic Field
             sage: f._derivative(y)   # with respect to y
             (3*t^2 + O(t^3))*x^2*y^2
             sage: f._derivative(t)   # with respect to t (recurses into base ring)
@@ -1458 +1459 @@
             ValueError: must specify which variable to differentiate with respect to
         """
         if var is None:
-            raise ValueError, "must specify which variable to differentiate with respect to"
+            raise ValueError("must specify which variable to differentiate with respect to")
 
         gens = list(self.parent().gens())
 
@@ -1481 +1482 @@
         d = polydict.PolyDict(d, self.parent().base_ring()(0), remove_zero=True)
         return MPolynomial_polydict(self.parent(), d)
 
+    def integral(self, var=None):
+        r"""
+        Integrates ``self`` with respect to variable ``var``.
+
+        .. NOTE::
+
+            The integral is always chosen so the constant term is 0.
+
+        If ``var`` is not one of the generators of this ring, integral(var)
+        is called recursively on each coefficient of this polynomial.
+
+        EXAMPLES:
+
+        On polynomials with rational coefficients::
+
+            sage: x, y = PolynomialRing(QQ, 'x, y').gens()
+            sage: ex = x*y + x - y
+            sage: it = ex.integral(x); it
+            1/2*x^2*y + 1/2*x^2 - x*y
+            sage: it.parent() == x.parent()
+            True
+
+        On polynomials with coefficients in power series::
+
+            sage: R.<t> = PowerSeriesRing(QQbar)
+            sage: S.<x, y> = PolynomialRing(R)
+            sage: f = (t^2 + O(t^3))*x^2*y^3 + (37*t^4 + O(t^5))*x^3
+            sage: f.parent()
+            Multivariate Polynomial Ring in x, y over Power Series Ring in t over Algebraic Field
+            sage: f.integral(x)   # with respect to x
+            (1/3*t^2 + O(t^3))*x^3*y^3 + (37/4*t^4 + O(t^5))*x^4
+            sage: f.integral(x).parent()
+            Multivariate Polynomial Ring in x, y over Power Series Ring in t over Algebraic Field
+
+            sage: f.integral(y)   # with respect to y
+            (1/4*t^2 + O(t^3))*x^2*y^4 + (37*t^4 + O(t^5))*x^3*y
+            sage: f.integral(t)   # with respect to t (recurses into base ring)
+            (1/3*t^3 + O(t^4))*x^2*y^3 + (37/5*t^5 + O(t^6))*x^3
+
+        TESTS::
+
+            sage: f.integral()    # can't figure out the variable
+            Traceback (most recent call last):
+            ...
+            ValueError: must specify which variable to integrate with respect to
+        """
+        if var is None:
+            raise ValueError("must specify which variable to integrate "
+                             "with respect to")
+
+        gens = list(self.parent().gens())
+
+        # check if var is one of the generators
+        try:
+            index = gens.index(var)
+        except ValueError:
+            # var is not a generator; do term-by-term integration recursively
+            # var may be, for example, a generator of the base ring
+            d = dict([(e, x.integral(var))
+                      for (e, x) in self.dict().iteritems()])
+            d = polydict.PolyDict(d, self.parent().base_ring()(0),
+                                  remove_zero=True)
+            return MPolynomial_polydict(self.parent(), d)
+
+        # integrate w.r.t. indicated variable
+        d = {}
+        v = polydict.ETuple({index:1}, len(gens))
+        for (exp, coeff) in self.dict().iteritems():
+            d[exp.eadd(v)] = coeff / (1+exp[index])
+        d = polydict.PolyDict(d, self.parent().base_ring()(0), remove_zero=True)
+        return MPolynomial_polydict(self.parent(), d)
+
     def factor(self, proof=True):
         r"""
         Compute the irreducible factorization of this polynomial.

sage/rings/polynomial/multi_polynomial_libsingular.pyx

@@ -4764 +4764 @@
         return new_MP(self._parent,p)
 
 
+    def integral(self, MPolynomial_libsingular var):
+        """
+        Integrates this polynomial with respect to the provided
+        variable.
+
+        One requires that `\QQ` is contained in the ring.
+
+        INPUT:
+
+        - ``variable`` - the integral is taken with respect to variable
+
+        EXAMPLES::
+
+            sage: R.<x, y> = PolynomialRing(QQ, 2)
+            sage: f = 3*x^3*y^2 + 5*y^2 + 3*x + 2
+            sage: f.integral(x)
+            3/4*x^4*y^2 + 5*x*y^2 + 3/2*x^2 + 2*x
+            sage: f.integral(y)
+            x^3*y^3 + 5/3*y^3 + 3*x*y + 2*y
+
+        TESTS::
+
+            sage: z, w = polygen(QQ, 'z, w')
+            sage: f.integral(z)
+            Traceback (most recent call last):
+            ...
+            TypeError: the variable is not in the same ring as self
+        """
+        if var is None:
+            raise ValueError("please specify a variable")
+
+        ring = var.parent()
+        if ring is not self._parent:
+            raise TypeError("the variable is not in the same ring as self")
+
+        if not ring.has_coerce_map_from(RationalField()):
+            raise TypeError("the ring must contain the rational numbers")
+
+        gens = ring.gens()
+
+        try:
+            index = gens.index(var)
+        except ValueError:
+            raise TypeError("not a variable in the same ring as self")
+
+        d = {}
+        v = ETuple({index:1}, len(gens))
+        for (exp, coeff) in self.dict().iteritems():
+            d[exp.eadd(v)] = coeff / (1+exp[index])
+        return MPolynomial_polydict(self.parent(), d)
+
+
     def resultant(self, MPolynomial_libsingular other, variable=None):
         """
         Compute the resultant of this polynomial and the first

sage/rings/polynomial/polynomial_element.pyx

@@ -2366 +2366 @@
         coeffs = self.list()
         return self._parent([n*coeffs[n] for n from 1 <= n <= degree])
 
-    def integral(self):
+    def integral(self,var=None):
         """
         Return the integral of this polynomial.
-        
-        .. note::
-
-           The integral is always chosen so the constant term is 0.
-        
+
+        By default, the integration variable is the variable of the
+        polynomial.
+
+        Otherwise, the integration variable is the optional parameter ``var``
+
+        .. NOTE::
+
+            The integral is always chosen so the constant term is 0.
+
         EXAMPLES::
-        
+
             sage: R.<x> = ZZ[]
             sage: R(0).integral()
             0
             sage: f = R(2).integral(); f
             2*x
-        
+
         Note that the integral lives over the fraction field of the
         scalar coefficients::
-        
+
             sage: f.parent()
             Univariate Polynomial Ring in x over Rational Field
             sage: R(0).integral().parent()
             Univariate Polynomial Ring in x over Rational Field
-        
+
             sage: f = x^3 + x - 2
             sage: g = f.integral(); g
             1/4*x^4 + 1/2*x^2 - 2*x
             sage: g.parent()
             Univariate Polynomial Ring in x over Rational Field
 
-        This shows that the issue at trac #7711 is resolved::
+        This shows that the issue at :trac:`7711` is resolved::
 
             sage: P.<x,z> = PolynomialRing(GF(2147483647))
             sage: Q.<y> = PolynomialRing(P)
@@ -2434 +2439 @@
             Univariate Polynomial Ring in x over Power Series Ring in c
             over Univariate Polynomial Ring in b over Multivariate Polynomial
             Ring in a1, a2 over Rational Field
-        """
+
+        Integration with respect to a variable in the base ring::
+
+            sage: R.<x> = QQ[]
+            sage: t = PolynomialRing(R,'t').gen()
+            sage: f = x*t +5*t^2
+            sage: f.integral(x)
+            5*x*t^2 + 1/2*x^2*t
+        """
+        if var is not None and var != self._parent.gen():
+            # call integral() recursively on coefficients
+            return self._parent([coeff.integral(var) for coeff in self.list()])
         cdef Py_ssize_t n, degree = self.degree()
         R = self.parent()
         Q = (self.constant_coefficient()/1).parent()

sage/rings/power_series_poly.pyx

@@ -862 +862 @@
         return PowerSeries_poly(self._parent, self.__f._derivative(),
                                 self.prec()-1, check=False)
 
-    def integral(self):
+    def integral(self,var=None):
         """
-        The integral of this power series with 0 constant term.
+        The integral of this power series
+
+        By default, the integration variable is the variable of the
+        power series.
+
+        Otherwise, the integration variable is the optional parameter ``var``
+
+        .. NOTE::
+
+            The integral is always chosen so the constant term is 0.
 
         EXAMPLES::
-        
+
             sage: k.<w> = QQ[[]]
             sage: (1+17*w+15*w^3+O(w^5)).integral()
-            w + 17/2*w^2 + 15/4*w^4 + O(w^6)        
+            w + 17/2*w^2 + 15/4*w^4 + O(w^6)
             sage: (w^3 + 4*w^4 + O(w^7)).integral()
             1/4*w^4 + 4/5*w^5 + O(w^8)
             sage: (3*w^2).integral()
             w^3
+
+        TESTS::
+
+            sage: t = PowerSeriesRing(QQ,'t').gen()
+            sage: f = t + 5*t^2 + 21*t^3
+            sage: g = f.integral() ; g
+            1/2*t^2 + 5/3*t^3 + 21/4*t^4
+            sage: g.parent()
+            Power Series Ring in t over Rational Field
+
+            sage: R.<x> = QQ[]
+            sage: t = PowerSeriesRing(R,'t').gen()
+            sage: f = x*t +5*t^2
+            sage: f.integral()
+            1/2*x*t^2 + 5/3*t^3
+            sage: f.integral(x)
+            1/2*x^2*t + 5*x*t^2
         """
-        return PowerSeries_poly(self._parent, self.__f.integral(),
-                                         self.prec()+1, check=False)
+        return PowerSeries_poly(self._parent, self.__f.integral(var),
+                                self.prec()+1, check=False)
 
     def reversion(self, precision=None):
         """