Ticket #11595: trac_11595-exact-eigenspaces.patch

File trac_11595-exact-eigenspaces.patch, 32.9 KB (added by rbeezer, 11 years ago)

  • sage/calculus/wester.py 

    # HG changeset patch
# User Rob Beezer <beezer@ups.edu>
# Date 1311566250 25200
# Node ID d5a48f59037fce6106dcdd8bac96df5da8ac5281
# Parent  8532a2ad1e558cbc91ddaaa6b7cc79956dd1e8ba
11595: update exact eigenspace routines

diff --git a/sage/calculus/wester.py b/sage/calculus/wester.py
    a b  
    468468
    469469    sage: # (YES) Find the eigenvalues of a 3x3 integer matrix.
    470470    sage: m = matrix(QQ, 3, [5,-3,-7, -2,1,2, 2,-3,-4])
    471     sage: m.eigenspaces()
     471    sage: m.eigenspaces_left()
    472472    [
    473473    (3, Vector space of degree 3 and dimension 1 over Rational Field
    474474    User basis matrix:

  • sage/graphs/generic_graph.py 

    diff --git a/sage/graphs/generic_graph.py b/sage/graphs/generic_graph.py
    a b  
    1424214242            M = self.kirchhoff_matrix()
    1424314243        else:
    1424414244            M = self.adjacency_matrix()
    14245         return M.right_eigenspaces(algebraic_multiplicity=False)
     14245        # could pass format='all' to get QQbar eigenvalues and eigenspaces
     14246        # which would be a change in default behavior
     14247        return M.right_eigenspaces(format='galois', algebraic_multiplicity=False)
    1424614248   
    1424714249    ### Automorphism and isomorphism
    1424814250

  • sage/matrix/matrix2.pyx 

    diff --git a/sage/matrix/matrix2.pyx b/sage/matrix/matrix2.pyx
    a b  
    43614361            if f.degree() == n:
    43624362                break
    43634363        return f
    4364            
    4365     def eigenspaces(self, var='a', even_if_inexact=None):
     4364
     4365    # Deprecated Aug 2008 Trac #3794
     4366    # Removed July 2011
     4367    # def eigenspaces(self, var='a', even_if_inexact=None):
     4368
     4369    def eigenspaces_left(self, format='all', var='a', algebraic_multiplicity=False):
    43664370        r"""
    4367         Deprecated: Instead of eigenspaces, use eigenspaces_left
    4368         """
    4369         # sage.misc.misc.deprecation("Use eigenspaces_left")
    4370         if even_if_inexact is not None:
    4371             sage.misc.misc.deprecation("The 'even_if_inexact' parameter is deprecated; a warning will be issued if called over an inexact ring.")
    4372         return self.eigenspaces_left(var=var)
    4373 
    4374 
    4375 
    4376     def eigenspaces_left(self, var='a', algebraic_multiplicity=False):
    4377         r"""
    4378         Compute left eigenspaces of a matrix.
    4379        
     4371        Compute the left eigenspaces of a matrix.
     4372
     4373        Note that ``eigenspaces_left()`` and ``left_eigenspaces()``
     4374        are identical methods.  Here "left" refers to the eigenvectors
     4375        being placed to the left of the matrix.
     4376
     4377        INPUT:
     4378
     4379        - ``self`` - a square matrix over an exact field.  For inexact
     4380          matrices consult the numerical or symbolic matrix classes.
     4381
     4382        - ``format`` - default: 'all'
     4383          - 'all' - attempts to create every eigenspace.  This will
     4384            always be possible for matrices with rational entries.
     4385          - 'galois' - for each irreducible factor of the characteristic
     4386            polynomial, a single eigenspace will be output for a
     4387            single root/eigenvalue for the irreducible factor.
     4388
     4389        - ``var`` - default: 'a' - variable name used to
     4390          represent elements of the root field of each
     4391          irreducible factor of the characteristic polynomial.
     4392          If var='a', then the root fields will be in terms of
     4393          a0, a1, a2, ...., where the numbering runs across all
     4394          the irreducible factors of the characteristic polynomial,
     4395          even for linear factors.
     4396
     4397        - ``algebraic_multiplicity`` - default: False - whether or
     4398          not to include the algebraic multiplicity of each eigenvalue
     4399          in the output.  See the discussion below.
     4400
     4401        OUTPUT:
     4402
    43804403        If algebraic_multiplicity=False, return a list of pairs (e, V)
    4381         where e runs through all eigenvalues (up to Galois conjugation) of
    4382         this matrix, and V is the corresponding left eigenspace.
    4383        
    4384         If algebraic_multiplicity=True, return a list of pairs (e, V, n)
     4404        where e is an eigenvalue of the matrix, and V is the corresponding
     4405        left eigenspace.  For Galois conjugates of eigenvalues, there
     4406        may be just one representative eigenspace, depending on the
     4407        ``format`` keyword.
     4408
     4409        If algebraic_multiplicity=True, return a list of triples (e, V, n)
    43854410        where e and V are as above and n is the algebraic multiplicity of
    4386         the eigenvalue. If the eigenvalues are given symbolically, as roots
    4387         of an irreducible factor of the characteristic polynomial, then the
    4388         algebraic multiplicity returned is the multiplicity of each
    4389         conjugate eigenvalue.
    4390        
    4391         The eigenspaces are returned sorted by the corresponding
    4392         characteristic polynomials, where polynomials are sorted in
    4393         dictionary order starting with constant terms.
    4394        
    4395         INPUT:
    4396        
    4397        
    4398         -  ``var`` - variable name used to represent elements
    4399            of the root field of each irreducible factor of the characteristic
    4400            polynomial I.e., if var='a', then the root fields will be in terms
    4401            of a0, a1, a2, ..., ak.
    4402        
    4403        
     4411        the eigenvalue.
     4412
    44044413        .. warning::
    44054414
    44064415           Uses a somewhat naive algorithm (simply factors the
    44074416           characteristic polynomial and computes kernels directly
    44084417           over the extension field).
    44094418
    4410         TODO:
    4411 
    4412         Maybe implement the better algorithm that is in
    4413         dual_eigenvector in sage/modular/hecke/module.py.
    4414        
    4415         EXAMPLES: We compute the left eigenspaces of a `3\times 3`
    4416         rational matrix.
    4417        
    4418         ::
    4419        
     4419        EXAMPLES:
     4420
     4421        We compute the left eigenspaces of a `3\times 3`
     4422        rational matrix. First, we request `all` of the eigenvalues,
     4423        so the results are in the field of algebraic numbers, `QQbar`.
     4424        Then we request just one eigenspace per irreducible factor of
     4425        the characteristic polynomial with the `galois` keyword.  ::
     4426
    44204427            sage: A = matrix(QQ,3,3,range(9)); A
    44214428            [0 1 2]
    44224429            [3 4 5]
    44234430            [6 7 8]
    4424             sage: es = A.eigenspaces_left(); es
     4431            sage: es = A.eigenspaces_left(format='all'); es
     4432            [
     4433            (0, Vector space of degree 3 and dimension 1 over Rational Field
     4434            User basis matrix:
     4435            [ 1 -2  1]),
     4436            (-1.348469228349535?, Vector space of degree 3 and dimension 1 over Algebraic Field
     4437            User basis matrix:
     4438            [                   1  0.3101020514433644? -0.3797958971132713?]),
     4439            (13.34846922834954?, Vector space of degree 3 and dimension 1 over Algebraic Field
     4440            User basis matrix:
     4441            [                 1 1.289897948556636? 1.579795897113272?])
     4442            ]
     4443
     4444            sage: es = A.eigenspaces_left(format='galois'); es
    44254445            [
    44264446            (0, Vector space of degree 3 and dimension 1 over Rational Field
    44274447            User basis matrix:
    … …  
    44304450            User basis matrix:
    44314451            [            1 1/15*a1 + 2/5 2/15*a1 - 1/5])
    44324452            ]
    4433             sage: es = A.eigenspaces_left(algebraic_multiplicity=True); es
     4453            sage: es = A.eigenspaces_left(format='galois', algebraic_multiplicity=True); es
    44344454            [
    44354455            (0, Vector space of degree 3 and dimension 1 over Rational Field
    44364456            User basis matrix:
    … …  
    44434463            sage: delta = e*v - v*A
    44444464            sage: abs(abs(delta)) < 1e-10
    44454465            True
    4446        
    4447         The same computation, but with implicit base change to a field::
    4448        
     4466
     4467        The same computation, but with implicit base change to a field.  ::
     4468
    44494469            sage: A = matrix(ZZ,3,range(9)); A
    44504470            [0 1 2]
    44514471            [3 4 5]
    44524472            [6 7 8]
    4453             sage: A.eigenspaces_left()
     4473            sage: A.eigenspaces_left(format='galois')
    44544474            [
    44554475            (0, Vector space of degree 3 and dimension 1 over Rational Field
    44564476            User basis matrix:
    … …  
    44594479            User basis matrix:
    44604480            [            1 1/15*a1 + 2/5 2/15*a1 - 1/5])
    44614481            ]
    4462        
     4482
    44634483        We compute the left eigenspaces of the matrix of the Hecke operator
    4464         `T_2` on level 43 modular symbols.
    4465        
    4466         ::
    4467        
     4484        `T_2` on level 43 modular symbols, both with all eigenvalues (the default)
     4485        and with one subspace per factor. ::
     4486
    44684487            sage: A = ModularSymbols(43).T(2).matrix(); A
    44694488            [ 3  0  0  0  0  0 -1]
    44704489            [ 0 -2  1  0  0  0  0]
    … …  
    44884507            User basis matrix:
    44894508            [ 0  1  0  1 -1  1 -1]
    44904509            [ 0  0  1  0 -1  2 -1], 2),
     4510            (-1.414213562373095?, Vector space of degree 7 and dimension 2 over Algebraic Field
     4511            User basis matrix:
     4512            [                  0                   1                   0                  -1 0.4142135623730951?                   1                  -1]
     4513            [                  0                   0                   1                   0                  -1                   0  2.414213562373095?], 2),
     4514            (1.414213562373095?, Vector space of degree 7 and dimension 2 over Algebraic Field
     4515            User basis matrix:
     4516            [                   0                    1                    0                   -1  -2.414213562373095?                    1                   -1]
     4517            [                   0                    0                    1                    0                   -1                    0 -0.4142135623730951?], 2)
     4518            ]
     4519            sage: A.eigenspaces_left(format='galois', algebraic_multiplicity=True)
     4520            [
     4521            (3, Vector space of degree 7 and dimension 1 over Rational Field
     4522            User basis matrix:
     4523            [   1    0  1/7    0 -1/7    0 -2/7], 1),
     4524            (-2, Vector space of degree 7 and dimension 2 over Rational Field
     4525            User basis matrix:
     4526            [ 0  1  0  1 -1  1 -1]
     4527            [ 0  0  1  0 -1  2 -1], 2),
    44914528            (a2, Vector space of degree 7 and dimension 2 over Number Field in a2 with defining polynomial x^2 - 2
    44924529            User basis matrix:
    44934530            [      0       1       0      -1 -a2 - 1       1      -1]
    44944531            [      0       0       1       0      -1       0 -a2 + 1], 2)
    44954532            ]
    4496        
    4497         Next we compute the left eigenspaces over the finite field of order
    4498         11::
    4499        
     4533
     4534        Next we compute the left eigenspaces over the finite field of order 11. ::
     4535
    45004536            sage: A = ModularSymbols(43, base_ring=GF(11), sign=1).T(2).matrix(); A
    45014537            [ 3  9  0  0]
    45024538            [ 0  9  0  1]
    … …  
    45064542            Finite Field of size 11
    45074543            sage: A.charpoly()
    45084544            x^4 + 10*x^3 + 3*x^2 + 2*x + 1
    4509             sage: A.eigenspaces_left(var = 'beta')
     4545            sage: A.eigenspaces_left(format='galois', var = 'beta')
    45104546            [
    45114547            (9, Vector space of degree 4 and dimension 1 over Finite Field of size 11
    45124548            User basis matrix:
    … …  
    45184554            User basis matrix:
    45194555            [           0            1            0 5*beta2 + 10])
    45204556            ]
    4521        
    4522         TESTS:
    4523 
    4524         Warnings are issued if the generic algorithm is used over
    4525         inexact fields. Garbage may result in these cases because of
    4526         numerical precision issues.
    4527        
    4528         ::
    4529        
    4530             sage: R=RealField(30)
    4531             sage: M=matrix(R,2,[2,1,1,1])
    4532             sage: M.eigenspaces_left() # random output from numerical issues
     4557
     4558        This method is only applicable to exact matrices.
     4559        The "eigenmatrix" routines for matrices with double-precision
     4560        floating-point entries (``RDF``, ``CDF``) are the best
     4561        alternative.  There are also "eigenmatrix" routines for
     4562        matrices with symbolic entries.  ::
     4563
     4564            sage: A = matrix(QQ, 3, 3, range(9))
     4565            sage: A.change_ring(RR).eigenspaces_left()
     4566            Traceback (most recent call last):
     4567            ...
     4568            NotImplementedError: eigenspaces cannot be computed reliably for inexact rings such as Real Field with 53 bits of precision,
     4569            consult numerical or symbolic matrix classes for other options
     4570
     4571            sage: em = A.change_ring(RDF).eigenmatrix_left()
     4572            sage: eigenvalues = em[0]; eigenvalues
     4573            [    13.3484692...                 0                 0]
     4574            [                0    -1.34846922...                 0]
     4575            [                0                 0 -6.2265089...e-16]
     4576            sage: eigenvectors = em[1]; eigenvectors
     4577            [ 0.440242867...  0.567868371...  0.695493875...]
     4578            [ 0.897878732...  0.278434036... -0.341010658...]
     4579            [ 0.408248290... -0.816496580...  0.408248290...]
     4580
     4581            sage: x, y = var('x y')
     4582            sage: S = matrix([[x, y], [y, 3*x^2]])
     4583            sage: em = S.eigenmatrix_left()
     4584            sage: eigenvalues = em[0]; eigenvalues
     4585            [-1/2*sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2) + 3/2*x^2 + 1/2*x                                                        0]
     4586            [                                                       0  3/2*x^2 + 1/2*x + 1/2*sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2)]
     4587            sage: eigenvectors = em[1]; eigenvectors
     4588            [                                                     1  1/2*(3*x^2 - x - sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2))/y]
     4589            [                                                     1 -1/2*(x - sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2) - 3*x^2)/y]
     4590
     4591        A request for ``'all'`` the eigenvalues, when it is not
     4592        possible, will raise an error.  Using the ``'galois'``
     4593        format option is more likely to be successful.  ::
     4594
     4595            sage: F.<b> = FiniteField(11^2)
     4596            sage: A = matrix(F, [[b + 1, b + 1], [10*b + 4, 5*b + 4]])
     4597            sage: A.eigenspaces_left(format='all')
     4598            Traceback (most recent call last):
     4599            ...
     4600            NotImplementedError: unable to construct eigenspaces for eigenvalues outside the base field,
     4601            try the keyword option: format='galois'
     4602
     4603            sage: A.eigenspaces_left(format='galois')
    45334604            [
    4534             (2.6180340, Vector space of degree 2 and dimension 0 over Real Field with 30 bits of precision
     4605            (a0, Vector space of degree 2 and dimension 1 over Univariate Quotient Polynomial Ring in a0 over Finite Field in b of size 11^2 with modulus x^2 + (5*b + 6)*x + 8*b + 10
    45354606            User basis matrix:
    4536             []),
    4537             (0.38196601, Vector space of degree 2 and dimension 0 over Real Field with 30 bits of precision
    4538             User basis matrix:
    4539             [])
     4607            [               1 6*b*a0 + 3*b + 1])
    45404608            ]
    4541             sage: R=ComplexField(30)
    4542             sage: N=matrix(R,2,[2,1,1,1])
    4543             sage: N.eigenspaces_left() # random output from numerical issues
    4544             [
    4545             (2.6180340, Vector space of degree 2 and dimension 0 over Complex Field with 30 bits of precision
    4546             User basis matrix:
    4547             []),
    4548             (0.38196601, Vector space of degree 2 and dimension 0 over Complex Field with 30 bits of precision
    4549             User basis matrix:
    4550             [])
    4551             ]
    4552         """
    4553         x = self.fetch('eigenspaces_left')
     4609
     4610        TESTS::
     4611
     4612            sage: B = matrix(QQ, 2, 3, range(6))
     4613            sage: B.eigenspaces_left()
     4614            Traceback (most recent call last):
     4615            ...
     4616            TypeError: matrix must be square, not 2 x 3
     4617
     4618            sage: B = matrix(QQ, 4, 4, range(16))
     4619            sage: B.eigenspaces_left(format='junk')
     4620            Traceback (most recent call last):
     4621            ...
     4622            ValueError: format keyword must be 'all' or 'galois', not junk
     4623
     4624            sage: B.eigenspaces_left(algebraic_multiplicity='garbage')
     4625            Traceback (most recent call last):
     4626            ...
     4627            ValueError: algebraic_multiplicity keyword must be True or False
     4628
     4629        """
     4630        if not format in ['all', 'galois']:
     4631            raise ValueError("format keyword must be 'all' or 'galois', not {0}".format(format))
     4632        if not algebraic_multiplicity in [True, False]:
     4633            raise ValueError('algebraic_multiplicity keyword must be True or False'.format(algebraic_multiplicity))
     4634        if not self.is_square():
     4635            raise TypeError('matrix must be square, not {0} x {1}'.format(self.nrows(), self.ncols()))
     4636        if not self.base_ring().is_exact():
     4637            msg = ("eigenspaces cannot be computed reliably for inexact rings such as {0},\n",
     4638                   "consult numerical or symbolic matrix classes for other options")
     4639            raise NotImplementedError(''.join(msg).format(self.base_ring()))
     4640
     4641        key = 'eigenspaces_left_' + format + '{0}'.format(var)
     4642        x = self.fetch(key)
    45544643        if not x is None:
    45554644            if algebraic_multiplicity:
    45564645                return x
    45574646            else:
    4558                 return  Sequence([(e[0],e[1]) for e in x], cr=True)
    4559 
    4560         if not self.base_ring().is_exact():
    4561             from warnings import warn
    4562             warn("Using generic algorithm for an inexact ring, which will probably give incorrect results due to numerical precision issues.")
    4563 
    4564 
    4565 
    4566         # minpoly is rarely implemented and is unreliable (leading to hangs) via linbox when implemented
    4567         # as of 2007-03-25.
    4568         #try:
    4569         #    G = self.minpoly().factor()  # can be computed faster when available.
    4570         #except NotImplementedError:
    4571         G = self.fcp()   # factored charpoly of self.
     4647                return Sequence([(e[0],e[1]) for e in x], cr=True, check=False)
     4648
     4649        # Possible improvements:
     4650        # algorithm for dual_eigenvector in sage/modular/hecke/module.py
     4651        # use minpoly when algebraic_multiplicity=False
     4652        # as of 2007-03-25 minpoly is unreliable via linbox
     4653
     4654        import sage.rings.qqbar
     4655        import sage.categories.homset
     4656        G = self.fcp()   # factored characteristic polynomial
    45724657        V = []
    4573         i = 0
     4658        i = -1 # variable name index, increments for each eigenvalue
    45744659        for h, e in G:
     4660            i = i + 1
    45754661            if h.degree() == 1:
    45764662                alpha = -h[0]/h[1]
    45774663                F = alpha.parent()
    45784664                if F != self.base_ring():
    45794665                    self = self.change_ring(F)
    45804666                A = self - alpha
     4667                W = A.kernel()
     4668                V.append((alpha, W.ambient_module().span_of_basis(W.basis()), e))
    45814669            else:
    4582                 F = h.root_field('%s%s'%(var,i))
     4670                F = h.root_field('{0}{1}'.format(var,i))
    45834671                alpha = F.gen(0)
    45844672                A = self.change_ring(F) - alpha
    4585             W = A.kernel()
    4586             i = i + 1
    4587             V.append((alpha, W.ambient_module().span_of_basis(W.basis()), e))
    4588         V = Sequence(V, cr=True)
    4589         self.cache('eigenspaces_left', V)
     4673                W = A.kernel()
     4674                WB = W.basis()
     4675                if format == 'galois':
     4676                    V.append((alpha, W.ambient_module().span_of_basis(WB), e))
     4677                elif format == 'all':
     4678                    try:
     4679                        alpha_conj = alpha.galois_conjugates(sage.rings.qqbar.QQbar)
     4680                    except (AttributeError, TypeError):
     4681                        msg = ("unable to construct eigenspaces for eigenvalues outside the base field,\n"
     4682                               "try the keyword option: format='galois'")
     4683                        raise NotImplementedError(''.join(msg))
     4684                    for ev in alpha_conj:
     4685                        m = sage.categories.homset.hom(alpha.parent(), ev.parent(), ev)
     4686                        space = (ev.parent())**self.nrows()
     4687                        evec_list = [(space)([m(i) for i in v]) for v in WB]
     4688                        V.append((ev, space.span_of_basis(evec_list, already_echelonized=True), e))
     4689        V = Sequence(V, cr=True, check=False)
     4690        self.cache(key, V)
    45904691        if algebraic_multiplicity:
    45914692            return V
    45924693        else:
    4593             return Sequence([(e[0],e[1]) for e in V], cr=True)
    4594 
    4595     def eigenspaces_right(self, var='a', algebraic_multiplicity=False):
     4694            return Sequence([(e[0],e[1]) for e in V], cr=True, check=False)
     4695
     4696    left_eigenspaces = eigenspaces_left
     4697
     4698    def eigenspaces_right(self, format='all', var='a', algebraic_multiplicity=False):
    45964699        r"""
    4597         Compute right eigenspaces of a matrix.
    4598        
     4700        Compute the right eigenspaces of a matrix.
     4701
     4702        Note that ``eigenspaces_right()`` and ``right_eigenspaces()``
     4703        are identical methods.  Here "right" refers to the eigenvectors
     4704        being placed to the right of the matrix.
     4705
     4706        INPUT:
     4707
     4708        - ``self`` - a square matrix over an exact field.  For inexact
     4709          matrices consult the numerical or symbolic matrix classes.
     4710
     4711        - ``format`` - default: 'all'
     4712          - 'all' - attempts to create every eigenspace.  This will
     4713            always be possible for matrices with rational entries.
     4714          - 'galois' - for each irreducible factor of the characteristic
     4715            polynomial, a single eigenspace will be output for a
     4716            single root/eigenvalue for the irreducible factor.
     4717
     4718        - ``var`` - default: 'a' - variable name used to
     4719          represent elements of the root field of each
     4720          irreducible factor of the characteristic polynomial.
     4721          If var='a', then the root fields will be in terms of
     4722          a0, a1, a2, ...., where the numbering runs across all
     4723          the irreducible factors of the characteristic polynomial,
     4724          even for linear factors.
     4725
     4726        - ``algebraic_multiplicity`` - default: False - whether or
     4727          not to include the algebraic multiplicity of each eigenvalue
     4728          in the output.  See the discussion below.
     4729
     4730        OUTPUT:
     4731
    45994732        If algebraic_multiplicity=False, return a list of pairs (e, V)
    4600         where e runs through all eigenvalues (up to Galois conjugation) of
    4601         this matrix, and V is the corresponding right eigenspace.
    4602        
    4603         If algebraic_multiplicity=True, return a list of pairs (e, V, n)
     4733        where e is an eigenvalue of the matrix, and V is the corresponding
     4734        left eigenspace.  For Galois conjugates of eigenvalues, there
     4735        may be just one representative eigenspace, depending on the
     4736        ``format`` keyword.
     4737
     4738        If algebraic_multiplicity=True, return a list of triples (e, V, n)
    46044739        where e and V are as above and n is the algebraic multiplicity of
    4605         the eigenvalue. If the eigenvalues are given symbolically, as roots
    4606         of an irreducible factor of the characteristic polynomial, then the
    4607         algebraic multiplicity returned is the multiplicity of each
    4608         conjugate eigenvalue.
    4609        
    4610         The eigenspaces are returned sorted by the corresponding
    4611         characteristic polynomials, where polynomials are sorted in
    4612         dictionary order starting with constant terms.
    4613        
    4614         INPUT:
    4615        
    4616        
    4617         -  ``var`` - variable name used to represent elements
    4618            of the root field of each irreducible factor of the characteristic
    4619            polynomial I.e., if var='a', then the root fields will be in terms
    4620            of a0, a1, a2, ..., ak.
    4621        
    4622        
     4740        the eigenvalue.
     4741
    46234742        .. warning::
    46244743
    46254744           Uses a somewhat naive algorithm (simply factors the
    46264745           characteristic polynomial and computes kernels directly
    46274746           over the extension field).
    46284747
    4629         TODO: Maybe implement the better algorithm that
    4630         is in dual_eigenvector in sage/modular/hecke/module.py.
    4631        
    4632         EXAMPLES: We compute the right eigenspaces of a `3\times 3`
    4633         rational matrix.
    4634        
    4635         ::
    4636        
    4637             sage: A = matrix(QQ,3,3,range(9)); A
     4748        EXAMPLES:
     4749
     4750        Right eigenspaces are computed from the left eigenspaces of the
     4751        transpose of the matrix.  As such, there is a greater collection
     4752        of illustrative examples at the :meth:`eigenspaces_left`.
     4753
     4754        We compute the right eigenspaces of a `3\times 3` rational matrix.  ::
     4755
     4756            sage: A = matrix(QQ, 3 ,3, range(9)); A
    46384757            [0 1 2]
    46394758            [3 4 5]
    46404759            [6 7 8]
    4641             sage: es = A.eigenspaces_right(); es
     4760            sage: A.eigenspaces_right()
     4761            [
     4762            (0, Vector space of degree 3 and dimension 1 over Rational Field
     4763            User basis matrix:
     4764            [ 1 -2  1]),
     4765            (-1.348469228349535?, Vector space of degree 3 and dimension 1 over Algebraic Field
     4766            User basis matrix:
     4767            [                   1  0.1303061543300932? -0.7393876913398137?]),
     4768            (13.34846922834954?, Vector space of degree 3 and dimension 1 over Algebraic Field
     4769            User basis matrix:
     4770            [                 1 3.069693845669907? 5.139387691339814?])
     4771            ]
     4772            sage: es = A.eigenspaces_right(format='galois'); es
    46424773            [
    46434774            (0, Vector space of degree 3 and dimension 1 over Rational Field
    46444775            User basis matrix:
    … …  
    46474778            User basis matrix:
    46484779            [           1 1/5*a1 + 2/5 2/5*a1 - 1/5])
    46494780            ]
    4650             sage: es = A.eigenspaces_right(algebraic_multiplicity=True); es
     4781            sage: es = A.eigenspaces_right(format='galois', algebraic_multiplicity=True); es
    46514782            [
    46524783            (0, Vector space of degree 3 and dimension 1 over Rational Field
    46534784            User basis matrix:
    … …  
    46604791            sage: delta = v*e - A*v
    46614792            sage: abs(abs(delta)) < 1e-10
    46624793            True
    4663        
     4794
    46644795        The same computation, but with implicit base change to a field::
    4665        
    4666             sage: A = matrix(ZZ,3,range(9)); A
     4796
     4797            sage: A = matrix(ZZ, 3, range(9)); A
    46674798            [0 1 2]
    46684799            [3 4 5]
    46694800            [6 7 8]
    4670             sage: A.eigenspaces_right()
     4801            sage: A.eigenspaces_right(format='galois')
    46714802            [
    46724803            (0, Vector space of degree 3 and dimension 1 over Rational Field
    46734804            User basis matrix:
    … …  
    46764807            User basis matrix:
    46774808            [           1 1/5*a1 + 2/5 2/5*a1 - 1/5])
    46784809            ]
    4679        
    4680         TESTS: Warnings are issued if the generic algorithm is used over
    4681         inexact fields. Garbage may result in these cases because of
    4682         numerical precision issues.
    4683        
    4684         ::
    4685        
    4686             sage: R=RealField(30)
    4687             sage: M=matrix(R,2,[2,1,1,1])
    4688             sage: M.eigenspaces_right() # random output from numerical issues
    4689             [(2.6180340,
    4690             Vector space of degree 2 and dimension 0 over Real Field with 30 bits of precision
    4691             User basis matrix:
    4692             []),
    4693             (0.38196601,
    4694             Vector space of degree 2 and dimension 0 over Real Field with 30 bits of precision
    4695             User basis matrix:
    4696             [])]
    4697             sage: R=ComplexField(30)
    4698             sage: N=matrix(R,2,[2,1,1,1])
    4699             sage: N.eigenspaces_right() # random output from numerical issues
    4700             [(2.6180340,
    4701             Vector space of degree 2 and dimension 0 over Complex Field with 30 bits of precision
    4702             User basis matrix:
    4703             []),
    4704             (0.38196601,
    4705             Vector space of degree 2 and dimension 0 over Complex Field with 30 bits of precision
    4706             User basis matrix:
    4707             [])]
    4708         """
    4709         return self.transpose().eigenspaces_left(var=var, algebraic_multiplicity=algebraic_multiplicity)
     4810
     4811        This method is only applicable to exact matrices.
     4812        The "eigenmatrix" routines for matrices with double-precision
     4813        floating-point entries (``RDF``, ``CDF``) are the best
     4814        alternative.  There are also "eigenmatrix" routines for
     4815        matrices with symbolic entries.  ::
     4816
     4817            sage: B = matrix(RR, 3, 3, range(9))
     4818            sage: B.eigenspaces_right()
     4819            Traceback (most recent call last):
     4820            ...
     4821            NotImplementedError: eigenspaces cannot be computed reliably for inexact rings such as Real Field with 53 bits of precision,
     4822            consult numerical or symbolic matrix classes for other options
     4823
     4824            sage: em = B.change_ring(RDF).eigenmatrix_right()
     4825            sage: eigenvalues = em[0]; eigenvalues
     4826            [     13.3484692...                  0                  0]
     4827            [                 0     -1.34846922...                  0]
     4828            [                 0                  0 -8.86256604...e-16]
     4829            sage: eigenvectors = em[1]; eigenvectors
     4830            [ 0.164763817...  0.799699663...  0.408248290...]
     4831            [ 0.505774475...  0.104205787... -0.816496580...]
     4832            [ 0.846785134... -0.591288087...  0.408248290...]
     4833
     4834            sage: x, y = var('x y')
     4835            sage: S = matrix([[x, y], [y, 3*x^2]])
     4836            sage: em = S.eigenmatrix_right()
     4837            sage: eigenvalues = em[0]; eigenvalues
     4838            [-1/2*sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2) + 3/2*x^2 + 1/2*x                                                        0]
     4839            [                                                       0  3/2*x^2 + 1/2*x + 1/2*sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2)]
     4840            sage: eigenvectors = em[1]; eigenvectors
     4841            [                                                     1                                                      1]
     4842            [ 1/2*(3*x^2 - x - sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2))/y -1/2*(x - sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2) - 3*x^2)/y]
     4843
     4844        TESTS::
     4845
     4846            sage: B = matrix(QQ, 2, 3, range(6))
     4847            sage: B.eigenspaces_right()
     4848            Traceback (most recent call last):
     4849            ...
     4850            TypeError: matrix must be square, not 2 x 3
     4851
     4852            sage: B = matrix(QQ, 4, 4, range(16))
     4853            sage: B.eigenspaces_right(format='junk')
     4854            Traceback (most recent call last):
     4855            ...
     4856            ValueError: format keyword must be 'all' or 'galois', not junk
     4857
     4858            sage: B.eigenspaces_right(algebraic_multiplicity='garbage')
     4859            Traceback (most recent call last):
     4860            ...
     4861            ValueError: algebraic_multiplicity keyword must be True or False
     4862        """
     4863        if not format in ['all', 'galois']:
     4864            raise ValueError("format keyword must be 'all' or 'galois', not {0}".format(format))
     4865        if not algebraic_multiplicity in [True, False]:
     4866            raise ValueError('algebraic_multiplicity keyword must be True or False'.format(algebraic_multiplicity))
     4867        if not self.is_square():
     4868            raise TypeError('matrix must be square, not {0} x {1}'.format(self.nrows(), self.ncols()))
     4869        if not self.base_ring().is_exact():
     4870            msg = ("eigenspaces cannot be computed reliably for inexact rings such as {0},\n",
     4871                   "consult numerical or symbolic matrix classes for other options")
     4872            raise NotImplementedError(''.join(msg).format(self.base_ring()))
     4873
     4874        key = 'eigenspaces_right_' + format + '{0}'.format(var)
     4875        x = self.fetch(key)
     4876        if not x is None:
     4877            if algebraic_multiplicity:
     4878                return x
     4879            else:
     4880                return Sequence([(e[0],e[1]) for e in x], cr=True, check=False)
     4881
     4882        V = self.transpose().eigenspaces_left(format=format, var=var, algebraic_multiplicity=True)
     4883
     4884        self.cache(key, V)
     4885        if algebraic_multiplicity:
     4886            return V
     4887        else:
     4888            return Sequence([(e[0],e[1]) for e in V], cr=True, check=False)
    47104889
    47114890    right_eigenspaces = eigenspaces_right
    47124891
    … …  
    48835062        V = []
    48845063        from sage.rings.qqbar import QQbar
    48855064        from sage.categories.homset import hom
    4886         eigenspaces = self.eigenspaces_left(algebraic_multiplicity=True)
     5065        eigenspaces = self.eigenspaces_left(format='galois', algebraic_multiplicity=True)
    48875066        evec_list=[]
    48885067        n = self._nrows
    48895068        evec_eval_list = []

  • sage/modules/free_module.py 

    diff --git a/sage/modules/free_module.py b/sage/modules/free_module.py
    a b  
    816816       
    817817        ::
    818818       
    819             sage: v = matrix(RDF, 3, range(9)).eigenspaces()[0][1].basis()[0]
     819            sage: v = matrix(RDF, 3, range(9)).eigenspaces_left()[0][1].basis()[0]
    820820            sage: v.complex_vector()
    821821            (...0.440242867..., ...0.567868371..., ...0.695493875...)
    822822            sage: v.complex_vector().parent().echelonized_basis_matrix()[0] * v[0]

  • sage/quadratic_forms/quadratic_form__local_representation_conditions.py 

    diff --git a/sage/quadratic_forms/quadratic_form__local_representation_conditions.py b/sage/quadratic_forms/quadratic_form__local_representation_conditions.py
    a b  
    143143        ## Compute the local conditions at the real numbers (i.e. "p = infinity")
    144144        ## ----------------------------------------------------------------------
    145145        M = Q.matrix()
    146         E = M.eigenspaces()
     146        E = M.eigenspaces_left()
    147147        M_eigenvalues = [E[i][0]  for i in range(len(E))]
    148148
    149149        pos_flag = infinity