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from sympy.external.gmpy import GROUND_TYPES
from sympy import Integer, Rational, S, sqrt, Matrix, symbols
from sympy import FF, ZZ, QQ, QQ_I, EXRAW
from sympy.polys.matrices.domainmatrix import DomainMatrix, DomainScalar, DM
from sympy.polys.matrices.exceptions import (
DMBadInputError, DMDomainError, DMShapeError, DMFormatError, DMNotAField,
DMNonSquareMatrixError, DMNonInvertibleMatrixError,
)
from sympy.polys.matrices.ddm import DDM
from sympy.polys.matrices.sdm import SDM
from sympy.testing.pytest import raises
def test_DM():
ddm = DDM([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A = DM([[1, 2], [3, 4]], ZZ)
if GROUND_TYPES != 'flint':
assert A.rep == ddm
else:
assert A.rep == ddm.to_dfm()
assert A.shape == (2, 2)
assert A.domain == ZZ
def test_DomainMatrix_init():
lol = [[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]]
dod = {0: {0: ZZ(1), 1:ZZ(2)}, 1: {0:ZZ(3), 1:ZZ(4)}}
ddm = DDM(lol, (2, 2), ZZ)
sdm = SDM(dod, (2, 2), ZZ)
A = DomainMatrix(lol, (2, 2), ZZ)
if GROUND_TYPES != 'flint':
assert A.rep == ddm
else:
assert A.rep == ddm.to_dfm()
assert A.shape == (2, 2)
assert A.domain == ZZ
A = DomainMatrix(dod, (2, 2), ZZ)
assert A.rep == sdm
assert A.shape == (2, 2)
assert A.domain == ZZ
raises(TypeError, lambda: DomainMatrix(ddm, (2, 2), ZZ))
raises(TypeError, lambda: DomainMatrix(sdm, (2, 2), ZZ))
raises(TypeError, lambda: DomainMatrix(Matrix([[1]]), (1, 1), ZZ))
for fmt, rep in [('sparse', sdm), ('dense', ddm)]:
if fmt == 'dense' and GROUND_TYPES == 'flint':
rep = rep.to_dfm()
A = DomainMatrix(lol, (2, 2), ZZ, fmt=fmt)
assert A.rep == rep
A = DomainMatrix(dod, (2, 2), ZZ, fmt=fmt)
assert A.rep == rep
raises(ValueError, lambda: DomainMatrix(lol, (2, 2), ZZ, fmt='invalid'))
raises(DMBadInputError, lambda: DomainMatrix([[ZZ(1), ZZ(2)]], (2, 2), ZZ))
def test_DomainMatrix_from_rep():
ddm = DDM([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A = DomainMatrix.from_rep(ddm)
# XXX: Should from_rep convert to DFM?
assert A.rep == ddm
assert A.shape == (2, 2)
assert A.domain == ZZ
sdm = SDM({0: {0: ZZ(1), 1:ZZ(2)}, 1: {0:ZZ(3), 1:ZZ(4)}}, (2, 2), ZZ)
A = DomainMatrix.from_rep(sdm)
assert A.rep == sdm
assert A.shape == (2, 2)
assert A.domain == ZZ
A = DomainMatrix([[ZZ(1)]], (1, 1), ZZ)
raises(TypeError, lambda: DomainMatrix.from_rep(A))
def test_DomainMatrix_from_list():
ddm = DDM([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A = DomainMatrix.from_list([[1, 2], [3, 4]], ZZ)
if GROUND_TYPES != 'flint':
assert A.rep == ddm
else:
assert A.rep == ddm.to_dfm()
assert A.shape == (2, 2)
assert A.domain == ZZ
dom = FF(7)
ddm = DDM([[dom(1), dom(2)], [dom(3), dom(4)]], (2, 2), dom)
A = DomainMatrix.from_list([[1, 2], [3, 4]], dom)
# Not a DFM because FF(7) is not supported by DFM
assert A.rep == ddm
assert A.shape == (2, 2)
assert A.domain == dom
ddm = DDM([[QQ(1, 2), QQ(3, 1)], [QQ(1, 4), QQ(5, 1)]], (2, 2), QQ)
A = DomainMatrix.from_list([[(1, 2), (3, 1)], [(1, 4), (5, 1)]], QQ)
if GROUND_TYPES != 'flint':
assert A.rep == ddm
else:
assert A.rep == ddm.to_dfm()
assert A.shape == (2, 2)
assert A.domain == QQ
def test_DomainMatrix_from_list_sympy():
ddm = DDM([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A = DomainMatrix.from_list_sympy(2, 2, [[1, 2], [3, 4]])
if GROUND_TYPES != 'flint':
assert A.rep == ddm
else:
assert A.rep == ddm.to_dfm()
assert A.shape == (2, 2)
assert A.domain == ZZ
K = QQ.algebraic_field(sqrt(2))
ddm = DDM(
[[K.convert(1 + sqrt(2)), K.convert(2 + sqrt(2))],
[K.convert(3 + sqrt(2)), K.convert(4 + sqrt(2))]],
(2, 2),
K
)
A = DomainMatrix.from_list_sympy(
2, 2, [[1 + sqrt(2), 2 + sqrt(2)], [3 + sqrt(2), 4 + sqrt(2)]],
extension=True)
assert A.rep == ddm
assert A.shape == (2, 2)
assert A.domain == K
def test_DomainMatrix_from_dict_sympy():
sdm = SDM({0: {0: QQ(1, 2)}, 1: {1: QQ(2, 3)}}, (2, 2), QQ)
sympy_dict = {0: {0: Rational(1, 2)}, 1: {1: Rational(2, 3)}}
A = DomainMatrix.from_dict_sympy(2, 2, sympy_dict)
assert A.rep == sdm
assert A.shape == (2, 2)
assert A.domain == QQ
fds = DomainMatrix.from_dict_sympy
raises(DMBadInputError, lambda: fds(2, 2, {3: {0: Rational(1, 2)}}))
raises(DMBadInputError, lambda: fds(2, 2, {0: {3: Rational(1, 2)}}))
def test_DomainMatrix_from_Matrix():
sdm = SDM({0: {0: ZZ(1), 1: ZZ(2)}, 1: {0: ZZ(3), 1: ZZ(4)}}, (2, 2), ZZ)
A = DomainMatrix.from_Matrix(Matrix([[1, 2], [3, 4]]))
assert A.rep == sdm
assert A.shape == (2, 2)
assert A.domain == ZZ
K = QQ.algebraic_field(sqrt(2))
sdm = SDM(
{0: {0: K.convert(1 + sqrt(2)), 1: K.convert(2 + sqrt(2))},
1: {0: K.convert(3 + sqrt(2)), 1: K.convert(4 + sqrt(2))}},
(2, 2),
K
)
A = DomainMatrix.from_Matrix(
Matrix([[1 + sqrt(2), 2 + sqrt(2)], [3 + sqrt(2), 4 + sqrt(2)]]),
extension=True)
assert A.rep == sdm
assert A.shape == (2, 2)
assert A.domain == K
A = DomainMatrix.from_Matrix(Matrix([[QQ(1, 2), QQ(3, 4)], [QQ(0, 1), QQ(0, 1)]]), fmt='dense')
ddm = DDM([[QQ(1, 2), QQ(3, 4)], [QQ(0, 1), QQ(0, 1)]], (2, 2), QQ)
if GROUND_TYPES != 'flint':
assert A.rep == ddm
else:
assert A.rep == ddm.to_dfm()
assert A.shape == (2, 2)
assert A.domain == QQ
def test_DomainMatrix_eq():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A == A
B = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(1)]], (2, 2), ZZ)
assert A != B
C = [[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]]
assert A != C
def test_DomainMatrix_unify_eq():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
B1 = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
B2 = DomainMatrix([[QQ(1), QQ(3)], [QQ(3), QQ(4)]], (2, 2), QQ)
B3 = DomainMatrix([[ZZ(1)]], (1, 1), ZZ)
assert A.unify_eq(B1) is True
assert A.unify_eq(B2) is False
assert A.unify_eq(B3) is False
def test_DomainMatrix_get_domain():
K, items = DomainMatrix.get_domain([1, 2, 3, 4])
assert items == [ZZ(1), ZZ(2), ZZ(3), ZZ(4)]
assert K == ZZ
K, items = DomainMatrix.get_domain([1, 2, 3, Rational(1, 2)])
assert items == [QQ(1), QQ(2), QQ(3), QQ(1, 2)]
assert K == QQ
def test_DomainMatrix_convert_to():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Aq = A.convert_to(QQ)
assert Aq == DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
def test_DomainMatrix_choose_domain():
A = [[1, 2], [3, 0]]
assert DM(A, QQ).choose_domain() == DM(A, ZZ)
assert DM(A, QQ).choose_domain(field=True) == DM(A, QQ)
assert DM(A, ZZ).choose_domain(field=True) == DM(A, QQ)
x = symbols('x')
B = [[1, x], [x**2, x**3]]
assert DM(B, QQ[x]).choose_domain(field=True) == DM(B, ZZ.frac_field(x))
def test_DomainMatrix_to_flat_nz():
Adm = DM([[1, 2], [3, 0]], ZZ)
Addm = Adm.rep.to_ddm()
Asdm = Adm.rep.to_sdm()
for A in [Adm, Addm, Asdm]:
elems, data = A.to_flat_nz()
assert A.from_flat_nz(elems, data, A.domain) == A
elemsq = [QQ(e) for e in elems]
assert A.from_flat_nz(elemsq, data, QQ) == A.convert_to(QQ)
elems2 = [2*e for e in elems]
assert A.from_flat_nz(elems2, data, A.domain) == 2*A
def test_DomainMatrix_to_sympy():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A.to_sympy() == A.convert_to(EXRAW)
def test_DomainMatrix_to_field():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Aq = A.to_field()
assert Aq == DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
def test_DomainMatrix_to_sparse():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A_sparse = A.to_sparse()
assert A_sparse.rep == {0: {0: 1, 1: 2}, 1: {0: 3, 1: 4}}
def test_DomainMatrix_to_dense():
A = DomainMatrix({0: {0: 1, 1: 2}, 1: {0: 3, 1: 4}}, (2, 2), ZZ)
A_dense = A.to_dense()
ddm = DDM([[1, 2], [3, 4]], (2, 2), ZZ)
if GROUND_TYPES != 'flint':
assert A_dense.rep == ddm
else:
assert A_dense.rep == ddm.to_dfm()
def test_DomainMatrix_unify():
Az = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Aq = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
assert Az.unify(Az) == (Az, Az)
assert Az.unify(Aq) == (Aq, Aq)
assert Aq.unify(Az) == (Aq, Aq)
assert Aq.unify(Aq) == (Aq, Aq)
As = DomainMatrix({0: {1: ZZ(1)}, 1:{0:ZZ(2)}}, (2, 2), ZZ)
Ad = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert As.unify(As) == (As, As)
assert Ad.unify(Ad) == (Ad, Ad)
Bs, Bd = As.unify(Ad, fmt='dense')
assert Bs.rep == DDM([[0, 1], [2, 0]], (2, 2), ZZ).to_dfm_or_ddm()
assert Bd.rep == DDM([[1, 2],[3, 4]], (2, 2), ZZ).to_dfm_or_ddm()
Bs, Bd = As.unify(Ad, fmt='sparse')
assert Bs.rep == SDM({0: {1: 1}, 1: {0: 2}}, (2, 2), ZZ)
assert Bd.rep == SDM({0: {0: 1, 1: 2}, 1: {0: 3, 1: 4}}, (2, 2), ZZ)
raises(ValueError, lambda: As.unify(Ad, fmt='invalid'))
def test_DomainMatrix_to_Matrix():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A_Matrix = Matrix([[1, 2], [3, 4]])
assert A.to_Matrix() == A_Matrix
assert A.to_sparse().to_Matrix() == A_Matrix
assert A.convert_to(QQ).to_Matrix() == A_Matrix
assert A.convert_to(QQ.algebraic_field(sqrt(2))).to_Matrix() == A_Matrix
def test_DomainMatrix_to_list():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A.to_list() == [[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]]
def test_DomainMatrix_to_list_flat():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A.to_list_flat() == [ZZ(1), ZZ(2), ZZ(3), ZZ(4)]
def test_DomainMatrix_flat():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A.flat() == [ZZ(1), ZZ(2), ZZ(3), ZZ(4)]
def test_DomainMatrix_from_list_flat():
nums = [ZZ(1), ZZ(2), ZZ(3), ZZ(4)]
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert DomainMatrix.from_list_flat(nums, (2, 2), ZZ) == A
assert DDM.from_list_flat(nums, (2, 2), ZZ) == A.rep.to_ddm()
assert SDM.from_list_flat(nums, (2, 2), ZZ) == A.rep.to_sdm()
assert A == A.from_list_flat(A.to_list_flat(), A.shape, A.domain)
raises(DMBadInputError, DomainMatrix.from_list_flat, nums, (2, 3), ZZ)
raises(DMBadInputError, DDM.from_list_flat, nums, (2, 3), ZZ)
raises(DMBadInputError, SDM.from_list_flat, nums, (2, 3), ZZ)
def test_DomainMatrix_to_dod():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A.to_dod() == {0: {0: ZZ(1), 1:ZZ(2)}, 1: {0:ZZ(3), 1:ZZ(4)}}
A = DomainMatrix([[ZZ(1), ZZ(0)], [ZZ(0), ZZ(4)]], (2, 2), ZZ)
assert A.to_dod() == {0: {0: ZZ(1)}, 1: {1: ZZ(4)}}
def test_DomainMatrix_from_dod():
items = {0: {0: ZZ(1), 1:ZZ(2)}, 1: {0:ZZ(3), 1:ZZ(4)}}
A = DM([[1, 2], [3, 4]], ZZ)
assert DomainMatrix.from_dod(items, (2, 2), ZZ) == A.to_sparse()
assert A.from_dod_like(items) == A
assert A.from_dod_like(items, QQ) == A.convert_to(QQ)
def test_DomainMatrix_to_dok():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A.to_dok() == {(0, 0):ZZ(1), (0, 1):ZZ(2), (1, 0):ZZ(3), (1, 1):ZZ(4)}
A = DomainMatrix([[ZZ(1), ZZ(0)], [ZZ(0), ZZ(4)]], (2, 2), ZZ)
dok = {(0, 0):ZZ(1), (1, 1):ZZ(4)}
assert A.to_dok() == dok
assert A.to_dense().to_dok() == dok
assert A.to_sparse().to_dok() == dok
assert A.rep.to_ddm().to_dok() == dok
assert A.rep.to_sdm().to_dok() == dok
def test_DomainMatrix_from_dok():
items = {(0, 0): ZZ(1), (1, 1): ZZ(2)}
A = DM([[1, 0], [0, 2]], ZZ)
assert DomainMatrix.from_dok(items, (2, 2), ZZ) == A.to_sparse()
assert DDM.from_dok(items, (2, 2), ZZ) == A.rep.to_ddm()
assert SDM.from_dok(items, (2, 2), ZZ) == A.rep.to_sdm()
def test_DomainMatrix_repr():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert repr(A) == 'DomainMatrix([[1, 2], [3, 4]], (2, 2), ZZ)'
def test_DomainMatrix_transpose():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
AT = DomainMatrix([[ZZ(1), ZZ(3)], [ZZ(2), ZZ(4)]], (2, 2), ZZ)
assert A.transpose() == AT
def test_DomainMatrix_is_zero_matrix():
A = DomainMatrix([[ZZ(1)]], (1, 1), ZZ)
B = DomainMatrix([[ZZ(0)]], (1, 1), ZZ)
assert A.is_zero_matrix is False
assert B.is_zero_matrix is True
def test_DomainMatrix_is_upper():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(0), ZZ(4)]], (2, 2), ZZ)
B = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A.is_upper is True
assert B.is_upper is False
def test_DomainMatrix_is_lower():
A = DomainMatrix([[ZZ(1), ZZ(0)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
B = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A.is_lower is True
assert B.is_lower is False
def test_DomainMatrix_is_diagonal():
A = DM([[1, 0], [0, 4]], ZZ)
B = DM([[1, 2], [3, 4]], ZZ)
assert A.is_diagonal is A.to_sparse().is_diagonal is True
assert B.is_diagonal is B.to_sparse().is_diagonal is False
def test_DomainMatrix_is_square():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
B = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)], [ZZ(5), ZZ(6)]], (3, 2), ZZ)
assert A.is_square is True
assert B.is_square is False
def test_DomainMatrix_diagonal():
A = DM([[1, 2], [3, 4]], ZZ)
assert A.diagonal() == A.to_sparse().diagonal() == [ZZ(1), ZZ(4)]
A = DM([[1, 2], [3, 4], [5, 6]], ZZ)
assert A.diagonal() == A.to_sparse().diagonal() == [ZZ(1), ZZ(4)]
A = DM([[1, 2, 3], [4, 5, 6]], ZZ)
assert A.diagonal() == A.to_sparse().diagonal() == [ZZ(1), ZZ(5)]
def test_DomainMatrix_rank():
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)], [QQ(6), QQ(8)]], (3, 2), QQ)
assert A.rank() == 2
def test_DomainMatrix_add():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
B = DomainMatrix([[ZZ(2), ZZ(4)], [ZZ(6), ZZ(8)]], (2, 2), ZZ)
assert A + A == A.add(A) == B
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
L = [[2, 3], [3, 4]]
raises(TypeError, lambda: A + L)
raises(TypeError, lambda: L + A)
A1 = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A2 = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
raises(DMShapeError, lambda: A1 + A2)
raises(DMShapeError, lambda: A2 + A1)
raises(DMShapeError, lambda: A1.add(A2))
raises(DMShapeError, lambda: A2.add(A1))
Az = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Aq = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
Asum = DomainMatrix([[QQ(2), QQ(4)], [QQ(6), QQ(8)]], (2, 2), QQ)
assert Az + Aq == Asum
assert Aq + Az == Asum
raises(DMDomainError, lambda: Az.add(Aq))
raises(DMDomainError, lambda: Aq.add(Az))
As = DomainMatrix({0: {1: ZZ(1)}, 1: {0: ZZ(2)}}, (2, 2), ZZ)
Ad = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Asd = As + Ad
Ads = Ad + As
assert Asd == DomainMatrix([[1, 3], [5, 4]], (2, 2), ZZ)
assert Asd.rep == DDM([[1, 3], [5, 4]], (2, 2), ZZ).to_dfm_or_ddm()
assert Ads == DomainMatrix([[1, 3], [5, 4]], (2, 2), ZZ)
assert Ads.rep == DDM([[1, 3], [5, 4]], (2, 2), ZZ).to_dfm_or_ddm()
raises(DMFormatError, lambda: As.add(Ad))
def test_DomainMatrix_sub():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
B = DomainMatrix([[ZZ(0), ZZ(0)], [ZZ(0), ZZ(0)]], (2, 2), ZZ)
assert A - A == A.sub(A) == B
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
L = [[2, 3], [3, 4]]
raises(TypeError, lambda: A - L)
raises(TypeError, lambda: L - A)
A1 = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A2 = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
raises(DMShapeError, lambda: A1 - A2)
raises(DMShapeError, lambda: A2 - A1)
raises(DMShapeError, lambda: A1.sub(A2))
raises(DMShapeError, lambda: A2.sub(A1))
Az = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Aq = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
Adiff = DomainMatrix([[QQ(0), QQ(0)], [QQ(0), QQ(0)]], (2, 2), QQ)
assert Az - Aq == Adiff
assert Aq - Az == Adiff
raises(DMDomainError, lambda: Az.sub(Aq))
raises(DMDomainError, lambda: Aq.sub(Az))
As = DomainMatrix({0: {1: ZZ(1)}, 1: {0: ZZ(2)}}, (2, 2), ZZ)
Ad = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Asd = As - Ad
Ads = Ad - As
assert Asd == DomainMatrix([[-1, -1], [-1, -4]], (2, 2), ZZ)
assert Asd.rep == DDM([[-1, -1], [-1, -4]], (2, 2), ZZ).to_dfm_or_ddm()
assert Asd == -Ads
assert Asd.rep == -Ads.rep
def test_DomainMatrix_neg():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Aneg = DomainMatrix([[ZZ(-1), ZZ(-2)], [ZZ(-3), ZZ(-4)]], (2, 2), ZZ)
assert -A == A.neg() == Aneg
def test_DomainMatrix_mul():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A2 = DomainMatrix([[ZZ(7), ZZ(10)], [ZZ(15), ZZ(22)]], (2, 2), ZZ)
assert A*A == A.matmul(A) == A2
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
L = [[1, 2], [3, 4]]
raises(TypeError, lambda: A * L)
raises(TypeError, lambda: L * A)
Az = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Aq = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
Aprod = DomainMatrix([[QQ(7), QQ(10)], [QQ(15), QQ(22)]], (2, 2), QQ)
assert Az * Aq == Aprod
assert Aq * Az == Aprod
raises(DMDomainError, lambda: Az.matmul(Aq))
raises(DMDomainError, lambda: Aq.matmul(Az))
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
AA = DomainMatrix([[ZZ(2), ZZ(4)], [ZZ(6), ZZ(8)]], (2, 2), ZZ)
x = ZZ(2)
assert A * x == x * A == A.mul(x) == AA
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
AA = DomainMatrix.zeros((2, 2), ZZ)
x = ZZ(0)
assert A * x == x * A == A.mul(x).to_sparse() == AA
As = DomainMatrix({0: {1: ZZ(1)}, 1: {0: ZZ(2)}}, (2, 2), ZZ)
Ad = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Asd = As * Ad
Ads = Ad * As
assert Asd == DomainMatrix([[3, 4], [2, 4]], (2, 2), ZZ)
assert Asd.rep == DDM([[3, 4], [2, 4]], (2, 2), ZZ).to_dfm_or_ddm()
assert Ads == DomainMatrix([[4, 1], [8, 3]], (2, 2), ZZ)
assert Ads.rep == DDM([[4, 1], [8, 3]], (2, 2), ZZ).to_dfm_or_ddm()
def test_DomainMatrix_mul_elementwise():
A = DomainMatrix([[ZZ(2), ZZ(2)], [ZZ(0), ZZ(0)]], (2, 2), ZZ)
B = DomainMatrix([[ZZ(4), ZZ(0)], [ZZ(3), ZZ(0)]], (2, 2), ZZ)
C = DomainMatrix([[ZZ(8), ZZ(0)], [ZZ(0), ZZ(0)]], (2, 2), ZZ)
assert A.mul_elementwise(B) == C
assert B.mul_elementwise(A) == C
def test_DomainMatrix_pow():
eye = DomainMatrix.eye(2, ZZ)
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
A2 = DomainMatrix([[ZZ(7), ZZ(10)], [ZZ(15), ZZ(22)]], (2, 2), ZZ)
A3 = DomainMatrix([[ZZ(37), ZZ(54)], [ZZ(81), ZZ(118)]], (2, 2), ZZ)
assert A**0 == A.pow(0) == eye
assert A**1 == A.pow(1) == A
assert A**2 == A.pow(2) == A2
assert A**3 == A.pow(3) == A3
raises(TypeError, lambda: A ** Rational(1, 2))
raises(NotImplementedError, lambda: A ** -1)
raises(NotImplementedError, lambda: A.pow(-1))
A = DomainMatrix.zeros((2, 1), ZZ)
raises(DMNonSquareMatrixError, lambda: A ** 1)
def test_DomainMatrix_clear_denoms():
A = DM([[(1,2),(1,3)],[(1,4),(1,5)]], QQ)
den_Z = DomainScalar(ZZ(60), ZZ)
Anum_Z = DM([[30, 20], [15, 12]], ZZ)
Anum_Q = Anum_Z.convert_to(QQ)
assert A.clear_denoms() == (den_Z, Anum_Q)
assert A.clear_denoms(convert=True) == (den_Z, Anum_Z)
assert A * den_Z == Anum_Q
assert A == Anum_Q / den_Z
def test_DomainMatrix_clear_denoms_rowwise():
A = DM([[(1,2),(1,3)],[(1,4),(1,5)]], QQ)
den_Z = DM([[6, 0], [0, 20]], ZZ).to_sparse()
Anum_Z = DM([[3, 2], [5, 4]], ZZ)
Anum_Q = DM([[3, 2], [5, 4]], QQ)
assert A.clear_denoms_rowwise() == (den_Z, Anum_Q)
assert A.clear_denoms_rowwise(convert=True) == (den_Z, Anum_Z)
assert den_Z * A == Anum_Q
assert A == den_Z.to_field().inv() * Anum_Q
A = DM([[(1,2),(1,3),0,0],[0,0,0,0], [(1,4),(1,5),(1,6),(1,7)]], QQ)
den_Z = DM([[6, 0, 0], [0, 1, 0], [0, 0, 420]], ZZ).to_sparse()
Anum_Z = DM([[3, 2, 0, 0], [0, 0, 0, 0], [105, 84, 70, 60]], ZZ)
Anum_Q = Anum_Z.convert_to(QQ)
assert A.clear_denoms_rowwise() == (den_Z, Anum_Q)
assert A.clear_denoms_rowwise(convert=True) == (den_Z, Anum_Z)
assert den_Z * A == Anum_Q
assert A == den_Z.to_field().inv() * Anum_Q
def test_DomainMatrix_cancel_denom():
A = DM([[2, 4], [6, 8]], ZZ)
assert A.cancel_denom(ZZ(1)) == (DM([[2, 4], [6, 8]], ZZ), ZZ(1))
assert A.cancel_denom(ZZ(3)) == (DM([[2, 4], [6, 8]], ZZ), ZZ(3))
assert A.cancel_denom(ZZ(4)) == (DM([[1, 2], [3, 4]], ZZ), ZZ(2))
A = DM([[1, 2], [3, 4]], ZZ)
assert A.cancel_denom(ZZ(2)) == (A, ZZ(2))
assert A.cancel_denom(ZZ(-2)) == (-A, ZZ(2))
# Test canonicalization of denominator over Gaussian rationals.
A = DM([[1, 2], [3, 4]], QQ_I)
assert A.cancel_denom(QQ_I(0,2)) == (QQ_I(0,-1)*A, QQ_I(2))
raises(ZeroDivisionError, lambda: A.cancel_denom(ZZ(0)))
def test_DomainMatrix_cancel_denom_elementwise():
A = DM([[2, 4], [6, 8]], ZZ)
numers, denoms = A.cancel_denom_elementwise(ZZ(1))
assert numers == DM([[2, 4], [6, 8]], ZZ)
assert denoms == DM([[1, 1], [1, 1]], ZZ)
numers, denoms = A.cancel_denom_elementwise(ZZ(4))
assert numers == DM([[1, 1], [3, 2]], ZZ)
assert denoms == DM([[2, 1], [2, 1]], ZZ)
raises(ZeroDivisionError, lambda: A.cancel_denom_elementwise(ZZ(0)))
def test_DomainMatrix_content_primitive():
A = DM([[2, 4], [6, 8]], ZZ)
A_primitive = DM([[1, 2], [3, 4]], ZZ)
A_content = ZZ(2)
assert A.content() == A_content
assert A.primitive() == (A_content, A_primitive)
def test_DomainMatrix_scc():
Ad = DomainMatrix([[ZZ(1), ZZ(2), ZZ(3)],
[ZZ(0), ZZ(1), ZZ(0)],
[ZZ(2), ZZ(0), ZZ(4)]], (3, 3), ZZ)
As = Ad.to_sparse()
Addm = Ad.rep
Asdm = As.rep
for A in [Ad, As, Addm, Asdm]:
assert Ad.scc() == [[1], [0, 2]]
A = DM([[ZZ(1), ZZ(2), ZZ(3)]], ZZ)
raises(DMNonSquareMatrixError, lambda: A.scc())
def test_DomainMatrix_rref():
# More tests in test_rref.py
A = DomainMatrix([], (0, 1), QQ)
assert A.rref() == (A, ())
A = DomainMatrix([[QQ(1)]], (1, 1), QQ)
assert A.rref() == (A, (0,))
A = DomainMatrix([[QQ(0)]], (1, 1), QQ)
assert A.rref() == (A, ())
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
Ar, pivots = A.rref()
assert Ar == DomainMatrix([[QQ(1), QQ(0)], [QQ(0), QQ(1)]], (2, 2), QQ)
assert pivots == (0, 1)
A = DomainMatrix([[QQ(0), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
Ar, pivots = A.rref()
assert Ar == DomainMatrix([[QQ(1), QQ(0)], [QQ(0), QQ(1)]], (2, 2), QQ)
assert pivots == (0, 1)
A = DomainMatrix([[QQ(0), QQ(2)], [QQ(0), QQ(4)]], (2, 2), QQ)
Ar, pivots = A.rref()
assert Ar == DomainMatrix([[QQ(0), QQ(1)], [QQ(0), QQ(0)]], (2, 2), QQ)
assert pivots == (1,)
Az = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
Ar, pivots = Az.rref()
assert Ar == DomainMatrix([[QQ(1), QQ(0)], [QQ(0), QQ(1)]], (2, 2), QQ)
assert pivots == (0, 1)
methods = ('auto', 'GJ', 'FF', 'CD', 'GJ_dense', 'FF_dense', 'CD_dense')
Az = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
for method in methods:
Ar, pivots = Az.rref(method=method)
assert Ar == DomainMatrix([[QQ(1), QQ(0)], [QQ(0), QQ(1)]], (2, 2), QQ)
assert pivots == (0, 1)
raises(ValueError, lambda: Az.rref(method='foo'))
raises(ValueError, lambda: Az.rref_den(method='foo'))
def test_DomainMatrix_columnspace():
A = DomainMatrix([[QQ(1), QQ(-1), QQ(1)], [QQ(2), QQ(-2), QQ(3)]], (2, 3), QQ)
Acol = DomainMatrix([[QQ(1), QQ(1)], [QQ(2), QQ(3)]], (2, 2), QQ)
assert A.columnspace() == Acol
Az = DomainMatrix([[ZZ(1), ZZ(-1), ZZ(1)], [ZZ(2), ZZ(-2), ZZ(3)]], (2, 3), ZZ)
raises(DMNotAField, lambda: Az.columnspace())
A = DomainMatrix([[QQ(1), QQ(-1), QQ(1)], [QQ(2), QQ(-2), QQ(3)]], (2, 3), QQ, fmt='sparse')
Acol = DomainMatrix({0: {0: QQ(1), 1: QQ(1)}, 1: {0: QQ(2), 1: QQ(3)}}, (2, 2), QQ)
assert A.columnspace() == Acol
def test_DomainMatrix_rowspace():
A = DomainMatrix([[QQ(1), QQ(-1), QQ(1)], [QQ(2), QQ(-2), QQ(3)]], (2, 3), QQ)
assert A.rowspace() == A
Az = DomainMatrix([[ZZ(1), ZZ(-1), ZZ(1)], [ZZ(2), ZZ(-2), ZZ(3)]], (2, 3), ZZ)
raises(DMNotAField, lambda: Az.rowspace())
A = DomainMatrix([[QQ(1), QQ(-1), QQ(1)], [QQ(2), QQ(-2), QQ(3)]], (2, 3), QQ, fmt='sparse')
assert A.rowspace() == A
def test_DomainMatrix_nullspace():
A = DomainMatrix([[QQ(1), QQ(1)], [QQ(1), QQ(1)]], (2, 2), QQ)
Anull = DomainMatrix([[QQ(-1), QQ(1)]], (1, 2), QQ)
assert A.nullspace() == Anull
A = DomainMatrix([[ZZ(1), ZZ(1)], [ZZ(1), ZZ(1)]], (2, 2), ZZ)
Anull = DomainMatrix([[ZZ(-1), ZZ(1)]], (1, 2), ZZ)
assert A.nullspace() == Anull
raises(DMNotAField, lambda: A.nullspace(divide_last=True))
A = DomainMatrix([[ZZ(2), ZZ(2)], [ZZ(2), ZZ(2)]], (2, 2), ZZ)
Anull = DomainMatrix([[ZZ(-2), ZZ(2)]], (1, 2), ZZ)
Arref, den, pivots = A.rref_den()
assert den == ZZ(2)
assert Arref.nullspace_from_rref() == Anull
assert Arref.nullspace_from_rref(pivots) == Anull
assert Arref.to_sparse().nullspace_from_rref() == Anull.to_sparse()
assert Arref.to_sparse().nullspace_from_rref(pivots) == Anull.to_sparse()
def test_DomainMatrix_solve():
# XXX: Maybe the _solve method should be changed...
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(2), QQ(4)]], (2, 2), QQ)
b = DomainMatrix([[QQ(1)], [QQ(2)]], (2, 1), QQ)
particular = DomainMatrix([[1, 0]], (1, 2), QQ)
nullspace = DomainMatrix([[-2, 1]], (1, 2), QQ)
assert A._solve(b) == (particular, nullspace)
b3 = DomainMatrix([[QQ(1)], [QQ(1)], [QQ(1)]], (3, 1), QQ)
raises(DMShapeError, lambda: A._solve(b3))
bz = DomainMatrix([[ZZ(1)], [ZZ(1)]], (2, 1), ZZ)
raises(DMNotAField, lambda: A._solve(bz))
def test_DomainMatrix_inv():
A = DomainMatrix([], (0, 0), QQ)
assert A.inv() == A
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
Ainv = DomainMatrix([[QQ(-2), QQ(1)], [QQ(3, 2), QQ(-1, 2)]], (2, 2), QQ)
assert A.inv() == Ainv
Az = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
raises(DMNotAField, lambda: Az.inv())
Ans = DomainMatrix([[QQ(1), QQ(2)]], (1, 2), QQ)
raises(DMNonSquareMatrixError, lambda: Ans.inv())
Aninv = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(6)]], (2, 2), QQ)
raises(DMNonInvertibleMatrixError, lambda: Aninv.inv())
def test_DomainMatrix_det():
A = DomainMatrix([], (0, 0), ZZ)
assert A.det() == 1
A = DomainMatrix([[1]], (1, 1), ZZ)
assert A.det() == 1
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A.det() == ZZ(-2)
A = DomainMatrix([[ZZ(1), ZZ(2), ZZ(3)], [ZZ(1), ZZ(2), ZZ(4)], [ZZ(1), ZZ(3), ZZ(5)]], (3, 3), ZZ)
assert A.det() == ZZ(-1)
A = DomainMatrix([[ZZ(1), ZZ(2), ZZ(3)], [ZZ(1), ZZ(2), ZZ(4)], [ZZ(1), ZZ(2), ZZ(5)]], (3, 3), ZZ)
assert A.det() == ZZ(0)
Ans = DomainMatrix([[QQ(1), QQ(2)]], (1, 2), QQ)
raises(DMNonSquareMatrixError, lambda: Ans.det())
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
assert A.det() == QQ(-2)
def test_DomainMatrix_eval_poly():
dM = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
p = [ZZ(1), ZZ(2), ZZ(3)]
result = DomainMatrix([[ZZ(12), ZZ(14)], [ZZ(21), ZZ(33)]], (2, 2), ZZ)
assert dM.eval_poly(p) == result == p[0]*dM**2 + p[1]*dM + p[2]*dM**0
assert dM.eval_poly([]) == dM.zeros(dM.shape, dM.domain)
assert dM.eval_poly([ZZ(2)]) == 2*dM.eye(2, dM.domain)
dM2 = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
raises(DMNonSquareMatrixError, lambda: dM2.eval_poly([ZZ(1)]))
def test_DomainMatrix_eval_poly_mul():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
b = DomainMatrix([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
p = [ZZ(1), ZZ(2), ZZ(3)]
result = DomainMatrix([[ZZ(40)], [ZZ(87)]], (2, 1), ZZ)
assert A.eval_poly_mul(p, b) == result == p[0]*A**2*b + p[1]*A*b + p[2]*b
dM = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
dM1 = DomainMatrix([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
raises(DMNonSquareMatrixError, lambda: dM1.eval_poly_mul([ZZ(1)], b))
b1 = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
raises(DMShapeError, lambda: dM.eval_poly_mul([ZZ(1)], b1))
bq = DomainMatrix([[QQ(1)], [QQ(2)]], (2, 1), QQ)
raises(DMDomainError, lambda: dM.eval_poly_mul([ZZ(1)], bq))
def _check_solve_den(A, b, xnum, xden):
# Examples for solve_den, solve_den_charpoly, solve_den_rref should use
# this so that all methods and types are tested.
case1 = (A, xnum, b)
case2 = (A.to_sparse(), xnum.to_sparse(), b.to_sparse())
for Ai, xnum_i, b_i in [case1, case2]:
# The key invariant for solve_den:
assert Ai*xnum_i == xden*b_i
# solve_den_rref can differ at least by a minus sign
answers = [(xnum_i, xden), (-xnum_i, -xden)]
assert Ai.solve_den(b) in answers
assert Ai.solve_den(b, method='rref') in answers
assert Ai.solve_den_rref(b) in answers
# charpoly can only be used if A is square and guarantees to return the
# actual determinant as a denominator.
m, n = Ai.shape
if m == n:
assert Ai.solve_den(b_i, method='charpoly') == (xnum_i, xden)
assert Ai.solve_den_charpoly(b_i) == (xnum_i, xden)
else:
raises(DMNonSquareMatrixError, lambda: Ai.solve_den_charpoly(b))
raises(DMNonSquareMatrixError, lambda: Ai.solve_den(b, method='charpoly'))
def test_DomainMatrix_solve_den():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
b = DomainMatrix([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
result = DomainMatrix([[ZZ(0)], [ZZ(-1)]], (2, 1), ZZ)
den = ZZ(-2)
_check_solve_den(A, b, result, den)
A = DomainMatrix([
[ZZ(1), ZZ(2), ZZ(3)],
[ZZ(1), ZZ(2), ZZ(4)],
[ZZ(1), ZZ(3), ZZ(5)]], (3, 3), ZZ)
b = DomainMatrix([[ZZ(1)], [ZZ(2)], [ZZ(3)]], (3, 1), ZZ)
result = DomainMatrix([[ZZ(2)], [ZZ(0)], [ZZ(-1)]], (3, 1), ZZ)
den = ZZ(-1)
_check_solve_den(A, b, result, den)
A = DomainMatrix([[ZZ(2)], [ZZ(2)]], (2, 1), ZZ)
b = DomainMatrix([[ZZ(3)], [ZZ(3)]], (2, 1), ZZ)
result = DomainMatrix([[ZZ(3)]], (1, 1), ZZ)
den = ZZ(2)
_check_solve_den(A, b, result, den)
def test_DomainMatrix_solve_den_charpoly():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
b = DomainMatrix([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
A1 = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
raises(DMNonSquareMatrixError, lambda: A1.solve_den_charpoly(b))
b1 = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
raises(DMShapeError, lambda: A.solve_den_charpoly(b1))
bq = DomainMatrix([[QQ(1)], [QQ(2)]], (2, 1), QQ)
raises(DMDomainError, lambda: A.solve_den_charpoly(bq))
def test_DomainMatrix_solve_den_charpoly_check():
# Test check
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(2), ZZ(4)]], (2, 2), ZZ)
b = DomainMatrix([[ZZ(1)], [ZZ(3)]], (2, 1), ZZ)
raises(DMNonInvertibleMatrixError, lambda: A.solve_den_charpoly(b))
adjAb = DomainMatrix([[ZZ(-2)], [ZZ(1)]], (2, 1), ZZ)
assert A.adjugate() * b == adjAb
assert A.solve_den_charpoly(b, check=False) == (adjAb, ZZ(0))
def test_DomainMatrix_solve_den_errors():
A = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
b = DomainMatrix([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
raises(DMShapeError, lambda: A.solve_den(b))
raises(DMShapeError, lambda: A.solve_den_rref(b))
A = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
b = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
raises(DMShapeError, lambda: A.solve_den(b))
raises(DMShapeError, lambda: A.solve_den_rref(b))
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
b1 = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
raises(DMShapeError, lambda: A.solve_den(b1))
A = DomainMatrix([[ZZ(2)]], (1, 1), ZZ)
b = DomainMatrix([[ZZ(2)]], (1, 1), ZZ)
raises(DMBadInputError, lambda: A.solve_den(b1, method='invalid'))
A = DomainMatrix([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
b = DomainMatrix([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
raises(DMNonSquareMatrixError, lambda: A.solve_den_charpoly(b))
def test_DomainMatrix_solve_den_rref_underdetermined():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(1), ZZ(2)]], (2, 2), ZZ)
b = DomainMatrix([[ZZ(1)], [ZZ(1)]], (2, 1), ZZ)
raises(DMNonInvertibleMatrixError, lambda: A.solve_den(b))
raises(DMNonInvertibleMatrixError, lambda: A.solve_den_rref(b))
def test_DomainMatrix_adj_poly_det():
A = DM([[ZZ(1), ZZ(2), ZZ(3)],
[ZZ(4), ZZ(5), ZZ(6)],
[ZZ(7), ZZ(8), ZZ(9)]], ZZ)
p, detA = A.adj_poly_det()
assert p == [ZZ(1), ZZ(-15), ZZ(-18)]
assert A.adjugate() == p[0]*A**2 + p[1]*A**1 + p[2]*A**0 == A.eval_poly(p)
assert A.det() == detA
A = DM([[ZZ(1), ZZ(2), ZZ(3)],
[ZZ(7), ZZ(8), ZZ(9)]], ZZ)
raises(DMNonSquareMatrixError, lambda: A.adj_poly_det())
def test_DomainMatrix_inv_den():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
den = ZZ(-2)
result = DomainMatrix([[ZZ(4), ZZ(-2)], [ZZ(-3), ZZ(1)]], (2, 2), ZZ)
assert A.inv_den() == (result, den)
def test_DomainMatrix_adjugate():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
result = DomainMatrix([[ZZ(4), ZZ(-2)], [ZZ(-3), ZZ(1)]], (2, 2), ZZ)
assert A.adjugate() == result
def test_DomainMatrix_adj_det():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
adjA = DomainMatrix([[ZZ(4), ZZ(-2)], [ZZ(-3), ZZ(1)]], (2, 2), ZZ)
assert A.adj_det() == (adjA, ZZ(-2))
def test_DomainMatrix_lu():
A = DomainMatrix([], (0, 0), QQ)
assert A.lu() == (A, A, [])
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
L = DomainMatrix([[QQ(1), QQ(0)], [QQ(3), QQ(1)]], (2, 2), QQ)
U = DomainMatrix([[QQ(1), QQ(2)], [QQ(0), QQ(-2)]], (2, 2), QQ)
swaps = []
assert A.lu() == (L, U, swaps)
A = DomainMatrix([[QQ(0), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
L = DomainMatrix([[QQ(1), QQ(0)], [QQ(0), QQ(1)]], (2, 2), QQ)
U = DomainMatrix([[QQ(3), QQ(4)], [QQ(0), QQ(2)]], (2, 2), QQ)
swaps = [(0, 1)]
assert A.lu() == (L, U, swaps)
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(2), QQ(4)]], (2, 2), QQ)
L = DomainMatrix([[QQ(1), QQ(0)], [QQ(2), QQ(1)]], (2, 2), QQ)
U = DomainMatrix([[QQ(1), QQ(2)], [QQ(0), QQ(0)]], (2, 2), QQ)
swaps = []
assert A.lu() == (L, U, swaps)
A = DomainMatrix([[QQ(0), QQ(2)], [QQ(0), QQ(4)]], (2, 2), QQ)
L = DomainMatrix([[QQ(1), QQ(0)], [QQ(0), QQ(1)]], (2, 2), QQ)
U = DomainMatrix([[QQ(0), QQ(2)], [QQ(0), QQ(4)]], (2, 2), QQ)
swaps = []
assert A.lu() == (L, U, swaps)
A = DomainMatrix([[QQ(1), QQ(2), QQ(3)], [QQ(4), QQ(5), QQ(6)]], (2, 3), QQ)
L = DomainMatrix([[QQ(1), QQ(0)], [QQ(4), QQ(1)]], (2, 2), QQ)
U = DomainMatrix([[QQ(1), QQ(2), QQ(3)], [QQ(0), QQ(-3), QQ(-6)]], (2, 3), QQ)
swaps = []
assert A.lu() == (L, U, swaps)
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)], [QQ(5), QQ(6)]], (3, 2), QQ)
L = DomainMatrix([
[QQ(1), QQ(0), QQ(0)],
[QQ(3), QQ(1), QQ(0)],
[QQ(5), QQ(2), QQ(1)]], (3, 3), QQ)
U = DomainMatrix([[QQ(1), QQ(2)], [QQ(0), QQ(-2)], [QQ(0), QQ(0)]], (3, 2), QQ)
swaps = []
assert A.lu() == (L, U, swaps)
A = [[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 1], [0, 0, 1, 2]]
L = [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 1, 1]]
U = [[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]]
to_dom = lambda rows, dom: [[dom(e) for e in row] for row in rows]
A = DomainMatrix(to_dom(A, QQ), (4, 4), QQ)
L = DomainMatrix(to_dom(L, QQ), (4, 4), QQ)
U = DomainMatrix(to_dom(U, QQ), (4, 4), QQ)
assert A.lu() == (L, U, [])
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
raises(DMNotAField, lambda: A.lu())
def test_DomainMatrix_lu_solve():
# Base case
A = b = x = DomainMatrix([], (0, 0), QQ)
assert A.lu_solve(b) == x
# Basic example
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
b = DomainMatrix([[QQ(1)], [QQ(2)]], (2, 1), QQ)
x = DomainMatrix([[QQ(0)], [QQ(1, 2)]], (2, 1), QQ)
assert A.lu_solve(b) == x
# Example with swaps
A = DomainMatrix([[QQ(0), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
b = DomainMatrix([[QQ(1)], [QQ(2)]], (2, 1), QQ)
x = DomainMatrix([[QQ(0)], [QQ(1, 2)]], (2, 1), QQ)
assert A.lu_solve(b) == x
# Non-invertible
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(2), QQ(4)]], (2, 2), QQ)
b = DomainMatrix([[QQ(1)], [QQ(2)]], (2, 1), QQ)
raises(DMNonInvertibleMatrixError, lambda: A.lu_solve(b))
# Overdetermined, consistent
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)], [QQ(5), QQ(6)]], (3, 2), QQ)
b = DomainMatrix([[QQ(1)], [QQ(2)], [QQ(3)]], (3, 1), QQ)
x = DomainMatrix([[QQ(0)], [QQ(1, 2)]], (2, 1), QQ)
assert A.lu_solve(b) == x
# Overdetermined, inconsistent
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)], [QQ(5), QQ(6)]], (3, 2), QQ)
b = DomainMatrix([[QQ(1)], [QQ(2)], [QQ(4)]], (3, 1), QQ)
raises(DMNonInvertibleMatrixError, lambda: A.lu_solve(b))
# Underdetermined
A = DomainMatrix([[QQ(1), QQ(2)]], (1, 2), QQ)
b = DomainMatrix([[QQ(1)]], (1, 1), QQ)
raises(NotImplementedError, lambda: A.lu_solve(b))
# Non-field
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
b = DomainMatrix([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
raises(DMNotAField, lambda: A.lu_solve(b))
# Shape mismatch
A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
b = DomainMatrix([[QQ(1), QQ(2)]], (1, 2), QQ)
raises(DMShapeError, lambda: A.lu_solve(b))
def test_DomainMatrix_charpoly():
A = DomainMatrix([], (0, 0), ZZ)
p = [ZZ(1)]
assert A.charpoly() == p
assert A.to_sparse().charpoly() == p
A = DomainMatrix([[1]], (1, 1), ZZ)
p = [ZZ(1), ZZ(-1)]
assert A.charpoly() == p
assert A.to_sparse().charpoly() == p
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
p = [ZZ(1), ZZ(-5), ZZ(-2)]
assert A.charpoly() == p
assert A.to_sparse().charpoly() == p
A = DomainMatrix([[ZZ(1), ZZ(2), ZZ(3)], [ZZ(4), ZZ(5), ZZ(6)], [ZZ(7), ZZ(8), ZZ(9)]], (3, 3), ZZ)
p = [ZZ(1), ZZ(-15), ZZ(-18), ZZ(0)]
assert A.charpoly() == p
assert A.to_sparse().charpoly() == p
A = DomainMatrix([[ZZ(0), ZZ(1), ZZ(0)],
[ZZ(1), ZZ(0), ZZ(1)],
[ZZ(0), ZZ(1), ZZ(0)]], (3, 3), ZZ)
p = [ZZ(1), ZZ(0), ZZ(-2), ZZ(0)]
assert A.charpoly() == p
assert A.to_sparse().charpoly() == p
A = DM([[17, 0, 30, 0, 0, 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[69, 0, 0, 0, 0, 86, 0, 0, 0, 0],
[23, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[ 0, 0, 0, 13, 0, 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0, 0, 0, 32, 0, 0],
[ 0, 0, 0, 0, 37, 67, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], ZZ)
p = ZZ.map([1, -17, -2070, 0, -771420, 0, 0, 0, 0, 0, 0])
assert A.charpoly() == p
assert A.to_sparse().charpoly() == p
Ans = DomainMatrix([[QQ(1), QQ(2)]], (1, 2), QQ)
raises(DMNonSquareMatrixError, lambda: Ans.charpoly())
def test_DomainMatrix_charpoly_factor_list():
A = DomainMatrix([], (0, 0), ZZ)
assert A.charpoly_factor_list() == []
A = DM([[1]], ZZ)
assert A.charpoly_factor_list() == [
([ZZ(1), ZZ(-1)], 1)
]
A = DM([[1, 2], [3, 4]], ZZ)
assert A.charpoly_factor_list() == [
([ZZ(1), ZZ(-5), ZZ(-2)], 1)
]
A = DM([[1, 2, 0], [3, 4, 0], [0, 0, 1]], ZZ)
assert A.charpoly_factor_list() == [
([ZZ(1), ZZ(-1)], 1),
([ZZ(1), ZZ(-5), ZZ(-2)], 1)
]
def test_DomainMatrix_eye():
A = DomainMatrix.eye(3, QQ)
assert A.rep == SDM.eye((3, 3), QQ)
assert A.shape == (3, 3)
assert A.domain == QQ
def test_DomainMatrix_zeros():
A = DomainMatrix.zeros((1, 2), QQ)
assert A.rep == SDM.zeros((1, 2), QQ)
assert A.shape == (1, 2)
assert A.domain == QQ
def test_DomainMatrix_ones():
A = DomainMatrix.ones((2, 3), QQ)
if GROUND_TYPES != 'flint':
assert A.rep == DDM.ones((2, 3), QQ)
else:
assert A.rep == SDM.ones((2, 3), QQ).to_dfm()
assert A.shape == (2, 3)
assert A.domain == QQ
def test_DomainMatrix_diag():
A = DomainMatrix({0:{0:ZZ(2)}, 1:{1:ZZ(3)}}, (2, 2), ZZ)
assert DomainMatrix.diag([ZZ(2), ZZ(3)], ZZ) == A
A = DomainMatrix({0:{0:ZZ(2)}, 1:{1:ZZ(3)}}, (3, 4), ZZ)
assert DomainMatrix.diag([ZZ(2), ZZ(3)], ZZ, (3, 4)) == A
def test_DomainMatrix_hstack():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
B = DomainMatrix([[ZZ(5), ZZ(6)], [ZZ(7), ZZ(8)]], (2, 2), ZZ)
C = DomainMatrix([[ZZ(9), ZZ(10)], [ZZ(11), ZZ(12)]], (2, 2), ZZ)
AB = DomainMatrix([
[ZZ(1), ZZ(2), ZZ(5), ZZ(6)],
[ZZ(3), ZZ(4), ZZ(7), ZZ(8)]], (2, 4), ZZ)
ABC = DomainMatrix([
[ZZ(1), ZZ(2), ZZ(5), ZZ(6), ZZ(9), ZZ(10)],
[ZZ(3), ZZ(4), ZZ(7), ZZ(8), ZZ(11), ZZ(12)]], (2, 6), ZZ)
assert A.hstack(B) == AB
assert A.hstack(B, C) == ABC
def test_DomainMatrix_vstack():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
B = DomainMatrix([[ZZ(5), ZZ(6)], [ZZ(7), ZZ(8)]], (2, 2), ZZ)
C = DomainMatrix([[ZZ(9), ZZ(10)], [ZZ(11), ZZ(12)]], (2, 2), ZZ)
AB = DomainMatrix([
[ZZ(1), ZZ(2)],
[ZZ(3), ZZ(4)],
[ZZ(5), ZZ(6)],
[ZZ(7), ZZ(8)]], (4, 2), ZZ)
ABC = DomainMatrix([
[ZZ(1), ZZ(2)],
[ZZ(3), ZZ(4)],
[ZZ(5), ZZ(6)],
[ZZ(7), ZZ(8)],
[ZZ(9), ZZ(10)],
[ZZ(11), ZZ(12)]], (6, 2), ZZ)
assert A.vstack(B) == AB
assert A.vstack(B, C) == ABC
def test_DomainMatrix_applyfunc():
A = DomainMatrix([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
B = DomainMatrix([[ZZ(2), ZZ(4)]], (1, 2), ZZ)
assert A.applyfunc(lambda x: 2*x) == B
def test_DomainMatrix_scalarmul():
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
lamda = DomainScalar(QQ(3)/QQ(2), QQ)
assert A * lamda == DomainMatrix([[QQ(3, 2), QQ(3)], [QQ(9, 2), QQ(6)]], (2, 2), QQ)
assert A * 2 == DomainMatrix([[ZZ(2), ZZ(4)], [ZZ(6), ZZ(8)]], (2, 2), ZZ)
assert 2 * A == DomainMatrix([[ZZ(2), ZZ(4)], [ZZ(6), ZZ(8)]], (2, 2), ZZ)
assert A * DomainScalar(ZZ(0), ZZ) == DomainMatrix({}, (2, 2), ZZ)
assert A * DomainScalar(ZZ(1), ZZ) == A
raises(TypeError, lambda: A * 1.5)
def test_DomainMatrix_truediv():
A = DomainMatrix.from_Matrix(Matrix([[1, 2], [3, 4]]))
lamda = DomainScalar(QQ(3)/QQ(2), QQ)
assert A / lamda == DomainMatrix({0: {0: QQ(2, 3), 1: QQ(4, 3)}, 1: {0: QQ(2), 1: QQ(8, 3)}}, (2, 2), QQ)
b = DomainScalar(ZZ(1), ZZ)
assert A / b == DomainMatrix({0: {0: QQ(1), 1: QQ(2)}, 1: {0: QQ(3), 1: QQ(4)}}, (2, 2), QQ)
assert A / 1 == DomainMatrix({0: {0: QQ(1), 1: QQ(2)}, 1: {0: QQ(3), 1: QQ(4)}}, (2, 2), QQ)
assert A / 2 == DomainMatrix({0: {0: QQ(1, 2), 1: QQ(1)}, 1: {0: QQ(3, 2), 1: QQ(2)}}, (2, 2), QQ)
raises(ZeroDivisionError, lambda: A / 0)
raises(TypeError, lambda: A / 1.5)
raises(ZeroDivisionError, lambda: A / DomainScalar(ZZ(0), ZZ))
A = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert A.to_field() / 2 == DomainMatrix([[QQ(1, 2), QQ(1)], [QQ(3, 2), QQ(2)]], (2, 2), QQ)
assert A / 2 == DomainMatrix([[QQ(1, 2), QQ(1)], [QQ(3, 2), QQ(2)]], (2, 2), QQ)
assert A.to_field() / QQ(2,3) == DomainMatrix([[QQ(3, 2), QQ(3)], [QQ(9, 2), QQ(6)]], (2, 2), QQ)
def test_DomainMatrix_getitem():
dM = DomainMatrix([
[ZZ(1), ZZ(2), ZZ(3)],
[ZZ(4), ZZ(5), ZZ(6)],
[ZZ(7), ZZ(8), ZZ(9)]], (3, 3), ZZ)
assert dM[1:,:-2] == DomainMatrix([[ZZ(4)], [ZZ(7)]], (2, 1), ZZ)
assert dM[2,:-2] == DomainMatrix([[ZZ(7)]], (1, 1), ZZ)
assert dM[:-2,:-2] == DomainMatrix([[ZZ(1)]], (1, 1), ZZ)
assert dM[:-1,0:2] == DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(4), ZZ(5)]], (2, 2), ZZ)
assert dM[:, -1] == DomainMatrix([[ZZ(3)], [ZZ(6)], [ZZ(9)]], (3, 1), ZZ)
assert dM[-1, :] == DomainMatrix([[ZZ(7), ZZ(8), ZZ(9)]], (1, 3), ZZ)
assert dM[::-1, :] == DomainMatrix([
[ZZ(7), ZZ(8), ZZ(9)],
[ZZ(4), ZZ(5), ZZ(6)],
[ZZ(1), ZZ(2), ZZ(3)]], (3, 3), ZZ)
raises(IndexError, lambda: dM[4, :-2])
raises(IndexError, lambda: dM[:-2, 4])
assert dM[1, 2] == DomainScalar(ZZ(6), ZZ)
assert dM[-2, 2] == DomainScalar(ZZ(6), ZZ)
assert dM[1, -2] == DomainScalar(ZZ(5), ZZ)
assert dM[-1, -3] == DomainScalar(ZZ(7), ZZ)
raises(IndexError, lambda: dM[3, 3])
raises(IndexError, lambda: dM[1, 4])
raises(IndexError, lambda: dM[-1, -4])
dM = DomainMatrix({0: {0: ZZ(1)}}, (10, 10), ZZ)
assert dM[5, 5] == DomainScalar(ZZ(0), ZZ)
assert dM[0, 0] == DomainScalar(ZZ(1), ZZ)
dM = DomainMatrix({1: {0: 1}}, (2,1), ZZ)
assert dM[0:, 0] == DomainMatrix({1: {0: 1}}, (2, 1), ZZ)
raises(IndexError, lambda: dM[3, 0])
dM = DomainMatrix({2: {2: ZZ(1)}, 4: {4: ZZ(1)}}, (5, 5), ZZ)
assert dM[:2,:2] == DomainMatrix({}, (2, 2), ZZ)
assert dM[2:,2:] == DomainMatrix({0: {0: 1}, 2: {2: 1}}, (3, 3), ZZ)
assert dM[3:,3:] == DomainMatrix({1: {1: 1}}, (2, 2), ZZ)
assert dM[2:, 6:] == DomainMatrix({}, (3, 0), ZZ)
def test_DomainMatrix_getitem_sympy():
dM = DomainMatrix({2: {2: ZZ(2)}, 4: {4: ZZ(1)}}, (5, 5), ZZ)
val1 = dM.getitem_sympy(0, 0)
assert val1 is S.Zero
val2 = dM.getitem_sympy(2, 2)
assert val2 == 2 and isinstance(val2, Integer)
def test_DomainMatrix_extract():
dM1 = DomainMatrix([
[ZZ(1), ZZ(2), ZZ(3)],
[ZZ(4), ZZ(5), ZZ(6)],
[ZZ(7), ZZ(8), ZZ(9)]], (3, 3), ZZ)
dM2 = DomainMatrix([
[ZZ(1), ZZ(3)],
[ZZ(7), ZZ(9)]], (2, 2), ZZ)
assert dM1.extract([0, 2], [0, 2]) == dM2
assert dM1.to_sparse().extract([0, 2], [0, 2]) == dM2.to_sparse()
assert dM1.extract([0, -1], [0, -1]) == dM2
assert dM1.to_sparse().extract([0, -1], [0, -1]) == dM2.to_sparse()
dM3 = DomainMatrix([
[ZZ(1), ZZ(2), ZZ(2)],
[ZZ(4), ZZ(5), ZZ(5)],
[ZZ(4), ZZ(5), ZZ(5)]], (3, 3), ZZ)
assert dM1.extract([0, 1, 1], [0, 1, 1]) == dM3
assert dM1.to_sparse().extract([0, 1, 1], [0, 1, 1]) == dM3.to_sparse()
empty = [
([], [], (0, 0)),
([1], [], (1, 0)),
([], [1], (0, 1)),
]
for rows, cols, size in empty:
assert dM1.extract(rows, cols) == DomainMatrix.zeros(size, ZZ).to_dense()
assert dM1.to_sparse().extract(rows, cols) == DomainMatrix.zeros(size, ZZ)
dM = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
bad_indices = [([2], [0]), ([0], [2]), ([-3], [0]), ([0], [-3])]
for rows, cols in bad_indices:
raises(IndexError, lambda: dM.extract(rows, cols))
raises(IndexError, lambda: dM.to_sparse().extract(rows, cols))
def test_DomainMatrix_setitem():
dM = DomainMatrix({2: {2: ZZ(1)}, 4: {4: ZZ(1)}}, (5, 5), ZZ)
dM[2, 2] = ZZ(2)
assert dM == DomainMatrix({2: {2: ZZ(2)}, 4: {4: ZZ(1)}}, (5, 5), ZZ)
def setitem(i, j, val):
dM[i, j] = val
raises(TypeError, lambda: setitem(2, 2, QQ(1, 2)))
raises(NotImplementedError, lambda: setitem(slice(1, 2), 2, ZZ(1)))
def test_DomainMatrix_pickling():
import pickle
dM = DomainMatrix({2: {2: ZZ(1)}, 4: {4: ZZ(1)}}, (5, 5), ZZ)
assert pickle.loads(pickle.dumps(dM)) == dM
dM = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
assert pickle.loads(pickle.dumps(dM)) == dM
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