from sympy.core.numbers import (Float, I, Rational)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import Abs
from sympy.polys.polytools import PurePoly
from sympy.matrices import \
Matrix, MutableSparseMatrix, ImmutableSparseMatrix, SparseMatrix, eye, \
ones, zeros, ShapeError, NonSquareMatrixError
from sympy.testing.pytest import raises
def test_sparse_creation():
a = SparseMatrix(2, 2, {(0, 0): [[1, 2], [3, 4]]})
assert a == SparseMatrix([[1, 2], [3, 4]])
a = SparseMatrix(2, 2, {(0, 0): [[1, 2]]})
assert a == SparseMatrix([[1, 2], [0, 0]])
a = SparseMatrix(2, 2, {(0, 0): [1, 2]})
assert a == SparseMatrix([[1, 0], [2, 0]])
def test_sparse_matrix():
def sparse_eye(n):
return SparseMatrix.eye(n)
def sparse_zeros(n):
return SparseMatrix.zeros(n)
# creation args
raises(TypeError, lambda: SparseMatrix(1, 2))
a = SparseMatrix((
(1, 0),
(0, 1)
))
assert SparseMatrix(a) == a
from sympy.matrices import MutableDenseMatrix
a = MutableSparseMatrix([])
b = MutableDenseMatrix([1, 2])
assert a.row_join(b) == b
assert a.col_join(b) == b
assert type(a.row_join(b)) == type(a)
assert type(a.col_join(b)) == type(a)
# make sure 0 x n matrices get stacked correctly
sparse_matrices = [SparseMatrix.zeros(0, n) for n in range(4)]
assert SparseMatrix.hstack(*sparse_matrices) == Matrix(0, 6, [])
sparse_matrices = [SparseMatrix.zeros(n, 0) for n in range(4)]
assert SparseMatrix.vstack(*sparse_matrices) == Matrix(6, 0, [])
# test element assignment
a = SparseMatrix((
(1, 0),
(0, 1)
))
a[3] = 4
assert a[1, 1] == 4
a[3] = 1
a[0, 0] = 2
assert a == SparseMatrix((
(2, 0),
(0, 1)
))
a[1, 0] = 5
assert a == SparseMatrix((
(2, 0),
(5, 1)
))
a[1, 1] = 0
assert a == SparseMatrix((
(2, 0),
(5, 0)
))
assert a.todok() == {(0, 0): 2, (1, 0): 5}
# test_multiplication
a = SparseMatrix((
(1, 2),
(3, 1),
(0, 6),
))
b = SparseMatrix((
(1, 2),
(3, 0),
))
c = a*b
assert c[0, 0] == 7
assert c[0, 1] == 2
assert c[1, 0] == 6
assert c[1, 1] == 6
assert c[2, 0] == 18
assert c[2, 1] == 0
try:
eval('c = a @ b')
except SyntaxError:
pass
else:
assert c[0, 0] == 7
assert c[0, 1] == 2
assert c[1, 0] == 6
assert c[1, 1] == 6
assert c[2, 0] == 18
assert c[2, 1] == 0
x = Symbol("x")
c = b * Symbol("x")
assert isinstance(c, SparseMatrix)
assert c[0, 0] == x
assert c[0, 1] == 2*x
assert c[1, 0] == 3*x
assert c[1, 1] == 0
c = 5 * b
assert isinstance(c, SparseMatrix)
assert c[0, 0] == 5
assert c[0, 1] == 2*5
assert c[1, 0] == 3*5
assert c[1, 1] == 0
#test_power
A = SparseMatrix([[2, 3], [4, 5]])
assert (A**5)[:] == [6140, 8097, 10796, 14237]
A = SparseMatrix([[2, 1, 3], [4, 2, 4], [6, 12, 1]])
assert (A**3)[:] == [290, 262, 251, 448, 440, 368, 702, 954, 433]
# test_creation
x = Symbol("x")
a = SparseMatrix([[x, 0], [0, 0]])
m = a
assert m.cols == m.rows
assert m.cols == 2
assert m[:] == [x, 0, 0, 0]
b = SparseMatrix(2, 2, [x, 0, 0, 0])
m = b
assert m.cols == m.rows
assert m.cols == 2
assert m[:] == [x, 0, 0, 0]
assert a == b
S = sparse_eye(3)
S.row_del(1)
assert S == SparseMatrix([
[1, 0, 0],
[0, 0, 1]])
S = sparse_eye(3)
S.col_del(1)
assert S == SparseMatrix([
[1, 0],
[0, 0],
[0, 1]])
S = SparseMatrix.eye(3)
S[2, 1] = 2
S.col_swap(1, 0)
assert S == SparseMatrix([
[0, 1, 0],
[1, 0, 0],
[2, 0, 1]])
S.row_swap(0, 1)
assert S == SparseMatrix([
[1, 0, 0],
[0, 1, 0],
[2, 0, 1]])
a = SparseMatrix(1, 2, [1, 2])
b = a.copy()
c = a.copy()
assert a[0] == 1
a.row_del(0)
assert a == SparseMatrix(0, 2, [])
b.col_del(1)
assert b == SparseMatrix(1, 1, [1])
assert SparseMatrix([[1, 2, 3], [1, 2], [1]]) == Matrix([
[1, 2, 3],
[1, 2, 0],
[1, 0, 0]])
assert SparseMatrix(4, 4, {(1, 1): sparse_eye(2)}) == Matrix([
[0, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 0]])
raises(ValueError, lambda: SparseMatrix(1, 1, {(1, 1): 1}))
assert SparseMatrix(1, 2, [1, 2]).tolist() == [[1, 2]]
assert SparseMatrix(2, 2, [1, [2, 3]]).tolist() == [[1, 0], [2, 3]]
raises(ValueError, lambda: SparseMatrix(2, 2, [1]))
raises(ValueError, lambda: SparseMatrix(1, 1, [[1, 2]]))
assert SparseMatrix([.1]).has(Float)
# autosizing
assert SparseMatrix(None, {(0, 1): 0}).shape == (0, 0)
assert SparseMatrix(None, {(0, 1): 1}).shape == (1, 2)
assert SparseMatrix(None, None, {(0, 1): 1}).shape == (1, 2)
raises(ValueError, lambda: SparseMatrix(None, 1, [[1, 2]]))
raises(ValueError, lambda: SparseMatrix(1, None, [[1, 2]]))
raises(ValueError, lambda: SparseMatrix(3, 3, {(0, 0): ones(2), (1, 1): 2}))
# test_determinant
x, y = Symbol('x'), Symbol('y')
assert SparseMatrix(1, 1, [0]).det() == 0
assert SparseMatrix([[1]]).det() == 1
assert SparseMatrix(((-3, 2), (8, -5))).det() == -1
assert SparseMatrix(((x, 1), (y, 2*y))).det() == 2*x*y - y
assert SparseMatrix(( (1, 1, 1),
(1, 2, 3),
(1, 3, 6) )).det() == 1
assert SparseMatrix(( ( 3, -2, 0, 5),
(-2, 1, -2, 2),
( 0, -2, 5, 0),
( 5, 0, 3, 4) )).det() == -289
assert SparseMatrix(( ( 1, 2, 3, 4),
( 5, 6, 7, 8),
( 9, 10, 11, 12),
(13, 14, 15, 16) )).det() == 0
assert SparseMatrix(( (3, 2, 0, 0, 0),
(0, 3, 2, 0, 0),
(0, 0, 3, 2, 0),
(0, 0, 0, 3, 2),
(2, 0, 0, 0, 3) )).det() == 275
assert SparseMatrix(( (1, 0, 1, 2, 12),
(2, 0, 1, 1, 4),
(2, 1, 1, -1, 3),
(3, 2, -1, 1, 8),
(1, 1, 1, 0, 6) )).det() == -55
assert SparseMatrix(( (-5, 2, 3, 4, 5),
( 1, -4, 3, 4, 5),
( 1, 2, -3, 4, 5),
( 1, 2, 3, -2, 5),
( 1, 2, 3, 4, -1) )).det() == 11664
assert SparseMatrix(( ( 3, 0, 0, 0),
(-2, 1, 0, 0),
( 0, -2, 5, 0),
( 5, 0, 3, 4) )).det() == 60
assert SparseMatrix(( ( 1, 0, 0, 0),
( 5, 0, 0, 0),
( 9, 10, 11, 0),
(13, 14, 15, 16) )).det() == 0
assert SparseMatrix(( (3, 2, 0, 0, 0),
(0, 3, 2, 0, 0),
(0, 0, 3, 2, 0),
(0, 0, 0, 3, 2),
(0, 0, 0, 0, 3) )).det() == 243
assert SparseMatrix(( ( 2, 7, -1, 3, 2),
( 0, 0, 1, 0, 1),
(-2, 0, 7, 0, 2),
(-3, -2, 4, 5, 3),
( 1, 0, 0, 0, 1) )).det() == 123
# test_slicing
m0 = sparse_eye(4)
assert m0[:3, :3] == sparse_eye(3)
assert m0[2:4, 0:2] == sparse_zeros(2)
m1 = SparseMatrix(3, 3, lambda i, j: i + j)
assert m1[0, :] == SparseMatrix(1, 3, (0, 1, 2))
assert m1[1:3, 1] == SparseMatrix(2, 1, (2, 3))
m2 = SparseMatrix(
[[0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11], [12, 13, 14, 15]])
assert m2[:, -1] == SparseMatrix(4, 1, [3, 7, 11, 15])
assert m2[-2:, :] == SparseMatrix([[8, 9, 10, 11], [12, 13, 14, 15]])
assert SparseMatrix([[1, 2], [3, 4]])[[1], [1]] == Matrix([[4]])
# test_submatrix_assignment
m = sparse_zeros(4)
m[2:4, 2:4] = sparse_eye(2)
assert m == SparseMatrix([(0, 0, 0, 0),
(0, 0, 0, 0),
(0, 0, 1, 0),
(0, 0, 0, 1)])
assert len(m.todok()) == 2
m[:2, :2] = sparse_eye(2)
assert m == sparse_eye(4)
m[:, 0] = SparseMatrix(4, 1, (1, 2, 3, 4))
assert m == SparseMatrix([(1, 0, 0, 0),
(2, 1, 0, 0),
(3, 0, 1, 0),
(4, 0, 0, 1)])
m[:, :] = sparse_zeros(4)
assert m == sparse_zeros(4)
m[:, :] = ((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16))
assert m == SparseMatrix((( 1, 2, 3, 4),
( 5, 6, 7, 8),
( 9, 10, 11, 12),
(13, 14, 15, 16)))
m[:2, 0] = [0, 0]
assert m == SparseMatrix((( 0, 2, 3, 4),
( 0, 6, 7, 8),
( 9, 10, 11, 12),
(13, 14, 15, 16)))
# test_reshape
m0 = sparse_eye(3)
assert m0.reshape(1, 9) == SparseMatrix(1, 9, (1, 0, 0, 0, 1, 0, 0, 0, 1))
m1 = SparseMatrix(3, 4, lambda i, j: i + j)
assert m1.reshape(4, 3) == \
SparseMatrix([(0, 1, 2), (3, 1, 2), (3, 4, 2), (3, 4, 5)])
assert m1.reshape(2, 6) == \
SparseMatrix([(0, 1, 2, 3, 1, 2), (3, 4, 2, 3, 4, 5)])
# test_applyfunc
m0 = sparse_eye(3)
assert m0.applyfunc(lambda x: 2*x) == sparse_eye(3)*2
assert m0.applyfunc(lambda x: 0 ) == sparse_zeros(3)
# test__eval_Abs
assert abs(SparseMatrix(((x, 1), (y, 2*y)))) == SparseMatrix(((Abs(x), 1), (Abs(y), 2*Abs(y))))
# test_LUdecomp
testmat = SparseMatrix([[ 0, 2, 5, 3],
[ 3, 3, 7, 4],
[ 8, 4, 0, 2],
[-2, 6, 3, 4]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == sparse_zeros(4)
testmat = SparseMatrix([[ 6, -2, 7, 4],
[ 0, 3, 6, 7],
[ 1, -2, 7, 4],
[-9, 2, 6, 3]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == sparse_zeros(4)
x, y, z = Symbol('x'), Symbol('y'), Symbol('z')
M = Matrix(((1, x, 1), (2, y, 0), (y, 0, z)))
L, U, p = M.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - M == sparse_zeros(3)
# test_LUsolve
A = SparseMatrix([[2, 3, 5],
[3, 6, 2],
[8, 3, 6]])
x = SparseMatrix(3, 1, [3, 7, 5])
b = A*x
soln = A.LUsolve(b)
assert soln == x
A = SparseMatrix([[0, -1, 2],
[5, 10, 7],
[8, 3, 4]])
x = SparseMatrix(3, 1, [-1, 2, 5])
b = A*x
soln = A.LUsolve(b)
assert soln == x
# test_inverse
A = sparse_eye(4)
assert A.inv() == sparse_eye(4)
assert A.inv(method="CH") == sparse_eye(4)
assert A.inv(method="LDL") == sparse_eye(4)
A = SparseMatrix([[2, 3, 5],
[3, 6, 2],
[7, 2, 6]])
Ainv = SparseMatrix(Matrix(A).inv())
assert A*Ainv == sparse_eye(3)
assert A.inv(method="CH") == Ainv
assert A.inv(method="LDL") == Ainv
A = SparseMatrix([[2, 3, 5],
[3, 6, 2],
[5, 2, 6]])
Ainv = SparseMatrix(Matrix(A).inv())
assert A*Ainv == sparse_eye(3)
assert A.inv(method="CH") == Ainv
assert A.inv(method="LDL") == Ainv
# test_cross
v1 = Matrix(1, 3, [1, 2, 3])
v2 = Matrix(1, 3, [3, 4, 5])
assert v1.cross(v2) == Matrix(1, 3, [-2, 4, -2])
assert v1.norm(2)**2 == 14
# conjugate
a = SparseMatrix(((1, 2 + I), (3, 4)))
assert a.C == SparseMatrix([
[1, 2 - I],
[3, 4]
])
# mul
assert a*Matrix(2, 2, [1, 0, 0, 1]) == a
assert a + Matrix(2, 2, [1, 1, 1, 1]) == SparseMatrix([
[2, 3 + I],
[4, 5]
])
# col join
assert a.col_join(sparse_eye(2)) == SparseMatrix([
[1, 2 + I],
[3, 4],
[1, 0],
[0, 1]
])
# row insert
assert a.row_insert(2, sparse_eye(2)) == SparseMatrix([
[1, 2 + I],
[3, 4],
[1, 0],
[0, 1]
])
# col insert
assert a.col_insert(2, SparseMatrix.zeros(2, 1)) == SparseMatrix([
[1, 2 + I, 0],
[3, 4, 0],
])
# symmetric
assert not a.is_symmetric(simplify=False)
# col op
M = SparseMatrix.eye(3)*2
M[1, 0] = -1
M.col_op(1, lambda v, i: v + 2*M[i, 0])
assert M == SparseMatrix([
[ 2, 4, 0],
[-1, 0, 0],
[ 0, 0, 2]
])
# fill
M = SparseMatrix.eye(3)
M.fill(2)
assert M == SparseMatrix([
[2, 2, 2],
[2, 2, 2],
[2, 2, 2],
])
# test_cofactor
assert sparse_eye(3) == sparse_eye(3).cofactor_matrix()
test = SparseMatrix([[1, 3, 2], [2, 6, 3], [2, 3, 6]])
assert test.cofactor_matrix() == \
SparseMatrix([[27, -6, -6], [-12, 2, 3], [-3, 1, 0]])
test = SparseMatrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
assert test.cofactor_matrix() == \
SparseMatrix([[-3, 6, -3], [6, -12, 6], [-3, 6, -3]])
# test_jacobian
x = Symbol('x')
y = Symbol('y')
L = SparseMatrix(1, 2, [x**2*y, 2*y**2 + x*y])
syms = [x, y]
assert L.jacobian(syms) == Matrix([[2*x*y, x**2], [y, 4*y + x]])
L = SparseMatrix(1, 2, [x, x**2*y**3])
assert L.jacobian(syms) == SparseMatrix([[1, 0], [2*x*y**3, x**2*3*y**2]])
# test_QR
A = Matrix([[1, 2], [2, 3]])
Q, S = A.QRdecomposition()
R = Rational
assert Q == Matrix([
[ 5**R(-1, 2), (R(2)/5)*(R(1)/5)**R(-1, 2)],
[2*5**R(-1, 2), (-R(1)/5)*(R(1)/5)**R(-1, 2)]])
assert S == Matrix([
[5**R(1, 2), 8*5**R(-1, 2)],
[ 0, (R(1)/5)**R(1, 2)]])
assert Q*S == A
assert Q.T * Q == sparse_eye(2)
R = Rational
# test nullspace
# first test reduced row-ech form
M = SparseMatrix([[5, 7, 2, 1],
[1, 6, 2, -1]])
out, tmp = M.rref()
assert out == Matrix([[1, 0, -R(2)/23, R(13)/23],
[0, 1, R(8)/23, R(-6)/23]])
M = SparseMatrix([[ 1, 3, 0, 2, 6, 3, 1],
[-2, -6, 0, -2, -8, 3, 1],
[ 3, 9, 0, 0, 6, 6, 2],
[-1, -3, 0, 1, 0, 9, 3]])
out, tmp = M.rref()
assert out == Matrix([[1, 3, 0, 0, 2, 0, 0],
[0, 0, 0, 1, 2, 0, 0],
[0, 0, 0, 0, 0, 1, R(1)/3],
[0, 0, 0, 0, 0, 0, 0]])
# now check the vectors
basis = M.nullspace()
assert basis[0] == Matrix([-3, 1, 0, 0, 0, 0, 0])
assert basis[1] == Matrix([0, 0, 1, 0, 0, 0, 0])
assert basis[2] == Matrix([-2, 0, 0, -2, 1, 0, 0])
assert basis[3] == Matrix([0, 0, 0, 0, 0, R(-1)/3, 1])
# test eigen
x = Symbol('x')
y = Symbol('y')
sparse_eye3 = sparse_eye(3)
assert sparse_eye3.charpoly(x) == PurePoly((x - 1)**3)
assert sparse_eye3.charpoly(y) == PurePoly((y - 1)**3)
# test values
M = Matrix([( 0, 1, -1),
( 1, 1, 0),
(-1, 0, 1)])
vals = M.eigenvals()
assert sorted(vals.keys()) == [-1, 1, 2]
R = Rational
M = Matrix([[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
assert M.eigenvects() == [(1, 3, [
Matrix([1, 0, 0]),
Matrix([0, 1, 0]),
Matrix([0, 0, 1])])]
M = Matrix([[5, 0, 2],
[3, 2, 0],
[0, 0, 1]])
assert M.eigenvects() == [(1, 1, [Matrix([R(-1)/2, R(3)/2, 1])]),
(2, 1, [Matrix([0, 1, 0])]),
(5, 1, [Matrix([1, 1, 0])])]
assert M.zeros(3, 5) == SparseMatrix(3, 5, {})
A = SparseMatrix(10, 10, {(0, 0): 18, (0, 9): 12, (1, 4): 18, (2, 7): 16, (3, 9): 12, (4, 2): 19, (5, 7): 16, (6, 2): 12, (9, 7): 18})
assert A.row_list() == [(0, 0, 18), (0, 9, 12), (1, 4, 18), (2, 7, 16), (3, 9, 12), (4, 2, 19), (5, 7, 16), (6, 2, 12), (9, 7, 18)]
assert A.col_list() == [(0, 0, 18), (4, 2, 19), (6, 2, 12), (1, 4, 18), (2, 7, 16), (5, 7, 16), (9, 7, 18), (0, 9, 12), (3, 9, 12)]
assert SparseMatrix.eye(2).nnz() == 2
def test_scalar_multiply():
assert SparseMatrix([[1, 2]]).scalar_multiply(3) == SparseMatrix([[3, 6]])
def test_transpose():
assert SparseMatrix(((1, 2), (3, 4))).transpose() == \
SparseMatrix(((1, 3), (2, 4)))
def test_trace():
assert SparseMatrix(((1, 2), (3, 4))).trace() == 5
assert SparseMatrix(((0, 0), (0, 4))).trace() == 4
def test_CL_RL():
assert SparseMatrix(((1, 2), (3, 4))).row_list() == \
[(0, 0, 1), (0, 1, 2), (1, 0, 3), (1, 1, 4)]
assert SparseMatrix(((1, 2), (3, 4))).col_list() == \
[(0, 0, 1), (1, 0, 3), (0, 1, 2), (1, 1, 4)]
def test_add():
assert SparseMatrix(((1, 0), (0, 1))) + SparseMatrix(((0, 1), (1, 0))) == \
SparseMatrix(((1, 1), (1, 1)))
a = SparseMatrix(100, 100, lambda i, j: int(j != 0 and i % j == 0))
b = SparseMatrix(100, 100, lambda i, j: int(i != 0 and j % i == 0))
assert (len(a.todok()) + len(b.todok()) - len((a + b).todok()) > 0)
def test_errors():
raises(ValueError, lambda: SparseMatrix(1.4, 2, lambda i, j: 0))
raises(TypeError, lambda: SparseMatrix([1, 2, 3], [1, 2]))
raises(ValueError, lambda: SparseMatrix([[1, 2], [3, 4]])[(1, 2, 3)])
raises(IndexError, lambda: SparseMatrix([[1, 2], [3, 4]])[5])
raises(ValueError, lambda: SparseMatrix([[1, 2], [3, 4]])[1, 2, 3])
raises(TypeError,
lambda: SparseMatrix([[1, 2], [3, 4]]).copyin_list([0, 1], set()))
raises(
IndexError, lambda: SparseMatrix([[1, 2], [3, 4]])[1, 2])
raises(TypeError, lambda: SparseMatrix([1, 2, 3]).cross(1))
raises(IndexError, lambda: SparseMatrix(1, 2, [1, 2])[3])
raises(ShapeError,
lambda: SparseMatrix(1, 2, [1, 2]) + SparseMatrix(2, 1, [2, 1]))
def test_len():
assert not SparseMatrix()
assert SparseMatrix() == SparseMatrix([])
assert SparseMatrix() == SparseMatrix([[]])
def test_sparse_zeros_sparse_eye():
assert SparseMatrix.eye(3) == eye(3, cls=SparseMatrix)
assert len(SparseMatrix.eye(3).todok()) == 3
assert SparseMatrix.zeros(3) == zeros(3, cls=SparseMatrix)
assert len(SparseMatrix.zeros(3).todok()) == 0
def test_copyin():
s = SparseMatrix(3, 3, {})
s[1, 0] = 1
assert s[:, 0] == SparseMatrix(Matrix([0, 1, 0]))
assert s[3] == 1
assert s[3: 4] == [1]
s[1, 1] = 42
assert s[1, 1] == 42
assert s[1, 1:] == SparseMatrix([[42, 0]])
s[1, 1:] = Matrix([[5, 6]])
assert s[1, :] == SparseMatrix([[1, 5, 6]])
s[1, 1:] = [[42, 43]]
assert s[1, :] == SparseMatrix([[1, 42, 43]])
s[0, 0] = 17
assert s[:, :1] == SparseMatrix([17, 1, 0])
s[0, 0] = [1, 1, 1]
assert s[:, 0] == SparseMatrix([1, 1, 1])
s[0, 0] = Matrix([1, 1, 1])
assert s[:, 0] == SparseMatrix([1, 1, 1])
s[0, 0] = SparseMatrix([1, 1, 1])
assert s[:, 0] == SparseMatrix([1, 1, 1])
def test_sparse_solve():
A = SparseMatrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
assert A.cholesky() == Matrix([
[ 5, 0, 0],
[ 3, 3, 0],
[-1, 1, 3]])
assert A.cholesky() * A.cholesky().T == Matrix([
[25, 15, -5],
[15, 18, 0],
[-5, 0, 11]])
A = SparseMatrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
L, D = A.LDLdecomposition()
assert 15*L == Matrix([
[15, 0, 0],
[ 9, 15, 0],
[-3, 5, 15]])
assert D == Matrix([
[25, 0, 0],
[ 0, 9, 0],
[ 0, 0, 9]])
assert L * D * L.T == A
A = SparseMatrix(((3, 0, 2), (0, 0, 1), (1, 2, 0)))
assert A.inv() * A == SparseMatrix(eye(3))
A = SparseMatrix([
[ 2, -1, 0],
[-1, 2, -1],
[ 0, 0, 2]])
ans = SparseMatrix([
[Rational(2, 3), Rational(1, 3), Rational(1, 6)],
[Rational(1, 3), Rational(2, 3), Rational(1, 3)],
[ 0, 0, S.Half]])
assert A.inv(method='CH') == ans
assert A.inv(method='LDL') == ans
assert A * ans == SparseMatrix(eye(3))
s = A.solve(A[:, 0], 'LDL')
assert A*s == A[:, 0]
s = A.solve(A[:, 0], 'CH')
assert A*s == A[:, 0]
A = A.col_join(A)
s = A.solve_least_squares(A[:, 0], 'CH')
assert A*s == A[:, 0]
s = A.solve_least_squares(A[:, 0], 'LDL')
assert A*s == A[:, 0]
def test_lower_triangular_solve():
raises(NonSquareMatrixError, lambda:
SparseMatrix([[1, 2]]).lower_triangular_solve(Matrix([[1, 2]])))
raises(ShapeError, lambda:
SparseMatrix([[1, 2], [0, 4]]).lower_triangular_solve(Matrix([1])))
raises(ValueError, lambda:
SparseMatrix([[1, 2], [3, 4]]).lower_triangular_solve(Matrix([[1, 2], [3, 4]])))
a, b, c, d = symbols('a:d')
u, v, w, x = symbols('u:x')
A = SparseMatrix([[a, 0], [c, d]])
B = MutableSparseMatrix([[u, v], [w, x]])
C = ImmutableSparseMatrix([[u, v], [w, x]])
sol = Matrix([[u/a, v/a], [(w - c*u/a)/d, (x - c*v/a)/d]])
assert A.lower_triangular_solve(B) == sol
assert A.lower_triangular_solve(C) == sol
def test_upper_triangular_solve():
raises(NonSquareMatrixError, lambda:
SparseMatrix([[1, 2]]).upper_triangular_solve(Matrix([[1, 2]])))
raises(ShapeError, lambda:
SparseMatrix([[1, 2], [0, 4]]).upper_triangular_solve(Matrix([1])))
raises(TypeError, lambda:
SparseMatrix([[1, 2], [3, 4]]).upper_triangular_solve(Matrix([[1, 2], [3, 4]])))
a, b, c, d = symbols('a:d')
u, v, w, x = symbols('u:x')
A = SparseMatrix([[a, b], [0, d]])
B = MutableSparseMatrix([[u, v], [w, x]])
C = ImmutableSparseMatrix([[u, v], [w, x]])
sol = Matrix([[(u - b*w/d)/a, (v - b*x/d)/a], [w/d, x/d]])
assert A.upper_triangular_solve(B) == sol
assert A.upper_triangular_solve(C) == sol
def test_diagonal_solve():
a, d = symbols('a d')
u, v, w, x = symbols('u:x')
A = SparseMatrix([[a, 0], [0, d]])
B = MutableSparseMatrix([[u, v], [w, x]])
C = ImmutableSparseMatrix([[u, v], [w, x]])
sol = Matrix([[u/a, v/a], [w/d, x/d]])
assert A.diagonal_solve(B) == sol
assert A.diagonal_solve(C) == sol
def test_hermitian():
x = Symbol('x')
a = SparseMatrix([[0, I], [-I, 0]])
assert a.is_hermitian
a = SparseMatrix([[1, I], [-I, 1]])
assert a.is_hermitian
a[0, 0] = 2*I
assert a.is_hermitian is False
a[0, 0] = x
assert a.is_hermitian is None
a[0, 1] = a[1, 0]*I
assert a.is_hermitian is False