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, Rational(1, 2)]])
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]
pytest.raises(ValueError,
lambda: SparseMatrix([[1, 0, 1], [0, 0, 1]]).solve([1, 1]))
pytest.raises(
ValueError,
lambda: SparseMatrix([[1, 0], [0, 0], [2, 1]]).solve([1, 1, 1]))
def test_sparse_matrix():
def sparse_eye(n):
return SparseMatrix.eye(n)
def sparse_zeros(n):
return SparseMatrix.zeros(n)
# creation args
pytest.raises(TypeError, lambda: SparseMatrix(1, 2))
pytest.raises(ValueError, lambda: SparseMatrix(2, 2, (1, 3, 4, 5, 6)))
a = SparseMatrix(((1, 0), (0, 1)))
assert SparseMatrix(a) == a
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)
# 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._smat == {(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
c = b * 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
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)
del S[1, :]
assert S == SparseMatrix([[1, 0, 0], [0, 0, 1]])
S = sparse_eye(3)
del S[:, 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]])
S.col_swap(0, 1)
assert S == SparseMatrix([[0, 1, 0], [1, 0, 0], [0, 2, 1]])
S.row_swap(0, 2)
assert S == SparseMatrix([[0, 2, 1], [1, 0, 0], [0, 1, 0]])
S.col_swap(0, 2)
assert S == SparseMatrix([[1, 2, 0], [0, 0, 1], [0, 1, 0]])
a = SparseMatrix(1, 2, [1, 2])
b = a.copy()
c = a.copy()
assert a[0] == 1
del a[0, :]
assert a == SparseMatrix(0, 2, [])
del b[:, 1]
assert b == SparseMatrix(1, 1, [1])
# test_determinant
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(((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._smat) == 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_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).permuteBkwd(p) - 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).permuteBkwd(p) - testmat == sparse_zeros(4)
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).permuteBkwd(p) - M == sparse_zeros(3)
# test_LUsolve
A = SparseMatrix([[2, 3, 5], [3, 6, 2], [8, 3, 6]])
B = SparseMatrix(3, 1, [3, 7, 5])
b = A * B
soln = A.LUsolve(b)
assert soln == B
A = SparseMatrix([[0, -1, 2], [5, 10, 7], [8, 3, 4]])
B = SparseMatrix(3, 1, [-1, 2, 5])
b = A * B
soln = A.LUsolve(b)
assert soln == B
# 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]])
assert a * 0 == Matrix([[0, 0], [0, 0]])
# col join
assert a.col_join(sparse_eye(2)) == SparseMatrix([[1, 2 + I], [3, 4],
[1, 0], [0, 1]])
A = SparseMatrix(ones(3))
B = eye(3)
assert A.col_join(B) == Matrix([[1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 0, 0],
[0, 1, 0], [0, 0, 1]])
# row join
A = SparseMatrix(((1, 0, 1), (0, 1, 0), (1, 1, 0)))
B = Matrix(((1, 0, 0), (0, 1, 0), (0, 0, 1)))
assert A.row_join(B) == Matrix([[1, 0, 1, 1, 0, 0], [0, 1, 0, 0, 1, 0],
[1, 1, 0, 0, 0, 1]])
# symmetric
assert not a.is_symmetric(simplify=False)
assert sparse_eye(3).is_symmetric(simplify=False)
# test_cofactor
assert sparse_eye(3) == sparse_eye(3).cofactorMatrix()
test = SparseMatrix([[1, 3, 2], [2, 6, 3], [2, 3, 6]])
assert test.cofactorMatrix() == \
SparseMatrix([[27, -6, -6], [-12, 2, 3], [-3, 1, 0]])
test = SparseMatrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
assert test.cofactorMatrix() == \
SparseMatrix([[-3, 6, -3], [6, -12, 6], [-3, 6, -3]])
# test_jacobian
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
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) == [-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
M = SparseMatrix.eye(3) * 2
M[1, 0] = -1
M.col_op(1, lambda v, i: v + 2 * M[i, 0])
assert M == Matrix([[2, 4, 0], [-1, 0, 0], [0, 0, 2]])
M = SparseMatrix.zeros(3)
M.fill(1)
assert M == ones(3)
assert SparseMatrix(ones(0, 3)).tolist() == []