def test_sparse_solve():
from sympy.matrices import SparseMatrix
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([[S(2) / 3, S(1) / 3, S(1) / 6],
[S(1) / 3, S(2) / 3, S(1) / 3], [0, 0, S(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]
def test_sparse_solve():
from sympy.matrices import SparseMatrix
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([
[S(2)/3, S(1)/3, S(1)/6],
[S(1)/3, S(2)/3, S(1)/3],
[ 0, 0, S(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]
def test_sparse_solve():
from sympy.matrices import SparseMatrix
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]
