def test_pinv_solve():
# Fully determined system (unique result, identical to other solvers).
A = Matrix([[1, 5], [7, 9]])
B = Matrix([12, 13])
assert A.pinv_solve(B) == A.cholesky_solve(B)
assert A.pinv_solve(B) == A.LDLsolve(B)
assert A.pinv_solve(B) == Matrix([sympify('-43/26'), sympify('71/26')])
assert A * A.pinv() * B == B
# Fully determined, with two-dimensional B matrix.
B = Matrix([[12, 13, 14], [15, 16, 17]])
assert A.pinv_solve(B) == A.cholesky_solve(B)
assert A.pinv_solve(B) == A.LDLsolve(B)
assert A.pinv_solve(B) == Matrix([[-33, -37, -41], [69, 75, 81]]) / 26
assert A * A.pinv() * B == B
# Underdetermined system (infinite results).
A = Matrix([[1, 0, 1], [0, 1, 1]])
B = Matrix([5, 7])
solution = A.pinv_solve(B)
w = {}
for s in solution.atoms(Symbol):
# Extract dummy symbols used in the solution.
w[s.name] = s
assert solution == Matrix(
[[w['w0_0'] / 3 + w['w1_0'] / 3 - w['w2_0'] / 3 + 1],
[w['w0_0'] / 3 + w['w1_0'] / 3 - w['w2_0'] / 3 + 3],
[-w['w0_0'] / 3 - w['w1_0'] / 3 + w['w2_0'] / 3 + 4]])
assert A * A.pinv() * B == B
# Overdetermined system (least squares results).
A = Matrix([[1, 0], [0, 0], [0, 1]])
B = Matrix([3, 2, 1])
assert A.pinv_solve(B) == Matrix([3, 1])
# Proof the solution is not exact.
assert A * A.pinv() * B != B
def test_LDLsolve():
A = Matrix([[2, 3, 5], [3, 6, 2], [8, 3, 6]])
x = Matrix(3, 1, [3, 7, 5])
b = A * x
soln = A.LDLsolve(b)
assert soln == x
A = Matrix([[0, -1, 2], [5, 10, 7], [8, 3, 4]])
x = Matrix(3, 1, [-1, 2, 5])
b = A * x
soln = A.LDLsolve(b)
assert soln == x
A = Matrix(((9, 3 * I), (-3 * I, 5)))
x = Matrix((-2, 1))
b = A * x
soln = A.LDLsolve(b)
assert expand_mul(soln) == x
A = Matrix(((9 * I, 3), (-3 + I, 5)))
x = Matrix((2 + 3 * I, -1))
b = A * x
soln = A.LDLsolve(b)
assert expand_mul(soln) == x
A = Matrix(((9, 3), (3, 9)))
x = Matrix((1, 1))
b = A * x
soln = A.LDLsolve(b)
assert expand_mul(soln) == x
A = Matrix([[-5, -3, -4], [-3, -7, 7]])
x = Matrix([[8], [7], [-2]])
b = A * x
raises(NotImplementedError, lambda: A.LDLsolve(b))
