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_cholesky_solve():
A = Matrix([[2, 3, 5], [3, 6, 2], [8, 3, 6]])
x = Matrix(3, 1, [3, 7, 5])
b = A * x
soln = A.cholesky_solve(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.cholesky_solve(b)
assert soln == x
A = Matrix(((1, 5), (5, 1)))
x = Matrix((4, -3))
b = A * x
soln = A.cholesky_solve(b)
assert soln == x
A = Matrix(((9, 3 * I), (-3 * I, 5)))
x = Matrix((-2, 1))
b = A * x
soln = A.cholesky_solve(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.cholesky_solve(b)
assert expand_mul(soln) == x
a00, a01, a11, b0, b1 = symbols('a00, a01, a11, b0, b1')
A = Matrix(((a00, a01), (a01, a11)))
b = Matrix((b0, b1))
x = A.cholesky_solve(b)
assert simplify(A * x) == b
def test_issue_17247_expression_blowup_30():
M = Matrix(
S('''[
[ -3/4, 45/32 - 37*I/16, 0, 0],
[-149/64 + 49*I/32, -177/128 - 1369*I/128, 0, -2063/256 + 541*I/128],
[ 0, 9/4 + 55*I/16, 2473/256 + 137*I/64, 0],
[ 0, 0, 0, -177/128 - 1369*I/128]]'''
))
assert M.cholesky_solve(ones(4, 1)) == Matrix(
S('''[
[ -32549314808672/3306971225785 - 17397006745216*I/3306971225785],
[ 67439348256/3306971225785 - 9167503335872*I/3306971225785],
[-15091965363354518272/21217636514687010905 + 16890163109293858304*I/21217636514687010905],
[ -11328/952745 + 87616*I/952745]]'''
))
