def test_inverse_P():
# Simple Test Case.
C = MultiPower(np.array([[-1, 0, 1], [0, 0, 0]]))
D = MultiPower(np.array([[-1, 0, 0], [0, 1, 0], [1, 0, 0]]))
# Creating a random object to run tests.
grob = Groebner([C, D])
# Create matrix
M = np.array([[0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5],
[0, 1, 2, 3, 4, 5]])
# Order of Column flips.
p = [1, 4, 3, 2, 0, 5]
# N is the matrix with the columns flipped.
N = M[:, p]
# Find the order of columns to flip back to.
pt = grob.inverse_P(p)
# Result done by hand.
pt_inv = [4, 0, 3, 2, 1, 5]
assert (np.allclose(M, N[:, pt])), "Matrix are not the same."
assert (all(pt == pt_inv)), "Wrong matrix order."
# Test Case 2:
A = np.random.random((5, 10))
Q, R, p = qr(A, pivoting=True)
pt = grob.inverse_P(p)
# We know that A[:,p] = QR, want to see if pt flips QR back to A.
assert (np.allclose(A, np.dot(Q, R)[:, pt]))
def testMultMatrix():
f1 = MultiPower(
np.array([[[5, 0, 0], [0, 0, 0], [0, 0, 0]],
[[0, -2, 0], [0, 0, 0], [0, 0, 0]],
[[1, 0, 0], [0, 0, 0], [0, 0, 0]]]))
f2 = MultiPower(
np.array([[[1, 0, 0], [0, 1, 0], [0, 0, 0]],
[[0, 0, 0], [0, 0, 0], [1, 0, 0]],
[[0, 0, 0], [0, 0, 0], [0, 0, 0]]]))
f3 = MultiPower(
np.array([[[0, 0, 0], [0, 0, 0], [3, 0, 0]],
[[0, -8, 0], [0, 0, 0], [0, 0, 0]],
[[0, 0, 0], [0, 0, 0], [0, 0, 0]]]))
F = [f1, f2, f3]
Gr = Groebner(F)
GB = Gr.solve()
VB = rf.vectorSpaceBasis(GB)
x = MultiPower(np.array([[0], [1]]))
y = MultiPower(np.array([[0, 1]]))
z = MultiPower(np.array([[[0, 1]]]))
mx_RealEig = [eig.real for eig in \
np.linalg.eigvals(rf.multMatrix(x, GB, VB)) if (eig.imag == 0)]
my_RealEig = [eig.real for eig in \
np.linalg.eigvals(rf.multMatrix(y, GB, VB)) if (eig.imag==0)]
mz_RealEig = [eig.real for eig in \
np.linalg.eigvals(rf.multMatrix(z, GB, VB)) if (eig.imag==0)]
assert (len(mx_RealEig) == 2)
assert (len(my_RealEig) == 2)
assert (len(mz_RealEig) == 2)
assert (np.allclose(mx_RealEig, [-1.100987715, .9657124563], atol=1.e-8))
assert (np.allclose(my_RealEig, [-2.878002536, -2.81249605], atol=1.e-8))
assert (np.allclose(mz_RealEig, [3.071618528, -2.821182227], atol=1.e-8))
def test_triangular_solve():
"""This tests the triangular_solve() method.
A visual graph of zeroes on the diagonal was also used to test this function.
"""
# Simple Test Case.
M = MultiPower(np.array([[-1, 0, 1], [0, 0, 0]]))
N = MultiPower(np.array([[-1, 0, 0], [0, 1, 0], [1, 0, 0]]))
# Creating a random object to run tests.
grob = Groebner([M, N])
A = np.array([[1, 2, 3, 4, 5], [0, 1, 2, 3, 4], [0, 0, 0, 1, 2]])
matrix = grob.triangular_solve(A)
answer = np.array([[1., 0., -1., 0., 1.], [0., 1., 2., 0., -2.],
[0., 0., 0., 1., 2.]])
assert (np.allclose(matrix, answer))
# Randomize test case: square matrix.
A = np.random.random((50, 50))
Q, R, p = qr(A, pivoting=True)
diagonal = np.diag(R)
r = np.sum(np.abs(diagonal) > 1e-10)
matrix = R[:r]
new_matrix = grob.triangular_solve(matrix)
true = sy.Matrix(new_matrix).rref()
x = sy.symbols('x')
f = sy.lambdify(x, true[0])
assert (np.allclose(new_matrix, f(1)))
# Randomize test case: shorter rows than column.
A = np.random.random((10, 50))
Q, R, p = qr(A, pivoting=True)
diagonal = np.diag(R)
r = np.sum(np.abs(diagonal) > 1e-10)
matrix = R[:r]
new_matrix = grob.triangular_solve(matrix)
true = sy.Matrix(new_matrix).rref()
x = sy.symbols('x')
f = sy.lambdify(x, true[0])
print(f(1))
print(new_matrix)
assert (np.allclose(new_matrix, f(1)))
# Randomize test case: longer rows than columns.
A = np.random.random((50, 10))
Q, R, p = qr(A, pivoting=True)
diagonal = np.diag(R)
r = np.sum(np.abs(diagonal) > 1e-10)
matrix = R[:r]
new_matrix = grob.triangular_solve(matrix)
true = sy.Matrix(new_matrix).rref()
x = sy.symbols('x')
f = sy.lambdify(x, true[0])
print(f(1))
print(new_matrix)
assert (np.allclose(new_matrix, f(1)))
def test_solve():
#First Test
A = MultiPower(np.array([[-10, 0], [0, 1], [1, 0]]))
B = MultiPower(np.array([[-26, 0, 0], [0, 0, 1], [0, 0, 0], [1, 0, 0]]))
C = MultiPower(
np.array([[-70, 0, 0, 0], [0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0],
[1, 0, 0, 0]]))
grob = Groebner([A, B, C])
X = MultiPower(np.array([[-2.], [1.]]))
Y = MultiPower(np.array([[-3., 1.]]))
x1, y1 = grob.solve()
assert (np.any([X == i and Y == j for i, j in permutations((x1, y1), 2)]))
#Second Test
A = MultiPower(
np.array([[[[
0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0
],
[
-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]],
[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]]],
[[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]],
[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]]]]))
B = MultiPower(
np.array([[[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]],
[[
-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]]],
[[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]],
[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]]]]))
C = MultiPower(
np.array([[[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 1
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]],
[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]]],
[[[
-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]],
[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]]]]))
grob = Groebner([A, B, C])
w1, x1, y1, z1 = grob.solve()
W = MultiPower(
np.array([[[[0.], [0.]], [[0.], [0.]], [[0.], [0.]], [[1.], [0.]]],
[[[0.], [-1.]], [[0.], [0.]], [[0.], [0.]], [[0.], [0.]]]]))
X = MultiPower(
np.array([[[[0., 0., 0., 0., 0., 1.], [-1., 0., 0., 0., 0., 0.]]]]))
Y = MultiPower(np.array([[[[0.], [0.], [1.]], [[-1.], [0.], [0.]]]]))
Z = MultiPower(
np.array([[[[0.], [0.]], [[0.], [0.]], [[0.], [1.]]],
[[[-1.], [0.]], [[0.], [0.]], [[0.], [0.]]]]))
assert (np.any([
W == i and X == j and Y == k and Z == l
for i, j, k, l in permutations((w1, x1, y1, z1), 4)
]))
#Third Test
A = MultiPower(np.array([[-1, 0, 1], [0, 0, 0]]))
B = MultiPower(np.array([[-1, 0, 0], [0, 1, 0], [1, 0, 0]]))
grob = Groebner([A, B])
x1, y1 = grob.solve()
assert (np.any([A == i and B == j for i, j in permutations((x1, y1), 2)]))
#Fourth Test
A = MultiPower(np.array([[-10, 0], [0, 1], [1, 0]]))
B = MultiPower(np.array([[-25, 0, 0], [0, 0, 1], [0, 0, 0], [1, 0, 0]]))
C = MultiPower(
np.array([[-70, 0, 0, 0], [0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0],
[1, 0, 0, 0]]))
grob = Groebner([A, B, C])
X = MultiPower(np.array([[1.]]))
x1 = grob.solve()
assert (X == x1[0])
#Fifth Test
A = MultiPower(np.array([[1, 1], [0, 0]]))
B = MultiPower(np.array([[1, 0], [1, 0]]))
C = MultiPower(np.array([[1, 0], [1, 0], [0, 1]]))
grob = Groebner([A, B, C])
X = MultiPower(np.array([[1.]]))
x1 = grob.solve()
assert (X == x1[0])
def test_init_():
C = MultiPower(np.array([[-1, 0, 1], [0, 0, 0]]))
D = MultiCheb(np.array([[-1, 0, 0], [0, 1, 0], [1, 0, 0]]))
with pytest.raises(ValueError):
grob = Groebner([C, D])
def test_reduce_matrix():
poly1 = MultiPower(np.array([[1., 0.], [0., 1.]]))
poly2 = MultiPower(np.array([[0., 0.], [1., 0.]]))
poly3 = MultiPower(np.array([[1., 0.], [12., -5.]]))
grob = Groebner([])
grob.new_polys = list((poly1, poly2, poly3))
grob.matrix_terms = []
grob.np_matrix = np.array([])
grob.term_set = set()
grob.lead_term_set = set()
grob._add_polys(grob.new_polys)
#This breaks becasue it hasn't been initialized.
#assert(grob.reduce_matrix())
#assert(len(grob.old_polys) == 2)
#assert(len(grob.new_polys) == 1)
poly1 = MultiPower(np.array([[1., 0.], [0., 0.]]))
poly2 = MultiPower(np.array([[0., 0.], [1., 0.]]))
poly3 = MultiPower(np.array([[1., 0.], [12., 0.]]))
grob = Groebner([])
grob.new_polys = list((poly1, poly2, poly3))
grob.matrix_terms = []
grob.np_matrix = np.array([])
grob.term_set = set()
grob.lead_term_set = set()
grob._add_polys(grob.new_polys)
#This breaks becasue it hasn't been initialized.
#assert(not grob.reduce_matrix())
#assert(len(grob.old_polys) == 3)
#assert(len(grob.new_polys) == 0)
poly1 = MultiPower(np.array([[1., -14.], [0., 2.]]))
poly2 = MultiPower(np.array([[0., 3.], [1., 6.]]))
poly3 = MultiPower(np.array([[1., 0.], [12., -5.]]))
grob = Groebner([])
grob.new_polys = list((poly1, poly2, poly3))
grob.matrix_terms = []
grob.np_matrix = np.array([])
grob.term_set = set()
grob.lead_term_set = set()
grob._add_polys(grob.new_polys)
assert (grob.reduce_matrix())
#assert(len(grob.old_polys) == 3)
assert (len(grob.new_polys) == 2)
def test_phi_criterion():
# Same as grob.solve(), but added true/false to test the phi's.
# *WARNING* MAKE SURE TO CHANGE solve_phi() test to match .solve method always!
def solve_phi(grob, phi=True):
polys = True
i = 1
while polys:
print("Starting Loop #" + str(i))
print("Initializing")
grob.initialize_np_matrix()
print(grob.np_matrix.shape)
print("ADDING PHI's")
grob.add_phi_to_matrix(False)
print(grob.np_matrix.shape)
print("ADDING r's")
grob.add_r_to_matrix()
print(grob2.np_matrix.shape)
polys = grob.reduce_matrix()
i += 1
print("WE WIN")
return grob.reduce_groebner_basis()
# Simple Test Case (Nothing gets added )
A = MultiPower(np.array([[-1, 0, 1], [0, 0, 0]]))
B = MultiPower(np.array([[-1, 0, 0], [0, 1, 0], [1, 0, 0]]))
grob1 = Groebner([A, B])
grob2 = Groebner([A, B])
x1, y1 = solve_phi(grob1, True)
x2, y2 = solve_phi(grob2, False)
assert (np.any([x2 == i and y2 == j for i, j in permutations((x1, y1), 2)
])), "Not the same basis!"
#Second Test
A = MultiPower(
np.array([[[[
0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0
],
[
-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]],
[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]]],
[[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]],
[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]]]]))
B = MultiPower(
np.array([[[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]],
[[
-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]]],
[[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]],
[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]]]]))
C = MultiPower(
np.array([[[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 1
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]],
[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]]],
[[[
-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]],
[[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0
],
[
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0
]]]]))
grob1 = Groebner([A, B, C])
grob2 = Groebner([A, B, C])
w1, x1, y1, z1 = solve_phi(grob1, True)
w2, x2, y2, z2 = solve_phi(grob2, False)
assert (np.any([
w2 == i and x2 == j and y2 == k and z2 == l
for i, j, k, l in permutations((w1, x1, y1, z1), 4)
]))
#Third Test
A = MultiPower(np.array([[-1, 0, 1], [0, 0, 0]]))
B = MultiPower(np.array([[-1, 0, 0], [0, 1, 0], [1, 0, 0]]))
grob1 = Groebner([A, B])
grob2 = Groebner([A, B])
x1, y1 = solve_phi(grob1, True)
x2, y2 = solve_phi(grob2, False)
assert (np.any([A == i and B == j for i, j in permutations((x1, y1), 2)]))
#Fourth Test
A = MultiPower(np.array([[-10, 0], [0, 1], [1, 0]]))
B = MultiPower(np.array([[-25, 0, 0], [0, 0, 1], [0, 0, 0], [1, 0, 0]]))
C = MultiPower(
np.array([[-70, 0, 0, 0], [0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0],
[1, 0, 0, 0]]))
grob1 = Groebner([A, B, C])
grob2 = Groebner([A, B, C])
x1 = solve_phi(grob1, True)
x2 = solve_phi(grob2, False)
assert (x2[0] == x1[0])
#Fifth Test
A = MultiPower(np.array([[1, 1], [0, 0]]))
B = MultiPower(np.array([[1, 0], [1, 0]]))
C = MultiPower(np.array([[1, 0], [1, 0], [0, 1]]))
grob1 = Groebner([A, B, C])
grob2 = Groebner([A, B, C])
x1 = solve_phi(grob1, True)
x2 = solve_phi(grob2, False)
assert (x2[0] == x1[0])