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 test_add():
a1 = np.arange(27).reshape((3, 3, 3))
Test2 = MultiPower(a1)
a2 = np.ones((3, 3, 3))
Test3 = MultiPower(a2)
addTest = Test2 + Test3
assert (addTest.coeff == (Test2.coeff + Test3.coeff)).all()
def test_mon_mult_random():
#test with random matrices
possible_dim = np.random.randint(1, 5, (1, 10))
dim = possible_dim[0, random.randint(1, 9)]
shape = list()
for i in range(dim):
shape.append(random.randint(2, 4))
matrix1 = np.random.randint(1, 11, (shape))
M1 = MultiPower(matrix1)
shape2 = list()
for i in range(dim):
shape2.append(random.randint(2, 4))
matrix2 = np.ones(shape2)
M2 = MultiPower(matrix2)
M3 = M1 * M2
for index, i in np.ndenumerate(M2.coeff):
if sum(index) == 0:
M4 = MultiPower.mon_mult(M1, index)
else:
M4 = M4 + MultiPower.mon_mult(M1, index)
if M3.shape != M4.shape:
new_M3, new_M4 = MultiPower.match_size(M3, M3, M4)
else:
new_M3, new_M4 = M3, M4
assert np.allclose(new_M3.coeff, new_M4.coeff)
def test_evaluate_at():
# Evaluate .5xyz + 2x + y + z at (4,2,1)
poly = MultiPower(
np.array([[[0, 1, 0], [1, 0, 0], [0, 0, 0]],
[[2, 0, 0], [0, .5, 0], [0, 0, 0]]]))
assert (poly.evaluate_at((4, 2, 1)) == 15)
def testRoots_4():
f1 = MultiPower(np.array([[5,-1],[1,0]]))
f2 = MultiPower(np.array([[1,-1],[-1,0]]))
root = rf.roots([f1, f2])[0]
assert(all(np.isclose(root, [-2,3])))
def calc_s(self, a, b):
'''
Calculates the S-polynomial of a,b
'''
lcm = self._lcm(a, b)
a_coeffs = np.zeros_like(a.coeff)
a_coeffs[tuple([i - j for i, j in zip(lcm, a.lead_term)
])] = 1. / (a.coeff[tuple(a.lead_term)])
b_coeffs = np.zeros_like(b.coeff)
b_coeffs[tuple([i - j for i, j in zip(lcm, b.lead_term)
])] = 1. / (b.coeff[tuple(b.lead_term)])
if isinstance(a, MultiPower) and isinstance(b, MultiPower):
b_ = MultiPower(b_coeffs)
a_ = MultiPower(a_coeffs)
elif isinstance(a, MultiCheb) and isinstance(b, MultiCheb):
b_ = MultiCheb(b_coeffs)
a_ = MultiCheb(a_coeffs)
else:
raise ValueError('Incompatiable polynomials')
a1 = a_ * a
b1 = b_ * b
s = a_ * a - b_ * b
#self.polys.append(s)
return s
def test_mult():
test1 = np.array([[0, 1], [2, 1]])
test2 = np.array([[2, 2], [3, 0]])
p1 = MultiPower(test1)
p2 = MultiPower(test2)
new_poly = p1 * p2
truth = MultiPower(np.array([[0, 2, 2], [4, 9, 2], [6, 3, 0]]))
assert np.allclose(new_poly.coeff, truth.coeff)
def testRoots_5():
f1 = MultiPower(np.array([[0,-1],[0,0],[1,0]]))
f2 = MultiPower(np.array([[1,-1],[1,0]]))
roots = rf.roots([f1, f2])
assert(all(np.isclose(roots[0], [-0.61803399, 0.38196601])))
assert(all(np.isclose(roots[1], [1.61803399, 2.61803399])))
def test_evaluate_at2():
# Evaluate -.5x^2y + 2xy^2 - 3z^2 + yz at (7.4,2.33,.25)
poly = MultiPower(
np.array([[[0, 0, -3], [0, 1, 0], [0, 0, 0]],
[[0, 0, 0], [0, 0, 0], [2, 0, 0]],
[[0, 0, 0], [-.5, 0, 0], [0, 0, 0]]]))
assert (np.isclose(poly.evaluate_at((7.4, 2.33, .25)), 16.94732))
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_vectorSpaceBasis():
f1 = MultiPower(np.array([[0,-1.5,.5],[-1.5,1.5,0],[1,0,0]]))
f2 = MultiPower(np.array([[0,0,0],[-1,0,1],[0,0,0]]))
f3 = MultiPower(np.array([[0,-1,0,1],[0,0,0,0],[0,0,0,0],[0,0,0,0]]))
G = [f1, f2, f3]
basis = rf.vectorSpaceBasis(G)[0]
trueBasis = [(0,0), (1,0), (0,1), (1,1), (0,2)]
assert (len(basis) == len(trueBasis)) and (m in basis for m in trueBasis), \
"Failed on MultiPower in 2 vars."
def testRoots_7(): # This test sometimes fails due to stability issues...
f1_coeff = np.zeros((3,3,3))
f1_coeff[(0,2,0)] = 1
f1_coeff[(1,1,0)] = -1
f1_coeff[(1,0,1)] = -2
f1 = MultiPower(f1_coeff)
f2_coeff = np.zeros((4,4,4))
f2_coeff[(0,3,0)] = 1
f2_coeff[(0,0,2)] = 1
f2_coeff[(0,0,0)] = 1
f2 = MultiPower(f2_coeff)
f3_coeff = np.zeros((3,3,3))
f3_coeff[(2,1,1)] = 1
f3_coeff[(0,1,1)] = -1
f3 = MultiPower(f3_coeff)
roots = rf.roots([f1, f2, f3])
values_at_roots = [[f1.evaluate_at(root) for root in roots],
[f2.evaluate_at(root) for root in roots],
[f3.evaluate_at(root) for root in roots]]
assert(np.all(np.isclose(values_at_roots, 0)))
def testReducePoly_3():
poly = MultiPower(
np.array([[0, -1, 0, 1], [0, 2, 0, 0], [0, 0, 1, 0], [1, 0, 0, 0]]))
g1 = MultiPower(np.array([[0, 0, 0], [-2, 0, 0], [1, 0, 0]]))
g2 = MultiPower(
np.array([[0, -1, 0, 1], [3, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]))
reduced = rf.reduce_poly(poly, [g1, g2])
assert (np.all(reduced.coeff == np.array([[0, 0, 0], [1, 2, 2]])))
def testReducePoly_4():
poly = MultiPower(np.array([[[-1,2,0],[0,0,0],[-3,0,0]],
[[0,0,0],[2,0,0],[0,0,0]],
[[0,0,0],[0,0,1],[0,0,0]]]))
d1 = MultiPower(np.array([[0,-3,0],
[0,0,0],
[1,0,0]]))
d2 = MultiPower(np.array([[0,0,0,1],
[4,0,0,0]]))
d3 = MultiPower(np.array([[[-1,1]]]))
reduced = rf.reduce_poly(poly, [d1, d2, d3])
assert(np.all(reduced.coeff == np.array([[[1],[0]],[[0],[2]]])))
def testRoots():
f1 = MultiPower(np.array([[0,-1.5,.5],[-1.5,1.5,0],[1,0,0]]), clean_zeros=False)
f2 = MultiPower(np.array([[0,0,0],[-1,0,1],[0,0,0]]), clean_zeros=False)
f3 = MultiPower(np.array([[0,-1,0,1],[0,0,0,0],[0,0,0,0],[0,0,0,0]]), clean_zeros=False)
roots = rf.roots([f1, f2, f3])
values_at_roots = np.array([[f1.evaluate_at(root) for root in roots],
[f2.evaluate_at(root) for root in roots],
[f3.evaluate_at(root) for root in roots]])
assert(np.all(values_at_roots==0))
def multMatrix(poly, GB, basis):
'''
Finds the matrix of the linear operator m_f on A = C[x_1,...,x_n]/I
where f is the polynomial argument. The linear operator m_f is defined
as m_f([g]) = [f]*[g] where [f] represents the coset of f in
A. Since m_f is a linear operator on A, it can be represented by its
matrix with respect to the vector space basis.
parameters
----------
poly : polynomial object
The polynomial f for which to find the matrix m_f.
GB: list of polynomial objects
Polynomials that make up a Groebner basis for the ideal
basis : list of tuples
The monomials that make up a basis for the vector space A
return
------
multOperatorMatrix : square numpy array
The matrix m_f
'''
# Reshape poly's coefficienet matrix if it is not in the same number
# of variables as the polynomials in the Groebner basis.
# (i.e. len(shape) is the number of variables the polynomial is in)
numVars = len(GB[0].shape)
polyVars = len(poly.coeff.shape)
if polyVars != numVars:
new_shape = [i for i in poly.coeff.shape]
for j in range(numVars-polyVars): new_shape.append(1)
if type(poly) is MultiPower:
poly = MultiPower(poly.coeff.reshape(tuple(new_shape)))
if type(poly) is MultiCheb:
poly = MultiCheb(poly.coeff.reshape(tuple(new_shape)))
dim = len(basis)
operatorMatrix = np.zeros((dim, dim))
for i in range(dim):
monomial = basis[i]
poly_ = poly.mon_mult(monomial)
operatorMatrix[:,i] = coordinateVector(poly_, GB, basis)
return operatorMatrix
def sm_to_poly(self, rows, reduced_matrix):
'''
Takes a list of indicies corresponding to the rows of the reduced matrix and
returns a list of polynomial objects
'''
shape = []
p_list = []
matrix_term_vals = [i.val for i in self.matrix_terms]
# Finds the maximum size needed for each of the poly coeff tensors
for i in range(len(matrix_term_vals[0])):
# add 1 to each to compensate for constant term
shape.append(max(matrix_term_vals, key=itemgetter(i))[i] + 1)
# Grabs each polynomial, makes coeff matrix and constructs object
for i in rows:
p = reduced_matrix[i]
p[np.where(abs(p) < global_accuracy)] = 0
coeff = np.zeros(shape)
for j, term in enumerate(matrix_term_vals):
coeff[term] = p[j]
if self.power:
poly = MultiPower(coeff)
p_list.append(poly)
else:
poly = MultiCheb(coeff)
p_list.append(poly)
return p_list
def testMultMatrix_2():
f1 = MultiPower(np.array([[0,-1.5,.5],[-1.5,1.5,0],[1,0,0]]))
f2 = MultiPower(np.array([[0,0,0],[-1,0,1],[0,0,0]]))
f3 = MultiPower(np.array([[0,-1,0,1],[0,0,0,0],[0,0,0,0],[0,0,0,0]]))
GB = [f1, f2, f3]
VB = rf.vectorSpaceBasis(GB)[0]
x = MultiPower(np.array([[0],[1]]))
y = MultiPower(np.array([[0,1]]))
mx_Eig = np.linalg.eigvals(rf.multMatrix(x, GB, VB))
my_Eig = np.linalg.eigvals(rf.multMatrix(y, GB, VB))
assert(len(mx_Eig) == 5)
assert(len(my_Eig) == 5)
assert(np.allclose(mx_Eig, [-1., 2., 1., 1., 0.]))
assert(np.allclose(my_Eig, [1., -1., 1., -1., 0.]))
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_mon_mult():
"""
Tests monomial multiplication using normal polynomial multiplication.
"""
#Simple 2D test cases
cheb1 = MultiCheb(np.array([[0, 0, 0], [0, 0, 0], [0, 0, 1]]))
mon1 = (1, 1)
result1 = cheb1.mon_mult(mon1)
truth1 = np.array([[0, 0, 0, 0], [0, 0.25, 0, 0.25], [0, 0, 0, 0],
[0, 0.25, 0, 0.25]])
assert np.allclose(result1.coeff, truth1)
#test with random matrices
cheb2 = np.random.randint(-9, 9, (4, 4))
C1 = MultiCheb(cheb2)
C2 = cheb2poly(C1)
C3 = MultiCheb.mon_mult(C1, (1, 1))
C4 = MultiPower.mon_mult(C2, (1, 1))
C5 = poly2cheb(C4)
assert np.allclose(C3.coeff, C5.coeff)
# test results of chebyshev mult compared to power multiplication
cheb3 = np.random.randn(5, 4)
c1 = MultiCheb(cheb3)
c2 = MultiCheb(np.ones((4, 2)))
for index, i in np.ndenumerate(c2.coeff):
if sum(index) == 0:
c3 = c1.mon_mult(index)
else:
c3 = c3 + c1.mon_mult(index)
p1 = cheb2poly(c1)
p2 = cheb2poly(c2)
p3 = p1 * p2
p4 = cheb2poly(c3)
assert np.allclose(p3.coeff, p4.coeff)
# test results of chebyshev mult compared to power multiplication in 3D
cheb4 = np.random.randn(3, 3, 3)
a1 = MultiCheb(cheb4)
a2 = MultiCheb(np.ones((3, 3, 3)))
for index, i in np.ndenumerate(a2.coeff):
if sum(index) == 0:
a3 = a1.mon_mult(index)
else:
a3 = a3 + a1.mon_mult(index)
q1 = cheb2poly(a1)
q2 = cheb2poly(a2)
q3 = q1 * q2
q4 = cheb2poly(a3)
assert np.allclose(q3.coeff, q4.coeff)
def test_vectorSpaceBasis_2():
f1 = MultiPower(np.array([[[0,0,1],[0,3/20,0],[0,0,0]],
[[0,0,0],[-3/40,1,0],[0,0,0]],
[[0,0,0],[0,0,0],[0,0,0]]]))
f2 = MultiPower(np.array([[[3/16,-5/2,0],[0,3/16,0],[0,0,0]],
[[0,0,1],[0,0,0],[0,0,0]],
[[0,0,0],[0,0,0],[0,0,0]]]))
f3 = MultiPower(np.array([[[0,1,1/2],[0,3/40,1],[0,0,0]],
[[-1/2,20/3,0],[-3/80,0,0],[0,0,0]],
[[0,0,0],[0,0,0],[0,0,0]]]))
f4 = MultiPower(np.array([[[3/32,-7/5,0,1],[-3/16,83/32,0,0],[0,0,0,0],[0,0,0,0]],
[[3/40,-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]]]))
f5 = 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]]]))
f6 = MultiPower(np.array([[[0,0,0],[0,0,0],[1,0,0]],
[[0,-8/3,0],[0,0,0],[0,0,0]],
[[0,0,0],[0,0,0],[0,0,0]]]))
G = [f1, f2, f3, f4, f5, f6]
basis = rf.vectorSpaceBasis(G)[0]
trueBasis = [(0,0,0),(1,0,0),(0,1,0),(1,1,0),(0,0,1),(0,0,2),(1,0,1),(0,1,1)]
assert (len(basis) == len(trueBasis)) and (m in basis for m in trueBasis), \
"Failed on MultiPower in 3 vars."
def testRoots_2():
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]]]))
roots = rf.roots([f1, f2, f3])
values_at_roots = np.array([[f1.evaluate_at(root) for root in roots],
[f2.evaluate_at(root) for root in roots],
[f3.evaluate_at(root) for root in roots]])
assert(np.all(abs(values_at_roots)<1.e-5))
def calc_r(self, m):
'''
Finds the r polynomial that has a leading monomial m
Returns the polynomial.
'''
for p in self.new_polys + self.old_polys:
l = list(p.lead_term)
if all([i <= j for i, j in zip(l, m)
]) and len(l) == len(m): #Checks to see if l divides m
c = [j - i for i, j in zip(l, m)]
if not l == m: #Make sure c isn't all 0
return p.mon_mult(c)
if self.power:
return MultiPower(np.array([0]))
else:
return MultiCheb(np.array([0]))
def testRoots_3():
# roots of [x^2-y, x^3-y+1]
f1 = MultiPower(np.array([[0,-1],[0,0],[1,0]]))
f2 = MultiPower(np.array([[1,-1],[0,0],[0,0],[1,0]]))
roots = rf.roots([f1, f2])
values_at_roots = np.array([[f1.evaluate_at(root) for root in roots],
[f2.evaluate_at(root) for root in roots]])
assert(np.all(values_at_roots==0))
def add_r_to_matrix(self):
'''
Makes Heap out of all monomials, and finds lcms to add them into the matrix
'''
for monomial in self.term_set:
m = list(monomial)
for p in self.polys:
l = list(p.lead_term)
if all([i <= j for i, j in zip(l, m)]) and len(l) == len(m):
c = [j - i for i, j in zip(l, m)]
c_coeff = np.zeros(np.array(self.largest_mon.val) + 1)
c_coeff[tuple(c)] = 1
if isinstance(p, MultiCheb):
c = MultiCheb(c_coeff)
elif isinstance(p, MultiPower):
c = MultiPower(c_coeff)
r = c * p
self.add_poly_to_matrix(r)
break
pass
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_mon_mult():
"""
Tests monomial multiplication using normal polynomial multiplication.
"""
mon = (1, 2)
Poly = MultiPower(
np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12],
[13, 14, 15, 16]]))
mon_matr = MultiPower(
np.array([[0, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 0], [0, 0, 0, 0]]))
P1 = mon_matr * Poly
P2 = MultiPower.mon_mult(Poly, mon)
mon2 = (0, 1, 1)
Poly2 = MultiPower(np.arange(1, 9).reshape(2, 2, 2))
mon_matr2 = MultiPower(np.array([[[0, 0], [0, 1]], [[0, 0], [0, 0]]]))
T1 = mon_matr2 * Poly2
T2 = MultiPower.mon_mult(Poly2, mon2)
assert np.allclose(P1.coeff, P2.coeff, atol=1.0e-10)
assert np.allclose(T1.coeff, T2.coeff, atol=1.0e-10)
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 testReducePoly_2():
poly = MultiPower(np.array([[-7], [2], [-13], [4]]))
g = MultiPower(np.array([[-2], [3], [1]]))
reduced = rf.reduce_poly(poly, [g])
assert (np.all(reduced.coeff == np.array([[-57.], [85.]])))
def testReducePoly():
poly = MultiPower(np.array([[-3], [2], [-4], [1]]))
g = MultiPower(np.array([[2], [1]]))
reduced = rf.reduce_poly(poly, [g])
assert (reduced.coeff == np.array([[-31.]]))
