def division(polys,
divisor_var=0,
max_cond_num=1.e6,
macaulay_zero_tol=1.e-12,
verbose=False,
polish=False,
return_all_roots=True):
'''Calculates the common zeros of polynomials using a division matrix.
Parameters
--------
polys: list of MultiCheb Polynomials
The polynomials for which the common roots are found.
divisor_var : int
What variable is being divided by. 0 is x, 1 is y, etc. Defaults to x.
max_cond_num : float
The maximum condition number of the Macaulay Matrix Reduction
macaulay_zero_tol : float
What is considered 0 in the macaulay matrix reduction.
verbose : bool
If True prints information about the solve.
polish: bool
If True runs a newton polish on the zeros before returning.
return_all_roots : bool
If True returns all the roots, otherwise just the ones in the unit box.
Returns
-----------
zeros : numpy array
The common roots of the polynomials. Each row is a root.
'''
#This first section creates the Macaulay Matrix with the monomials that don't have
#the divisor variable in the first columns.
polys, transform, is_projected = polys, lambda x: x, False
if len(polys) == 1:
from yroots.OneDimension import solve
return transform(solve(polys[0], MSmatrix=0))
power = is_power(polys)
dim = polys[0].dim
matrix_degree = np.sum([poly.degree for poly in polys]) - len(polys) + 1
poly_coeff_list = []
for poly in polys:
poly_coeff_list = add_polys(matrix_degree, poly, poly_coeff_list)
matrix, matrix_terms, cuts = create_matrix(poly_coeff_list, matrix_degree,
dim, divisor_var)
if verbose:
np.set_printoptions(suppress=False, linewidth=200)
print('\nStarting Macaulay Matrix\n', matrix)
print(
'\nColumns in Macaulay Matrix\nFirst element in tuple is degree of x monomial, Second element is degree of y monomial \n',
matrix_terms)
print(
'\nLocation of Cuts in the Macaulay Matrix into [ Mb | M1* | M2* ]\n',
cuts)
#If bottom left is zero only does the first QR reduction on top part of matrix (for speed). Otherwise does it on the whole thing
if np.allclose(matrix[cuts[0]:, :cuts[0]], 0):
matrix, matrix_terms = rrqr_reduceMacaulay(
matrix,
matrix_terms,
cuts,
max_cond_num=max_cond_num,
macaulay_zero_tol=macaulay_zero_tol)
else:
matrix, matrix_terms = rrqr_reduceMacaulay(
matrix,
matrix_terms,
cuts,
max_cond_num=max_cond_num,
macaulay_zero_tol=macaulay_zero_tol)
if isinstance(matrix, int):
return -1
VB = matrix_terms[matrix.shape[0]:]
if verbose:
np.set_printoptions(suppress=True, linewidth=200)
print("\nFinal Macaulay Matrix\n", matrix)
print("\nColumns in Macaulay Matrix\n", matrix_terms)
#------------> chebyshev
if not power:
#Builds the inverse matrix. The terms are the vector basis as well as y^k/x terms for all k. Reducing
#this matrix allows the y^k/x terms to be reduced back into the vector basis.
x_pows_over_y = matrix_terms[np.where(
matrix_terms[:, divisor_var] == 0)[0]]
x_pows_over_y[:, divisor_var] = -np.ones(x_pows_over_y.shape[0],
dtype='int')
inv_matrix_terms = np.vstack((x_pows_over_y, VB))
inv_matrix = np.zeros([len(x_pows_over_y), len(inv_matrix_terms)])
#A bunch of different dictionaries are used below for speed purposes and to prevent repeat calculations.
#A dictionary of term in inv_matrix_terms to their spot in inv_matrix_terms.
inv_spot_dict = dict()
spot = 0
for term in inv_matrix_terms:
inv_spot_dict[tuple(term)] = spot
spot += 1
#A dictionary of terms on the diagonal to their reduction in the vector basis.
diag_reduction_dict = dict()
for i in range(matrix.shape[0]):
term = matrix_terms[i]
diag_reduction_dict[tuple(term)] = matrix[i][-len(VB):]
#A dictionary of terms to the terms in their quotient when divided by x. (symbolically)
divisor_terms_dict = dict()
for term in matrix_terms:
divisor_terms_dict[tuple(term)] = get_divisor_terms(
term, divisor_var)
#A dictionary of terms to their quotient when divided by x. (in the vector basis)
term_divide_dict = dict()
for term in matrix_terms[-len(VB):]:
term_divide_dict[tuple(term)] = divide_term(
term, inv_matrix_terms, inv_spot_dict, diag_reduction_dict,
len(VB), divisor_terms_dict)
#Builds the inv_matrix by dividing the rows of matrix by x.
for i in range(cuts[0]):
inv_matrix[i] = divide_row(matrix[i][-len(VB):],
matrix_terms[-len(VB):],
term_divide_dict, len(inv_matrix_terms))
spot = matrix_terms[i]
spot[divisor_var] -= 1
inv_matrix[i][inv_spot_dict[tuple(spot)]] += 1
#Reduces the inv_matrix to solve for the y^k/x terms in the vector basis.
Q, R = qr(inv_matrix)
if np.linalg.cond(R[:, :R.shape[0]]) > max_cond_num:
return -1
inv_solutions = np.hstack((np.eye(R.shape[0]),
solve_triangular(R[:, :R.shape[0]],
R[:, R.shape[0]:])))
#A dictionary of term in the vector basis to their spot in the vector basis.
VB_spot_dict = dict()
spot = 0
for row in VB:
VB_spot_dict[tuple(row)] = spot
spot += 1
#A dictionary of terms of type y^k/x to their reduction in the vector basis.
inv_reduction_dict = dict()
for i in range(len(inv_solutions)):
inv_reduction_dict[tuple(
inv_matrix_terms[i])] = inv_solutions[i][len(inv_solutions):]
#Builds the division matrix and finds the eigenvalues and eigenvectors.
division_matrix = build_division_matrix(VB, VB_spot_dict,
diag_reduction_dict,
inv_reduction_dict,
divisor_terms_dict)
#<---------end Chebyshev
else:
#--------->Power
basisDict = makeBasisDict(matrix, matrix_terms, VB)
#Dictionary of terms in the vector basis their spots in the matrix.
VBdict = {}
spot = 0
for row in VB:
VBdict[tuple(row)] = spot
spot += 1
# Build division matrix
division_matrix = np.zeros((len(VB), len(VB)))
for i in range(VB.shape[0]):
var = np.zeros(dim)
var[divisor_var] = 1
term = tuple(VB[i] - var)
if term in VBdict:
division_matrix[VBdict[term]][i] += 1
else:
division_matrix[:, i] -= basisDict[term]
#<----------end Power
vals, vecs = eig(division_matrix, left=True, right=False)
#conjugate because scipy gives the conjugate eigenvector
vecs = vecs.conj()
# if len(vals) > len(np.unique(np.round(vals, 10))):
# return -1
# eigenvalue_cond = np.linalg.cond(vecs)
# if eigenvalue_cond*tol > 1:
# return -1
# if verbose:
# print("\nDivision Matrix\n", np.round(division_matrix[::-1,::-1], 2))
# print("\nLeft Eigenvectors (as rows)\n", vecs.T)
# if not power:
# if np.max(np.abs(vals)) > 1.e6:
# return -1
#Calculates the zeros, the x values from the eigenvalues and the y values from the eigenvectors.
zeros = list()
for i in range(len(vals)):
# if power and abs(vecs[-1][i]) < 1.e-3:
# #This root has magnitude greater than 1, will possibly generate a false root due to instability
# continue
# if np.abs(vals[i]) < 1.e-5:
# continue
root = np.zeros(dim, dtype=complex)
for spot in range(0, divisor_var):
root[spot] = vecs[-(2 + spot)][i] / vecs[-1][i]
for spot in range(divisor_var + 1, dim):
root[spot] = vecs[-(1 + spot)][i] / vecs[-1][i]
root[divisor_var] = 1 / vals[i]
# conditions = condeigv(division_matrix.T)
# if np.abs(vals[i]) > 1:
# print(root, conditions[i])
if polish:
root = newton_polish(polys, root)
#throw out bad roots in cheb
if not power:
if np.any([abs(poly(root)) > 1.e-1 for poly in polys]):
continue
zeros.append(root)
if return_all_roots:
return transform(np.array(zeros))
else:
# only return roots in the unit complex hyperbox
zeros = transform(np.array(zeros))
return zeros[np.all(np.abs(zeros) <= 1, axis=0)]
def multiplication(polys,
max_cond_num,
macaulay_zero_tol,
verbose=False,
MSmatrix=0,
return_all_roots=True):
'''
Finds the roots of the given list of multidimensional polynomials using a multiplication matrix.
Parameters
----------
polys : list of polynomial objects
Polynomials to find the common roots of.
verbose : bool
Prints information about how the roots are computed.
MSmatrix : int
Controls which Moller-Stetter matrix is constructed. The options are:
0 (default) -- The Moller-Stetter matrix of a random polynomial
Some positive integer i < dimension -- The Moller-Stetter matrix of x_i
return_all_roots : bool
If True returns all the roots, otherwise just the ones in the unit box.
max_cond_num : float
The maximum condition number of the Macaulay Matrix Reduction
macaulay_zero_tol : float
What is considered 0 in the macaulay matrix reduction.
returns
-------
roots : numpy array
The common roots of the polynomials. Each row is a root.
Raises
------
ConditioningError if MSMultMatrix(...) raises a ConditioningError.
'''
#We don't want to use Linear Projection right now
# polys, transform, is_projected = polys, lambda x:x, False
if len(polys) == 1:
from yroots.OneDimension import solve
return transform(solve(polys[0], MSmatrix=0))
poly_type = is_power(polys, return_string=True)
dim = polys[0].dim
if MSmatrix not in list(range(dim + 1)):
raise ValueError(
'MSmatrix must be 0 (random polynomial), or the index of a variable'
)
#By Bezout's Theorem. Useful for making sure that the reduced Macaulay Matrix is as we expect
degrees = [poly.degree for poly in polys]
max_number_of_roots = np.prod(degrees)
try:
m_f, var_dict, basisDict, VB = MSMultMatrix(
polys,
poly_type,
verbose=verbose,
MSmatrix=MSmatrix,
max_cond_num=max_cond_num,
macaulay_zero_tol=macaulay_zero_tol)
except ConditioningError as e:
raise e
if verbose:
print("\nM_f:\n", m_f[::-1, ::-1])
# Get list of indexes of single variables and store vars that were not
# in the vector space basis.
var_spots = []
removed_var_order = []
removed_var_spots = []
var_mask = []
for order, spot in enumerate(get_var_list(dim)):
if spot in var_dict:
var_spots.append(var_dict[tuple(spot)])
var_mask.append(True)
else:
removed_var_order.append(order)
removed_var_spots.append(spot)
var_mask.append(False)
# Get left eigenvectors (come in conjugate pairs)
vals, vecs = eig(m_f, left=True, right=False)
if verbose:
print('\nLeft Eigenvectors (as rows)\n', vecs.T)
print('\nEigenvals\n', vals)
zeros_spot = var_dict[tuple(0 for i in range(dim))]
#throw out roots that were calculated unstably
# vecs = vecs[:,np.abs(vecs[zeros_spot]) > 1.e-10]
if verbose:
print('\nVariable Spots in the Vector\n', var_spots)
print('\nEigeinvecs at the Variable Spots:\n', vecs[var_spots])
print('\nConstant Term Spot in the Vector\n', zeros_spot)
print('\nEigeinvecs at the Constant Term\n', vecs[zeros_spot])
roots = np.zeros([len(vecs), dim], dtype=complex)
roots[:, var_mask] = (vecs[var_spots] / vecs[zeros_spot]).T
#Compute the removed variables
for order, spot in zip(removed_var_order, removed_var_spots):
for coeff, pows in zip(basisDict[spot], VB):
temp = coeff
for place, num in enumerate(pows):
if num > 0:
temp *= roots[:, place]**num
roots[:, order] -= temp
#Check if too many roots
assert roots.shape[
0] <= max_number_of_roots, "Found too many roots,{}/{}/{}:{}".format(
roots.shape, max_number_of_roots, degrees, roots)
if return_all_roots:
roots = np.array(roots, dtype=complex)
# #print(roots)
# REMARK: We don't always have good information about the derivatives,
# so we can't use Newton polishing on our roots.
# for i in range(len(roots)):
# roots[i] = newton_polish(polys,roots[i],niter=100,tol=1e-20)
# #print(roots)
return roots
else:
# only return roots in the unit complex hyperbox
return roots[np.all(np.abs(roots) <= 1, axis=0)]
def MacaulayReduction(initial_poly_list,
max_cond_num,
macaulay_zero_tol,
verbose=False):
"""Reduces the Macaulay matrix to find a vector basis for the system of polynomials.
Parameters
--------
initial_poly_list: list
The polynomials in the system we are solving.
max_cond_num : float
The maximum condition number of the Macaulay Matrix Reduction
macaulay_zero_tol : float
What is considered 0 in the macaulay matrix reduction.
verbose : bool
Prints information about how the roots are computed.
Returns
-----------
basisDict : dict
A dictionary of terms not in the vector basis a matrixes of things in the vector basis that the term
can be reduced to.
VB : numpy array
The terms in the vector basis, each row being a term.
varsToRemove : list
The variables to remove from the basis because we have linear polysnomials
Raises
------
ConditioningError if rrqr_reduceMacaulay(...) raises a ConditioningError.
"""
power = is_power(initial_poly_list)
dim = initial_poly_list[0].dim
poly_coeff_list = []
degree = find_degree(initial_poly_list)
linear_polys = [poly for poly in initial_poly_list if poly.degree == 1]
nonlinear_polys = [poly for poly in initial_poly_list if poly.degree != 1]
#Choose which variables to remove if things are linear, and add linear polys to matrix
if len(linear_polys) == 1: #one linear
varsToRemove = [
np.argmax(np.abs(linear_polys[0].coeff[get_var_list(dim)]))
]
poly_coeff_list = add_polys(degree, linear_polys[0], poly_coeff_list)
elif len(linear_polys) > 1: #multiple linear
#get the row rededuced linear coefficients
A, Pc = nullspace(linear_polys)
varsToRemove = Pc[:len(A)].copy()
#add to macaulay matrix
for row in A:
#reconstruct a polynomial for each row
coeff = np.zeros([2] * dim)
coeff[get_var_list(dim)] = row[:-1]
coeff[tuple([0] * dim)] = row[-1]
if power:
poly = MultiPower(coeff)
else:
poly = MultiCheb(coeff)
poly_coeff_list = add_polys(degree, poly, poly_coeff_list)
else: #no linear
varsToRemove = []
#add nonlinear polys to poly_coeff_list
for poly in nonlinear_polys:
poly_coeff_list = add_polys(degree, poly, poly_coeff_list)
#Creates the matrix
matrix, matrix_terms, cuts = create_matrix(poly_coeff_list, degree, dim,
varsToRemove)
if verbose:
np.set_printoptions(suppress=False, linewidth=200)
print('\nStarting Macaulay Matrix\n', matrix)
print(
'\nColumns in Macaulay Matrix\nFirst element in tuple is degree of x, Second element is degree of y\n',
matrix_terms)
print(
'\nLocation of Cuts in the Macaulay Matrix into [ Mb | M1* | M2* ]\n',
cuts)
try:
matrix, matrix_terms = rrqr_reduceMacaulay(
matrix,
matrix_terms,
cuts,
max_cond_num=max_cond_num,
macaulay_zero_tol=macaulay_zero_tol)
except ConditioningError as e:
raise e
# TODO: rrqr_reduceMacaulay2 is not working when expected.
# if np.allclose(matrix[cuts[0]:,:cuts[0]], 0):
# matrix, matrix_terms = rrqr_reduceMacaulay2(matrix, matrix_terms, cuts, accuracy = accuracy)
# else:
# matrix, matrix_terms = rrqr_reduceMacaulay(matrix, matrix_terms, cuts, accuracy = accuracy)
if verbose:
np.set_printoptions(suppress=True, linewidth=200)
print("\nFinal Macaulay Matrix\n", matrix)
print("\nColumns in Macaulay Matrix\n", matrix_terms)
VB = matrix_terms[matrix.shape[0]:]
basisDict = makeBasisDict(matrix, matrix_terms, VB, power)
return basisDict, VB, varsToRemove
def solve(polys,
MSmatrix=0,
eigvals=True,
verbose=False,
return_all_roots=True,
max_cond_num=1.e6,
macaulay_zero_tol=1.e-12):
'''
Finds the roots of the given list of polynomials.
Parameters
----------
polys : list of polynomial objects
Polynomials to find the common roots of.
MSmatrix : int
Controls which Moller-Stetter matrix is constructed
For a univariate polynomial, the options are:
0 (default) -- The companion or colleague matrix, rotated 180 degrees
1 -- The unrotated companion or colleague matrix
-1 -- The inverse of the companion or colleague matrix
For a multivariate polynomial, the options are:
0 (default) -- The Moller-Stetter matrix of a random polynomial
Some positive integer i <= dimension -- The Moller-Stetter matrix of x_i, where variables are index from x1, ..., xn
Some negative integer i >= -dimension -- The Moller-Stetter matrix of x_i-inverse
eigvals : bool
Whether to compute roots of univariate polynomials from eigenvalues (True) or eigenvectors (False).
Roots of multivariate polynomials are always comptued from eigenvectors
verbose : bool
Prints information about how the roots are computed.
return_all_roots : bool
If True returns all the roots, otherwise just the ones in the unit box.
max_cond_num : float
The maximum condition number of the Macaulay Matrix Reduction
macaulay_zero_tol : float
What is considered 0 in the macaulay matrix reduction.
returns
-------
roots : numpy array
The common roots of the polynomials. Each row is a root.
'''
polys = match_poly_dimensions(polys)
# Determine polynomial type and dimension of the system
poly_type = is_power(polys, return_string=True)
dim = polys[0].dim
if dim == 1:
if len(polys) == 1:
return oneD.solve(polys[0],
MSmatrix=MSmatrix,
eigvals=eigvals,
verbose=verbose)
else:
zeros = np.unique(
oneD.solve(polys[0],
MSmatrix=MSmatrix,
eigvals=eigvals,
verbose=verbose))
#Finds the roots of each succesive polynomial and checks which roots are common.
for poly in polys[1:]:
if len(zeros) == 0:
break
zeros2 = np.unique(
oneD.solve(poly,
MSmatrix=MSmatrix,
eigvals=eigvals,
verbose=verbose))
common = list()
tol = 1.e-10
for zero in zeros2:
spot = np.where(np.abs(zeros - zero) < tol)
if len(spot[0]) > 0:
common.append(zero)
zeros = common
return zeros
else:
if MSmatrix < 0:
return division(polys,
verbose=verbose,
divisor_var=-MSmatrix - 1,
return_all_roots=return_all_roots,
max_cond_num=max_cond_num,
macaulay_zero_tol=macaulay_zero_tol)
else:
return multiplication(polys,
verbose=verbose,
MSmatrix=MSmatrix,
return_all_roots=return_all_roots,
max_cond_num=max_cond_num,
macaulay_zero_tol=macaulay_zero_tol)
def build_macaulay(initial_poly_list, verbose=False):
"""Constructs the unreduced Macaulay matrix. Removes linear polynomials by
substituting in for a number of variables equal to the number of linear
polynomials.
Parameters
--------
initial_poly_list: list
The polynomials in the system we are solving.
verbose : bool
Prints information about how the roots are computed.
Returns
-----------
matrix : 2d ndarray
The Macaulay matrix
matrix_terms : 2d integer ndarray
Array containing the ordered basis, where the ith row contains the
exponent/degree of the ith basis monomial
cut : int
Where to cut the Macaulay matrix for the highest-degree monomials
varsToRemove : list
The variables removed with removing linear polynomials
A : 2d ndarray
A matrix giving the linear relations between the removed variables and
the remaining variables
Pc : 1d integer ndarray
Array containing the order of the variables as the appear in the columns
of A
"""
power = is_power(initial_poly_list)
dim = initial_poly_list[0].dim
poly_coeff_list = []
degree = find_degree(initial_poly_list)
linear_polys = [poly for poly in initial_poly_list if poly.degree == 1]
nonlinear_polys = [poly for poly in initial_poly_list if poly.degree != 1]
#Choose which variables to remove if things are linear, and add linear polys to matrix
if len(linear_polys) >= 1: #Linear polys involved
#get the row rededuced linear coefficients
A, Pc = nullspace(linear_polys)
varsToRemove = Pc[:len(A)].copy()
#add to macaulay matrix
for row in A:
#reconstruct a polynomial for each row
coeff = np.zeros([2] * dim)
coeff[tuple(get_var_list(dim))] = row[:-1]
coeff[tuple([0] * dim)] = row[-1]
if not power:
poly = MultiCheb(coeff)
else:
poly = MultiPower(coeff)
poly_coeff_list = add_polys(degree, poly, poly_coeff_list)
else: #no linear
A, Pc = None, None
varsToRemove = []
#add nonlinear polys to poly_coeff_list
for poly in nonlinear_polys:
poly_coeff_list = add_polys(degree, poly, poly_coeff_list)
#Creates the matrix
# return (*create_matrix(poly_coeff_list, degree, dim, varsToRemove), A, Pc)
return create_matrix(poly_coeff_list, degree, dim, varsToRemove)
def multiplication(polys,
max_cond_num,
verbose=False,
return_all_roots=True,
method='svd'):
'''
Finds the roots of the given list of multidimensional polynomials using a multiplication matrix.
Parameters
----------
polys : list of polynomial objects
Polynomials to find the common roots of.
max_cond_num : float
The maximum condition number of the Macaulay Matrix Reduction
verbose : bool
Prints information about how the roots are computed.
return_all_roots : bool
If True returns all the roots, otherwise just the ones in the unit box.
returns
-------
roots : numpy array
The common roots of the polynomials. Each row is a root.
Raises
------
ConditioningError if reduce_macaulay() raises a ConditioningError.
TooManyRoots if the macaulay matrix returns more roots than the Bezout bound.
'''
#We don't want to use Linear Projection right now
# polys, transform, is_projected = polys, lambda x:x, False
if len(polys) == 1:
from yroots.OneDimension import solve
return transform(solve(polys[0], MSmatrix=0))
poly_type = is_power(polys, return_string=True)
dim = polys[0].dim
#By Bezout's Theorem. Useful for making sure that the reduced Macaulay Matrix is as we expect
degrees = [poly.degree for poly in polys]
max_number_of_roots = np.prod(degrees)
matrix, matrix_terms, cut = build_macaulay(polys, verbose)
# Attempt to reduce the Macaulay matrix
if method == 'qrt':
try:
E, Q, cond, cond_back = reduce_macaulay_qrt(
matrix, cut, max_cond_num)
except ConditioningError as e:
raise e
elif method == 'tvb':
try:
E, Q, cond, cond_back = reduce_macaulay_tvb(
matrix, cut, max_cond_num)
except ConditioningError as e:
raise e
elif method == 'svd':
try:
E, Q, cond, cond_back = reduce_macaulay_svd(
matrix, cut, max_cond_num)
except ConditioningError as e:
raise e
# Construct the Möller-Stetter matrices
# M is a 3d array containing the multiplication-by-x_i matrix in M[...,i]
if poly_type == "MultiCheb":
if method == 'qrt' or method == 'svd':
M = ms_matrices_cheb(E, Q, matrix_terms, dim)
elif method == 'tvb':
M = ms_matrices_p_cheb(E, Q, matrix_terms, dim, cut)
else:
if method == 'qrt' or method == 'svd':
M = ms_matrices(E, Q, matrix_terms, dim)
elif method == 'tvb':
M = ms_matrices_p(E, Q, matrix_terms, dim, cut)
# Compute the roots using eigenvalues of the Möller-Stetter matrices
roots, cond_eig = msroots(M)
# Check if too many roots
if roots.shape[0] > max_number_of_roots:
raise TooManyRoots("Found too many roots,{}/{}/{}:{}".format(
roots.shape, max_number_of_roots, degrees, roots))
if return_all_roots:
return roots
else:
# only return roots in the unit complex hyperbox
return roots[np.all(np.abs(roots) <= 1, axis=0)]
def multiplication(polys, max_cond_num, verbose=False, return_all_roots=True,method='svd'):
'''
Finds the roots of the given list of multidimensional polynomials using a multiplication matrix.
Parameters
----------
polys : list of polynomial objects
Polynomials to find the common roots of.
max_cond_num : float
The maximum condition number of the Macaulay Matrix Reduction
verbose : bool
Prints information about how the roots are computed.
return_all_roots : bool
If True returns all the roots, otherwise just the ones in the unit box.
returns
-------
roots : numpy array
The common roots of the polynomials. Each row is a root.
'''
#We don't want to use Linear Projection right now
# polys, transform, is_projected = polys, lambda x:x, False
if len(polys) == 1:
from yroots.OneDimension import solve
return transform(solve(polys[0], MSmatrix=0))
poly_type = is_power(polys, return_string = True)
dim = polys[0].dim
#By Bezout's Theorem. Useful for making sure that the reduced Macaulay Matrix is as we expect
bezout_bound = np.prod([poly.degree for poly in polys])
matrix, matrix_terms, cut = build_macaulay(polys, verbose)
roots = np.array([])
# If cut is zero, then all the polynomials are linear and we solve
# using solve_linear.
if cut == 0:
roots, cond = solve_linear([p.coeff for p in polys])
# Make sure roots is a 2D array.
roots = np.array([roots])
else:
# Attempt to reduce the Macaulay matrix
if method == 'svd':
res = reduce_macaulay_svd(matrix,cut,bezout_bound,max_cond_num)
if res[0] is None:
return res
E,Q = res
elif method == 'qrt':
res = reduce_macaulay_qrt(matrix,cut,bezout_bound,max_cond_num)
if res[0] is None:
return res
E,Q = res
elif method == 'tvb':
res = reduce_macaulay_tvb(matrix,cut,bezout_bound,max_cond_num)
if res[0] is None:
return res
E,Q = res
else:
raise ValueError("Method must be one of 'svd','qrt' or 'tvb'")
# Construct the Möller-Stetter matrices
# M is a 3d array containing the multiplication-by-x_i matrix in M[...,i]
if poly_type == "MultiCheb":
if method == 'qrt' or method == 'svd':
M = ms_matrices_cheb(E,Q,matrix_terms,dim)
elif method == 'tvb':
M = ms_matrices_p_cheb(E,Q,matrix_terms,dim,cut)
else:
if method == 'qrt' or method == 'svd':
M = ms_matrices(E,Q,matrix_terms,dim)
elif method == 'tvb':
M = ms_matrices_p(E,Q,matrix_terms,dim,cut)
# Compute the roots using eigenvalues of the Möller-Stetter matrices
roots = msroots(M)
if return_all_roots:
return roots
else:
# only return roots in the unit complex hyperbox
return roots[[np.all(np.abs(root) <= 1) for root in roots]]
def divisionNew(polys,
divisor_var=0,
tol=1.e-12,
verbose=False,
polish=False,
return_all_roots=True):
'''Calculates the common zeros of polynomials using a division matrix.
Parameters
--------
polys: list of MultiCheb Polynomials
The polynomials for which the common roots are found.
divisor_var : int
What variable is being divided by. 0 is x, 1 is y, etc. Defaults to x.
tol : float
The tolerance parameter for the Macaulay Reduce.
verbose : bool
If True prints information about the solve.
polish: bool
If True runs a newton polish on the zeros before returning.
Returns
-----------
zeros : numpy array
The common roots of the polynomials. Each row is a root.
'''
#This first section creates the Macaulay Matrix with the monomials that don't have
#the divisor variable in the first columns.
polys, transform, is_projected = LinearProjection.remove_linear(
polys, 1e-4, 1e-8)
if len(polys) == 1:
from yroots.OneDimension import solve
return transform(solve(polys[0], MSmatrix=0))
power = is_power(polys)
if power:
raise ValueError("This only works for Chebyshev polynomials")
dim = polys[0].dim
matrix_degree = np.sum([poly.degree for poly in polys]) - len(polys) + 1
poly_coeff_list = []
for poly in polys:
poly_coeff_list = add_polys(matrix_degree, poly, poly_coeff_list)
matrix, matrix_terms, cuts = create_matrix(poly_coeff_list, matrix_degree,\
dim, divisor_var)
x_pows_over_y = matrix_terms[:cuts[0]].copy()
x_pows_over_y[:, divisor_var] = -np.ones(cuts[0], dtype='int')
inv_matrix_terms = np.vstack((x_pows_over_y, matrix_terms))
num_rows = matrix.shape[0]
inv_matrix = np.zeros([num_rows + matrix.shape[0], len(inv_matrix_terms)])
cuts = tuple([cuts[0], cuts[0] + cuts[1]])
#A dictionary of term in inv_matrix_terms to their spot in inv_matrix_terms.
inv_spot_dict = dict()
spot = 0
for term in inv_matrix_terms:
inv_spot_dict[tuple(term)] = spot
spot += 1
#A dictionary of terms to the terms in their quotient when divided by x. (symbolically)
divisor_terms_dict = dict()
for term in matrix_terms:
divisor_terms_dict[tuple(term)] = get_divisor_terms(term, divisor_var)
#A dictionary of terms to their quotient when divided by x. (in the vector basis)
term_divide_dict = dict()
for term in matrix_terms:
term_divide_dict[tuple(term)] = divide_term(term, inv_matrix_terms,
inv_spot_dict,
divisor_terms_dict)
#Builds the inv_matrix by dividing the rows of matrix by x.
for i in range(num_rows):
inv_matrix[i] = divide_row(matrix[i], matrix_terms, term_divide_dict,
len(inv_matrix_terms))
inv_matrix[num_rows:, cuts[0]:] = matrix
matrix, matrix_terms, perm = rrqr_reduceMacaulay(inv_matrix,
inv_matrix_terms,
cuts,
accuracy=tol,
return_perm=True)
if isinstance(matrix, int):
return -1
for term in term_divide_dict:
term_divide_dict[tuple(term)] = term_divide_dict[tuple(term)][perm]
VB = matrix_terms[matrix.shape[0]:]
basisDict = makeBasisDict2(matrix, matrix_terms)
# print(len(VB))
#Dictionary of terms in the vector basis their spots in the matrix.
VBdict = {}
spot = 0
for row in VB:
VBdict[tuple(row)] = spot
spot += 1
#Builds the division matrix and finds the eigenvalues and eigenvectors.
division_matrix = build_division_matrix(VB, VBdict, basisDict,
term_divide_dict, matrix_terms)
vals, vecs = eig(division_matrix, left=True, right=False)
#conjugate because scipy gives the conjugate eigenvector
vecs = vecs.conj()
if len(vals) > len(np.unique(np.round(vals, 10))):
return -1
vals2, vecs2 = eig(vecs)
sorted_vals2 = np.sort(np.abs(vals2)) #Sorted smallest to biggest
if sorted_vals2[0] < sorted_vals2[-1] * tol:
return -1
if verbose:
print("\nDivision Matrix\n", np.round(division_matrix[::-1, ::-1], 2))
print("\nLeft Eigenvectors (as rows)\n", vecs.T)
if np.max(np.abs(vals)) > 1.e6:
return -1
#Calculates the zeros, the x values from the eigenvalues and the y values from the eigenvectors.
zeros = list()
for i in range(len(vals)):
if np.abs(vals[i]) < 1.e-5:
continue
root = np.zeros(dim, dtype=complex)
for spot in range(0, divisor_var):
root[spot] = vecs[-(2 + spot)][i] / vecs[-1][i]
for spot in range(divisor_var + 1, dim):
root[spot] = vecs[-(1 + spot)][i] / vecs[-1][i]
root[divisor_var] = 1 / vals[i]
if polish:
root = newton_polish(polys, root, tol=tol)
#throw out bad roots in cheb
if np.any([abs(poly(root)) > 1.e-1 for poly in polys]):
continue
zeros.append(root)
if return_all_roots:
return transform(np.array(zeros))
else:
# only return roots in the unit complex hyperbox
zeros = transform(np.array(zeros))
return zeros[np.all(np.abs(zeros) <= 1, axis=0)]