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poly_commn_roots.py
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/
poly_commn_roots.py
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"""
poly_commn_roots.py: polynomial common roots finding
Copyright (C) 2017 Hanjie Pan
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
Correspondence concerning LEAP should be addressed as follows:
Email: hanjie [Dot] pan [At] epfl [Dot] ch
Postal address: EPFL-IC-LCAV
Station 14
1015 Lausanne
Switzerland
"""
from __future__ import division
import numpy as np
import numexpr as ne
from scipy import linalg
import sympy
def find_roots(coef1, coef2):
"""
Find the common roots of two bivariate polynomials with coefficients specified by
two 2D arrays.
the variation along the first dimension (i.e., columns) is in the increasing order of y.
the variation along the second dimension (i.e., rows) is in the increasing order of x.
:param coef1: polynomial coefficients the first polynomial for the annihilation along rows
:param coef2: polynomial coefficients the second polynomial for the annihilation along cols
:return:
"""
# assert coef_col.shape[0] >= coef_row.shape[0] and coef_row.shape[1] >= coef_col.shape[1]
if coef1.shape[1] < coef2.shape[1]:
# swap input coefficients
coef1, coef2 = coef2, coef1
x, y = sympy.symbols('x, y') # build symbols
# collect both polynomials as a function of x; y will be included in the coefficients
poly1 = 0
poly2 = 0
max_row_degree_x = coef1.shape[1] - 1
max_row_degree_y = coef1.shape[0] - 1
for x_count in range(max_row_degree_x + 1):
for y_count in range(max_row_degree_y + 1):
if np.abs(coef1[y_count, x_count]) > 1e-10:
poly1 += coef1[y_count, x_count] * x ** (max_row_degree_x - x_count) * \
y ** (max_row_degree_y - y_count)
else:
coef1[y_count, x_count] = 0
max_col_degree_x = coef2.shape[1] - 1
max_col_degree_y = coef2.shape[0] - 1
for x_count in range(max_col_degree_x + 1):
for y_count in range(max_col_degree_y + 1):
if np.abs(coef2[y_count, x_count]) > 1e-10:
poly2 += coef2[y_count, x_count] * x ** (max_col_degree_x - x_count) * \
y ** (max_col_degree_y - y_count)
else:
coef2[y_count, x_count] = 0
poly1_x = sympy.Poly(poly1, x)
poly2_x = sympy.Poly(poly2, x)
K = max_row_degree_x # highest power of the first polynomial (in x)
L = max_col_degree_x # highest power of the second polynomial (in x)
if coef1.shape[0] == 1: # i.e., independent of variable y
x_roots_all = np.roots(coef1.squeeze())
eval_poly2 = sympy.lambdify(x, poly2)
x_roots = []
y_roots = []
for x_loop in x_roots_all:
y_roots_loop = np.roots(np.array(sympy.Poly(eval_poly2(x_loop), y).all_coeffs(), dtype=complex))
y_roots.append(y_roots_loop)
x_roots.append(np.tile(x_loop, y_roots_loop.size))
coef_validate = coef2
elif coef2.shape[1] == 1: # i.e., independent of variable x
y_roots_all = np.roots(coef2.squeeze())
eval_poly1 = sympy.lambdify(y, poly1)
x_roots = []
y_roots = []
for y_loop in y_roots_all:
x_roots_loop = np.roots(np.array(sympy.Poly(eval_poly1(y_loop), x).all_coeffs(), dtype=complex))
x_roots.append(x_roots_loop)
y_roots.append(np.tile(y_loop, x_roots_loop.size))
coef_validate = coef1
else:
if L >= 1:
toep1_r = np.hstack((poly1_x.all_coeffs()[::-1], np.zeros(L - 1)))
toep1_r = np.concatenate((toep1_r, np.zeros(L + K - toep1_r.size)))
toep1_c = np.concatenate(([poly1_x.all_coeffs()[-1]], np.zeros(L - 1)))
else: # for the case with L == 0
toep1_r = np.zeros((0, L + K))
toep1_c = np.zeros((0, 0))
if K >= 1:
toep2_r = np.hstack((poly2_x.all_coeffs()[::-1], np.zeros(K - 1)))
toep2_r = np.concatenate((toep2_r, np.zeros(L + K - toep2_r.size)))
toep2_c = np.concatenate(([poly2_x.all_coeffs()[-1]], np.zeros(K - 1)))
else: # for the case with K == 0
toep2_r = np.zeros((0, L + K))
toep2_c = np.zeros((0, 0))
blk_mtx1 = linalg.toeplitz(toep1_c, toep1_r)
blk_mtx2 = linalg.toeplitz(toep2_c, toep2_r)
if blk_mtx1.size != 0 and blk_mtx2.size != 0:
# for debugging only
# print('blk_mtx1 size: {0}, blk_mtx2_size: {1}'.format(blk_mtx1.shape, blk_mtx2.shape))
mtx = np.vstack((blk_mtx1, blk_mtx2))
elif blk_mtx1.size == 0 and blk_mtx2.size != 0:
mtx = blk_mtx2
elif blk_mtx1.size != 0 and blk_mtx2.size == 0:
mtx = blk_mtx1
else:
mtx = np.zeros((0, 0))
max_y_degree1 = coef1.shape[0] - 1
max_y_degree2 = coef2.shape[0] - 1
max_poly_degree = np.int(max_y_degree1 * L + max_y_degree2 * K)
num_samples = (max_poly_degree + 1) * 4 # <= 4 is the over-sampling factor used to determined the poly coef.
# randomly generate y-values
# y_vals = np.random.randn(num_samples, 1) + \
# 1j * np.random.randn(num_samples, 1)
y_vals = np.exp(1j * 2 * np.pi / num_samples * np.arange(num_samples))[:, np.newaxis]
y_powers = np.reshape(np.arange(max_poly_degree + 1)[::-1], (1, -1), order='F')
Y = ne.evaluate('y_vals ** y_powers')
# compute resultant, which is the determinant of mtx.
# it is a polynomial in terms of variable y
func_resultant = sympy.lambdify(y, sympy.Matrix(mtx))
det_As = np.array([linalg.det(np.array(func_resultant(y_roots_loop), dtype=complex))
for y_roots_loop in y_vals.squeeze()], dtype=complex)
coef_resultant = linalg.lstsq(Y, det_As)[0]
# trim out very small coefficients
eps = np.max(np.abs(coef_resultant)) * 1e-3
coef_resultant[np.abs(coef_resultant) < eps] = 0
y_roots_all = np.roots(coef_resultant)
# use the root values for y to find the root values for x
# check if poly1_x or poly2_x are constant w.r.t. x
if len(poly1_x.all_coeffs()) > 1:
func_loop = sympy.lambdify(y, poly1_x.all_coeffs())
coef_validate = coef2
elif len(poly2_x.all_coeffs()) > 1:
func_loop = sympy.lambdify(y, poly2_x.all_coeffs())
coef_validate = coef1
else:
raise RuntimeError('Neither polynomials contain x')
x_roots = []
y_roots = []
for loop in range(y_roots_all.size):
y_roots_loop = y_roots_all[loop]
x_roots_loop = np.roots(func_loop(y_roots_loop))
# check if there're duplicated roots
x_roots_loop = eliminate_duplicate_roots(x_roots_loop)
for roots_loop in x_roots_loop:
x_roots.append(roots_loop)
for roots_loop in np.tile(y_roots_loop, x_roots_loop.size):
y_roots.append(roots_loop)
x_roots, y_roots = np.array(x_roots), np.array(y_roots)
x_roots, y_roots = eliminate_duplicate_roots_2d(x_roots, y_roots)
# validate based on the polynomial values of the other polynomila
# that is not used in the last step to get the roots
poly_val = np.log10(np.abs(
check_error(coef_validate / linalg.norm(coef_validate.flatten()),
x_roots, y_roots)))
# if the error is 2 orders larger than the smallest error, then we discard the root
if np.array(poly_val).size > 1:
valid_idx = np.bitwise_or(poly_val < np.min(poly_val) + 2, poly_val < -3)
x_roots = x_roots[valid_idx]
y_roots = y_roots[valid_idx]
return x_roots, y_roots
def check_error(coef, x, y):
val = 0
max_degree_x = coef.shape[1] - 1
max_degree_y = coef.shape[0] - 1
for x_count in range(max_degree_x + 1):
for y_count in range(max_degree_y + 1):
val += coef[y_count, x_count] * x ** (max_degree_x - x_count) * \
y ** (max_degree_y - y_count)
return val
def eliminate_duplicate_roots(all_roots):
total_roots = all_roots.size
flags = np.ones(total_roots, dtype=bool)
for loop_outer in range(total_roots - 1):
root1 = all_roots[loop_outer]
# compute the difference
flags[loop_outer + 1 +
np.where(np.abs(root1 - all_roots[loop_outer + 1:])
< 1e-2)[0]] = False
return all_roots[flags]
def eliminate_duplicate_roots_2d(all_roots1, all_roots2):
total_roots = all_roots1.size
flags = np.ones(total_roots, dtype=bool)
for loop_outer in range(total_roots - 1):
root1 = all_roots1[loop_outer]
root2 = all_roots2[loop_outer]
# compute the difference
flags[loop_outer + 1 +
np.where(np.abs(
np.abs(root1 - all_roots1[loop_outer + 1:]) +
1j * np.abs(root2 - all_roots2[loop_outer + 1:])) < 1e-3
)[0]] = False
return all_roots1[flags], all_roots2[flags]
if __name__ == '__main__':
coef_row = np.random.randn(3, 4)
coef_col = np.random.randn(4, 3)
x_roots, y_roots = find_roots(coef_row, coef_col)
# print x_roots, y_roots
print(np.abs(check_error(coef_row, x_roots, y_roots)))