Source code for sympy.polys.fglmtools

"""Implementation of matrix FGLM Groebner basis conversion algorithm. """

from __future__ import print_function, division

from sympy.polys.monomials import monomial_mul, monomial_div
from sympy.core.compatibility import xrange

[docs]def matrix_fglm(F, ring, O_to): """ Converts the reduced Groebner basis ``F`` of a zero-dimensional ideal w.r.t. ``O_from`` to a reduced Groebner basis w.r.t. ``O_to``. References ========== J.C. Faugere, P. Gianni, D. Lazard, T. Mora (1994). Efficient Computation of Zero-dimensional Groebner Bases by Change of Ordering """ domain = ring.domain ngens = ring.ngens ring_to = ring.clone(order=O_to) old_basis = _basis(F, ring) M = _representing_matrices(old_basis, F, ring) # V contains the normalforms (wrt O_from) of S S = [ring.zero_monom] V = [[domain.one] + [domain.zero] * (len(old_basis) - 1)] G = [] L = [(i, 0) for i in xrange(ngens)] # (i, j) corresponds to x_i * S[j] L.sort(key=lambda k_l: O_to(_incr_k(S[k_l[1]], k_l[0])), reverse=True) t = L.pop() P = _identity_matrix(len(old_basis), domain) while True: s = len(S) v = _matrix_mul(M[t[0]], V[t[1]]) _lambda = _matrix_mul(P, v) if all(_lambda[i] == domain.zero for i in xrange(s, len(old_basis))): # there is a linear combination of v by V lt = ring.term_new(_incr_k(S[t[1]], t[0]), domain.one) rest = ring.from_dict(dict([ (S[i], _lambda[i]) for i in xrange(s) ])) g = (lt - rest).set_ring(ring_to) if g: G.append(g) else: # v is linearly independant from V P = _update(s, _lambda, P) S.append(_incr_k(S[t[1]], t[0])) V.append(v) L.extend([(i, s) for i in xrange(ngens)]) L = list(set(L)) L.sort(key=lambda k_l: O_to(_incr_k(S[k_l[1]], k_l[0])), reverse=True) L = [(k, l) for (k, l) in L if all(monomial_div(_incr_k(S[l], k), g.LM) is None for g in G)] if not L: G = [ g.monic() for g in G ] return sorted(G, key=lambda g: O_to(g.LM), reverse=True) t = L.pop()
def _incr_k(m, k): return tuple(list(m[:k]) + [m[k] + 1] + list(m[k + 1:])) def _identity_matrix(n, domain): M = [[domain.zero]*n for _ in xrange(n)] for i in xrange(n): M[i][i] = domain.one return M def _matrix_mul(M, v): return [sum([row[i] * v[i] for i in xrange(len(v))]) for row in M] def _update(s, _lambda, P): """ Update ``P`` such that for the updated `P'` `P' v = e_{s}`. """ k = min([j for j in xrange(s, len(_lambda)) if _lambda[j] != 0]) for r in xrange(len(_lambda)): if r != k: P[r] = [P[r][j] - (P[k][j] * _lambda[r]) / _lambda[k] for j in xrange(len(P[r]))] P[k] = [P[k][j] / _lambda[k] for j in xrange(len(P[k]))] P[k], P[s] = P[s], P[k] return P def _representing_matrices(basis, G, ring): """ Compute the matrices corresponding to the linear maps `m \mapsto x_i m` for all variables `x_i`. """ domain = ring.domain u = ring.ngens-1 def var(i): return tuple([0] * i + [1] + [0] * (u - i)) def representing_matrix(m): M = [[domain.zero] * len(basis) for _ in xrange(len(basis))] for i, v in enumerate(basis): r = ring.term_new(monomial_mul(m, v), domain.one).rem(G) for monom, coeff in r.terms(): j = basis.index(monom) M[j][i] = coeff return M return [representing_matrix(var(i)) for i in xrange(u + 1)] def _basis(G, ring): """ Computes a list of monomials which are not divisible by the leading monomials wrt to ``O`` of ``G``. These monomials are a basis of `K[X_1, \ldots, X_n]/(G)`. """ order = ring.order leading_monomials = [g.LM for g in G] candidates = [ring.zero_monom] basis = [] while candidates: t = candidates.pop() basis.append(t) new_candidates = [_incr_k(t, k) for k in xrange(ring.ngens) if all(monomial_div(_incr_k(t, k), lmg) is None for lmg in leading_monomials)] candidates.extend(new_candidates) candidates.sort(key=lambda m: order(m), reverse=True) basis = list(set(basis)) return sorted(basis, key=lambda m: order(m))