def matrix_fglm(F, u, O_from, O_to, K):
"""
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
J.C. Faugere's lecture notes:
http://www-salsa.lip6.fr/~jcf/Papers/2010_MPRI5e.pdf
"""
old_basis = _basis(F, u, O_from, K)
M = _representing_matrices(old_basis, F, u, O_from, K)
# V contains the normalforms (wrt O_from) of S
S = [(0,) * (u + 1)]
V = [[K.one] + [K.zero] * (len(old_basis) - 1)]
G = []
L = [(i, 0) for i in xrange(u + 1)] # (i, j) corresponds to x_i * S[j]
L.sort(key=lambda (k, l): O_to(_incr_k(S[l], k)), reverse=True)
t = L.pop()
P = _identity_matrix(len(old_basis), K)
while True:
s = len(S)
v = _matrix_mul(M[t[0]], V[t[1]], K)
_lambda = _matrix_mul(P, v, K)
if all(_lambda[i] == K.zero for i in xrange(s, len(old_basis))):
# there is a linear combination of v by V
lt = [(_incr_k(S[t[1]], t[0]), K.one)]
rest = sdp_strip(sdp_sort([(S[i], _lambda[i]) for i in xrange(s)], O_to))
g = sdp_sub(lt, rest, u, O_to, K)
if g != []:
G.append(g)
else:
# v is linearly independant from V
P = _update(s, _lambda, P, K)
S.append(_incr_k(S[t[1]], t[0]))
V.append(v)
L.extend([(i, s) for i in xrange(u + 1)])
L = list(set(L))
L.sort(key=lambda (k, l): O_to(_incr_k(S[l], k)), reverse=True)
L = [(k, l) for (k, l) in L if all(monomial_div(_incr_k(S[l], k), sdp_LM(g, u)) is None for g in G)]
if not L:
G = [sdp_monic(g, K) for g in G]
return sorted(G, key=lambda g: O_to(sdp_LM(g, u)), reverse=True)
t = L.pop()
def sdm_from_dict(d, O):
"""
Create an sdm from a dictionary.
Here ``O`` is the monomial order to use.
>>> from sympy.polys.distributedmodules import sdm_from_dict
>>> from sympy.polys import QQ, lex
>>> dic = {(1, 1, 0): QQ(1), (1, 0, 0): QQ(2), (0, 1, 0): QQ(0)}
>>> sdm_from_dict(dic, lex)
[((1, 1, 0), 1/1), ((1, 0, 0), 2/1)]
"""
return sdp_strip(sdp_from_dict(d, O))
def test_sdp_strip():
assert sdp_strip([((2, 2), 0), ((1, 1), 1), ((0, 0), 0)]) == [((1, 1), 1)]
def matrix_fglm(F, u, O_from, O_to, K):
"""
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
J.C. Faugere's lecture notes:
http://www-salsa.lip6.fr/~jcf/Papers/2010_MPRI5e.pdf
"""
old_basis = _basis(F, u, O_from, K)
M = _representing_matrices(old_basis, F, u, O_from, K)
# V contains the normalforms (wrt O_from) of S
S = [(0, ) * (u + 1)]
V = [[K.one] + [K.zero] * (len(old_basis) - 1)]
G = []
L = [(i, 0) for i in xrange(u + 1)] # (i, j) corresponds to x_i * S[j]
L.sort(key=lambda (k, l): O_to(_incr_k(S[l], k)), reverse=True)
t = L.pop()
P = _identity_matrix(len(old_basis), K)
while True:
s = len(S)
v = _matrix_mul(M[t[0]], V[t[1]], K)
_lambda = _matrix_mul(P, v, K)
if all([_lambda[i] == K.zero for i in xrange(s, len(old_basis))]):
# there is a linear combination of v by V
lt = [(_incr_k(S[t[1]], t[0]), K.one)]
rest = sdp_strip(
sdp_sort([(S[i], _lambda[i]) for i in xrange(s)], O_to))
g = sdp_sub(lt, rest, u, O_to, K)
if g != []:
G.append(g)
else:
# v is linearly independant from V
P = _update(s, _lambda, P, K)
S.append(_incr_k(S[t[1]], t[0]))
V.append(v)
L.extend([(i, s) for i in xrange(u + 1)])
L = list(set(L))
L.sort(key=lambda (k, l): O_to(_incr_k(S[l], k)), reverse=True)
L = [(k, l) for (k, l) in L if \
all([monomial_div(_incr_k(S[l], k), sdp_LM(g, u)) is None for g in G])]
if L == []:
return sorted(G, key=lambda g: O_to(sdp_LM(g, u)), reverse=True)
t = L.pop()