def generate(R, P, G, B):
while R:
h = normal(f[R.pop()], G | P)
if h is not None:
k, LM = h
G0 = set(g for g in G if monomial_div(f[g].LM, LM))
P0 = set(p for p in P if monomial_div(f[p].LM, LM))
G, P, R = G - G0, P - P0 | set([k]), R | G0 | P0
for i, j in set(B):
if i in G0 or j in G0:
del B[(i, j)]
G |= P
for i in G:
for j in P:
if i == j:
continue
if i < j:
k = (i, j)
else:
k = (j, i)
if k not in B:
B[k] = monomial_lcm(f[i].LM, f[j].LM)
G = set([normal(f[g], G - set([g]))[0] for g in G])
return R, P, G, B
def generate(R, P, G, B):
while R:
h = normal(f[R.pop()], G | P)
if h is not None:
k, LM = h
G0 = set(g for g in G if monomial_div(f[g].LM, LM))
P0 = set(p for p in P if monomial_div(f[p].LM, LM))
G, P, R = G - G0, P - P0 | set([k]), R | G0 | P0
for i, j in set(B):
if i in G0 or j in G0:
del B[(i, j)]
G |= P
for i in G:
for j in P:
if i == j:
continue
if i < j:
k = (i, j)
else:
k = (j, i)
if not B.has_key(k):
B[k] = monomial_lcm(f[i].LM, f[j].LM)
G = set([ normal(f[g], G - set([g]))[0] for g in G ])
return R, P, G, B
def poly_lcm(f, g, *symbols):
"""Computes least common multiple of two polynomials.
Given two univariate polynomials, the LCM is computed via
f*g = gcd(f, g)*lcm(f, g) formula. In multivariate case, we
compute the unique generator of the intersection of the two
ideals, generated by f and g. This is done by computing a
Groebner basis, with respect to any lexicographic ordering,
of t*f and (1 - t)*g, where t is an unrelated symbol and
filtering out solution that does not contain t.
For more information on the implemented algorithm refer to:
[1] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and
Algorithms, Springer, Second Edition, 1997, pp. 187
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
symbols, flags = f.symbols, f.flags
if f.is_monomial and g.is_monomial:
monom = monomial_lcm(f.LM, g.LM)
fc, gc = f.LC, g.LC
if fc.is_Rational and gc.is_Rational:
coeff = Integer(ilcm(fc.p, gc.p))
else:
coeff = S.One
return Poly((coeff, monom), *symbols, **flags)
fc, f = f.as_primitive()
gc, g = g.as_primitive()
lcm = ilcm(int(fc), int(gc))
if f.is_multivariate:
t = Symbol('t', dummy=True)
lex = {'order': 'lex'}
f_monoms = [(1, ) + monom for monom in f.monoms]
F = Poly((f.coeffs, f_monoms), t, *symbols, **lex)
g_monoms = [ (0,) + monom for monom in g.monoms ] + \
[ (1,) + monom for monom in g.monoms ]
g_coeffs = list(g.coeffs) + [-coeff for coeff in g.coeffs]
G = Poly(dict(zip(g_monoms, g_coeffs)), t, *symbols, **lex)
def independent(h):
return all(not monom[0] for monom in h.monoms)
H = [h for h in poly_groebner((F, G)) if independent(h)]
if lcm != 1:
h_coeffs = [coeff * lcm for coeff in H[0].coeffs]
else:
h_coeffs = H[0].coeffs
h_monoms = [monom[1:] for monom in H[0].monoms]
return Poly(dict(zip(h_monoms, h_coeffs)), *symbols, **flags)
else:
h = poly_div(f * g, poly_gcd(f, g))[0]
if lcm != 1:
return h.mul_term(lcm / h.LC)
else:
return h.as_monic()
def poly_lcm(f, g, *symbols):
"""Computes least common multiple of two polynomials.
Given two univariate polynomials, the LCM is computed via
f*g = gcd(f, g)*lcm(f, g) formula. In multivariate case, we
compute the unique generator of the intersection of the two
ideals, generated by f and g. This is done by computing a
Groebner basis, with respect to any lexicographic ordering,
of t*f and (1 - t)*g, where t is an unrelated symbol and
filtering out solution that does not contain t.
For more information on the implemented algorithm refer to:
[1] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and
Algorithms, Springer, Second Edition, 1997, pp. 187
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
symbols, flags = f.symbols, f.flags
if f.is_monomial and g.is_monomial:
monom = monomial_lcm(f.LM, g.LM)
fc, gc = f.LC, g.LC
if fc.is_Rational and gc.is_Rational:
coeff = Integer(ilcm(fc.p, gc.p))
else:
coeff = S.One
return Poly((coeff, monom), *symbols, **flags)
fc, f = f.as_primitive()
gc, g = g.as_primitive()
lcm = ilcm(int(fc), int(gc))
if f.is_multivariate:
t = Symbol('t', dummy=True)
lex = { 'order' : 'lex' }
f_monoms = [ (1,) + monom for monom in f.monoms ]
F = Poly((f.coeffs, f_monoms), t, *symbols, **lex)
g_monoms = [ (0,) + monom for monom in g.monoms ] + \
[ (1,) + monom for monom in g.monoms ]
g_coeffs = list(g.coeffs) + [ -coeff for coeff in g.coeffs ]
G = Poly(dict(zip(g_monoms, g_coeffs)), t, *symbols, **lex)
def independent(h):
return all(not monom[0] for monom in h.monoms)
H = [ h for h in poly_groebner((F, G)) if independent(h) ]
if lcm != 1:
h_coeffs = [ coeff*lcm for coeff in H[0].coeffs ]
else:
h_coeffs = H[0].coeffs
h_monoms = [ monom[1:] for monom in H[0].monoms ]
return Poly(dict(zip(h_monoms, h_coeffs)), *symbols, **flags)
else:
h = poly_div(f * g, poly_gcd(f, g))[0]
if lcm != 1:
return h.mul_term(lcm / h.LC)
else:
return h.as_monic()
