def poly_half_gcdex(f, g, *symbols):
"""Half extended Euclidean algorithm.
Efficiently computes gcd(f, g) and one of the coefficients
in extended Euclidean algorithm. Formally, given univariate
polynomials f and g over an Euclidean domain, computes s
and h, such that h = gcd(f, g) and s*f = h (mod g).
For more information on the implemented algorithm refer to:
[1] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlang, 2005
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
if f.is_multivariate:
raise MultivariatePolyError(f)
symbols, flags = f.symbols, f.flags
a = Poly(S.One, *symbols, **flags)
b = Poly((), *symbols, **flags)
while not g.is_zero:
q, r = poly_div(f, g)
f, g = g, r
c = a - q * b
a, b = b, c
return a.div_term(f.LC), f.as_monic()
def poly_half_gcdex(f, g, *symbols):
"""Half extended Euclidean algorithm.
Efficiently computes gcd(f, g) and one of the coefficients
in extended Euclidean algorithm. Formally, given univariate
polynomials f and g over an Euclidean domain, computes s
and h, such that h = gcd(f, g) and s*f = h (mod g).
For more information on the implemented algorithm refer to:
[1] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlang, 2005
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
if f.is_multivariate:
raise MultivariatePolyError(f)
symbols, flags = f.symbols, f.flags
a = Poly(S.One, *symbols, **flags)
b = Poly((), *symbols, **flags)
while not g.is_zero:
q, r = poly_div(f, g)
f, g = g, r
c = a - q*b
a, b = b, c
return a.div_term(f.LC), f.as_monic()
def poly_groebner(f, *symbols, **flags):
"""Computes reduced Groebner basis for a set of polynomials.
Given a set of multivariate polynomials F, find another set G,
such that Ideal F = Ideal G and G is a reduced Groebner basis.
The resulting basis is unique and has monic generators.
Groebner bases can be used to choose specific generators for a
polynomial ideal. Because these bases are unique you can check
for ideal equality by comparing the Groebner bases. To see if
one polynomial lies in an ideal, divide by the elements in the
base and see if the remainder vanishes.
They can also be used to solve systems of polynomial equations
as, by choosing lexicographic ordering, you can eliminate one
variable at a time, provided that the ideal is zero-dimensional
(finite number of solutions).
>>> from sympy import *
>>> x,y = symbols('xy')
>>> G = poly_groebner([x**2 + y**3, y**2-x], x, y, order='lex')
>>> [ g.as_basic() for g in G ]
[x - y**2, y**3 + y**4]
For more information on the implemented algorithm refer to:
[1] N.K. Bose, B. Buchberger, J.P. Guiver, Multidimensional
Systems Theory and Applications, Springer, 2003, pp. 98+
[2] A. Giovini, T. Mora, "One sugar cube, please" or Selection
strategies in Buchberger algorithm, Proc. ISSAC '91, ACM
[3] I.A. Ajwa, Z. Liu, P.S. Wang, Groebner Bases Algorithm,
http://citeseer.ist.psu.edu/ajwa95grbner.html, 1995
[4] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and
Algorithms, Springer, Second Edition, 1997, pp. 62
"""
if isinstance(f, (tuple, list, set)):
f, g = f[0], list(f[1:])
if not isinstance(f, Poly):
f = Poly(f, *symbols, **flags)
elif symbols or flags:
raise SymbolsError("Redundant symbols or flags were given")
f, g = f.unify_with(g)
symbols, flags = f.symbols, f.flags
else:
if not isinstance(f, Poly):
f = Poly(f, *symbols, **flags)
elif symbols or flags:
raise SymbolsError("Redundant symbols or flags were given")
return [f.as_monic()]
compare = monomial_cmp(flags.get('order'))
f = [h for h in [f] + g if h]
if not f:
return [Poly((), *symbols, **flags)]
R, P, G, B, F = set(), set(), set(), {}, {}
for i, h in enumerate(f):
F[h] = i
R.add(i)
def normal(g, H):
h = poly_div(g, [f[i] for i in H])[1]
if h.is_zero:
return None
else:
if not h in F:
F[h] = len(f)
f.append(h)
return F[h], h.LM
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
R, P, G, B = generate(R, P, G, B)
while B:
k, M = B.items()[0]
for l, N in B.iteritems():
if compare(M, N) == 1:
k, M = l, N
del B[k]
i, j = k[0], k[1]
p, q = f[i], f[j]
p_LM, q_LM = p.LM, q.LM
if M == monomial_mul(p_LM, q_LM):
continue
criterion = False
for g in G:
if g == i or g == j:
continue
if (min(i, g), max(i, g)) not in B:
continue
if (min(j, g), max(j, g)) not in B:
continue
if not monomial_div(M, f[g].LM):
continue
criterion = True
break
if criterion:
continue
p = p.mul_term(1 / p.LC, monomial_div(M, p_LM))
q = q.mul_term(1 / q.LC, monomial_div(M, q_LM))
h = normal(p - q, G)
if h is not None:
k, LM = h
G0 = set(g for g in G if monomial_div(f[g].LM, LM))
R, P, G = G0, set([k]), G - G0
for i, j in set(B):
if i in G0 or j in G0:
del B[(i, j)]
R, P, G, B = generate(R, P, G, B)
G = [f[g].as_monic() for g in G]
G = sorted(G, compare, lambda p: p.LM)
return list(reversed(G))
def poly_groebner(f, *symbols, **flags):
"""Computes reduced Groebner basis for a set of polynomials.
Given a set of multivariate polynomials F, find another set G,
such that Ideal F = Ideal G and G is a reduced Groebner basis.
The resulting basis is unique and has monic generators.
Groebner bases can be used to choose specific generators for a
polynomial ideal. Because these bases are unique you can check
for ideal equality by comparing the Groebner bases. To see if
one polynomial lies in an ideal, divide by the elements in the
base and see if the remainder vanishes.
They can also be used to solve systems of polynomial equations
as, by choosing lexicographic ordering, you can eliminate one
variable at a time, provided that the ideal is zero-dimensional
(finite number of solutions).
>>> from sympy import *
>>> x,y = symbols('xy')
>>> G = poly_groebner([x**2 + y**3, y**2-x], x, y, order='lex')
>>> [ g.as_basic() for g in G ]
[x - y**2, y**3 + y**4]
For more information on the implemented algorithm refer to:
[1] N.K. Bose, B. Buchberger, J.P. Guiver, Multidimensional
Systems Theory and Applications, Springer, 2003, pp. 98+
[2] A. Giovini, T. Mora, "One sugar cube, please" or Selection
strategies in Buchberger algorithm, Proc. ISSAC '91, ACM
[3] I.A. Ajwa, Z. Liu, P.S. Wang, Groebner Bases Algorithm,
http://citeseer.ist.psu.edu/ajwa95grbner.html, 1995
[4] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and
Algorithms, Springer, Second Edition, 1997, pp. 62
"""
if isinstance(f, (tuple, list, set)):
f, g = f[0], list(f[1:])
if not isinstance(f, Poly):
f = Poly(f, *symbols, **flags)
elif symbols or flags:
raise SymbolsError("Redundant symbols or flags were given")
f, g = f.unify_with(g)
symbols, flags = f.symbols, f.flags
else:
if not isinstance(f, Poly):
f = Poly(f, *symbols, **flags)
elif symbols or flags:
raise SymbolsError("Redundant symbols or flags were given")
return [f.as_monic()]
compare = monomial_cmp(flags.get('order'))
f = [ h for h in [f] + g if h ]
if not f:
return [Poly((), *symbols, **flags)]
R, P, G, B, F = set(), set(), set(), {}, {}
for i, h in enumerate(f):
F[h] = i; R.add(i)
def normal(g, H):
h = poly_div(g, [ f[i] for i in H ])[1]
if h.is_zero:
return None
else:
if not F.has_key(h):
F[h] = len(f)
f.append(h)
return F[h], h.LM
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
R, P, G, B = generate(R, P, G, B)
while B:
k, M = B.items()[0]
for l, N in B.iteritems():
if compare(M, N) == 1:
k, M = l, N
del B[k]
i, j = k[0], k[1]
p, q = f[i], f[j]
p_LM, q_LM = p.LM, q.LM
if M == monomial_mul(p_LM, q_LM):
continue
criterion = False
for g in G:
if g == i or g == j:
continue
if not B.has_key((min(i, g), max(i, g))):
continue
if not B.has_key((min(j, g), max(j, g))):
continue
if not monomial_div(M, f[g].LM):
continue
criterion = True
break
if criterion:
continue
p = p.mul_term(1/p.LC, monomial_div(M, p_LM))
q = q.mul_term(1/q.LC, monomial_div(M, q_LM))
h = normal(p - q, G)
if h is not None:
k, LM = h
G0 = set(g for g in G if monomial_div(f[g].LM, LM))
R, P, G = G0, set([k]), G - G0
for i, j in set(B):
if i in G0 or j in G0:
del B[(i, j)]
R, P, G, B = generate(R, P, G, B)
G = [ f[g].as_monic() for g in G ]
G = sorted(G, compare, lambda p: p.LM)
return list(reversed(G))