def poly_reduce(f, g, *symbols):
"""Removes common content from a pair of polynomials.
>>> from sympy import *
>>> x = Symbol('x')
>>> f = Poly(2930944*x**6 + 2198208*x**4 + 549552*x**2 + 45796, x)
>>> g = Poly(17585664*x**5 + 8792832*x**3 + 1099104*x, x)
>>> F, G = poly_reduce(f, g)
>>> F
Poly(64*x**6 + 48*x**4 + 12*x**2 + 1, x)
>>> G
Poly(384*x**5 + 192*x**3 + 24*x, x)
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
fc = int(f.content)
gc = int(g.content)
cont = igcd(fc, gc)
if cont != 1:
f = f.div_term(cont)
g = g.div_term(cont)
return f, g
def poly_reduce(f, g, *symbols):
"""Removes common content from a pair of polynomials.
>>> from sympy import *
>>> x = Symbol('x')
>>> f = Poly(2930944*x**6 + 2198208*x**4 + 549552*x**2 + 45796, x)
>>> g = Poly(17585664*x**5 + 8792832*x**3 + 1099104*x, x)
>>> F, G = poly_reduce(f, g)
>>> F
Poly(64*x**6 + 48*x**4 + 12*x**2 + 1, x)
>>> G
Poly(384*x**5 + 192*x**3 + 24*x, x)
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
fc = int(f.content)
gc = int(g.content)
cont = igcd(fc, gc)
if cont != 1:
f = f.div_term(cont)
g = g.div_term(cont)
return f, g
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_div(f, g, *symbols):
"""Generalized polynomial division with remainder.
Given polynomial f and a set of polynomials g = (g_1, ..., g_n)
compute a set of quotients q = (q_1, ..., q_n) and remainder r
such that f = q_1*f_1 + ... + q_n*f_n + r, where r = 0 or r is
a completely reduced polynomial with respect to g.
In particular g can be a tuple, list or a singleton. All g_i
and f can be given as Poly class instances or as expressions.
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. 62
[2] I.A. Ajwa, Z. Liu, P.S. Wang, Groebner Bases Algorithm,
http://citeseer.ist.psu.edu/ajwa95grbner.html, 1995
"""
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
r = Poly((), *symbols, **flags)
if isinstance(g, Basic):
if g.is_constant:
if g.is_zero:
raise ZeroDivisionError
elif g.is_one:
return f, r
else:
return f.div_term(g.LC), r
if g.is_monomial:
LC, LM = g.lead_term
q_coeffs, q_monoms = [], []
r_coeffs, r_monoms = [], []
for coeff, monom in f.iter_terms():
quotient = monomial_div(monom, LM)
if quotient is not None:
coeff /= LC
q_coeffs.append(Poly.cancel(coeff))
q_monoms.append(quotient)
else:
r_coeffs.append(coeff)
r_monoms.append(monom)
return (Poly((q_coeffs, q_monoms), *symbols,
**flags), Poly((r_coeffs, r_monoms), *symbols,
**flags))
g, q = [g], [r]
else:
q = [r] * len(g)
while not f.is_zero:
for i, h in enumerate(g):
monom = monomial_div(f.LM, h.LM)
if monom is not None:
coeff = Poly.cancel(f.LC / h.LC)
q[i] = q[i].add_term(coeff, monom)
f -= h.mul_term(coeff, monom)
break
else:
r = r.add_term(*f.LT)
f = f.kill_lead_term()
if len(q) != 1:
return q, r
else:
return q[0], r
def poly_div(f, g, *symbols):
"""Generalized polynomial division with remainder.
Given polynomial f and a set of polynomials g = (g_1, ..., g_n)
compute a set of quotients q = (q_1, ..., q_n) and remainder r
such that f = q_1*f_1 + ... + q_n*f_n + r, where r = 0 or r is
a completely reduced polynomial with respect to g.
In particular g can be a tuple, list or a singleton. All g_i
and f can be given as Poly class instances or as expressions.
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. 62
[2] I.A. Ajwa, Z. Liu, P.S. Wang, Groebner Bases Algorithm,
http://citeseer.ist.psu.edu/ajwa95grbner.html, 1995
"""
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
r = Poly((), *symbols, **flags)
if isinstance(g, Basic):
if g.is_constant:
if g.is_zero:
raise ZeroDivisionError
elif g.is_one:
return f, r
else:
return f.div_term(g.LC), r
if g.is_monomial:
LC, LM = g.lead_term
q_coeffs, q_monoms = [], []
r_coeffs, r_monoms = [], []
for coeff, monom in f.iter_terms():
quotient = monomial_div(monom, LM)
if quotient is not None:
coeff /= LC
q_coeffs.append(Poly.cancel(coeff))
q_monoms.append(quotient)
else:
r_coeffs.append(coeff)
r_monoms.append(monom)
return (Poly((q_coeffs, q_monoms), *symbols, **flags),
Poly((r_coeffs, r_monoms), *symbols, **flags))
g, q = [g], [r]
else:
q = [r] * len(g)
while not f.is_zero:
for i, h in enumerate(g):
monom = monomial_div(f.LM, h.LM)
if monom is not None:
coeff = Poly.cancel(f.LC / h.LC)
q[i] = q[i].add_term(coeff, monom)
f -= h.mul_term(coeff, monom)
break
else:
r = r.add_term(*f.LT)
f = f.kill_lead_term()
if len(q) != 1:
return q, r
else:
return q[0], r
