def poly_pdiv(f, g, *symbols):
"""Univariate polynomial pseudo-division with remainder.
Given univariate polynomials f and g over an integral domain D[x]
applying classical division algorithm to LC(g)**(d + 1) * f and g
where d = max(-1, deg(f) - deg(g)), compute polynomials q and r
such that LC(g)**(d + 1)*f = g*q + r and r = 0 or deg(r) < deg(g).
Polynomials q and r are called the pseudo-quotient of f by g and
the pseudo-remainder of f modulo g respectively.
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
q, r = Poly((), *symbols, **flags), f
coeff, N = g.LC, f.degree - g.degree + 1
while not r.is_zero:
M = r.degree - g.degree
if M < 0:
break
else:
T, N = (r.LC, (M, )), N - 1
q = q.mul_term(coeff).add_term(*T)
r = r.mul_term(coeff) - g.mul_term(*T)
return (q.mul_term(coeff**N), r.mul_term(coeff**N))
def poly_pdiv(f, g, *symbols):
"""Univariate polynomial pseudo-division with remainder.
Given univariate polynomials f and g over an integral domain D[x]
applying classical division algorithm to LC(g)**(d + 1) * f and g
where d = max(-1, deg(f) - deg(g)), compute polynomials q and r
such that LC(g)**(d + 1)*f = g*q + r and r = 0 or deg(r) < deg(g).
Polynomials q and r are called the pseudo-quotient of f by g and
the pseudo-remainder of f modulo g respectively.
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
q, r = Poly((), *symbols, **flags), f
coeff, N = g.LC, f.degree - g.degree + 1
while not r.is_zero:
M = r.degree - g.degree
if M < 0:
break
else:
T, N = (r.LC, (M,)), N - 1
q = q.mul_term(coeff).add_term(*T)
r = r.mul_term(coeff)-g.mul_term(*T)
return (q.mul_term(coeff**N), r.mul_term(coeff**N))
def poly_gcd(f, g, *symbols):
"""Compute greatest common divisor of two polynomials.
Given two univariate polynomials, subresultants are used
to compute the GCD. In multivariate case Groebner basis
approach is used together with f*g = gcd(f, g)*lcm(f, g)
well known formula.
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_zero and g.is_zero:
return f
if f.is_constant:
if f.is_zero:
cont, g = g.as_primitive()
return g.mul_term(cont / g.LC)
if f.is_one:
return f
if g.is_constant:
if g.is_zero:
cont, f = f.as_primitive()
return f.mul_term(cont / f.LC)
if g.is_one:
return g
if f.is_monomial and g.is_monomial:
monom = monomial_gcd(f.LM, g.LM)
fc, gc = f.LC, g.LC
if fc.is_Rational and gc.is_Rational:
coeff = Integer(igcd(fc.p, gc.p))
else:
coeff = S.One
return Poly((coeff, monom), *symbols, **flags)
cf, f = f.as_primitive()
cg, g = g.as_primitive()
gcd = igcd(int(cf), int(cg))
if f.is_multivariate:
h = poly_div(f * g, poly_lcm(f, g))[0]
else:
h = poly_subresultants(f, g, res=False)[-1]
if gcd != 1:
return h.mul_term(gcd / h.LC)
else:
return h.as_monic()
def poly_gcd(f, g, *symbols):
"""Compute greatest common divisor of two polynomials.
Given two univariate polynomials, subresultants are used
to compute the GCD. In multivariate case Groebner basis
approach is used together with f*g = gcd(f, g)*lcm(f, g)
well known formula.
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_zero and g.is_zero:
return f
if f.is_constant:
if f.is_zero:
cont, g = g.as_primitive()
return g.mul_term(cont / g.LC)
if f.is_one:
return f
if g.is_constant:
if g.is_zero:
cont, f = f.as_primitive()
return f.mul_term(cont / f.LC)
if g.is_one:
return g
if f.is_monomial and g.is_monomial:
monom = monomial_gcd(f.LM, g.LM)
fc, gc = f.LC, g.LC
if fc.is_Rational and gc.is_Rational:
coeff = Integer(igcd(fc.p, gc.p))
else:
coeff = S.One
return Poly((coeff, monom), *symbols, **flags)
cf, f = f.as_primitive()
cg, g = g.as_primitive()
gcd = igcd(int(cf), int(cg))
if f.is_multivariate:
h = poly_div(f*g, poly_lcm(f, g))[0]
else:
h = poly_subresultants(f, g)[-1]
if gcd != 1:
return h.mul_term(gcd / h.LC)
else:
return h.as_monic()