def poly_sqf(f, *symbols):
"""Compute square-free decomposition of an univariate polynomial.
Given an univariate polynomial f over an unique factorization domain
returns tuple (f_1, f_2, ..., f_n), where all A_i are co-prime and
square-free polynomials and f = f_1 * f_2**2 * ... * f_n**n.
>>> from sympy import *
>>> x,y = symbols('xy')
>>> p, q = poly_sqf(x*(x+1)**2, x)
>>> p.as_basic()
x
>>> q.as_basic()
1 + x
For more information on the implemented algorithm refer to:
[1] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlang, 2005
[2] J. von zur Gathen, J. Gerhard, Modern Computer Algebra,
Second Edition, Cambridge University Press, 2003
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
if f.is_multivariate:
raise MultivariatePolyError(f)
coeff, f = f.as_primitive()
sqf = []
h = f.diff()
g = poly_gcd(f, h)
p = poly_div(f, g)[0]
q = poly_div(h, g)[0]
p, q = poly_reduce(p, q)
while True:
h = q - p.diff()
if h.is_zero:
break
g = poly_gcd(p, h)
sqf.append(g)
p = poly_div(p, g)[0]
q = poly_div(h, g)[0]
p, q = poly_reduce(p, q)
sqf.append(p)
head, tail = sqf[0], sqf[1:]
head = head.mul_term(coeff)
return [head] + tail
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_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_sqf(f, *symbols):
"""Compute square-free decomposition of an univariate polynomial.
Given an univariate polynomial f over an unique factorization domain
returns tuple (f_1, f_2, ..., f_n), where all A_i are co-prime and
square-free polynomials and f = f_1 * f_2**2 * ... * f_n**n.
>>> from sympy import *
>>> x,y = symbols('xy')
>>> p, q = poly_sqf(x*(x+1)**2, x)
>>> p.as_basic()
x
>>> q.as_basic()
1 + x
For more information on the implemented algorithm refer to:
[1] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlang, 2005
[2] J. von zur Gathen, J. Gerhard, Modern Computer Algebra,
Second Edition, Cambridge University Press, 2003
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
if f.is_multivariate:
raise MultivariatePolyError(f)
coeff, f = f.as_primitive()
sqf = []
h = f.diff()
g = poly_gcd(f, h)
p = poly_div(f, g)[0]
q = poly_div(h, g)[0]
p, q = poly_reduce(p, q)
while True:
h = q - p.diff()
if h.is_zero:
break
g = poly_gcd(p, h)
sqf.append(g)
p = poly_div(p, g)[0]
q = poly_div(h, g)[0]
p, q = poly_reduce(p, q)
sqf.append(p)
head, tail = sqf[0], sqf[1:]
head = head.mul_term(coeff)
return [head] + tail
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()
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()