def factor(f, var=None, order=None):
"""Factorization of polynomials over the rationals.
A thin wrapper that returns a SymPy expression by multiplying the
different factors coming from L{factor_.factor}
"""
# Re-assemble factors:
return Mul(*[ff.sympy_expr for ff in factor_.factor(f, var, order)])
def factorization(poly, linear=False):
"""Returns a list with polynomial factors over rationals or, if
'linear' flag is set over Q(i). This simple handler should
be merged with original factor() method.
>>> from sympy import *
>>> x, y = symbols('xy')
>>> factorization(x**2 - y**2)
set([1, x - y, x + y])
>>> factorization(x**2 + 1)
set([1, 1 + x**2])
>>> factorization(x**2 + 1, linear=True)
set([1, x - I, I + x])
"""
factored = set([ q.as_basic() for q in factor_.factor(poly) ])
if not linear:
return factored
else:
factors = []
for factor in factored:
symbols = factor.atoms(Symbol)
if len(symbols) == 1:
x = symbols.pop()
else:
factors += [ factor ]
continue
linearities = [ x - r for r in roots(factor, x) ]
if not linearities:
factors += [ factor ]
else:
unfactorable = quo(factor, Basic.Mul(*linearities))
if not isinstance(unfactorable, Basic.Number):
factors += [ unfactorable ]
factors += linearities
return set(factors)
def roots(f, var=None):
"""Compute the roots of a univariate polynomial.
Usage:
======
The input f is assumed to be a univariate polynomial, either
as SymPy expression or as instance of Polynomial. In the
latter case, you can optionally specify the variable with
'var'.
The output is a list of all found roots with multiplicity.
Examples:
=========
>>> x, y = symbols('xy')
>>> roots(x**2 - 1)
[1, -1]
>>> roots(x - y, x)
[y]
Also see L{factor_.factor}, L{quadratic}, L{cubic}. L{n-poly},
L{count_real_roots}.
"""
from sympy.polynomials import factor_
if not isinstance(f, Polynomial):
f = Polynomial(f, var=var, order=None)
if len(f.var) == 0:
return []
if len(f.var) > 1:
raise PolynomialException('Multivariate polynomials not supported.')
# Determine type of coeffs (for factorization purposes)
symbols = f.sympy_expr.atoms(type=Symbol)
symbols = filter(lambda a: not a in f.var, symbols)
if symbols:
coeff = 'sym'
else:
coeff = coeff_ring(get_numbers(f.sympy_expr))
if coeff == 'rat':
denom, f = f.as_integer()
coeff = 'int'
if coeff == 'int':
content, f = f.as_primitive()
# Hack to get some additional cases right:
result = n_poly(f)
if result is not None:
return result
factors = factor_.factor(f)
else: # It's not possible to factorize.
factors = [f]
# Now check for roots in each factor.
result = []
for p in factors:
n = p.coeffs[0][1] # Degree of the factor.
if n == 0: # We have a constant factor.
pass
elif n == 1:
if len(p.coeffs) == 2:
result += [-(p.coeffs[1][0] / p.coeffs[0][0])]
else:
result += [S.Zero]
elif n == 2:
result += quadratic(p)
elif n == 3:
result += cubic(p)
else:
res = n_poly(p)
if res is not None:
result += res
# With symbols, __nonzero__ returns a StrictInequality, Exception.
try: result.sort()
except: pass
return result