def test_zzx_factor():
f = [1, 0, 0, 1, 1]
for i in xrange(0, 20):
assert zzx_factor(f) == (1, [(f, 1)])
assert zzx_factor([2,4,2]) == \
(2, [([1, 1], 2)])
assert zzx_factor([1,-6,11,-6]) == \
(1, [([1,-3], 1),
([1,-2], 1),
([1,-1], 1)])
assert zzx_factor([-1,0,0,0,1,0,0]) == \
(-1, [([1,-1], 1),
([1, 1], 1),
([1, 0], 2),
([1, 0, 1], 1)])
assert zzx_factor(zzx_from_dict({10:1, 0:-1})) == \
(1, [([1,-1], 1),
([1, 1], 1),
([1,-1, 1,-1, 1], 1),
([1, 1, 1, 1, 1], 1)])
def test_zzx_factor():
f = [1,0,0,1,1]
for i in xrange(0, 20):
assert zzx_factor(f) == (1, [(f, 1)])
assert zzx_factor([2,4,2]) == \
(2, [([1, 1], 2)])
assert zzx_factor([1,-6,11,-6]) == \
(1, [([1,-3], 1),
([1,-2], 1),
([1,-1], 1)])
assert zzx_factor([-1,0,0,0,1,0,0]) == \
(-1, [([1,-1], 1),
([1, 1], 1),
([1, 0], 2),
([1, 0, 1], 1)])
assert zzx_factor(zzx_from_dict({10:1, 0:-1})) == \
(1, [([1,-1], 1),
([1, 1], 1),
([1,-1, 1,-1, 1], 1),
([1, 1, 1, 1, 1], 1)])
def poly_factors(f, *symbols, **flags):
"""Factor polynomials over rationals.
>>> from sympy import *
>>> x, y = symbols("x y")
>>> poly_factors(x**2 - y**2, x, y)
(1, [(Poly(x - y, x, y), 1), (Poly(x + y, x, y), 1)])
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
denom, f = f.as_integer()
if f.is_univariate:
coeffs = map(int, f.iter_all_coeffs())
content, factors = zzx_factor(coeffs)
for i in xrange(len(factors)):
factor, k = factors[i]
n = zzx_degree(factor)
terms = {}
for j, coeff in enumerate(factor):
if coeff != 0:
terms[(n - j, )] = Integer(coeff)
factors[i] = Poly(terms, *f.symbols), k
else:
content, factors = kronecker_mv(f, **flags)
return Rational(content, denom), factors
def poly_factors(f, *symbols, **flags):
"""Factor polynomials over rationals.
>>> from sympy import *
>>> x, y = symbols("x y")
>>> poly_factors(x**2 - y**2, x, y)
(1, [(Poly(x - y, x, y), 1), (Poly(x + y, x, y), 1)])
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
denom, f = f.as_integer()
if f.is_univariate:
coeffs = map(int, f.iter_all_coeffs())
content, factors = zzx_factor(coeffs)
for i in xrange(len(factors)):
factor, k = factors[i]
n = zzx_degree(factor)
terms = {}
for j, coeff in enumerate(factor):
if coeff != 0:
terms[(n-j,)] = Integer(coeff)
factors[i] = Poly(terms, *f.symbols), k
else:
content, factors = kronecker_mv(f, **flags)
return Rational(content, denom), factors
def test_zzx_factor():
assert zzx_factor([ ]) == (0, [])
assert zzx_factor([7]) == (7, [])
f = [1,0,0,1,1]
for i in xrange(0, 20):
assert zzx_factor(f) == (1, [(f, 1)])
assert zzx_factor([2,4]) == \
(2, [([1, 2], 1)])
assert zzx_factor([1,2,2]) == \
(1, [([1,2,2], 1)])
assert zzx_factor([18,12,2]) == \
(2, [([3, 1], 2)])
assert zzx_factor([-9,0,1]) == \
(-1, [([3,-1], 1),
([3, 1], 1)])
assert zzx_factor_sqf([-9,0,1]) == \
(-1, [[3,-1],
[3, 1]])
assert zzx_factor([1,-6,11,-6]) == \
(1, [([1,-3], 1),
([1,-2], 1),
([1,-1], 1)])
assert zzx_factor_sqf([1,-6,11,-6]) == \
(1, [[1,-3],
[1,-2],
[1,-1]])
assert zzx_factor([-1,0,0,0,1,0,0]) == \
(-1, [([1,-1], 1),
([1, 1], 1),
([1, 0], 2),
([1, 0, 1], 1)])
f = [1080, 5184, 2099, 744, 2736, -648, 129, 0, -324]
assert zzx_factor(f) == \
(1, [([5, 24, 9, 0, 12], 1),
([216, 0, 31, 0, -27], 1)])
f = [-29802322387695312500000000000000000000,
0, 0, 0, 0,
2980232238769531250000000000000000,
0, 0, 0, 0,
1743435859680175781250000000000,
0, 0, 0, 0,
114142894744873046875000000,
0, 0, 0, 0,
-210106372833251953125,
0, 0, 0, 0,
95367431640625]
assert zzx_factor(f) == \
(-95367431640625, [([5, -1], 1),
([100, 10, -1], 2),
([625, 125, 25, 5, 1], 1),
([10000, -3000, 400, -20, 1], 2),
([10000, 2000, 400, 30, 1], 2)])
f = zzx_from_dict({10:1, 0:-1})
F_0 = zzx_factor(f, cyclotomic=True)
F_1 = zzx_factor(f, cyclotomic=False)
assert F_0 == F_1 == \
(1, [([1,-1], 1),
([1, 1], 1),
([1,-1, 1,-1, 1], 1),
([1, 1, 1, 1, 1], 1)])
f = zzx_from_dict({10:1, 0:1})
F_0 = zzx_factor(f, cyclotomic=True)
F_1 = zzx_factor(f, cyclotomic=False)
assert F_0 == F_1 == \
(1, [([1, 0, 1], 1),
([1, 0, -1, 0, 1, 0, -1, 0, 1], 1)])
def kronecker_mv(f, **flags):
"""Kronecker method for Z[X] polynomials.
NOTE: This function is very slow even on small input.
Use debug=True flag to see its progress, if any.
"""
symbols = f.symbols
def mv_int_div(f, g):
q = Poly((), *symbols)
r = Poly((), *symbols)
while not f.is_zero:
lc_f, lc_g = f.LC, g.LC
dv = lc_f % lc_g
cf = lc_f / lc_g
monom = monomial_div(f.LM, g.LM)
if dv == 0 and monom is not None:
q = q.add_term(cf, monom)
f -= g.mul_term(cf, monom)
else:
r = r.add_term(*f.LT)
f = f.kill_lead_term()
return q, r
def combinations(lisp, m):
def recursion(fa, lisp, m):
if m == 0:
yield fa
else:
for i, fa2 in enumerate(lisp[0:len(lisp) + 1 - m]):
for el in recursion(zzx_mul(fa2, fa), list(lisp[i + 1:]),
m - 1):
yield el
for i, fa in enumerate(lisp[0:len(lisp) + 1 - m]):
for el in recursion(fa, list(lisp[i + 1:]), m - 1):
yield el
debug = flags.get('debug', False)
cont, f = f.as_primitive()
N = len(symbols)
max_exp = {}
for v in symbols:
max_exp[v] = 0
for coeff, monom in f.iter_terms():
for v, exp in zip(symbols, monom):
if exp > max_exp[v]:
max_exp[v] = exp
symbols = sorted(symbols, reverse=True, key=lambda v: max_exp[v])
f = Poly(f, *symbols)
d = max_exp[symbols[0]] + 1
terms, exps = {}, []
for i in xrange(0, len(symbols)):
exps.append(d**i)
for coeff, monom in f.iter_terms():
exp = 0
for i, expi in enumerate(monom):
exp += expi * exps[i]
terms[exp] = int(coeff)
g, factors = zzx_from_dict(terms), []
try:
for ff, k in zzx_factor(g)[1]:
for i in xrange(0, k):
factors.append(ff)
except OverflowError:
raise PolynomialError(
"input too large for multivariate Kronecker method")
const, result, tested = 1, [], []
if debug: print "KRONECKER-MV: Z[x] #factors = %i ..." % (len(factors))
for k in range(1, len(factors) // 2 + 1):
for h in combinations(factors, k):
if h in tested:
continue
n = zzx_degree(h)
terms = {}
for coeff in h:
if not coeff:
n = n - 1
continue
else:
coeff = Integer(coeff)
y_deg, n = n, n - 1
monom = [0] * N
for i in xrange(N):
v_deg = y_deg % d
y_deg = (y_deg - v_deg) // d
monom[i] = v_deg
monom = tuple(monom)
if terms.has_key(monom):
terms[monom] += coeff
else:
terms[monom] = coeff
cand = Poly(terms, *symbols)
if cand.is_one:
continue
if cand.LC.is_negative:
cand = -cand
q, r = mv_int_div(f, cand)
if r.is_zero:
if debug: print "KRONECKER-MV: Z[X] factor found %s" % cand
result.append(cand)
f = q
else:
tested.append(h)
if f.is_constant:
const, f = f.LC, Poly(1, *symbols)
break
if f.is_one:
break
if not f.is_one:
if debug: print "KRONECKER-MV: Z[X] factor found %s" % f
result.append(f)
factors = {}
for ff in result:
if factors.has_key(ff):
factors[ff] += 1
else:
factors[ff] = 1
return cont * const, sorted(factors.items())
def test_zzx_factor():
assert zzx_factor([]) == (0, [])
assert zzx_factor([7]) == (7, [])
f = [1, 0, 0, 1, 1]
for i in xrange(0, 20):
assert zzx_factor(f) == (1, [(f, 1)])
assert zzx_factor([2,4]) == \
(2, [([1, 2], 1)])
assert zzx_factor([1,2,2]) == \
(1, [([1,2,2], 1)])
assert zzx_factor([18,12,2]) == \
(2, [([3, 1], 2)])
assert zzx_factor([-9,0,1]) == \
(-1, [([3,-1], 1),
([3, 1], 1)])
assert zzx_factor_sqf([-9,0,1]) == \
(-1, [[3,-1],
[3, 1]])
assert zzx_factor([1,-6,11,-6]) == \
(1, [([1,-3], 1),
([1,-2], 1),
([1,-1], 1)])
assert zzx_factor_sqf([1,-6,11,-6]) == \
(1, [[1,-3],
[1,-2],
[1,-1]])
assert zzx_factor([-1,0,0,0,1,0,0]) == \
(-1, [([1,-1], 1),
([1, 1], 1),
([1, 0], 2),
([1, 0, 1], 1)])
f = [1080, 5184, 2099, 744, 2736, -648, 129, 0, -324]
assert zzx_factor(f) == \
(1, [([5, 24, 9, 0, 12], 1),
([216, 0, 31, 0, -27], 1)])
f = [
-29802322387695312500000000000000000000, 0, 0, 0, 0,
2980232238769531250000000000000000, 0, 0, 0, 0,
1743435859680175781250000000000, 0, 0, 0, 0,
114142894744873046875000000, 0, 0, 0, 0, -210106372833251953125, 0, 0,
0, 0, 95367431640625
]
assert zzx_factor(f) == \
(-95367431640625, [([5, -1], 1),
([100, 10, -1], 2),
([625, 125, 25, 5, 1], 1),
([10000, -3000, 400, -20, 1], 2),
([10000, 2000, 400, 30, 1], 2)])
f = zzx_from_dict({10: 1, 0: -1})
F_0 = zzx_factor(f, cyclotomic=True)
F_1 = zzx_factor(f, cyclotomic=False)
assert F_0 == F_1 == \
(1, [([1,-1], 1),
([1, 1], 1),
([1,-1, 1,-1, 1], 1),
([1, 1, 1, 1, 1], 1)])
f = zzx_from_dict({10: 1, 0: 1})
F_0 = zzx_factor(f, cyclotomic=True)
F_1 = zzx_factor(f, cyclotomic=False)
assert F_0 == F_1 == \
(1, [([1, 0, 1], 1),
([1, 0, -1, 0, 1, 0, -1, 0, 1], 1)])
def kronecker_mv(f, **flags):
"""Kronecker method for Z[X] polynomials.
NOTE: This function is very slow even on small input.
Use debug=True flag to see its progress, if any.
"""
symbols = f.symbols
def mv_int_div(f, g):
q = Poly((), *symbols)
r = Poly((), *symbols)
while not f.is_zero:
lc_f, lc_g = f.LC, g.LC
dv = lc_f % lc_g
cf = lc_f / lc_g
monom = monomial_div(f.LM, g.LM)
if dv == 0 and monom is not None:
q = q.add_term(cf, monom)
f -= g.mul_term(cf, monom)
else:
r = r.add_term(*f.LT)
f = f.kill_lead_term()
return q, r
def combinations(lisp, m):
def recursion(fa, lisp, m):
if m == 0:
yield fa
else:
for i, fa2 in enumerate(lisp[0 : len(lisp) + 1 - m]):
for el in recursion(zzx_mul(fa2, fa), list(lisp[i + 1:]), m - 1):
yield el
for i, fa in enumerate(lisp[0 : len(lisp) + 1 - m]):
for el in recursion(fa, list(lisp[i + 1:]), m - 1):
yield el
debug = flags.get('debug', False)
cont, f = f.as_primitive()
N = len(symbols)
max_exp = {}
for v in symbols:
max_exp[v] = 0
for coeff, monom in f.iter_terms():
for v, exp in zip(symbols, monom):
if exp > max_exp[v]:
max_exp[v] = exp
symbols = sorted(symbols, reverse=True,
key=lambda v: max_exp[v])
f = Poly(f, *symbols)
d = max_exp[symbols[0]] + 1
terms, exps = {}, []
for i in xrange(0, len(symbols)):
exps.append(d**i)
for coeff, monom in f.iter_terms():
exp = 0
for i, expi in enumerate(monom):
exp += expi * exps[i]
terms[exp] = int(coeff)
g, factors = zzx_from_dict(terms), []
try:
for ff, k in zzx_factor(g)[1]:
for i in xrange(0, k):
factors.append(ff)
except OverflowError:
raise PolynomialError("input too large for multivariate Kronecker method")
const, result, tested = 1, [], []
if debug: print "KRONECKER-MV: Z[x] #factors = %i ..." % (len(factors))
for k in range(1, len(factors)//2 + 1):
for h in combinations(factors, k):
if h in tested:
continue
n = zzx_degree(h)
terms = {}
for coeff in h:
if not coeff:
n = n-1
continue
else:
coeff = Integer(coeff)
y_deg, n = n, n-1
monom = [0] * N
for i in xrange(N):
v_deg = y_deg % d
y_deg = (y_deg - v_deg) // d
monom[i] = v_deg
monom = tuple(monom)
if terms.has_key(monom):
terms[monom] += coeff
else:
terms[monom] = coeff
cand = Poly(terms, *symbols)
if cand.is_one:
continue
if cand.LC.is_negative:
cand = -cand;
q, r = mv_int_div(f, cand)
if r.is_zero:
if debug: print "KRONECKER-MV: Z[X] factor found %s" % cand
result.append(cand)
f = q
else:
tested.append(h)
if f.is_constant:
const, f = f.LC, Poly(1, *symbols)
break
if f.is_one:
break
if not f.is_one:
if debug: print "KRONECKER-MV: Z[X] factor found %s" % f
result.append(f)
factors = {}
for ff in result:
if factors.has_key(ff):
factors[ff] += 1
else:
factors[ff] = 1
return cont*const, sorted(factors.items())