def content(self):
"""Return the content of a Polynomial.
Usage:
======
Returns the content, that is, the positive greatest common
divisor of the (integer) coefficients.
Examples:
=========
>>> x, y = symbols('xy')
>>> f = Polynomial(6*x + 20*y + 4*x*y)
>>> f.content()
2
Also see L{as_primitive}, L{leading_coeff}.
"""
result = 0
for term in self.coeffs:
if not isinstance(term[0], Integer):
print "%s is no integer coefficient!" % term[0]
return S.Zero
result = abs(numbers.gcd(result, abs(int(term[0]))))
return Integer(result)
def pollard_rho(n, max_iters=5, seed=1234):
"""Use Pollard's rho method to try to extract a factor of n. The
returned factor may be a composite number. A maximum of max_iters
iterations are performed; if no factor is found, None is returned.
The rho algorithm is a Monte Carlo method whose outcome can
be affected by changing the random seed value.
References
==========
Richard Crandall & Carl Pomerance (2005), "Prime Numbers:
A Computational Perspective", Springer, 2nd edition, 229-231
"""
random.seed(seed + max_iters)
for i in range(max_iters):
a = random.randint(1, n-3)
s = random.randint(0, n-1)
U = V = s
F = lambda x: (x**2 + a) % a
while 1:
U = F(U)
V = F(F(V))
g = numbers.gcd(U-V, n)
if g == 1:
continue
if g == n:
break
return g
return None
def pollard_pm1(n, B=10, seed=1234):
"""Use Pollard's p-1 method to try to extract a factor of n. The
returned factor may be a composite number. The search is performed
up to a smoothness bound B; if no factor is found, None is
returned.
The p-1 algorithm is a Monte Carlo method whose outcome can
be affected by changing the random seed value.
Example usage
=============
With the default smoothness bound, this number can't be cracked:
>>> pollard_pm1(21477639576571)
Increasing the smoothness bound helps:
>>> pollard_pm1(21477639576571, 2000)
4410317L
References
==========
Richard Crandall & Carl Pomerance (2005), "Prime Numbers:
A Computational Perspective", Springer, 2nd edition, 236-238
"""
from math import log
random.seed(seed + B)
a = random.randint(2, n-1)
for p in sieve.primerange(2, B):
e = int(log(B, p))
a = pow(a, p**e, n)
g = numbers.gcd(a-1, n)
if 1 < g < n:
return g
else:
return None
def A(n, j):
if j == 1:
return Float(1)
s = Float(0)
pi = pi_float()
for h in xrange(1, j):
if gcd(h,j) == 1:
s += cos((g(h,j)-2*h*n)*pi/j)
return s
def as_integer(self):
"""Return the polynomial with integer coefficients.
Usage:
======
Starting from an instance of Polynomial with only rational
coefficients, this function multiplies it with the common
denominator. The result is a tuple consisting of the
factor applied to the coefficients and a new instance of
Polynomial in the integers.
Example:
========
>>> x, y = symbols('xy')
>>> f = Polynomial(x/6 + y/4 + x*y/3)
>>> denominator, f = f.as_integer()
>>> print denominator
12
>>> print f
2*x + 3*y + 4*x*y
>>> denominator, f = f.as_integer()
>>> print denominator
1
>>> print f
2*x + 3*y + 4*x*y
Also see L{as_monic}, L{as_primitive}.
"""
denom = S.One
for term in self.coeffs:
if not isinstance(term[0], Rational):
print "%s is no rational coefficient!" % term[0]
return S.Zero, self
else:
# Compute the least common multiple of the denominators:
denom = term[0].q*denom/numbers.gcd(int(denom), int(term[0].q))
if denom is S.One:
return S.One, self
else:
return (denom,
Polynomial(coeffs=tuple([((term[0]*denom),) + term[1:]
for term in self.coeffs]),
var=self.var, order=self.order))