def npartitions(n, verbose=False):
"""
Calculate the partition function P(n), i.e. the number of ways that
n can be written as a sum of positive integers.
P(n) is computed using the Hardy-Ramanujan-Rademacher formula,
described e.g. at http://mathworld.wolfram.com/PartitionFunctionP.html
The correctness of this implementation has been tested for 10**n
up to n = 8.
"""
n = int(n)
if n < 0: return 0
if n <= 5: return [1, 1, 2, 3, 5, 7][n]
# Estimate number of bits in p(n). This formula could be tidied
pbits = int((math.pi*(2*n/3.)**0.5-math.log(4*n))/math.log(10)+1)*\
math.log(10,2)
prec = p = int(pbits*1.1 + 100)
s = fzero
M = max(6, int(0.24*n**0.5+4))
sq23pi = mpf_mul(mpf_sqrt(from_rational(2,3,p), p), mpf_pi(p), p)
sqrt8 = mpf_sqrt(from_int(8), p)
for q in xrange(1, M):
a = A(n,q,p)
d = D(n,q,p, sq23pi, sqrt8)
s = mpf_add(s, mpf_mul(a, d), prec)
if verbose:
print "step", q, "of", M, to_str(a, 10), to_str(d, 10)
# On average, the terms decrease rapidly in magnitude. Dynamically
# reducing the precision greatly improves performance.
p = bitcount(abs(to_int(d))) + 50
np = to_int(mpf_add(s, fhalf, prec))
return int(np)
def D(n, j, prec, sq23pi, sqrt8):
"""
Compute the sinh term in the outer sum of the HRR formula.
The constants sqrt(2/3*pi) and sqrt(8) must be precomputed.
"""
j = from_int(j)
pi = mpf_pi(prec)
a = mpf_div(sq23pi, j, prec)
b = mpf_sub(from_int(n), from_rational(1,24,prec), prec)
c = mpf_sqrt(b, prec)
ch, sh = cosh_sinh(mpf_mul(a,c), prec)
D = mpf_div(mpf_sqrt(j,prec), mpf_mul(mpf_mul(sqrt8,b),pi), prec)
E = mpf_sub(mpf_mul(a,ch), mpf_div(sh,c,prec), prec)
return mpf_mul(D, E)
def _create_evalf_table():
global evalf_table
evalf_table = {
C.Symbol : evalf_symbol,
C.Dummy : evalf_symbol,
C.Real : lambda x, prec, options: (x._mpf_, None, prec, None),
C.Rational : lambda x, prec, options: (from_rational(x.p, x.q, prec), None, prec, None),
C.Integer : lambda x, prec, options: (from_int(x.p, prec), None, prec, None),
C.Zero : lambda x, prec, options: (None, None, prec, None),
C.One : lambda x, prec, options: (fone, None, prec, None),
C.Half : lambda x, prec, options: (fhalf, None, prec, None),
C.Pi : lambda x, prec, options: (mpf_pi(prec), None, prec, None),
C.Exp1 : lambda x, prec, options: (mpf_e(prec), None, prec, None),
C.ImaginaryUnit : lambda x, prec, options: (None, fone, None, prec),
C.NegativeOne : lambda x, prec, options: (fnone, None, prec, None),
C.exp : lambda x, prec, options: evalf_pow(C.Pow(S.Exp1, x.args[0],
evaluate=False), prec, options),
C.cos : evalf_trig,
C.sin : evalf_trig,
C.Add : evalf_add,
C.Mul : evalf_mul,
C.Pow : evalf_pow,
C.log : evalf_log,
C.atan : evalf_atan,
C.abs : evalf_abs,
C.re : evalf_re,
C.im : evalf_im,
C.floor : evalf_floor,
C.ceiling : evalf_ceiling,
C.Integral : evalf_integral,
C.Sum : evalf_sum,
C.Piecewise : evalf_piecewise,
C.bernoulli : evalf_bernoulli,
}
def _as_mpf_val(self, prec):
return mlib.from_rational(self.p, self.q, prec, rnd)
def _mpmath_(self, prec, rnd):
return mpmath.make_mpf(mlib.from_rational(self.p, self.q, prec, rnd))
def _as_mpf_val(self, prec):
return mlib.from_rational(self.p, self.q, prec, rnd)