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 __add__(self, other):
if (other is S.NaN) or (self is NaN):
return S.NaN
if isinstance(other, Number):
rhs, prec = other._as_mpf_op(self._prec)
return Real._new(mlib.mpf_add(self._mpf_, rhs, prec, rnd), prec)
return Number.__add__(self, other)
def evalf_log(expr, prec, options):
arg = expr.args[0]
workprec = prec+10
xre, xim, xacc, _ = evalf(arg, workprec, options)
if xim:
# XXX: use get_abs etc instead
re = evalf_log(C.log(C.abs(arg, evaluate=False), evaluate=False), prec, options)
im = mpf_atan2(xim, xre or fzero, prec)
return re[0], im, re[2], prec
imaginary_term = (mpf_cmp(xre, fzero) < 0)
re = mpf_log(mpf_abs(xre), prec, round_nearest)
size = fastlog(re)
if prec - size > workprec:
# We actually need to compute 1+x accurately, not x
arg = C.Add(S.NegativeOne,arg,evaluate=False)
xre, xim, xre_acc, xim_acc = evalf_add(arg, prec, options)
prec2 = workprec - fastlog(xre)
re = mpf_log(mpf_add(xre, fone, prec2), prec, round_nearest)
re_acc = prec
if imaginary_term:
return re, mpf_pi(prec), re_acc, prec
else:
return re, None, re_acc, None
def __add__(self, other):
if (other is S.NaN) or (self is NaN):
return S.NaN
if isinstance(other, Number):
rhs, prec = other._as_mpf_op(self._prec)
return Real._new(mlib.mpf_add(self._mpf_, rhs, prec, rnd), prec)
return Number.__add__(self, other)
def __add__(self, other):
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if (other is S.NaN) or (self is NaN):
return S.NaN
if isinstance(other, Number):
rhs, prec = other._as_mpf_op(self._prec)
return Real._new(mlib.mpf_add(self._mpf_, rhs, prec, rnd), prec)
return Number.__add__(self, other)
def __add__(self, other):
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if (other is S.NaN) or (self is NaN):
return S.NaN
if isinstance(other, Number):
rhs, prec = other._as_mpf_op(self._prec)
return Real._new(mlib.mpf_add(self._mpf_, rhs, prec, rnd), prec)
return Number.__add__(self, other)
def A(n, j, prec):
"""Compute the inner sum in the HRR formula."""
if j == 1:
return fone
s = fzero
pi = pi_fixed(prec)
for h in xrange(1, j):
if igcd(h,j) != 1:
continue
# & with mask to compute fractional part of fixed-point number
one = 1 << prec
onemask = one - 1
half = one >> 1
g = 0
if j >= 3:
for k in xrange(1, j):
t = h*k*one//j
if t > 0: frac = t & onemask
else: frac = -((-t) & onemask)
g += k*(frac - half)
g = ((g - 2*h*n*one)*pi//j) >> prec
s = mpf_add(s, mpf_cos(from_man_exp(g, -prec), prec), prec)
return s