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 __mul__(self, other):
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if isinstance(other, Number):
rhs, prec = other._as_mpf_op(self._prec)
return Real._new(mlib.mpf_mul(self._mpf_, rhs, prec, rnd), prec)
return Number.__mul__(self, other)
def __mul__(self, other):
try:
other = _sympify(other)
except SympifyError:
return NotImplemented
if isinstance(other, Number):
rhs, prec = other._as_mpf_op(self._prec)
return Real._new(mlib.mpf_mul(self._mpf_, rhs, prec, rnd), prec)
return Number.__mul__(self, other)
def __mul__(self, other):
if isinstance(other, Number):
rhs, prec = other._as_mpf_op(self._prec)
return Real._new(mlib.mpf_mul(self._mpf_, rhs, prec, rnd), prec)
return Number.__mul__(self, other)
def evalf_mul(v, prec, options):
args = v.args
# With guard digits, multiplication in the real case does not destroy
# accuracy. This is also true in the complex case when considering the
# total accuracy; however accuracy for the real or imaginary parts
# separately may be lower.
acc = prec
target_prec = prec
# XXX: big overestimate
prec = prec + len(args) + 5
direction = 0
# Empty product is 1
man, exp, bc = MP_BASE(1), 0, 1
direction = 0
complex_factors = []
# First, we multiply all pure real or pure imaginary numbers.
# direction tells us that the result should be multiplied by
# i**direction
for arg in args:
re, im, a, aim = evalf(arg, prec, options)
if re and im:
complex_factors.append((re, im, a, aim))
continue
elif re:
s, m, e, b = re
elif im:
a = aim
direction += 1
s, m, e, b = im
else:
return None, None, None, None
direction += 2*s
man *= m
exp += e
bc += b
if bc > 3*prec:
man >>= prec
exp += prec
acc = min(acc, a)
sign = (direction & 2) >> 1
v = normalize(sign, man, exp, bitcount(man), prec, round_nearest)
if complex_factors:
# Multiply first complex number by the existing real scalar
re, im, re_acc, im_acc = complex_factors[0]
re = mpf_mul(re, v, prec)
im = mpf_mul(im, v, prec)
re_acc = min(re_acc, acc)
im_acc = min(im_acc, acc)
# Multiply consecutive complex factors
complex_factors = complex_factors[1:]
for wre, wim, wre_acc, wim_acc in complex_factors:
re, im, re_acc, im_acc = cmul((re, re_acc), (im,im_acc),
(wre,wre_acc), (wim,wim_acc), prec, target_prec)
if options.get('verbose'):
print "MUL: obtained accuracy", re_acc, im_acc, "expected", target_prec
# multiply by i
if direction & 1:
return mpf_neg(im), re, re_acc, im_acc
else:
return re, im, re_acc, im_acc
else:
# multiply by i
if direction & 1:
return None, v, None, acc
else:
return v, None, acc, None
if (prec-target_prec) > options.get('maxprec', DEFAULT_MAXPREC):
return re, im, re_accuracy, im_accuracy
prec = prec + max(10+2**i, diff)
options['maxprec'] = min(oldmaxprec, 2*prec)
if options.get('verbose'):
print "ADD: restarting with prec", prec
i += 1
finally:
options['maxprec'] = oldmaxprec
# Helper for complex multiplication
# XXX: should be able to multiply directly, and use complex_accuracy
# to obtain the final accuracy
def cmul((a, aacc), (b, bacc), (c, cacc), (d, dacc), prec, target_prec):
A, Aacc = mpf_mul(a,c,prec), min(aacc, cacc)
B, Bacc = mpf_mul(mpf_neg(b),d,prec), min(bacc, dacc)
C, Cacc = mpf_mul(a,d,prec), min(aacc, dacc)
D, Dacc = mpf_mul(b,c,prec), min(bacc, cacc)
re, re_accuracy = add_terms([(A, Aacc), (B, Bacc)], prec, target_prec)
im, im_accuracy = add_terms([(C, Cacc), (D, Cacc)], prec, target_prec)
return re, im, re_accuracy, im_accuracy
def evalf_mul(v, prec, options):
args = v.args
# With guard digits, multiplication in the real case does not destroy
# accuracy. This is also true in the complex case when considering the
# total accuracy; however accuracy for the real or imaginary parts
# separately may be lower.
acc = prec
target_prec = prec
def __mul__(self, other):
if isinstance(other, Number):
rhs, prec = other._as_mpf_op(self._prec)
return Real._new(mlib.mpf_mul(self._mpf_, rhs, prec, rnd), prec)
return Number.__mul__(self, other)
def evalf_mul(v, prec, options):
args = v.args
# With guard digits, multiplication in the real case does not destroy
# accuracy. This is also true in the complex case when considering the
# total accuracy; however accuracy for the real or imaginary parts
# separately may be lower.
acc = prec
target_prec = prec
# XXX: big overestimate
prec = prec + len(args) + 5
direction = 0
# Empty product is 1
man, exp, bc = MP_BASE(1), 0, 1
direction = 0
complex_factors = []
# First, we multiply all pure real or pure imaginary numbers.
# direction tells us that the result should be multiplied by
# i**direction
for arg in args:
re, im, re_acc, im_acc = evalf(arg, prec, options)
if re and im:
complex_factors.append((re, im, re_acc, im_acc))
continue
elif re:
(s, m, e, b), w_acc = re, re_acc
elif im:
(s, m, e, b), w_acc = im, im_acc
direction += 1
else:
return None, None, None, None
direction += 2*s
man *= m
exp += e
bc += b
if bc > 3*prec:
man >>= prec
exp += prec
acc = min(acc, w_acc)
sign = (direction & 2) >> 1
v = normalize(sign, man, exp, bitcount(man), prec, round_nearest)
if complex_factors:
# make existing real scalar look like an imaginary and
# multiply by the remaining complex numbers
re, im = v, (0, MP_BASE(0), 0, 0)
for wre, wim, wre_acc, wim_acc in complex_factors:
# acc is the overall accuracy of the product; we aren't
# computing exact accuracies of the product.
acc = min(acc,
complex_accuracy((wre, wim, wre_acc, wim_acc)))
A = mpf_mul(re, wre, prec)
B = mpf_mul(mpf_neg(im), wim, prec)
C = mpf_mul(re, wim, prec)
D = mpf_mul(im, wre, prec)
re, xre_acc = add_terms([(A, acc), (B, acc)], prec, target_prec)
im, xim_acc = add_terms([(C, acc), (D, acc)], prec, target_prec)
if options.get('verbose'):
print "MUL: wanted", target_prec, "accurate bits, got", acc
# multiply by i
if direction & 1:
return mpf_neg(im), re, acc, acc
else:
return re, im, acc, acc
else:
# multiply by i
if direction & 1:
return None, v, None, acc
else:
return v, None, acc, None