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 evalf_trig(v, prec, options):
"""
This function handles sin and cos of real arguments.
TODO: should also handle tan and complex arguments.
"""
if v.func is C.cos:
func = mpf_cos
elif v.func is C.sin:
func = mpf_sin
else:
raise NotImplementedError
arg = v.args[0]
# 20 extra bits is possibly overkill. It does make the need
# to restart very unlikely
xprec = prec + 20
re, im, re_accuracy, im_accuracy = evalf(arg, xprec, options)
if im:
raise NotImplementedError
if not re:
if v.func is C.cos:
return fone, None, prec, None
elif v.func is C.sin:
return None, None, None, None
else:
raise NotImplementedError
# For trigonometric functions, we are interested in the
# fixed-point (absolute) accuracy of the argument.
xsize = fastlog(re)
# Magnitude <= 1.0. OK to compute directly, because there is no
# danger of hitting the first root of cos (with sin, magnitude
# <= 2.0 would actually be ok)
if xsize < 1:
return func(re, prec, round_nearest), None, prec, None
# Very large
if xsize >= 10:
xprec = prec + xsize
re, im, re_accuracy, im_accuracy = evalf(arg, xprec, options)
# Need to repeat in case the argument is very close to a
# multiple of pi (or pi/2), hitting close to a root
while 1:
y = func(re, prec, round_nearest)
ysize = fastlog(y)
gap = -ysize
accuracy = (xprec - xsize) - gap
if accuracy < prec:
if options.get('verbose'):
print "SIN/COS", accuracy, "wanted", prec, "gap", gap
print to_str(y,10)
if xprec > options.get('maxprec', DEFAULT_MAXPREC):
return y, None, accuracy, None
xprec += gap
re, im, re_accuracy, im_accuracy = evalf(arg, xprec, options)
continue
else:
return y, None, prec, None
def _print_Real(self, expr):
prec = expr._prec
if prec < 5:
dps = 0
else:
dps = prec_to_dps(expr._prec)
return mlib.to_str(expr._mpf_, dps, strip_zeros=False)
def _print_Real(self, expr):
prec = expr._prec
if prec < 5:
dps = 0
else:
dps = prec_to_dps(expr._prec)
return mlib.to_str(expr._mpf_, dps, strip_zeros=False)
def _print_Real(self, expr):
prec = expr._prec
if prec < 5:
dps = 0
else:
dps = prec_to_dps(expr._prec)
if self._settings["full_prec"] == True:
strip = False
elif self._settings["full_prec"] == False:
strip = True
elif self._settings["full_prec"] == "auto":
strip = self._print_level > 1
return mlib.to_str(expr._mpf_, dps, strip_zeros=strip)
def _print_Real(self, expr):
prec = expr._prec
if prec < 5:
dps = 0
else:
dps = prec_to_dps(expr._prec)
if self._settings["full_prec"] == True:
strip = False
elif self._settings["full_prec"] == False:
strip = True
elif self._settings["full_prec"] == "auto":
strip = self._print_level > 1
return mlib.to_str(expr._mpf_, dps, strip_zeros=strip)
def evalf(x, prec, options):
try:
r = evalf_table[x.func](x, prec, options)
except KeyError:
#r = finalize_complex(x._eval_evalf(prec)._mpf_, fzero, prec)
try:
# Fall back to ordinary evalf if possible
if 'subs' in options:
x = x.subs(options['subs'])
r = x._eval_evalf(prec)._mpf_, None, prec, None
except AttributeError:
raise NotImplementedError
if options.get("verbose"):
print "### input", x
print "### output", to_str(r[0] or fzero, 50)
#print "### raw", r[0], r[2]
print
if options.get("chop"):
r = chop_parts(r, prec)
if options.get("strict"):
check_target(x, r, prec)
return r
def evalf(x, prec, options):
try:
r = evalf_table[x.func](x, prec, options)
except KeyError:
#r = finalize_complex(x._eval_evalf(prec)._mpf_, fzero, prec)
try:
# Fall back to ordinary evalf if possible
if 'subs' in options:
x = x.subs(options['subs'])
r = x._eval_evalf(prec)._mpf_, None, prec, None
except AttributeError:
raise NotImplementedError
if options.get("verbose"):
print "### input", x
print "### output", to_str(r[0] or fzero, 50)
#print "### raw", r[0], r[2]
print
if options.get("chop"):
r = chop_parts(r, prec)
if options.get("strict"):
check_target(x, r, prec)
return r
def _print_Real(self, expr):
# Based off of that in StrPrinter
dps = prec_to_dps(expr._prec)
str_real = mlib.to_str(expr._mpf_, dps, strip_zeros=True)
# Must always have a mul symbol (as 2.5 10^{20} just looks odd)
seperator = r" \times "
if self._settings["mul_symbol"] is not None:
seperator = self._settings["mul_symbol_latex"]
if "e" in str_real:
(mant, exp) = str_real.split("e")
if exp[0] == "+":
exp = exp[1:]
return r"%s%s10^{%s}" % (mant, seperator, exp)
elif str_real == "+inf":
return r"\infty"
elif str_real == "-inf":
return r"- \intfy"
else:
return str_real
def _print_Real(self, expr):
dps = prec_to_dps(expr._prec)
r = mlib.to_str(expr._mpf_, repr_dps(expr._prec))
return "%s('%s', prec=%i)" % (expr.__class__.__name__, r, dps)
def _print_Real(self, expr):
dps = prec_to_dps(expr._prec)
r = mlib.to_str(expr._mpf_, repr_dps(expr._prec))
return "%s('%s', prec=%i)" % (expr.__class__.__name__, r, dps)