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 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 complex arguments.
TODO: should also handle tan of 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_acc, im_acc = evalf(arg, xprec, options)
if im:
if 'subs' in options:
v = v.subs(options['subs'])
return evalf(v._eval_evalf(prec), prec, options)
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, rnd), None, prec, None
# Very large
if xsize >= 10:
xprec = prec + xsize
re, im, re_acc, im_acc = 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, rnd)
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_acc, im_acc = evalf(arg, xprec, options)
continue
else:
return y, None, prec, None
def evalf_trig(v, prec, options):
"""
This function handles sin and cos of complex arguments.
TODO: should also handle tan of 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_acc, im_acc = evalf(arg, xprec, options)
if im:
if "subs" in options:
v = v.subs(options["subs"])
return evalf(v._eval_evalf(prec), prec, options)
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, rnd), None, prec, None
# Very large
if xsize >= 10:
xprec = prec + xsize
re, im, re_acc, im_acc = 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, rnd)
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_acc, im_acc = evalf(arg, xprec, options)
continue
else:
return y, None, prec, None
def print_boys_test(maxm):
l1 = []
l2 = []
ms = [S(m) for m in xrange(maxm + 1)]
ts = [
S(0),
S(10) ** (-20),
S(10) ** (-10),
S(10) ** (-7),
S(10) ** (-5),
S(10) ** (-2),
S(10) ** (-1),
5 * S(10) ** (-1),
S(1),
Rational(3, 2),
S(2),
S(10),
S(10) ** 2,
S(10) ** 5,
S(10) ** 10,
S(10) ** 20,
]
for m in ms:
for t in ts:
r = boys_mpmath(m, t).evalf(30, maxn=1000000, strict=True)
s = to_str(r._mpf_, 20, strip_zeros=False, min_fixed=0, max_fixed=0)
# print m, t, s
l1.append("boys_function(%i,%.1e)" % (m, t))
l2.append(s)
print "result = np.array([%s])" % ", ".join(l1)
print "check = np.array([%s])" % ", ".join(l2)
def print_boys_inc(maxm, resolution):
# Computation of all the boys function data
boys_fn_data = []
for m in xrange(maxm + 1):
l = []
boys_fn_data.append(l)
m = sympify(m)
counter = 0
while True:
t = Rational(counter, resolution)
f = boys_mpmath(m, t)
f_numer = f.evalf(30, maxn=1000, strict=True)
f_str = to_str(f_numer._mpf_,
16,
strip_zeros=False,
min_fixed=0,
max_fixed=0)
l.append(f_str)
ft = boys_function_tail(m, t)
ft_numer = ft.evalf(30, maxn=1000, strict=True)
counter += 1
error = abs(float(ft) - float(f))
if counter % 100 == 0 or error == 0:
print '%2i %5i % 20.12e % 20.12e' % (m, counter, f, error)
if error == 0:
break
f = file('boys_inc.cpp', 'w')
print >> f
print >> f
print >> f, '#define BOYS_RESOLUTION', resolution
print >> f, '#define BOYS_MAX_DATA', maxm
print >> f
for m in xrange(maxm + 1):
print >> f, 'static double boys_fn_data_%i[%i] = {' % (
m, len(boys_fn_data[m]))
print >> f, wrap_strings(boys_fn_data[m], 22, 10)
print >> f, '};'
print >> f
print >> f, 'static double *boys_fn_data[BOYS_MAX_DATA+1] = {'
print >> f, wrap_strings(['boys_fn_data_%i' % m for m in xrange(maxm + 1)],
16, 4)
print >> f, '};'
print >> f
print >> f, 'static long boys_sizes[BOYS_MAX_DATA+1] = {'
print >> f, wrap_strings(
[str(len(boys_fn_data[m])) for m in xrange(maxm + 1)], 6, 8)
print >> f, '};'
print >> f
f.close()
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):
from sympy import re as re_, im as im_
try:
rf = evalf_table[x.func]
r = rf(x, prec, options)
except KeyError:
try:
# Fall back to ordinary evalf if possible
if "subs" in options:
x = x.subs(evalf_subs(prec, options["subs"]))
re, im = x._eval_evalf(prec).as_real_imag()
if re.has(re_) or im.has(im_):
raise NotImplementedError
if re == 0:
re = None
reprec = None
else:
re = re._to_mpmath(prec, allow_ints=False)._mpf_
reprec = prec
if im == 0:
im = None
imprec = None
else:
im = im._to_mpmath(prec, allow_ints=False)._mpf_
imprec = prec
r = re, im, reprec, imprec
except AttributeError:
raise NotImplementedError
if options.get("verbose"):
print "### input", x
print "### output", to_str(r[0] or fzero, 50)
print "### raw", r # r[0], r[2]
print
chop = options.get("chop", False)
if chop:
if chop is True:
chop_prec = prec
else:
# convert (approximately) from given tolerance;
# the formula here will will make 1e-i rounds to 0 for
# i in the range +/-27 while 2e-i will not be chopped
chop_prec = int(round(-3.321 * math.log10(chop) + 2.5))
if chop_prec == 3:
chop_prec -= 1
r = chop_parts(r, chop_prec)
if options.get("strict"):
check_target(x, r, prec)
return r
def evalf(x, prec, options):
from sympy import re as re_, im as im_
try:
rf = evalf_table[x.func]
r = rf(x, prec, options)
except KeyError:
try:
# Fall back to ordinary evalf if possible
if 'subs' in options:
x = x.subs(evalf_subs(prec, options['subs']))
re, im = x._eval_evalf(prec).as_real_imag()
if re.has(re_) or im.has(im_):
raise NotImplementedError
if re == 0:
re = None
reprec = None
else:
re = re._to_mpmath(prec, allow_ints=False)._mpf_
reprec = prec
if im == 0:
im = None
imprec = None
else:
im = im._to_mpmath(prec, allow_ints=False)._mpf_
imprec = prec
r = re, im, reprec, imprec
except AttributeError:
raise NotImplementedError
if options.get("verbose"):
print("### input", x)
print("### output", to_str(r[0] or fzero, 50))
print("### raw", r ) # r[0], r[2]
print()
chop = options.get('chop', False)
if chop:
if chop is True:
chop_prec = prec
else:
# convert (approximately) from given tolerance;
# the formula here will will make 1e-i rounds to 0 for
# i in the range +/-27 while 2e-i will not be chopped
chop_prec = int(round(-3.321*math.log10(chop) + 2.5))
if chop_prec == 3:
chop_prec -= 1
r = chop_parts(r, chop_prec)
if options.get("strict"):
check_target(x, r, prec)
return r
def print_boys_inc(maxm, resolution):
# Computation of all the boys function data
boys_fn_data = []
for m in xrange(maxm + 1):
l = []
boys_fn_data.append(l)
m = sympify(m)
counter = 0
while True:
t = Rational(counter, resolution)
f = boys_mpmath(m, t)
f_numer = f.evalf(30, maxn=1000, strict=True)
f_str = to_str(f_numer._mpf_, 16, strip_zeros=False, min_fixed=0, max_fixed=0)
l.append(f_str)
ft = boys_function_tail(m, t)
ft_numer = ft.evalf(30, maxn=1000, strict=True)
counter += 1
error = abs(float(ft) - float(f))
if counter % 100 == 0 or error == 0:
print "%2i %5i % 20.12e % 20.12e" % (m, counter, f, error)
if error == 0:
break
f = file("boys_inc.cpp", "w")
print >> f
print >> f
print >> f, "#define BOYS_RESOLUTION", resolution
print >> f, "#define BOYS_MAX_DATA", maxm
print >> f
for m in xrange(maxm + 1):
print >> f, "static double boys_fn_data_%i[%i] = {" % (m, len(boys_fn_data[m]))
print >> f, wrap_strings(boys_fn_data[m], 22, 10)
print >> f, "};"
print >> f
print >> f, "static double *boys_fn_data[BOYS_MAX_DATA+1] = {"
print >> f, wrap_strings(["boys_fn_data_%i" % m for m in xrange(maxm + 1)], 16, 4)
print >> f, "};"
print >> f
print >> f, "static long boys_sizes[BOYS_MAX_DATA+1] = {"
print >> f, wrap_strings([str(len(boys_fn_data[m])) for m in xrange(maxm + 1)], 6, 8)
print >> f, "};"
print >> f
f.close()
def _print_Float(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
rv = mlib.to_str(expr._mpf_, dps, strip_zeros=strip)
if rv.startswith('-.0'):
rv = '-0.' + rv[3:]
elif rv.startswith('.0'):
rv = '0.' + rv[2:]
return rv
def _print_Float(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
rv = mlib.to_str(expr._mpf_, dps, strip_zeros=strip)
if rv.startswith('-.0'):
rv = '-0.' + rv[3:]
elif rv.startswith('.0'):
rv = '0.' + rv[2:]
return rv
def print_boys_test(maxm):
l1 = []
l2 = []
ms = [S(m) for m in xrange(maxm+1)]
ts = [S(0), S(10)**(-20), S(10)**(-10), S(10)**(-7), S(10)**(-5),
S(10)**(-2), S(10)**(-1), 5*S(10)**(-1), S(1), Rational(3,2), S(2),
S(10), S(10)**2, S(10)**5, S(10)**10, S(10)**20]
for m in ms:
for t in ts:
r = boys_mpmath(m, t).evalf(30, maxn=1000000, strict=True)
s = to_str(r._mpf_, 20, strip_zeros=False, min_fixed=0, max_fixed=0)
#print m, t, s
l1.append('boys_function(%i,%.1e)' % (m, t))
l2.append(s)
print 'result = np.array([%s])' % ', '.join(l1)
print 'check = np.array([%s])' % ', '.join(l2)
def evalf(x, prec, options):
from sympy import re as re_, im as im_
try:
rf = evalf_table[x.func]
r = rf(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(evalf_subs(prec, options['subs']))
re, im = x._eval_evalf(prec).as_real_imag()
if re.has(re_) or im.has(im_):
raise NotImplementedError
if re == 0:
re = None
reprec = None
else:
re = re._to_mpmath(prec, allow_ints=False)._mpf_
reprec = prec
if im == 0:
im = None
imprec = None
else:
im = im._to_mpmath(prec, allow_ints=False)._mpf_
imprec = prec
r = re, im, reprec, imprec
except AttributeError:
raise NotImplementedError
if options.get("verbose"):
print "### input", x
print "### output", to_str(r[0] or fzero, 50)
print "### raw", r#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):
from sympy import re as re_, im as im_
try:
rf = evalf_table[x.func]
r = rf(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(evalf_subs(prec, options['subs']))
re, im = x._eval_evalf(prec).as_real_imag()
if re.has(re_) or im.has(im_):
raise NotImplementedError
if re == 0:
re = None
reprec = None
else:
re = re._to_mpmath(prec, allow_ints=False)._mpf_
reprec = prec
if im == 0:
im = None
imprec = None
else:
im = im._to_mpmath(prec, allow_ints=False)._mpf_
imprec = prec
r = re, im, reprec, imprec
except AttributeError:
raise NotImplementedError
if options.get("verbose"):
print "### input", x
print "### output", to_str(r[0] or fzero, 50)
print "### raw", r #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:
rf = evalf_table[x.func]
r = rf(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#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:
rf = evalf_table[x.func]
r = rf(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 #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_Float(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)
separator = r" \times "
if self._settings['mul_symbol'] is not None:
separator = 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, separator, exp)
elif str_real == "+inf":
return r"\infty"
elif str_real == "-inf":
return r"- \infty"
else:
return str_real
def _print_Float(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)
separator = r" \times "
if self._settings['mul_symbol'] is not None:
separator = 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, separator, exp)
elif str_real == "+inf":
return r"\infty"
elif str_real == "-inf":
return r"- \infty"
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)
