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, rnd)
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, _, _ = evalf_add(arg, prec, options)
prec2 = workprec - fastlog(xre)
re = mpf_log(mpf_add(xre, fone, prec2), prec, rnd)
re_acc = prec
if imaginary_term:
return re, mpf_pi(prec), re_acc, prec
else:
return re, None, re_acc, None
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 _create_evalf_table():
global evalf_table
evalf_table = {
C.Symbol: evalf_symbol,
C.Dummy: evalf_symbol,
C.Float: 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.NaN: lambda x, prec, options: (fnan, 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 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, rnd)
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, _, _ = evalf_add(arg, prec, options)
prec2 = workprec - fastlog(xre)
# xre is now x - 1 so we add 1 back here to calculate x
re = mpf_log(mpf_abs(mpf_add(xre, fone, prec2)), prec, rnd)
re_acc = prec
if imaginary_term:
return re, mpf_pi(prec), re_acc, prec
else:
return re, None, re_acc, None
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 = mpf_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 _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 = mpf_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.Float: 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.NaN : lambda x, prec, options: (fnan, 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.Product: evalf_prod,
C.Piecewise: evalf_piecewise,
C.bernoulli: evalf_bernoulli,
}
def _as_mpf_val(self, prec):
return mpf_pi(prec)
def _as_mpf_val(self, prec):
return mpf_pi(prec)