def _calc_spouge_coefficients(a, prec):
"""
Calculate Spouge coefficients for approximation with parameter a.
Return a list of big integers representing the coefficients in
fixed-point form with a precision of prec bits.
"""
# We'll store the coefficients as fixed-point numbers but calculate
# them as Floats for convenience. The initial terms are huge, so we
# need to allocate extra bits to ensure full accuracy. The integer
# part of the largest term has size ~= exp(a) or 2**(1.4*a)
floatprec = prec + int(a*1.4)
Float.store()
Float.setprec(floatprec)
c = [0] * a
b = exp(a-1)
e = exp(1)
c[0] = make_fixed(sqrt(2*pi_float()), prec)
for k in range(1, a):
# print "%02f" % (100.0 * k / a), "% done"
c[k] = make_fixed(((-1)**(k-1) * (a-k)**k) * b / sqrt(a-k), prec)
# Divide off e and k instead of computing exp and k! from scratch
b = b / (e * k)
Float.revert()
return c
def zeta(s):
"""
zeta(s) -- calculate the Riemann zeta function of a real or complex
argument s.
"""
Float.store()
Float._prec += 8
si = s
s = ComplexFloat(s)
if s.real < 0:
# Reflection formula (XXX: gets bad around the zeros)
pi = pi_float()
y = power(2, s) * power(pi, s-1) * sin(pi*s/2) * gamma(1-s) * zeta(1-s)
else:
p = Float._prec
n = int((p + 2.28*abs(float(s.imag)))/2.54) + 3
d = _zeta_coefs(n)
if isinstance(si, (int, long)):
t = 0
for k in range(n):
t += (((-1)**k * (d[k] - d[n])) << p) // (k+1)**si
y = (Float((t, -p)) / -d[n]) / (Float(1) - Float(2)**(1-si))
else:
t = Float(0)
for k in range(n):
t += (-1)**k * Float(d[k]-d[n]) * exp(-_logk(k+1)*s)
y = (t / -d[n]) / (Float(1) - exp(log(2)*(1-s)))
Float.revert()
if isinstance(y, ComplexFloat) and s.imag == 0:
return +y.real
else:
return +y
def atan2(y, x):
"""atan2(y,x) has the same magnitude as atan(y/x) but
accounts for the signs of y and x"""
if y < 0:
return -atan2(-y, x)
if not x and not y:
return Float(0)
if y > 0 and x == 0:
Float._prec += 2
t = pi_float() / 2
Float._prec -= 2
return t
Float._prec += 2
if x > 0:
a = atan(y / x)
else:
a = pi_float() - atan(-y / x)
Float._prec -= 2
return +a
def atan(x):
"""Compute atan(x) for a real number x"""
if not isinstance(x, Float):
x = Float(x)
if x < -0.6:
return _with_extraprec(2, lambda: -atan(-x))
if x < 0.6:
return _atan_series_1(x)
if x < 1.5:
return _atan_series_2(x)
# For large x, use atan(x) = pi/2 - atan(1/x)
Float._prec += 4
prec = Float._prec
if x.exp > 10 * prec:
t = pi_float() / 2 # XXX
else:
t = pi_float() / 2 - atan(Float(1) / x)
Float._prec -= 4
return +t
def erf(x):
x = ComplexFloat(x)
if x == 0: return Float(0)
if x.real < 0: return -erf(-x)
Float.store()
Float._prec += 10
y = lower_gamma(0.5, x**2) / sqrt(pi_float())
if x.imag == 0:
y = y.real
Float.revert()
return +y
def gamma(x):
"""
gamma(x) -- calculate the gamma function of a real or complex
number x.
x must not be a negative integer or 0
"""
Float.store()
Float._prec += 2
if isinstance(x, complex):
x = ComplexFloat(x)
elif not isinstance(x, (Float, ComplexFloat, Rational, int, long)):
x = Float(x)
if isinstance(x, (ComplexFloat, complex)):
re, im = x.real, x.imag
else:
re, im = x, 0
# For negative x (or positive x close to the pole at x = 0),
# we use the reflection formula
if re < 0.25:
if re == int(re) and im == 0:
raise ZeroDivisionError, "gamma function pole"
Float._prec += 3
p = pi_float()
g = p / (sin(p*x) * gamma(1-x))
else:
x -= 1
prec, a, c = _get_spouge_coefficients(Float.getprec()+7)
s = _spouge_sum(x, prec, a, c)
if not isinstance(x, (Float, ComplexFloat)):
x = Float(x)
# TODO: higher precision may be needed here when the precision
# and/or size of x are extremely large
Float._prec += 10
g = exp(log(x+a)*(x+Float(0.5))) * exp(-x-a) * s
Float.revert()
return +g
def evalf(expr):
"""
evalf(expr) attempts to evaluate a SymPy expression to a Float or
ComplexFloat with an error smaller than 10**(-Float.getdps())
"""
if isinstance(expr, (Float, ComplexFloat)):
return expr
elif isinstance(expr, (int, float)):
return Float(expr)
elif isinstance(expr, complex):
return ComplexFloat(expr)
expr = Basic.sympify(expr)
if isinstance(expr, (Rational)):
y = Float(expr)
elif isinstance(expr, Real):
y = Float(str(expr))
elif expr is I:
y = ComplexFloat(0,1)
elif expr is pi:
y = constants.pi_float()
elif expr is E:
y = functions.exp(1)
elif isinstance(expr, Mul):
factors = expr[:]
workprec = Float.getprec() + 1 + len(factors)
Float.store()
Float.setprec(workprec)
y = Float(1)
for f in factors:
y *= evalf(f)
Float.revert()
elif isinstance(expr, Pow):
base, expt = expr[:]
workprec = Float.getprec() + 8 # may need more
Float.store()
Float.setprec(workprec)
base = evalf(base)
expt = evalf(expt)
if expt == 0.5:
y = functions.sqrt(base)
else:
y = functions.exp(functions.log(base) * expt)
Float.revert()
elif isinstance(expr, Basic.exp):
Float.store()
Float.setprec(Float.getprec() + 3)
#XXX: how is it possible, that this works:
x = evalf(expr[0])
#and this too:
#x = evalf(expr[1])
#?? (Try to uncomment it and you'll see)
y = functions.exp(x)
Float.revert()
elif isinstance(expr, Add):
# TODO: this doesn't yet work as it should.
# We need some way to handle sums whose results are
# very close to 0, and when necessary, repeat the
# summation with higher precision
reqprec = Float.getprec()
Float.store()
Float.setprec(10)
terms = expr[:]
approxterms = [abs(evalf(x)) for x in terms]
min_mag = min(x.exp for x in approxterms)
max_mag = max(x.exp+bitcount(x.man) for x in approxterms)
Float.setprec(reqprec - 10 + max_mag - min_mag + 1 + len(terms))
workprec = Float.getdps()
y = 0
for t in terms:
y += evalf(t)
Float.revert()
else:
# print expr, expr.__class__
raise NotImplementedError
# print expr, y
return +y