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from __future__ import print_function
from ..libmp.backend import xrange
from .functions import defun, defun_wrapped, defun_static
@defun
def stieltjes(ctx, n, a=1):
n = ctx.convert(n)
a = ctx.convert(a)
if n < 0:
return ctx.bad_domain("Stieltjes constants defined for n >= 0")
if hasattr(ctx, "stieltjes_cache"):
stieltjes_cache = ctx.stieltjes_cache
else:
stieltjes_cache = ctx.stieltjes_cache = {}
if a == 1:
if n == 0:
return +ctx.euler
if n in stieltjes_cache:
prec, s = stieltjes_cache[n]
if prec >= ctx.prec:
return +s
mag = 1
def f(x):
xa = x/a
v = (xa-ctx.j)*ctx.ln(a-ctx.j*x)**n/(1+xa**2)/(ctx.exp(2*ctx.pi*x)-1)
return ctx._re(v) / mag
orig = ctx.prec
try:
# Normalize integrand by approx. magnitude to
# speed up quadrature (which uses absolute error)
if n > 50:
ctx.prec = 20
mag = ctx.quad(f, [0,ctx.inf], maxdegree=3)
ctx.prec = orig + 10 + int(n**0.5)
s = ctx.quad(f, [0,ctx.inf], maxdegree=20)
v = ctx.ln(a)**n/(2*a) - ctx.ln(a)**(n+1)/(n+1) + 2*s/a*mag
finally:
ctx.prec = orig
if a == 1 and ctx.isint(n):
stieltjes_cache[n] = (ctx.prec, v)
return +v
@defun_wrapped
def siegeltheta(ctx, t, derivative=0):
d = int(derivative)
if (t == ctx.inf or t == ctx.ninf):
if d < 2:
if t == ctx.ninf and d == 0:
return ctx.ninf
return ctx.inf
else:
return ctx.zero
if d == 0:
if ctx._im(t):
# XXX: cancellation occurs
a = ctx.loggamma(0.25+0.5j*t)
b = ctx.loggamma(0.25-0.5j*t)
return -ctx.ln(ctx.pi)/2*t - 0.5j*(a-b)
else:
if ctx.isinf(t):
return t
return ctx._im(ctx.loggamma(0.25+0.5j*t)) - ctx.ln(ctx.pi)/2*t
if d > 0:
a = (-0.5j)**(d-1)*ctx.polygamma(d-1, 0.25-0.5j*t)
b = (0.5j)**(d-1)*ctx.polygamma(d-1, 0.25+0.5j*t)
if ctx._im(t):
if d == 1:
return -0.5*ctx.log(ctx.pi)+0.25*(a+b)
else:
return 0.25*(a+b)
else:
if d == 1:
return ctx._re(-0.5*ctx.log(ctx.pi)+0.25*(a+b))
else:
return ctx._re(0.25*(a+b))
@defun_wrapped
def grampoint(ctx, n):
# asymptotic expansion, from
# http://mathworld.wolfram.com/GramPoint.html
g = 2*ctx.pi*ctx.exp(1+ctx.lambertw((8*n+1)/(8*ctx.e)))
return ctx.findroot(lambda t: ctx.siegeltheta(t)-ctx.pi*n, g)
@defun_wrapped
def siegelz(ctx, t, **kwargs):
d = int(kwargs.get("derivative", 0))
t = ctx.convert(t)
t1 = ctx._re(t)
t2 = ctx._im(t)
prec = ctx.prec
try:
if abs(t1) > 500*prec and t2**2 < t1:
v = ctx.rs_z(t, d)
if ctx._is_real_type(t):
return ctx._re(v)
return v
except NotImplementedError:
pass
ctx.prec += 21
e1 = ctx.expj(ctx.siegeltheta(t))
z = ctx.zeta(0.5+ctx.j*t)
if d == 0:
v = e1*z
ctx.prec=prec
if ctx._is_real_type(t):
return ctx._re(v)
return +v
z1 = ctx.zeta(0.5+ctx.j*t, derivative=1)
theta1 = ctx.siegeltheta(t, derivative=1)
if d == 1:
v = ctx.j*e1*(z1+z*theta1)
ctx.prec=prec
if ctx._is_real_type(t):
return ctx._re(v)
return +v
z2 = ctx.zeta(0.5+ctx.j*t, derivative=2)
theta2 = ctx.siegeltheta(t, derivative=2)
comb1 = theta1**2-ctx.j*theta2
if d == 2:
def terms():
return [2*z1*theta1, z2, z*comb1]
v = ctx.sum_accurately(terms, 1)
v = -e1*v
ctx.prec = prec
if ctx._is_real_type(t):
return ctx._re(v)
return +v
ctx.prec += 10
z3 = ctx.zeta(0.5+ctx.j*t, derivative=3)
theta3 = ctx.siegeltheta(t, derivative=3)
comb2 = theta1**3-3*ctx.j*theta1*theta2-theta3
if d == 3:
def terms():
return [3*theta1*z2, 3*z1*comb1, z3+z*comb2]
v = ctx.sum_accurately(terms, 1)
v = -ctx.j*e1*v
ctx.prec = prec
if ctx._is_real_type(t):
return ctx._re(v)
return +v
z4 = ctx.zeta(0.5+ctx.j*t, derivative=4)
theta4 = ctx.siegeltheta(t, derivative=4)
def terms():
return [theta1**4, -6*ctx.j*theta1**2*theta2, -3*theta2**2,
-4*theta1*theta3, ctx.j*theta4]
comb3 = ctx.sum_accurately(terms, 1)
if d == 4:
def terms():
return [6*theta1**2*z2, -6*ctx.j*z2*theta2, 4*theta1*z3,
4*z1*comb2, z4, z*comb3]
v = ctx.sum_accurately(terms, 1)
v = e1*v
ctx.prec = prec
if ctx._is_real_type(t):
return ctx._re(v)
return +v
if d > 4:
h = lambda x: ctx.siegelz(x, derivative=4)
return ctx.diff(h, t, n=d-4)
_zeta_zeros = [
14.134725142,21.022039639,25.010857580,30.424876126,32.935061588,
37.586178159,40.918719012,43.327073281,48.005150881,49.773832478,
52.970321478,56.446247697,59.347044003,60.831778525,65.112544048,
67.079810529,69.546401711,72.067157674,75.704690699,77.144840069,
79.337375020,82.910380854,84.735492981,87.425274613,88.809111208,
92.491899271,94.651344041,95.870634228,98.831194218,101.317851006,
103.725538040,105.446623052,107.168611184,111.029535543,111.874659177,
114.320220915,116.226680321,118.790782866,121.370125002,122.946829294,
124.256818554,127.516683880,129.578704200,131.087688531,133.497737203,
134.756509753,138.116042055,139.736208952,141.123707404,143.111845808,
146.000982487,147.422765343,150.053520421,150.925257612,153.024693811,
156.112909294,157.597591818,158.849988171,161.188964138,163.030709687,
165.537069188,167.184439978,169.094515416,169.911976479,173.411536520,
174.754191523,176.441434298,178.377407776,179.916484020,182.207078484,
184.874467848,185.598783678,187.228922584,189.416158656,192.026656361,
193.079726604,195.265396680,196.876481841,198.015309676,201.264751944,
202.493594514,204.189671803,205.394697202,207.906258888,209.576509717,
211.690862595,213.347919360,214.547044783,216.169538508,219.067596349,
220.714918839,221.430705555,224.007000255,224.983324670,227.421444280,
229.337413306,231.250188700,231.987235253,233.693404179,236.524229666,
]
def _load_zeta_zeros(url):
import urllib
d = urllib.urlopen(url)
L = [float(x) for x in d.readlines()]
# Sanity check
assert round(L[0]) == 14
_zeta_zeros[:] = L
@defun
def oldzetazero(ctx, n, url='http://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros1'):
n = int(n)
if n < 0:
return ctx.zetazero(-n).conjugate()
if n == 0:
raise ValueError("n must be nonzero")
if n > len(_zeta_zeros) and n <= 100000:
_load_zeta_zeros(url)
if n > len(_zeta_zeros):
raise NotImplementedError("n too large for zetazeros")
return ctx.mpc(0.5, ctx.findroot(ctx.siegelz, _zeta_zeros[n-1]))
@defun_wrapped
def riemannr(ctx, x):
if x == 0:
return ctx.zero
# Check if a simple asymptotic estimate is accurate enough
if abs(x) > 1000:
a = ctx.li(x)
b = 0.5*ctx.li(ctx.sqrt(x))
if abs(b) < abs(a)*ctx.eps:
return a
if abs(x) < 0.01:
# XXX
ctx.prec += int(-ctx.log(abs(x),2))
# Sum Gram's series
s = t = ctx.one
u = ctx.ln(x)
k = 1
while abs(t) > abs(s)*ctx.eps:
t = t * u / k
s += t / (k * ctx._zeta_int(k+1))
k += 1
return s
@defun_static
def primepi(ctx, x):
x = int(x)
if x < 2:
return 0
return len(ctx.list_primes(x))
# TODO: fix the interface wrt contexts
@defun_wrapped
def primepi2(ctx, x):
x = int(x)
if x < 2:
return ctx._iv.zero
if x < 2657:
return ctx._iv.mpf(ctx.primepi(x))
mid = ctx.li(x)
# Schoenfeld's estimate for x >= 2657, assuming RH
err = ctx.sqrt(x,rounding='u')*ctx.ln(x,rounding='u')/8/ctx.pi(rounding='d')
a = ctx.floor((ctx._iv.mpf(mid)-err).a, rounding='d')
b = ctx.ceil((ctx._iv.mpf(mid)+err).b, rounding='u')
return ctx._iv.mpf([a,b])
@defun_wrapped
def primezeta(ctx, s):
if ctx.isnan(s):
return s
if ctx.re(s) <= 0:
raise ValueError("prime zeta function defined only for re(s) > 0")
if s == 1:
return ctx.inf
if s == 0.5:
return ctx.mpc(ctx.ninf, ctx.pi)
r = ctx.re(s)
if r > ctx.prec:
return 0.5**s
else:
wp = ctx.prec + int(r)
def terms():
orig = ctx.prec
# zeta ~ 1+eps; need to set precision
# to get logarithm accurately
k = 0
while 1:
k += 1
u = ctx.moebius(k)
if not u:
continue
ctx.prec = wp
t = u*ctx.ln(ctx.zeta(k*s))/k
if not t:
return
#print ctx.prec, ctx.nstr(t)
ctx.prec = orig
yield t
return ctx.sum_accurately(terms)
# TODO: for bernpoly and eulerpoly, ensure that all exact zeros are covered
@defun_wrapped
def bernpoly(ctx, n, z):
# Slow implementation:
#return sum(ctx.binomial(n,k)*ctx.bernoulli(k)*z**(n-k) for k in xrange(0,n+1))
n = int(n)
if n < 0:
raise ValueError("Bernoulli polynomials only defined for n >= 0")
if z == 0 or (z == 1 and n > 1):
return ctx.bernoulli(n)
if z == 0.5:
return (ctx.ldexp(1,1-n)-1)*ctx.bernoulli(n)
if n <= 3:
if n == 0: return z ** 0
if n == 1: return z - 0.5
if n == 2: return (6*z*(z-1)+1)/6
if n == 3: return z*(z*(z-1.5)+0.5)
if ctx.isinf(z):
return z ** n
if ctx.isnan(z):
return z
if abs(z) > 2:
def terms():
t = ctx.one
yield t
r = ctx.one/z
k = 1
while k <= n:
t = t*(n+1-k)/k*r
if not (k > 2 and k & 1):
yield t*ctx.bernoulli(k)
k += 1
return ctx.sum_accurately(terms) * z**n
else:
def terms():
yield ctx.bernoulli(n)
t = ctx.one
k = 1
while k <= n:
t = t*(n+1-k)/k * z
m = n-k
if not (m > 2 and m & 1):
yield t*ctx.bernoulli(m)
k += 1
return ctx.sum_accurately(terms)
@defun_wrapped
def eulerpoly(ctx, n, z):
n = int(n)
if n < 0:
raise ValueError("Euler polynomials only defined for n >= 0")
if n <= 2:
if n == 0: return z ** 0
if n == 1: return z - 0.5
if n == 2: return z*(z-1)
if ctx.isinf(z):
return z**n
if ctx.isnan(z):
return z
m = n+1
if z == 0:
return -2*(ctx.ldexp(1,m)-1)*ctx.bernoulli(m)/m * z**0
if z == 1:
return 2*(ctx.ldexp(1,m)-1)*ctx.bernoulli(m)/m * z**0
if z == 0.5:
if n % 2:
return ctx.zero
# Use exact code for Euler numbers
if n < 100 or n*ctx.mag(0.46839865*n) < ctx.prec*0.25:
return ctx.ldexp(ctx._eulernum(n), -n)
# http://functions.wolfram.com/Polynomials/EulerE2/06/01/02/01/0002/
def terms():
t = ctx.one
k = 0
w = ctx.ldexp(1,n+2)
while 1:
v = n-k+1
if not (v > 2 and v & 1):
yield (2-w)*ctx.bernoulli(v)*t
k += 1
if k > n:
break
t = t*z*(n-k+2)/k
w *= 0.5
return ctx.sum_accurately(terms) / m
@defun
def eulernum(ctx, n, exact=False):
n = int(n)
if exact:
return int(ctx._eulernum(n))
if n < 100:
return ctx.mpf(ctx._eulernum(n))
if n % 2:
return ctx.zero
return ctx.ldexp(ctx.eulerpoly(n,0.5), n)
# TODO: this should be implemented low-level
def polylog_series(ctx, s, z):
tol = +ctx.eps
l = ctx.zero
k = 1
zk = z
while 1:
term = zk / k**s
l += term
if abs(term) < tol:
break
zk *= z
k += 1
return l
def polylog_continuation(ctx, n, z):
if n < 0:
return z*0
twopij = 2j * ctx.pi
a = -twopij**n/ctx.fac(n) * ctx.bernpoly(n, ctx.ln(z)/twopij)
if ctx._is_real_type(z) and z < 0:
a = ctx._re(a)
if ctx._im(z) < 0 or (ctx._im(z) == 0 and ctx._re(z) >= 1):
a -= twopij*ctx.ln(z)**(n-1)/ctx.fac(n-1)
return a
def polylog_unitcircle(ctx, n, z):
tol = +ctx.eps
if n > 1:
l = ctx.zero
logz = ctx.ln(z)
logmz = ctx.one
m = 0
while 1:
if (n-m) != 1:
term = ctx.zeta(n-m) * logmz / ctx.fac(m)
if term and abs(term) < tol:
break
l += term
logmz *= logz
m += 1
l += ctx.ln(z)**(n-1)/ctx.fac(n-1)*(ctx.harmonic(n-1)-ctx.ln(-ctx.ln(z)))
elif n < 1: # else
l = ctx.fac(-n)*(-ctx.ln(z))**(n-1)
logz = ctx.ln(z)
logkz = ctx.one
k = 0
while 1:
b = ctx.bernoulli(k-n+1)
if b:
term = b*logkz/(ctx.fac(k)*(k-n+1))
if abs(term) < tol:
break
l -= term
logkz *= logz
k += 1
else:
raise ValueError
if ctx._is_real_type(z) and z < 0:
l = ctx._re(l)
return l
def polylog_general(ctx, s, z):
v = ctx.zero
u = ctx.ln(z)
if not abs(u) < 5: # theoretically |u| < 2*pi
j = ctx.j
v = 1-s
y = ctx.ln(-z)/(2*ctx.pi*j)
return ctx.gamma(v)*(j**v*ctx.zeta(v,0.5+y) + j**-v*ctx.zeta(v,0.5-y))/(2*ctx.pi)**v
t = 1
k = 0
while 1:
term = ctx.zeta(s-k) * t
if abs(term) < ctx.eps:
break
v += term
k += 1
t *= u
t /= k
return ctx.gamma(1-s)*(-u)**(s-1) + v
@defun_wrapped
def polylog(ctx, s, z):
s = ctx.convert(s)
z = ctx.convert(z)
if z == 1:
return ctx.zeta(s)
if z == -1:
return -ctx.altzeta(s)
if s == 0:
return z/(1-z)
if s == 1:
return -ctx.ln(1-z)
if s == -1:
return z/(1-z)**2
if abs(z) <= 0.75 or (not ctx.isint(s) and abs(z) < 0.9):
return polylog_series(ctx, s, z)
if abs(z) >= 1.4 and ctx.isint(s):
return (-1)**(s+1)*polylog_series(ctx, s, 1/z) + polylog_continuation(ctx, int(ctx.re(s)), z)
if ctx.isint(s):
return polylog_unitcircle(ctx, int(ctx.re(s)), z)
return polylog_general(ctx, s, z)
@defun_wrapped
def clsin(ctx, s, z, pi=False):
if ctx.isint(s) and s < 0 and int(s) % 2 == 1:
return z*0
if pi:
a = ctx.expjpi(z)
else:
a = ctx.expj(z)
if ctx._is_real_type(z) and ctx._is_real_type(s):
return ctx.im(ctx.polylog(s,a))
b = 1/a
return (-0.5j)*(ctx.polylog(s,a) - ctx.polylog(s,b))
@defun_wrapped
def clcos(ctx, s, z, pi=False):
if ctx.isint(s) and s < 0 and int(s) % 2 == 0:
return z*0
if pi:
a = ctx.expjpi(z)
else:
a = ctx.expj(z)
if ctx._is_real_type(z) and ctx._is_real_type(s):
return ctx.re(ctx.polylog(s,a))
b = 1/a
return 0.5*(ctx.polylog(s,a) + ctx.polylog(s,b))
@defun
def altzeta(ctx, s, **kwargs):
try:
return ctx._altzeta(s, **kwargs)
except NotImplementedError:
return ctx._altzeta_generic(s)
@defun_wrapped
def _altzeta_generic(ctx, s):
if s == 1:
return ctx.ln2 + 0*s
return -ctx.powm1(2, 1-s) * ctx.zeta(s)
@defun
def zeta(ctx, s, a=1, derivative=0, method=None, **kwargs):
d = int(derivative)
if a == 1 and not (d or method):
try:
return ctx._zeta(s, **kwargs)
except NotImplementedError:
pass
s = ctx.convert(s)
prec = ctx.prec
method = kwargs.get('method')
verbose = kwargs.get('verbose')
if (not s) and (not derivative):
return ctx.mpf(0.5) - ctx._convert_param(a)[0]
if a == 1 and method != 'euler-maclaurin':
im = abs(ctx._im(s))
re = abs(ctx._re(s))
#if (im < prec or method == 'borwein') and not derivative:
# try:
# if verbose:
# print "zeta: Attempting to use the Borwein algorithm"
# return ctx._zeta(s, **kwargs)
# except NotImplementedError:
# if verbose:
# print "zeta: Could not use the Borwein algorithm"
# pass
if abs(im) > 500*prec and 10*re < prec and derivative <= 4 or \
method == 'riemann-siegel':
try: # py2.4 compatible try block
try:
if verbose:
print("zeta: Attempting to use the Riemann-Siegel algorithm")
return ctx.rs_zeta(s, derivative, **kwargs)
except NotImplementedError:
if verbose:
print("zeta: Could not use the Riemann-Siegel algorithm")
pass
finally:
ctx.prec = prec
if s == 1:
return ctx.inf
abss = abs(s)
if abss == ctx.inf:
if ctx.re(s) == ctx.inf:
if d == 0:
return ctx.one
return ctx.zero
return s*0
elif ctx.isnan(abss):
return 1/s
if ctx.re(s) > 2*ctx.prec and a == 1 and not derivative:
return ctx.one + ctx.power(2, -s)
return +ctx._hurwitz(s, a, d, **kwargs)
@defun
def _hurwitz(ctx, s, a=1, d=0, **kwargs):
prec = ctx.prec
verbose = kwargs.get('verbose')
try:
extraprec = 10
ctx.prec += extraprec
# We strongly want to special-case rational a
a, atype = ctx._convert_param(a)
if ctx.re(s) < 0:
if verbose:
print("zeta: Attempting reflection formula")
try:
return _hurwitz_reflection(ctx, s, a, d, atype)
except NotImplementedError:
pass
if verbose:
print("zeta: Reflection formula failed")
if verbose:
print("zeta: Using the Euler-Maclaurin algorithm")
while 1:
ctx.prec = prec + extraprec
T1, T2 = _hurwitz_em(ctx, s, a, d, prec+10, verbose)
cancellation = ctx.mag(T1) - ctx.mag(T1+T2)
if verbose:
print("Term 1:", T1)
print("Term 2:", T2)
print("Cancellation:", cancellation, "bits")
if cancellation < extraprec:
return T1 + T2
else:
extraprec = max(2*extraprec, min(cancellation + 5, 100*prec))
if extraprec > kwargs.get('maxprec', 100*prec):
raise ctx.NoConvergence("zeta: too much cancellation")
finally:
ctx.prec = prec
def _hurwitz_reflection(ctx, s, a, d, atype):
# TODO: implement for derivatives
if d != 0:
raise NotImplementedError
res = ctx.re(s)
negs = -s
# Integer reflection formula
if ctx.isnpint(s):
n = int(res)
if n <= 0:
return ctx.bernpoly(1-n, a) / (n-1)
if not (atype == 'Q' or atype == 'Z'):
raise NotImplementedError
t = 1-s
# We now require a to be standardized
v = 0
shift = 0
b = a
while ctx.re(b) > 1:
b -= 1
v -= b**negs
shift -= 1
while ctx.re(b) <= 0:
v += b**negs
b += 1
shift += 1
# Rational reflection formula
try:
p, q = a._mpq_
except:
assert a == int(a)
p = int(a)
q = 1
p += shift*q
assert 1 <= p <= q
g = ctx.fsum(ctx.cospi(t/2-2*k*b)*ctx._hurwitz(t,(k,q)) \
for k in range(1,q+1))
g *= 2*ctx.gamma(t)/(2*ctx.pi*q)**t
v += g
return v
def _hurwitz_em(ctx, s, a, d, prec, verbose):
# May not be converted at this point
a = ctx.convert(a)
tol = -prec
# Estimate number of terms for Euler-Maclaurin summation; could be improved
M1 = 0
M2 = prec // 3
N = M2
lsum = 0
# This speeds up the recurrence for derivatives
if ctx.isint(s):
s = int(ctx._re(s))
s1 = s-1
while 1:
# Truncated L-series
l = ctx._zetasum(s, M1+a, M2-M1-1, [d])[0][0]
#if d:
# l = ctx.fsum((-ctx.ln(n+a))**d * (n+a)**negs for n in range(M1,M2))
#else:
# l = ctx.fsum((n+a)**negs for n in range(M1,M2))
lsum += l
M2a = M2+a
logM2a = ctx.ln(M2a)
logM2ad = logM2a**d
logs = [logM2ad]
logr = 1/logM2a
rM2a = 1/M2a
M2as = M2a**(-s)
if d:
tailsum = ctx.gammainc(d+1, s1*logM2a) / s1**(d+1)
else:
tailsum = 1/((s1)*(M2a)**s1)
tailsum += 0.5 * logM2ad * M2as
U = [1]
r = M2as
fact = 2
for j in range(1, N+1):
# TODO: the following could perhaps be tidied a bit
j2 = 2*j
if j == 1:
upds = [1]
else:
upds = [j2-2, j2-1]
for m in upds:
D = min(m,d+1)
if m <= d:
logs.append(logs[-1] * logr)
Un = [0]*(D+1)
for i in xrange(D): Un[i] = (1-m-s)*U[i]
for i in xrange(1,D+1): Un[i] += (d-(i-1))*U[i-1]
U = Un
r *= rM2a
t = ctx.fdot(U, logs) * r * ctx.bernoulli(j2)/(-fact)
tailsum += t
if ctx.mag(t) < tol:
return lsum, (-1)**d * tailsum
fact *= (j2+1)*(j2+2)
if verbose:
print("Sum range:", M1, M2, "term magnitude", ctx.mag(t), "tolerance", tol)
M1, M2 = M2, M2*2
if ctx.re(s) < 0:
N += N//2
@defun
def _zetasum(ctx, s, a, n, derivatives=[0], reflect=False):
"""
Returns [xd0,xd1,...,xdr], [yd0,yd1,...ydr] where
xdk = D^k ( 1/a^s + 1/(a+1)^s + ... + 1/(a+n)^s )
ydk = D^k conj( 1/a^(1-s) + 1/(a+1)^(1-s) + ... + 1/(a+n)^(1-s) )
D^k = kth derivative with respect to s, k ranges over the given list of
derivatives (which should consist of either a single element
or a range 0,1,...r). If reflect=False, the ydks are not computed.
"""
#print "zetasum", s, a, n
# don't use the fixed-point code if there are large exponentials
if abs(ctx.re(s)) < 0.5 * ctx.prec:
try:
return ctx._zetasum_fast(s, a, n, derivatives, reflect)
except NotImplementedError:
pass
negs = ctx.fneg(s, exact=True)
have_derivatives = derivatives != [0]
have_one_derivative = len(derivatives) == 1
if not reflect:
if not have_derivatives:
return [ctx.fsum((a+k)**negs for k in xrange(n+1))], []
if have_one_derivative:
d = derivatives[0]
x = ctx.fsum(ctx.ln(a+k)**d * (a+k)**negs for k in xrange(n+1))
return [(-1)**d * x], []
maxd = max(derivatives)
if not have_one_derivative:
derivatives = range(maxd+1)
xs = [ctx.zero for d in derivatives]
if reflect:
ys = [ctx.zero for d in derivatives]
else:
ys = []
for k in xrange(n+1):
w = a + k
xterm = w ** negs
if reflect:
yterm = ctx.conj(ctx.one / (w * xterm))
if have_derivatives:
logw = -ctx.ln(w)
if have_one_derivative:
logw = logw ** maxd
xs[0] += xterm * logw
if reflect:
ys[0] += yterm * logw
else:
t = ctx.one
for d in derivatives:
xs[d] += xterm * t
if reflect:
ys[d] += yterm * t
t *= logw
else:
xs[0] += xterm
if reflect:
ys[0] += yterm
return xs, ys
@defun
def dirichlet(ctx, s, chi=[1], derivative=0):
s = ctx.convert(s)
q = len(chi)
d = int(derivative)
if d > 2:
raise NotImplementedError("arbitrary order derivatives")
prec = ctx.prec
try:
ctx.prec += 10
if s == 1:
have_pole = True
for x in chi:
if x and x != 1:
have_pole = False
h = +ctx.eps
ctx.prec *= 2*(d+1)
s += h
if have_pole:
return +ctx.inf
z = ctx.zero
for p in range(1,q+1):
if chi[p%q]:
if d == 1:
z += chi[p%q] * (ctx.zeta(s, (p,q), 1) - \
ctx.zeta(s, (p,q))*ctx.log(q))
else:
z += chi[p%q] * ctx.zeta(s, (p,q))
z /= q**s
finally:
ctx.prec = prec
return +z
def secondzeta_main_term(ctx, s, a, **kwargs):
tol = ctx.eps
f = lambda n: ctx.gammainc(0.5*s, a*gamm**2, regularized=True)*gamm**(-s)
totsum = term = ctx.zero
mg = ctx.inf
n = 0
while mg > tol:
totsum += term
n += 1
gamm = ctx.im(ctx.zetazero_memoized(n))
term = f(n)
mg = abs(term)
err = 0
if kwargs.get("error"):
sg = ctx.re(s)
err = 0.5*ctx.pi**(-1)*max(1,sg)*a**(sg-0.5)*ctx.log(gamm/(2*ctx.pi))*\
ctx.gammainc(-0.5, a*gamm**2)/abs(ctx.gamma(s/2))
err = abs(err)
return +totsum, err, n
def secondzeta_prime_term(ctx, s, a, **kwargs):
tol = ctx.eps
f = lambda n: ctx.gammainc(0.5*(1-s),0.25*ctx.log(n)**2 * a**(-1))*\
((0.5*ctx.log(n))**(s-1))*ctx.mangoldt(n)/ctx.sqrt(n)/\
(2*ctx.gamma(0.5*s)*ctx.sqrt(ctx.pi))
totsum = term = ctx.zero
mg = ctx.inf
n = 1
while mg > tol or n < 9:
totsum += term
n += 1
term = f(n)
if term == 0:
mg = ctx.inf
else:
mg = abs(term)
if kwargs.get("error"):
err = mg
return +totsum, err, n
def secondzeta_exp_term(ctx, s, a):
if ctx.isint(s) and ctx.re(s) <= 0:
m = int(round(ctx.re(s)))
if not m & 1:
return ctx.mpf('-0.25')**(-m//2)
tol = ctx.eps
f = lambda n: (0.25*a)**n/((n+0.5*s)*ctx.fac(n))
totsum = ctx.zero
term = f(0)
mg = ctx.inf
n = 0
while mg > tol:
totsum += term
n += 1
term = f(n)
mg = abs(term)
v = a**(0.5*s)*totsum/ctx.gamma(0.5*s)
return v
def secondzeta_singular_term(ctx, s, a, **kwargs):
factor = a**(0.5*(s-1))/(4*ctx.sqrt(ctx.pi)*ctx.gamma(0.5*s))
extraprec = ctx.mag(factor)
ctx.prec += extraprec
factor = a**(0.5*(s-1))/(4*ctx.sqrt(ctx.pi)*ctx.gamma(0.5*s))
tol = ctx.eps
f = lambda n: ctx.bernpoly(n,0.75)*(4*ctx.sqrt(a))**n*\
ctx.gamma(0.5*n)/((s+n-1)*ctx.fac(n))
totsum = ctx.zero
mg1 = ctx.inf
n = 1
term = f(n)
mg2 = abs(term)
while mg2 > tol and mg2 <= mg1:
totsum += term
n += 1
term = f(n)
totsum += term
n +=1
term = f(n)
mg1 = mg2
mg2 = abs(term)
totsum += term
pole = -2*(s-1)**(-2)+(ctx.euler+ctx.log(16*ctx.pi**2*a))*(s-1)**(-1)
st = factor*(pole+totsum)
err = 0
if kwargs.get("error"):
if not ((mg2 > tol) and (mg2 <= mg1)):
if mg2 <= tol:
err = ctx.mpf(10)**int(ctx.log(abs(factor*tol),10))
if mg2 > mg1:
err = ctx.mpf(10)**int(ctx.log(abs(factor*mg1),10))
err = max(err, ctx.eps*1.)
ctx.prec -= extraprec
return +st, err
@defun
def secondzeta(ctx, s, a = 0.015, **kwargs):
r"""
Evaluates the secondary zeta function `Z(s)`, defined for
`\mathrm{Re}(s)>1` by
.. math ::
Z(s) = \sum_{n=1}^{\infty} \frac{1}{\tau_n^s}
where `\frac12+i\tau_n` runs through the zeros of `\zeta(s)` with
imaginary part positive.
`Z(s)` extends to a meromorphic function on `\mathbb{C}` with a
double pole at `s=1` and simple poles at the points `-2n` for
`n=0`, 1, 2, ...
**Examples**
>>> from mpmath import *
>>> mp.pretty = True; mp.dps = 15
>>> secondzeta(2)
0.023104993115419
>>> xi = lambda s: 0.5*s*(s-1)*pi**(-0.5*s)*gamma(0.5*s)*zeta(s)
>>> Xi = lambda t: xi(0.5+t*j)
>>> chop(-0.5*diff(Xi,0,n=2)/Xi(0))
0.023104993115419
We may ask for an approximate error value::
>>> secondzeta(0.5+100j, error=True)
((-0.216272011276718 - 0.844952708937228j), 2.22044604925031e-16)
The function has poles at the negative odd integers,
and dyadic rational values at the negative even integers::
>>> mp.dps = 30
>>> secondzeta(-8)
-0.67236328125
>>> secondzeta(-7)
+inf
**Implementation notes**
The function is computed as sum of four terms `Z(s)=A(s)-P(s)+E(s)-S(s)`
respectively main, prime, exponential and singular terms.
The main term `A(s)` is computed from the zeros of zeta.
The prime term depends on the von Mangoldt function.
The singular term is responsible for the poles of the function.
The four terms depends on a small parameter `a`. We may change the
value of `a`. Theoretically this has no effect on the sum of the four
terms, but in practice may be important.
A smaller value of the parameter `a` makes `A(s)` depend on
a smaller number of zeros of zeta, but `P(s)` uses more values of
von Mangoldt function.
We may also add a verbose option to obtain data about the
values of the four terms.
>>> mp.dps = 10
>>> secondzeta(0.5 + 40j, error=True, verbose=True)
main term = (-30190318549.138656312556 - 13964804384.624622876523j)
computed using 19 zeros of zeta
prime term = (132717176.89212754625045 + 188980555.17563978290601j)
computed using 9 values of the von Mangoldt function
exponential term = (542447428666.07179812536 + 362434922978.80192435203j)
singular term = (512124392939.98154322355 + 348281138038.65531023921j)
((0.059471043 + 0.3463514534j), 1.455191523e-11)
>>> secondzeta(0.5 + 40j, a=0.04, error=True, verbose=True)
main term = (-151962888.19606243907725 - 217930683.90210294051982j)
computed using 9 zeros of zeta
prime term = (2476659342.3038722372461 + 28711581821.921627163136j)
computed using 37 values of the von Mangoldt function
exponential term = (178506047114.7838188264 + 819674143244.45677330576j)
singular term = (175877424884.22441310708 + 790744630738.28669174871j)
((0.059471043 + 0.3463514534j), 1.455191523e-11)
Notice the great cancellation between the four terms. Changing `a`, the
four terms are very different numbers but the cancellation gives
the good value of Z(s).
**References**
A. Voros, Zeta functions for the Riemann zeros, Ann. Institute Fourier,
53, (2003) 665--699.
A. Voros, Zeta functions over Zeros of Zeta Functions, Lecture Notes
of the Unione Matematica Italiana, Springer, 2009.
"""
s = ctx.convert(s)
a = ctx.convert(a)
tol = ctx.eps
if ctx.isint(s) and ctx.re(s) <= 1:
if abs(s-1) < tol*1000:
return ctx.inf
m = int(round(ctx.re(s)))
if m & 1:
return ctx.inf
else:
return ((-1)**(-m//2)*\
ctx.fraction(8-ctx.eulernum(-m,exact=True),2**(-m+3)))
prec = ctx.prec
try:
t3 = secondzeta_exp_term(ctx, s, a)
extraprec = max(ctx.mag(t3),0)
ctx.prec += extraprec + 3
t1, r1, gt = secondzeta_main_term(ctx,s,a,error='True', verbose='True')
t2, r2, pt = secondzeta_prime_term(ctx,s,a,error='True', verbose='True')
t4, r4 = secondzeta_singular_term(ctx,s,a,error='True')
t3 = secondzeta_exp_term(ctx, s, a)
err = r1+r2+r4
t = t1-t2+t3-t4
if kwargs.get("verbose"):
print('main term =', t1)
print(' computed using', gt, 'zeros of zeta')
print('prime term =', t2)
print(' computed using', pt, 'values of the von Mangoldt function')
print('exponential term =', t3)
print('singular term =', t4)
finally:
ctx.prec = prec
if kwargs.get("error"):
w = max(ctx.mag(abs(t)),0)
err = max(err*2**w, ctx.eps*1.*2**w)
return +t, err
return +t
@defun_wrapped
def lerchphi(ctx, z, s, a):
r"""
Gives the Lerch transcendent, defined for `|z| < 1` and
`\Re{a} > 0` by
.. math ::
\Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(a+k)^s}
and generally by the recurrence `\Phi(z,s,a) = z \Phi(z,s,a+1) + a^{-s}`
along with the integral representation valid for `\Re{a} > 0`
.. math ::
\Phi(z,s,a) = \frac{1}{2 a^s} +
\int_0^{\infty} \frac{z^t}{(a+t)^s} dt -
2 \int_0^{\infty} \frac{\sin(t \log z - s
\operatorname{arctan}(t/a)}{(a^2 + t^2)^{s/2}
(e^{2 \pi t}-1)} dt.
The Lerch transcendent generalizes the Hurwitz zeta function :func:`zeta`
(`z = 1`) and the polylogarithm :func:`polylog` (`a = 1`).
**Examples**
Several evaluations in terms of simpler functions::
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> lerchphi(-1,2,0.5); 4*catalan
3.663862376708876060218414
3.663862376708876060218414
>>> diff(lerchphi, (-1,-2,1), (0,1,0)); 7*zeta(3)/(4*pi**2)
0.2131391994087528954617607
0.2131391994087528954617607
>>> lerchphi(-4,1,1); log(5)/4
0.4023594781085250936501898
0.4023594781085250936501898
>>> lerchphi(-3+2j,1,0.5); 2*atanh(sqrt(-3+2j))/sqrt(-3+2j)
(1.142423447120257137774002 + 0.2118232380980201350495795j)
(1.142423447120257137774002 + 0.2118232380980201350495795j)
Evaluation works for complex arguments and `|z| \ge 1`::
>>> lerchphi(1+2j, 3-j, 4+2j)
(0.002025009957009908600539469 + 0.003327897536813558807438089j)
>>> lerchphi(-2,2,-2.5)
-12.28676272353094275265944
>>> lerchphi(10,10,10)
(-4.462130727102185701817349e-11 - 1.575172198981096218823481e-12j)
>>> lerchphi(10,10,-10.5)
(112658784011940.5605789002 - 498113185.5756221777743631j)
Some degenerate cases::
>>> lerchphi(0,1,2)
0.5
>>> lerchphi(0,1,-2)
-0.5
Reduction to simpler functions::
>>> lerchphi(1, 4.25+1j, 1)
(1.044674457556746668033975 - 0.04674508654012658932271226j)
>>> zeta(4.25+1j)
(1.044674457556746668033975 - 0.04674508654012658932271226j)
>>> lerchphi(1 - 0.5**10, 4.25+1j, 1)
(1.044629338021507546737197 - 0.04667768813963388181708101j)
>>> lerchphi(3, 4, 1)
(1.249503297023366545192592 - 0.2314252413375664776474462j)
>>> polylog(4, 3) / 3
(1.249503297023366545192592 - 0.2314252413375664776474462j)
>>> lerchphi(3, 4, 1 - 0.5**10)
(1.253978063946663945672674 - 0.2316736622836535468765376j)
**References**
1. [DLMF]_ section 25.14
"""
if z == 0:
return a ** (-s)
# Faster, but these cases are useful for testing right now
if z == 1:
return ctx.zeta(s, a)
if a == 1:
return ctx.polylog(s, z) / z
if ctx.re(a) < 1:
if ctx.isnpint(a):
raise ValueError("Lerch transcendent complex infinity")
m = int(ctx.ceil(1-ctx.re(a)))
v = ctx.zero
zpow = ctx.one
for n in xrange(m):
v += zpow / (a+n)**s
zpow *= z
return zpow * ctx.lerchphi(z,s, a+m) + v
g = ctx.ln(z)
v = 1/(2*a**s) + ctx.gammainc(1-s, -a*g) * (-g)**(s-1) / z**a
h = s / 2
r = 2*ctx.pi
f = lambda t: ctx.sin(s*ctx.atan(t/a)-t*g) / \
((a**2+t**2)**h * ctx.expm1(r*t))
v += 2*ctx.quad(f, [0, ctx.inf])
if not ctx.im(z) and not ctx.im(s) and not ctx.im(a) and ctx.re(z) < 1:
v = ctx.chop(v)
return v
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