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# -*-cython-*-
#
# Implementation of the Lambert W function [1]. Based on the MPMath
# implementation [2], and documentaion [3].
#
# Copyright: Yosef Meller, 2009
# Author email: mellerf@netvision.net.il
#
# Distributed under the same license as SciPy
#
# References:
# [1] On the Lambert W function, Adv. Comp. Math. 5 (1996) 329-359,
# available online: http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf
# [2] mpmath source code, Subversion revision 990
# http://code.google.com/p/mpmath/source/browse/trunk/mpmath/functions.py?spec=svn994&r=992
# [3] mpmath source code, Subversion revision 994
# http://code.google.com/p/mpmath/source/browse/trunk/mpmath/function_docs.py?spec=svn994&r=994
# NaN checking as per suggestions of the cython-users list,
# http://groups.google.com/group/cython-users/browse_thread/thread/ff03eed8221bc36d
# TODO: use a series expansion when extremely close to the branch point
# at `-1/e` and make sure that the proper branch is chosen there
import cython
cimport sf_error
cdef extern from "math.h":
double exp(double x) nogil
double log(double x) nogil
from _complexstuff cimport *
# Heavy lifting is here:
@cython.cdivision(True)
cdef inline double complex lambertw_scalar(double complex z, long k, double tol) nogil:
"""
This is just the implementation of W for a single input z.
See the docstring for lambertw() below for the full description.
"""
# Comments copied verbatim from [2] are marked with '>'
if zisnan(z):
return z
# Return value:
cdef double complex w
#> We must be extremely careful near the singularities at -1/e and 0
cdef double u
u = exp(-1)
cdef double absz
absz = zabs(z)
if absz <= u:
if z == 0:
#> w(0,0) = 0; for all other branches we hit the pole
if k == 0:
return z
sf_error.error("lambertw", sf_error.SINGULAR, NULL)
return -inf
if k == 0:
w = z # Initial guess for iteration
#> For small real z < 0, the -1 branch beaves roughly like log(-z)
elif k == -1 and z.imag ==0 and z.real < 0:
w = log(-z.real)
#> Use a simple asymptotic approximation.
else:
w = zlog(z)
#> The branches are roughly logarithmic. This approximation
#> gets better for large |k|; need to check that this always
#> works for k ~= -1, 0, 1.
if k: w = w + k*2*pi*1j
elif k == 0 and z.imag and zabs(z) <= 0.7:
#> Both the W(z) ~= z and W(z) ~= ln(z) approximations break
#> down around z ~= -0.5 (converging to the wrong branch), so patch
#> with a constant approximation (adjusted for sign)
if zabs(z+0.5) < 0.1:
if z.imag > 0:
w = 0.7 + 0.7j
else:
w = 0.7 - 0.7j
else:
w = z
else:
if z.real == inf:
if k == 0:
return z
else:
return z + 2*k*pi*1j
if z.real == -inf:
return (-z) + (2*k+1)*pi*1j
#> Simple asymptotic approximation as above
w = zlog(z)
if k: w = w + k*2*pi*1j
#> Use Halley iteration to solve w*exp(w) = z
cdef double complex ew, wew, wewz, wn
cdef int i
for i in range(100):
ew = zexp(w)
wew = w*ew
wewz = wew-z
wn = w - wewz / (wew + ew - (w + 2)*wewz/(2*w + 2))
if zabs(wn-w) < tol*zabs(wn):
return wn
else:
w = wn
sf_error.error("lambertw", sf_error.SLOW,
"iteration failed to converge: %g + %gj",
<double>z.real, <double>z.imag)
return nan
|
