mirror of
https://github.com/lcn2/calc.git
synced 2025-08-16 01:03:29 +03:00
a31078bbec
Some folks might think: “you still use RCS”?!? And we will say, hey, at least we switched from SCCS to RCS back in … I think it was around 1994 ... at least we are keeping up! :-) :-) :-) Logs say that SCCS version 18 became RCS version 19 on 1994 March 18. RCS served us well. But now it is time to move on. And so we are switching to git. Calc releases produce a lot of file changes. In the 125 releases of calc since 1996, when I started managing calc releases, there have been 15473 file mods!
285 lines
8.7 KiB
Plaintext
285 lines
8.7 KiB
Plaintext
/*
|
|
* lambertw - Lambert's W-function
|
|
*
|
|
* Copyright (C) 2013 Christoph Zurnieden
|
|
*
|
|
* Calc is open software; you can redistribute it and/or modify it under
|
|
* the terms of the version 2.1 of the GNU Lesser General Public License
|
|
* as published by the Free Software Foundation.
|
|
*
|
|
* Calc is distributed in the hope that it will be useful, but WITHOUT
|
|
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
|
|
* or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
|
|
* Public License for more details.
|
|
*
|
|
* A copy of version 2.1 of the GNU Lesser General Public License is
|
|
* distributed with calc under the filename COPYING-LGPL. You should have
|
|
* received a copy with calc; if not, write to Free Software Foundation, Inc.
|
|
* 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
|
|
*
|
|
* Under source code control: 2013/08/11 01:31:28
|
|
* File existed as early as: 2013
|
|
*/
|
|
|
|
|
|
static resource_debug_level;
|
|
resource_debug_level = config("resource_debug", 0);
|
|
|
|
|
|
/*
|
|
|
|
R. M. Corless and G. H. Gonnet and D. E. G. Hare and D. J. Jeffrey and
|
|
D. E. Knuth, "On the Lambert W Function", Advances n Computational
|
|
Mathematics, 329--359, (1996)
|
|
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.112.6117
|
|
|
|
D. J. Jeffrey, D. E. G. Hare, R. M. Corless, "Unwinding the branches of the
|
|
Lambert W function", The Mathematical Scientist, 21, pp 1-7, (1996)
|
|
http://www.apmaths.uwo.ca/~djeffrey/Offprints/wbranch.pdf
|
|
|
|
Darko Verebic, "Having Fun with Lambert W(x) Function"
|
|
arXiv:1003.1628v1, March 2010, http://arxiv.org/abs/1003.1628
|
|
|
|
Winitzki, S. "Uniform Approximations for Transcendental Functions",
|
|
In Part 1 of Computational Science and its Applications - ICCSA 2003,
|
|
Lecture Notes in Computer Science, Vol. 2667, Springer-Verlag,
|
|
Berlin, 2003, 780-789. DOI 10.1007/3-540-44839-X_82
|
|
A copy may be found by Google.
|
|
|
|
|
|
*/
|
|
static true = 1;
|
|
static false = 0;
|
|
|
|
/* Branch 0, Winitzki (2003) , the well known Taylor series*/
|
|
define __CZ__lambertw_0(z,eps){
|
|
local a=2.344e0, b=0.8842e0, c=0.9294e0, d=0.5106e0, e=-1.213e0;
|
|
local y=sqrt(2*exp(1)*z+2);
|
|
return (2*ln(1+b*y)-ln(1+c*ln(1+d*y))+e)/(1+1/(2*ln(1+b*y)+2*a));
|
|
}
|
|
/* branch -1 */
|
|
define __CZ__lambertw_m1(z,eps){
|
|
local wn k;
|
|
/* Cut-off found in Maxima */
|
|
if(z < 0.3) return __CZ__lambertw_app(z,eps);
|
|
wn = z;
|
|
/* Verebic (2010) eqs. 16-18*/
|
|
for(k=0;k<10;k++){
|
|
wn = ln(-z)-ln(-wn);
|
|
}
|
|
return wn;
|
|
}
|
|
|
|
/*
|
|
generic approximation
|
|
|
|
series for 1+W((z-2)/(2 e))
|
|
|
|
Corless et al (1996) (4.22)
|
|
Verebic (2010) eqs. 35-37; more coefficients given at the end of sect. 3.1
|
|
or online
|
|
http://www.wolframalpha.com/input/?
|
|
i=taylor+%28+1%2Bproductlog%28+%28z-2%29%2F%282*e%29+%29+%29
|
|
or by using the function lambertw_series_print() after running
|
|
lambertw_series(z,eps,branch,terms) at least once with the wanted number of
|
|
terms and z = 1 (which might throw an error because the series will not
|
|
converge in anybodies lifetime for something that far from the branchpoint).
|
|
|
|
|
|
*/
|
|
define __CZ__lambertw_app(z,eps){
|
|
local b0=-1, b1=1, b2=-1/3, b3=11/72;
|
|
local y=sqrt(2*exp(1)*z+2);
|
|
return b0 + ( y * (b1 + (y * (b2 + (b3 * y)))));
|
|
}
|
|
|
|
static __CZ__Ws_a;
|
|
static __CZ__Ws_c;
|
|
static __CZ__Ws_len=0;
|
|
|
|
define lambertw_series_print(){
|
|
local k;
|
|
for(k=0;k<__CZ__Ws_len;k++){
|
|
print num(__CZ__Ws_c[k]):"/":den(__CZ__Ws_c[k]):"*p^":k;
|
|
}
|
|
}
|
|
|
|
/*
|
|
The series is fast but only if _very_ close to the branchpoint
|
|
The exact branch must be given explicitly, e.g.:
|
|
|
|
; lambertw(-exp(-1)+.001)-lambertw_series(-exp(-1)+.001,epsilon()*1e-10,0)
|
|
-0.14758879113205794065490184399030194122136720202792-
|
|
0.00000000000000000000000000000000000000000000000000i
|
|
; lambertw(-exp(-1)+.001)-lambertw_series(-exp(-1)+.001,epsilon()*1e-10,1)
|
|
0.00000000000000000000000000000000000000000000000000-
|
|
0.00000000000000000000000000000000000000000000000000i
|
|
*/
|
|
define lambertw_series(z,eps,branch,terms){
|
|
local k l limit tmp sum A C P PP epslocal;
|
|
if(!isnull(terms))
|
|
limit = terms;
|
|
else
|
|
limit = 100;
|
|
|
|
if(isnull(eps))
|
|
eps = epsilon(epsilon()*1e-10);
|
|
epslocal = epsilon(eps);
|
|
|
|
P = sqrt(2*(exp(1)*z+1));
|
|
if(branch != 0) P = -P;
|
|
tmp=0;sum=0;PP=P;
|
|
|
|
__CZ__Ws_a = mat[limit+1];
|
|
__CZ__Ws_c = mat[limit+1];
|
|
__CZ__Ws_len = limit;
|
|
/*
|
|
c0 = -1; c1 = 1
|
|
a0 = 2; a1 =-1
|
|
*/
|
|
__CZ__Ws_c[0] = -1; __CZ__Ws_c[1] = 1;
|
|
__CZ__Ws_a[0] = 2; __CZ__Ws_a[1] = -1;
|
|
sum += __CZ__Ws_c[0];
|
|
sum += __CZ__Ws_c[1] * P;
|
|
PP *= P;
|
|
for(k=2;k<limit;k++){
|
|
for(l=2;l<k;l++){
|
|
__CZ__Ws_a[k] += __CZ__Ws_c[l]*__CZ__Ws_c[k+1-l];
|
|
}
|
|
|
|
__CZ__Ws_c[k] = (k-1) * ( __CZ__Ws_c[k-2]/2
|
|
+__CZ__Ws_a[k-2]/4)/
|
|
(k+1)-__CZ__Ws_a[k]/2-__CZ__Ws_c[k-1]/(k+1);
|
|
tmp = __CZ__Ws_c[k] * PP;
|
|
sum += tmp;
|
|
if(abs(tmp) <= eps){
|
|
epsilon(epslocal);
|
|
return sum;
|
|
}
|
|
PP *= P;
|
|
}
|
|
epsilon(epslocal);
|
|
return
|
|
newerror(strcat("lambertw_series: does not converge in ",
|
|
str(limit)," terms" ));
|
|
}
|
|
|
|
/* */
|
|
define lambertw(z,branch){
|
|
local eps epslarge ret branchpoint bparea w we ew w1e wn k places m1e;
|
|
local closeness;
|
|
|
|
eps = epsilon();
|
|
if(branch == 0){
|
|
if(!im(z)){
|
|
if(abs(z) <= eps) return 0;
|
|
if(abs(z-exp(1)) <= eps) return 1;
|
|
if(abs(z - (-ln(2)/2)) <= eps ) return -ln(2);
|
|
if(abs(z - (-pi()/2)) <= eps ) return 1i*pi()/2;
|
|
}
|
|
}
|
|
|
|
branchpoint = -exp(-1);
|
|
bparea = .2;
|
|
if(branch == 0){
|
|
if(!im(z) && abs(z-branchpoint) == 0) return -1;
|
|
ret = __CZ__lambertw_0(z,eps);
|
|
/* Yeah, C&P, I know, sorry */
|
|
##ret = ln(z) + 2*pi()*1i*branch - ln(ln(z)+2*pi()*1i*branch);
|
|
}
|
|
else if(branch == 1){
|
|
if(im(z)<0 && abs(z-branchpoint) <= bparea)
|
|
ret = __CZ__lambertw_app(z,eps);
|
|
/* Does calc have a goto? Oh, it does! */
|
|
ret =ln(z) + 2*pi()*1i*branch - ln(ln(z)+2*pi()*1i*branch);
|
|
}
|
|
else if(branch == -1){##print "-1";
|
|
if(!im(z) && abs(z-branchpoint) == 0) return -1;
|
|
if(!im(z) && z>branchpoint && z < 0){##print "0";
|
|
ret = __CZ__lambertw_m1(z,eps);}
|
|
if(im(z)>=0 && abs(z-branchpoint) <= bparea){##print "1";
|
|
ret = __CZ__lambertw_app(z,eps);}
|
|
ret =ln(z) + 2*pi()*1i*branch - ln(ln(z)+2*pi()*1i*branch);
|
|
}
|
|
else
|
|
ret = ln(z) + 2*pi()*1i*branch - ln(ln(z)+2*pi()*1i*branch);
|
|
|
|
/*
|
|
Such a high precision is only needed _very_ close to the branchpoint
|
|
and might even be insufficient if z has not been computed with
|
|
sufficient precision itself (M below was calculated by Mathematica and also
|
|
with the series above with epsilon(1e-200)):
|
|
; epsilon(1e-50)
|
|
0.00000000000000000001
|
|
; display(50)
|
|
20
|
|
; M=-0.9999999999999999999999997668356018402875796636464119050387
|
|
; lambertw(-exp(-1)+1e-50,0)-M
|
|
-0.00000000000000000000000002678416515423276355643684
|
|
; epsilon(1e-60)
|
|
0.0000000000000000000000000000000000000000000000000
|
|
; A=-exp(-1)+1e-50
|
|
; epsilon(1e-50)
|
|
0.00000000000000000000000000000000000000000000000000
|
|
; lambertw(A,0)-M
|
|
-0.00000000000000000000000000000000000231185460220585
|
|
; lambertw_series(A,epsilon(),0)-M
|
|
-0.00000000000000000000000000000000000132145133161626
|
|
; epsilon(1e-100)
|
|
0.00000000000000000000000000000000000000000000000001
|
|
; A=-exp(-1)+1e-50
|
|
; epsilon(1e-65)
|
|
0.00000000000000000000000000000000000000000000000000
|
|
; lambertw_series(A,epsilon(),0)-M
|
|
0.00000000000000000000000000000000000000000000000000
|
|
; lambertw_series(-exp(-1)+1e-50,epsilon(),0)-M
|
|
-0.00000000000000000000000000000000000000002959444084
|
|
; epsilon(1e-74)
|
|
0.00000000000000000000000000000000000000000000000000
|
|
; lambertw_series(-exp(-1)+1e-50,epsilon(),0)-M
|
|
-0.00000000000000000000000000000000000000000000000006
|
|
*/
|
|
closeness = abs(z-branchpoint);
|
|
if( closeness< 1){
|
|
if(closeness != 0)
|
|
eps = epsilon(epsilon()*( closeness));
|
|
else
|
|
eps = epsilon(epsilon()^2);
|
|
}
|
|
else
|
|
eps = epsilon(epsilon()*1e-2);
|
|
|
|
|
|
epslarge =epsilon();
|
|
|
|
places = highbit(1 + int(1/epslarge)) + 1;
|
|
w = ret;
|
|
for(k=0;k<100;k++){
|
|
ew = exp(w);
|
|
we = w*ew;
|
|
if(abs(we-z)<= 4*epslarge*abs(z))break;
|
|
w1e = (1+w)*ew;
|
|
wn = bround(w- ((we - z) / ( w1e - ( (w+2)*(we-z) )/(2*w+2) ) ),places++) ;
|
|
if( abs(wn - w) <= epslarge*abs(wn)) break;
|
|
else w = wn;
|
|
}
|
|
|
|
if(k==100){
|
|
epsilon(eps);
|
|
return newerror("lambertw: Halley iteration does not converge");
|
|
}
|
|
/* The Maxima coders added a check if the iteration converged to the correct
|
|
branch. This coder deems it superfluous. */
|
|
|
|
epsilon(eps);
|
|
return wn;
|
|
}
|
|
|
|
|
|
config("resource_debug", resource_debug_level),;
|
|
if (config("resource_debug") & 3) {
|
|
print "lambertw(z,branch)";
|
|
print "lambertw_series(z,eps,branch,terms)";
|
|
print "lambertw_series_print()";
|
|
}
|