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ellipke.m    117 lines (107 with data), 3.3 kB

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## Copyright (C) 2001 David Billinghurst <David.Billinghurst@riotinto.com>
## Copyright (C) 2001 Paul Kienzle <pkienzle@users.sf.net>
## Copyright (C) 2003 Jaakko Ruohio
##
## This program is free software; you can redistribute it and/or modify it under
## the terms of the GNU General Public License as published by the Free Software
## Foundation; either version 3 of the License, or (at your option) any later
## version.
##
## This program 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 General Public License for more
## details.
##
## You should have received a copy of the GNU General Public License along with
## this program; if not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{k}, @var{e}] =} ellipke (@var{m}[,@var{tol}])
## Compute complete elliptic integral of first K(@var{m}) and second E(@var{m}).
##
## @var{m} is either real array or scalar with 0 <= m <= 1
##
## @var{tol} will be ignored (@sc{Matlab} uses this to allow faster, less
## accurate approximation)
##
## Ref: Abramowitz, Milton and Stegun, Irene A. Handbook of Mathematical
## Functions, Dover, 1965, Chapter 17.
## @seealso{ellipj}
## @end deftypefn
function [k,e] = ellipke( m )
if (nargin < 1 || nargin > 2)
print_usage;
endif
k = e = zeros(size(m));
m = m(:);
if any(~isreal(m))
error("ellipke must have real m");
endif
if any(m>1)
error("ellipke must have m <= 1");
endif
Nmax = 16;
idx = find(m == 1);
if (!isempty(idx))
k(idx) = Inf;
e(idx) = 1.0;
endif
idx = find(m == -Inf);
if (!isempty(idx))
k(idx) = 0.0;
e(idx) = Inf;
endif
## Arithmetic-Geometric Mean (AGM) algorithm
## ( Abramowitz and Stegun, Section 17.6 )
idx = find(m != 1 & m != -Inf);
if (!isempty(idx))
idx_neg = find(m < 0 & m != -Inf);
mult_k = 1./sqrt(1-m(idx_neg));
mult_e = sqrt(1-m(idx_neg));
m(idx_neg) = -m(idx_neg)./(1-m(idx_neg));
a = ones(length(idx),1);
b = sqrt(1.0-m(idx));
c = sqrt(m(idx));
f = 0.5;
sum = f*c.*c;
for n = 2:Nmax
t = (a+b)/2;
c = (a-b)/2;
b = sqrt(a.*b);
a = t;
f = f * 2;
sum = sum + f*c.*c;
if all(c./a < eps), break; endif
endfor
if n >= Nmax, error("ellipke: not enough workspace"); endif
k(idx) = 0.5*pi./a;
e(idx) = 0.5*pi.*(1.0-sum)./a;
k(idx_neg) = mult_k.*k(idx_neg);
e(idx_neg) = mult_e.*e(idx_neg);
endif
endfunction
## Test complete elliptic functions of first and second kind
## against "exact" solution from Mathematica 3.0
%!test
%! m = [0.0; 0.01; 0.1; 0.5; 0.9; 0.99; 1.0 ];
%! [k,e] = ellipke(m);
%!
%! # K(1.0) is really infinity - see below
%! K = [
%! 1.5707963267948966192;
%! 1.5747455615173559527;
%! 1.6124413487202193982;
%! 1.8540746773013719184;
%! 2.5780921133481731882;
%! 3.6956373629898746778;
%! 0.0 ];
%! E = [
%! 1.5707963267948966192;
%! 1.5668619420216682912;
%! 1.5307576368977632025;
%! 1.3506438810476755025;
%! 1.1047747327040733261;
%! 1.0159935450252239356;
%! 1.0 ];
%! if k(7)==Inf, k(7)=0.0; endif;
%! assert(K,k,8*eps);
%! assert(E,e,8*eps);