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ellipke.m

## Copyright (C) 2001 David Billinghurst
##
## 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 2 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, write to the Free Software
## Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA

## Compute:
##     complete elliptic integral of first K(m) 
##     complete elliptic integral of second E(m)    
##
## usage: [k,e] = ellipke(m[,tol])
## 
## m is either real array or scalar with 0 <= m <= 1
## 
## tol  Ignored. 
##      (Matlab uses this to allow faster, less accurate approximation)
##
## Ref: Abramowitz, Milton and Stegun, Irene A
##      Handbook of Mathematical Functions, Dover, 1965
##      Chapter 17
##
## See also: ellipj

## Author: David Billinghurst <David.Billinghurst@riotinto.com>
## Created: 31 January 2001
## 2001-02-01 Paul Kienzle
##   * vectorized
##   * included function name in error messages

function [k,e] = ellipke( m )

  if (nargin < 1 || nargin > 2)
    usage("[k, e] = ellipke (m)");
  endif

  k = e = zeros(size(m));
  m = m(:);
  if any(~isreal(m))
    error("ellipke must have real m"); 
  endif
  if any(m<0) || any(m>1)
    error("ellipke must have m in the range [0,1]");
  endif

  Nmax = 16;
  dfi = do_fortran_indexing;
  unwind_protect
    do_fortran_indexing = 1;

    idx = find(m == 1);
    if (!isempty(idx))
      k(idx) = Inf;
      e(idx) = 1.0;
    endif

    ## Arithmetic-Geometric Mean (AGM) algorithm
    ## ( Abramowitz and Stegun, Section 17.6 )
    idx = find(m != 1);
    if (!isempty(idx))
      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;
    endif
  unwind_protect_cleanup
    do_fortran_indexing = dfi;
  end_unwind_protect

endfunction

%!test
%! ## Test complete elliptic functions of first and second kind
%! ## against "exact" solution from Mathematica 3.0
%! ##
%! ## David Billinghurst <David.Billinghurst@riotinto.com>
%! ## 1 February 2001
%! m = ;
%! [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);

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