## 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 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

## Author: David Billinghurst <David.Billinghurst@riotinto.com>
## Created: 31 January 2001
## 2001-02-01 Paul Kienzle
##   * vectorized
##   * included function name in error messages
## 2003-1-18 Jaakko Ruohio
##   * extended for m < 0

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
%! ## 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 = [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);
