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########################################################################
##
## Copyright (C) 2001-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## This file is part of Octave.
##
## Octave 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.
##
## Octave 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 Octave; see the file COPYING. If not, see
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {@var{k} =} ellipke (@var{m})
## @deftypefnx {} {@var{k} =} ellipke (@var{m}, @var{tol})
## @deftypefnx {} {[@var{k}, @var{e}] =} ellipke (@dots{})
## Compute complete elliptic integrals of the first K(@var{m}) and second
## E(@var{m}) kind.
##
## @var{m} must be a scalar or real array with -Inf @leq{} @var{m} @leq{} 1.
##
## The optional input @var{tol} controls the stopping tolerance of the
## algorithm and defaults to @code{eps (class (@var{m}))}. The tolerance can
## be increased to compute a faster, less accurate approximation.
##
## When called with one output only elliptic integrals of the first kind are
## returned.
##
## Mathematical Note:
##
## Elliptic integrals of the first kind are defined as
##
## @tex
## $$
## {\rm K} (m) = \int_0^1 {dt \over \sqrt{(1 - t^2) (1 - m t^2)}}
## $$
## @end tex
## @ifnottex
##
## @example
## @group
## 1
## / dt
## K (m) = | ------------------------------
## / sqrt ((1 - t^2)*(1 - m*t^2))
## 0
## @end group
## @end example
##
## @end ifnottex
##
## Elliptic integrals of the second kind are defined as
##
## @tex
## $$
## {\rm E} (m) = \int_0^1 {\sqrt{1 - m t^2} \over \sqrt{1 - t^2}} dt
## $$
## @end tex
## @ifnottex
##
## @example
## @group
## 1
## / sqrt (1 - m*t^2)
## E (m) = | ------------------ dt
## / sqrt (1 - t^2)
## 0
## @end group
## @end example
##
## @end ifnottex
##
## Reference: Milton @nospell{Abramowitz} and Irene A. @nospell{Stegun},
## @cite{Handbook of Mathematical Functions}, Chapter 17, Dover, 1965.
## @seealso{ellipj}
## @end deftypefn
function [k, e] = ellipke (m, tol = [])
if (nargin < 1)
print_usage ();
endif
sz = size (m);
if (! isreal (m))
error ("ellipke: M must be real");
elseif (any (m > 1))
error ("ellipke: M must be <= 1");
endif
if (isempty (tol))
tol = eps (class (m));
elseif (! (isreal (tol) && isscalar (tol) && tol > 0))
error ("ellipke: TOL must be a real scalar > 0");
endif
k = e = zeros (sz);
## Handle extreme values
idx_1 = (m == 1);
k(idx_1) = Inf;
e(idx_1) = 1;
idx_neginf = (m == -Inf);
k(idx_neginf) = 0;
e(idx_neginf) = Inf;
## Arithmetic-Geometric Mean (AGM) algorithm
## ( Abramowitz and Stegun, Section 17.6 )
Nmax = 16;
idx = ! idx_1 & ! idx_neginf;
if (any (idx))
idx_neg = find (m < 0 & ! idx_neginf);
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));
b = sqrt (1 - m(idx));
a = ones (size (b));
c = sqrt (m(idx));
f = 0.5;
sum = f*c.^2;
n = 2;
do
t = (a + b)/2;
c = (a - b)/2;
b = sqrt (a .* b);
a = t;
f *= 2;
sum += f*c.^2;
until (all (c./a < tol) || (++n > Nmax))
if (n >= Nmax)
error ("ellipke: algorithm did not converge in %d iterations", Nmax);
endif
k(idx) = 0.5*pi ./ a;
e(idx) = 0.5*pi*(1 - 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, 0.0];
%! [k,e] = ellipke (m);
%!
%! k_exp = [1.5707963267948966192, 1.5747455615173559527
%! 1.6124413487202193982, 1.8540746773013719184
%! 2.5780921133481731882, 3.6956373629898746778
%! Inf , 1.5707963267948966192 ];
%! e_exp = [1.5707963267948966192, 1.5668619420216682912
%! 1.5307576368977632025, 1.3506438810476755025
%! 1.1047747327040733261, 1.0159935450252239356
%! 1.0 , 1.5707963267948966192 ];
%! assert (k, k_exp, 8*eps);
%! assert (e, e_exp, 8*eps);
## Test against A&S Table 17.1
%!test
%! m = [0:5:50]'/100;
%! k_exp = [1.570796326794897;
%! 1.591003453790792;
%! 1.612441348720219;
%! 1.635256732264580;
%! 1.659623598610528;
%! 1.685750354812596;
%! 1.713889448178791;
%! 1.744350597225613;
%! 1.777519371491253;
%! 1.813883936816983;
%! 1.854074677301372 ];
%! e_exp = [1.570796327;
%! 1.550973352;
%! 1.530757637;
%! 1.510121831;
%! 1.489035058;
%! 1.467462209;
%! 1.445363064;
%! 1.422691133;
%! 1.399392139;
%! 1.375401972;
%! 1.350643881 ];
%! [k,e] = ellipke (m);
%! assert (k, k_exp, 1e-15);
%! assert (e, e_exp, 1e-8);
## Test negative values against "exact" solution from Mathematica.
%! m = [-0.01; -1; -5; -100; -1000; -Inf];
%! [k,e] = ellipke (m);
%!
%! k_exp = [1.5668912730681963584;
%! 1.3110287771460599052;
%! 0.9555039270640439337;
%! 0.3682192486091410329;
%! 0.1530293349884987857;
%! 0];
%! e_exp = [1.5747159850169884130;
%! 1.9100988945138560089;
%! 2.8301982463458773125;
%! 10.209260919814572009;
%! 31.707204053711259719;
%! Inf ];
%! assert (k, k_exp, 8*eps);
%! assert (e, e_exp, 8*eps (e_exp));
## Test input validation
%!error <Invalid call> ellipke ()
%!error <M must be real> ellipke (1i)
%!error <M must be .= 1> ellipke (2)
%!error <TOL must be a real scalar . 0> ellipke (1, i)
%!error <TOL must be a real scalar . 0> ellipke (1, [1 1])
%!error <TOL must be a real scalar . 0> ellipke (1, -1)
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