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Add argument to ellipj for Matlab compatibility * ellipj.cc: Allow third argument for compatibility, ignored for now.
author Mike Miller <mtmiller@ieee.org>
date Sun, 28 Apr 2013 19:35:10 -0400
parents a3fdd6041e64
children e2de3c8882be
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/*

Copyright (C) 2001 Leopoldo Cerbaro <redbliss@libero.it>

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
<http://www.gnu.org/licenses/>.

*/

#ifdef HAVE_CONFIG_H
#include <config.h>
#endif

#include "defun.h"
#include "error.h"
#include "lo-ieee.h"

static void
gripe_ellipj_arg (const char *arg)
{
  error ("ellipj: expecting scalar or matrix as %s argument", arg);
}

static void
sncndn (double u, double m, double& sn, double& cn, double& dn, double& err)
{
  static const int Nmax = 16;
  double m1, t=0, si_u, co_u, se_u, ta_u, b, c[Nmax], a[Nmax], phi;
  int n, Nn, ii;

  if (m < 0 || m > 1)
    {
      warning ("ellipj: expecting 0 <= m <= 1"); /* -lc- */
      sn = cn = dn = lo_ieee_nan_value ();
      return;
    }

  double sqrt_eps = sqrt (std::numeric_limits<double>::epsilon ());
  if (m < sqrt_eps)
    {
      /*  # For small m, ( Abramowitz and Stegun, Section 16.13 ) */
      si_u = sin (u);
      co_u = cos (u);
      t = 0.25*m*(u - si_u*co_u);
      sn = si_u - t * co_u;
      cn = co_u + t * si_u;
      dn = 1 - 0.5*m*si_u*si_u;
    }
  else if ((1 - m) < sqrt_eps)
    {
      /*  For m1 = (1-m) small ( Abramowitz and Stegun, Section 16.15 ) */
      m1 = 1 - m;
      si_u = sinh (u);
      co_u = cosh (u);
      ta_u = tanh (u);
      se_u = 1/co_u;
      sn = ta_u + 0.25*m1*(si_u*co_u - u)*se_u*se_u;
      cn = se_u - 0.25*m1*(si_u*co_u - u)*ta_u*se_u;
      dn = se_u + 0.25*m1*(si_u*co_u + u)*ta_u*se_u;
    }
  else
    {
      /*
      //  Arithmetic-Geometric Mean (AGM) algorithm
      //    ( Abramowitz and Stegun, Section 16.4 )
      */

      a[0] = 1;
      b    = sqrt (1 - m);
      c[0] = sqrt (m);
      for (n = 1; n < Nmax; ++n)
        {
          a[n] = (a[n - 1] + b)/2;
          c[n] = (a[n - 1] - b)/2;
          b = sqrt (a[n - 1]*b);
          if (c[n]/a[n] < std::numeric_limits<double>::epsilon ()) break;
        }
      if (n >= Nmax - 1)
        {
          err = 1;
          return;
        }
      Nn = n;
      for (ii = 1; n > 0; ii = ii*2, --n) ; // ii = pow(2,Nn)
      phi = ii*a[Nn]*u;
      for (n = Nn; n > 0; --n)
        {
          t = phi;
          phi = (asin ((c[n]/a[n])* sin (phi)) + phi)/2;
        }
      sn = sin (phi);
      cn = cos (phi);
      dn = cn/cos (t - phi);
    }
}

static void
sncndn (Complex& u, double m, Complex& sn, Complex& cn, Complex& dn,
        double& err)
{
  double m1 = 1 - m, ss1, cc1, dd1;

  sncndn (imag (u), m1, ss1, cc1, dd1, err);
  if (real (u) == 0)
    {
      /* u is pure imag: Jacoby imag. transf. */
      sn = Complex (0, ss1/cc1);
      cn = 1/cc1;         //    cn.imag = 0;
      dn = dd1/cc1;       //    dn.imag = 0;
    }
  else
    {
      /* u is generic complex */
      double ss, cc, dd, ddd;

      sncndn (real (u), m, ss, cc, dd, err);
      ddd = cc1*cc1 + m*ss*ss*ss1*ss1;
      sn = Complex (ss*dd1/ddd, cc*dd*ss1*cc1/ddd);
      cn = Complex (cc*cc1/ddd, -ss*dd*ss1*dd1/ddd);
      dn = Complex (dd*cc1*dd1/ddd, -m*ss*cc*ss1/ddd);
    }
}

DEFUN (ellipj, args, nargout,
  "-*- texinfo -*-\n\
@deftypefn  {Built-in Function} {[@var{sn}, @var{cn}, @var{dn}, @var{err}] =} ellipj (@var{u}, @var{m})\n\
@deftypefnx {Built-in Function} {[@var{sn}, @var{cn}, @var{dn}, @var{err}] =} ellipj (@var{u}, @var{m}, @var{tol})\n\
Compute the Jacobi elliptic functions @var{sn}, @var{cn}, and @var{dn}\n\
of complex argument @var{u} and real parameter @var{m}.\n\
\n\
If @var{m} is a scalar, the results are the same size as @var{u}.\n\
If @var{u} is a scalar, the results are the same size as @var{m}.\n\
If @var{u} is a column vector and @var{m} is a row vector, the\n\
results are matrices with @code{length (@var{u})} rows and\n\
@code{length (@var{m})} columns.  Otherwise, @var{u} and\n\
@var{m} must conform and the results will be the same size.\n\
\n\
The value of @var{u} may be complex.\n\
The value of @var{m} must be 0 <= m <= 1.\n\
\n\
@var{tol} is currently ignored (@sc{Matlab} uses this to allow faster,\n\
less accurate approximation).\n\
\n\
If requested, @var{err} contains the following status information\n\
and is the same size as the result.\n\
\n\
@enumerate 0\n\
@item\n\
Normal return.\n\
@item\n\
Error---no computation, algorithm termination condition not met,\n\
return @code{NaN}.\n\
@end enumerate\n\
 Ref: Abramowitz, Milton and Stegun, Irene A\n\
      Handbook of Mathematical Functions, Dover, 1965\n\
      Chapter 16 (Sections 16.4, 16.13 and 16.15)\n\
@seealso{ellipke}\n\
@end deftypefn")
{
  octave_value_list retval;

  int nargin = args.length ();

  if (nargin < 2 || nargin > 3)
    {
      print_usage ();
      return retval;
    }

  octave_value u_arg = args(0);
  octave_value m_arg = args(1);

  if (m_arg.is_scalar_type ())
    {
      double m = args(1).double_value ();

      if (error_state)
        {
          gripe_ellipj_arg ("second");
          return retval;
        }

      if (u_arg.is_scalar_type ())
        {
          if (u_arg.is_real_type ())
            {  // u real
              double u = args(0).double_value ();

              if (error_state)
                {
                  gripe_ellipj_arg ("first");
                  return retval;
                }
              double sn, cn, dn;
              double err = 0;

              sncndn (u, m, sn, cn, dn, err);
              retval (0) = sn;
              retval (1) = cn;
              retval (2) = dn;
              if (nargout > 3) retval(3) =  err;
            }
          else
            {  // u complex
              Complex u = u_arg.complex_value ();

              if (error_state)
                {
                  gripe_ellipj_arg ("second");
                  return retval;
                }

              Complex sn, cn, dn;
              double err;

              sncndn (u, m, sn, cn, dn, err);

              retval (0) = sn;
              retval (1) = cn;
              retval (2) = dn;
              if (nargout > 3) retval(3) = err;
            }
        }
      else
        {  /* u is matrix ( m is scalar ) */
          ComplexMatrix u = u_arg.complex_matrix_value ();

          if (error_state)
            {
              gripe_ellipj_arg ("first");
              return retval;
            }

          octave_idx_type nr = u.rows ();
          octave_idx_type nc = u.cols ();

          ComplexMatrix sn (nr, nc), cn (nr, nc), dn (nr, nc);
          Matrix err (nr, nc);

          for (octave_idx_type j = 0; j < nc; j++)
            for (octave_idx_type i = 0; i < nr; i++)
              sncndn (u(i,j), m, sn(i,j), cn(i,j), dn(i,j), err(i,j));

          retval (0) = sn;
          retval (1) = cn;
          retval (2) = dn;
          if (nargout > 3) retval(3) = err;
        }
    }
  else
    {
      Matrix m = args(1).matrix_value ();

      if (error_state)
        {
          gripe_ellipj_arg ("second");
          return retval;
        }

      octave_idx_type mr = m.rows ();
      octave_idx_type mc = m.cols ();

      if (u_arg.is_scalar_type ())
        {    /* u is scalar */
          octave_idx_type nr = m.rows ();
          octave_idx_type nc = m.cols ();
          Matrix err (nr, nc);

          if (u_arg.is_real_type ())
            {
              double u = u_arg.double_value ();

              if (error_state)
                {
                  gripe_ellipj_arg ("first");
                  return retval;
                }

              Matrix sn (nr, nc), cn (nr, nc), dn (nr, nc);
              for (octave_idx_type j = 0; j < nc; j++)
                for (octave_idx_type i = 0; i < nr; i++)
                  sncndn (u, m(i,j), sn(i,j), cn(i,j), dn(i,j), err(i,j));

              retval (0) = sn;
              retval (1) = cn;
              retval (2) = dn;
              if (nargout > 3)  retval(3) = err;
            }
          else
            {
              Complex u = u_arg.complex_value ();
              if (error_state)
                {
                  gripe_ellipj_arg ("first");
                  return retval;
                }

              ComplexMatrix sn (nr, nc), cn (nr, nc), dn (nr, nc);
              for (octave_idx_type j = 0; j < nc; j++)
                for (octave_idx_type i = 0; i < nr; i++)
                  sncndn (u, m(i,j), sn(i,j), cn(i,j), dn(i,j), err(i,j));
              retval (0) = sn;
              retval (1) = cn;
              retval (2) = dn;
              if (nargout > 3)  retval(3) = err;
            }
        }
      else
        {    // u is matrix  (m is matrix)
          if (u_arg.is_real_type ())
            {  // u real matrix

              Matrix u = u_arg.matrix_value ();
              if (error_state)
                {
                  gripe_ellipj_arg ("first ");
                  return retval;
                }

              octave_idx_type ur = u.rows ();
              octave_idx_type uc = u.cols ();

              if (mr == 1 && uc == 1)
                {  // u column, m row
                  RowVector rm = m.row (0);
                  ColumnVector cu = u.column (0);

                  Matrix sn (ur, mc), cn (ur, mc), dn (ur, mc);
                  Matrix err (ur,mc);

                  for (octave_idx_type j = 0; j < mc; j++)
                    for (octave_idx_type i = 0; i < ur; i++)
                      sncndn (cu(i), rm(j), sn(i,j), cn(i,j), dn(i,j), err(i,j));

                  retval (0) = sn;
                  retval (1) = cn;
                  retval (2) = dn;
                  if (nargout > 3)  retval(3) = err;
                }
              else if (ur == mr && uc == mc)
                {
                  Matrix sn (ur, mc), cn (ur, mc), dn (ur, mc);
                  Matrix err (ur,mc);

                  for (octave_idx_type j = 0; j < uc; j++)
                    for (octave_idx_type i = 0; i < ur; i++)
                      sncndn (u(i,j), m(i,j), sn(i,j), cn(i,j), dn(i,j), err(i,j));

                  retval (0) = sn;
                  retval (1) = cn;
                  retval (2) = dn;
                  if (nargout > 3)  retval(3) = err;
                }
              else
                error ("u m invalid");
            }
          else
            {  // u complex matrix
              ComplexMatrix u = u_arg.complex_matrix_value ();
              if (error_state)
                {
                  gripe_ellipj_arg ("second");
                  return retval;
                }

              octave_idx_type ur = u.rows ();
              octave_idx_type uc = u.cols ();

              if (mr == 1 && uc == 1)
                {
                  RowVector rm = m.row (0);
                  ComplexColumnVector cu = u.column (0);

                  ComplexMatrix sn (ur, mc), cn (ur, mc), dn (ur, mc);
                  Matrix err (ur,mc);

                  for (octave_idx_type j = 0; j < mc; j++)
                    for (octave_idx_type i = 0; i < ur; i++)
                      sncndn (cu(i), rm(j), sn(i,j), cn(i,j), dn(i,j), err(i,j));

                  retval (0) = sn;
                  retval (1) = cn;
                  retval (2) = dn;
                  if (nargout > 3)  retval(3) = err;
                }
              else if (ur == mr && uc == mc)
                {
                  ComplexMatrix sn (ur, mc), cn (ur, mc), dn (ur, mc);
                  Matrix err (ur,mc);

                  for (octave_idx_type j = 0; j < uc; j++)
                    for (octave_idx_type i = 0; i < ur; i++)
                      sncndn (u(i,j), m(i,j), sn(i,j), cn(i,j), dn(i,j), err(i,j));

                  retval (0) = sn;
                  retval (1) = cn;
                  retval (2) = dn;
                  if (nargout > 3)  retval(3) = err;
                }
              else
                error ("u m invalid");
            }
        }
    }  // m matrix

  return retval;
}

/*
## demos taken from inst/ellipj.m

%!demo
%! N = 150;
%! % m = [1-logspace(0,log(eps),N-1), 1]; ## m near 1
%! % m = [0, logspace(log(eps),0,N-1)];   ## m near 0
%!   m = linspace(0,1,N);                 ## m equally spaced
%! u = linspace(-20,20,N);
%! M = ones(length(u),1) * m;
%! U = u' * ones(1, length(m));
%! [sn, cn, dn] = ellipj(U,M);
%!
%! %% Plotting
%! c = colormap(hot(64));
%! data = {sn,cn,dn};
%! dname = {"sn","cn","dn"};
%! for i=1:3
%!   subplot(1,3,i);
%!   data{i}(data{i} > 1) = 1;
%!   data{i}(data{i} < -1) = -1;
%!   image(m,u,32*data{i}+32);
%!   title(dname{i});
%! end
%! colormap(c);

%!demo
%! N = 200;
%! % m = [1-logspace(0,log(eps),N-1), 1]; ## m near 1
%! % m = [0, logspace(log(eps),0,N-1)];   ## m near 0
%!   m = linspace(0,1,N);                 ## m equally spaced
%! u = linspace(0,20,5);
%! M = ones(length(u),1) * m;
%! U = u' * ones(1, length(m));
%! [sn, cn, dn] = ellipj(U,M);
%!
%! %% Plotting
%! data = {sn,cn,dn};
%! dname = {"sn","cn","dn"};
%! for i=1:3
%!   subplot(1,3,i);
%!   plot(m, data{i});
%!   title(dname{i});
%!   grid on;
%! end
*/

/*
## tests taken from inst/test_sncndn.m

%!test
%! k = (tan(pi/8.))^2; m = k*k;
%! SN = [
%! -1. + I * 0. ,  -0.8392965923 + 0. * I
%! -1. + I * 0.2 ,  -0.8559363407 + 0.108250955 * I
%! -1. + I * 0.4 ,  -0.906529758 + 0.2204040232 * I
%! -1. + I * 0.6 ,  -0.9931306727 + 0.3403783409 * I
%! -1. + I * 0.8 ,  -1.119268095 + 0.4720784944 * I
%! -1. + I * 1. ,  -1.29010951 + 0.6192468708 * I
%! -1. + I * 1.2 ,  -1.512691987 + 0.7850890595 * I
%! -1. + I * 1.4 ,  -1.796200374 + 0.9714821804 * I
%! -1. + I * 1.6 ,  -2.152201882 + 1.177446413 * I
%! -1. + I * 1.8 ,  -2.594547417 + 1.396378892 * I
%! -1. + I * 2. ,  -3.138145339 + 1.611394819 * I
%! -0.8 + I * 0. ,  -0.7158157937 + 0. * I
%! -0.8 + I * 0.2 ,  -0.7301746722 + 0.1394690862 * I
%! -0.8 + I * 0.4 ,  -0.7738940898 + 0.2841710966 * I
%! -0.8 + I * 0.6 ,  -0.8489542135 + 0.4394411376 * I
%! -0.8 + I * 0.8 ,  -0.9588386397 + 0.6107824358 * I
%! -0.8 + I * 1. ,  -1.108848724 + 0.8038415767 * I
%! -0.8 + I * 1.2 ,  -1.306629972 + 1.024193359 * I
%! -0.8 + I * 1.4 ,  -1.563010199 + 1.276740951 * I
%! -0.8 + I * 1.6 ,  -1.893274688 + 1.564345558 * I
%! -0.8 + I * 1.8 ,  -2.318944084 + 1.88491973 * I
%! -0.8 + I * 2. ,  -2.869716809 + 2.225506523 * I
%! -0.6 + I * 0. ,  -0.5638287208 + 0. * I
%! -0.6 + I * 0.2 ,  -0.5752723012 + 0.1654722474 * I
%! -0.6 + I * 0.4 ,  -0.610164314 + 0.3374004736 * I
%! -0.6 + I * 0.6 ,  -0.6702507087 + 0.5224614298 * I
%! -0.6 + I * 0.8 ,  -0.7586657365 + 0.7277663879 * I
%! -0.6 + I * 1. ,  -0.8803349115 + 0.9610513652 * I
%! -0.6 + I * 1.2 ,  -1.042696526 + 1.230800819 * I
%! -0.6 + I * 1.4 ,  -1.256964505 + 1.546195843 * I
%! -0.6 + I * 1.6 ,  -1.540333527 + 1.916612621 * I
%! -0.6 + I * 1.8 ,  -1.919816065 + 2.349972151 * I
%! -0.6 + I * 2. ,  -2.438761841 + 2.848129496 * I
%! -0.4 + I * 0. ,  -0.3891382858 + 0. * I
%! -0.4 + I * 0.2 ,  -0.3971152026 + 0.1850563793 * I
%! -0.4 + I * 0.4 ,  -0.4214662882 + 0.3775700801 * I
%! -0.4 + I * 0.6 ,  -0.4635087491 + 0.5853434119 * I
%! -0.4 + I * 0.8 ,  -0.5256432877 + 0.8168992398 * I
%! -0.4 + I * 1. ,  -0.611733177 + 1.081923504 * I
%! -0.4 + I * 1.2 ,  -0.7278102331 + 1.391822501 * I
%! -0.4 + I * 1.4 ,  -0.8833807998 + 1.760456461 * I
%! -0.4 + I * 1.6 ,  -1.093891878 + 2.205107766 * I
%! -0.4 + I * 1.8 ,  -1.385545188 + 2.747638761 * I
%! -0.4 + I * 2. ,  -1.805081271 + 3.41525351 * I
%! -0.2 + I * 0. ,  -0.1986311721 + 0. * I
%! -0.2 + I * 0.2 ,  -0.2027299916 + 0.1972398665 * I
%! -0.2 + I * 0.4 ,  -0.2152524522 + 0.402598347 * I
%! -0.2 + I * 0.6 ,  -0.2369100139 + 0.6246336356 * I
%! -0.2 + I * 0.8 ,  -0.2690115146 + 0.8728455227 * I
%! -0.2 + I * 1. ,  -0.3136938773 + 1.158323088 * I
%! -0.2 + I * 1.2 ,  -0.3743615191 + 1.494672508 * I
%! -0.2 + I * 1.4 ,  -0.4565255082 + 1.899466033 * I
%! -0.2 + I * 1.6 ,  -0.5694611346 + 2.39667232 * I
%! -0.2 + I * 1.8 ,  -0.7296612675 + 3.020990664 * I
%! -0.2 + I * 2. ,  -0.9685726188 + 3.826022536 * I
%! 0. + I * 0. ,  0. + 0. * I
%! 0. + I * 0.2 ,  0. + 0.201376364 * I
%! 0. + I * 0.4 ,  0. + 0.4111029248 * I
%! 0. + I * 0.6 ,  0. + 0.6380048435 * I
%! 0. + I * 0.8 ,  0. + 0.8919321473 * I
%! 0. + I * 1. ,  0. + 1.184486615 * I
%! 0. + I * 1.2 ,  0. + 1.530096023 * I
%! 0. + I * 1.4 ,  0. + 1.947754612 * I
%! 0. + I * 1.6 ,  0. + 2.464074356 * I
%! 0. + I * 1.8 ,  0. + 3.119049475 * I
%! 0. + I * 2. ,  0. + 3.97786237 * I
%! 0.2 + I * 0. ,  0.1986311721 + 0. * I
%! 0.2 + I * 0.2 ,  0.2027299916 + 0.1972398665 * I
%! 0.2 + I * 0.4 ,  0.2152524522 + 0.402598347 * I
%! 0.2 + I * 0.6 ,  0.2369100139 + 0.6246336356 * I
%! 0.2 + I * 0.8 ,  0.2690115146 + 0.8728455227 * I
%! 0.2 + I * 1. ,  0.3136938773 + 1.158323088 * I
%! 0.2 + I * 1.2 ,  0.3743615191 + 1.494672508 * I
%! 0.2 + I * 1.4 ,  0.4565255082 + 1.899466033 * I
%! 0.2 + I * 1.6 ,  0.5694611346 + 2.39667232 * I
%! 0.2 + I * 1.8 ,  0.7296612675 + 3.020990664 * I
%! 0.2 + I * 2. ,  0.9685726188 + 3.826022536 * I
%! 0.4 + I * 0. ,  0.3891382858 + 0. * I
%! 0.4 + I * 0.2 ,  0.3971152026 + 0.1850563793 * I
%! 0.4 + I * 0.4 ,  0.4214662882 + 0.3775700801 * I
%! 0.4 + I * 0.6 ,  0.4635087491 + 0.5853434119 * I
%! 0.4 + I * 0.8 ,  0.5256432877 + 0.8168992398 * I
%! 0.4 + I * 1. ,  0.611733177 + 1.081923504 * I
%! 0.4 + I * 1.2 ,  0.7278102331 + 1.391822501 * I
%! 0.4 + I * 1.4 ,  0.8833807998 + 1.760456461 * I
%! 0.4 + I * 1.6 ,  1.093891878 + 2.205107766 * I
%! 0.4 + I * 1.8 ,  1.385545188 + 2.747638761 * I
%! 0.4 + I * 2. ,  1.805081271 + 3.41525351 * I
%! 0.6 + I * 0. ,  0.5638287208 + 0. * I
%! 0.6 + I * 0.2 ,  0.5752723012 + 0.1654722474 * I
%! 0.6 + I * 0.4 ,  0.610164314 + 0.3374004736 * I
%! 0.6 + I * 0.6 ,  0.6702507087 + 0.5224614298 * I
%! 0.6 + I * 0.8 ,  0.7586657365 + 0.7277663879 * I
%! 0.6 + I * 1. ,  0.8803349115 + 0.9610513652 * I
%! 0.6 + I * 1.2 ,  1.042696526 + 1.230800819 * I
%! 0.6 + I * 1.4 ,  1.256964505 + 1.546195843 * I
%! 0.6 + I * 1.6 ,  1.540333527 + 1.916612621 * I
%! 0.6 + I * 1.8 ,  1.919816065 + 2.349972151 * I
%! 0.6 + I * 2. ,  2.438761841 + 2.848129496 * I
%! 0.8 + I * 0. ,  0.7158157937 + 0. * I
%! 0.8 + I * 0.2 ,  0.7301746722 + 0.1394690862 * I
%! 0.8 + I * 0.4 ,  0.7738940898 + 0.2841710966 * I
%! 0.8 + I * 0.6 ,  0.8489542135 + 0.4394411376 * I
%! 0.8 + I * 0.8 ,  0.9588386397 + 0.6107824358 * I
%! 0.8 + I * 1. ,  1.108848724 + 0.8038415767 * I
%! 0.8 + I * 1.2 ,  1.306629972 + 1.024193359 * I
%! 0.8 + I * 1.4 ,  1.563010199 + 1.276740951 * I
%! 0.8 + I * 1.6 ,  1.893274688 + 1.564345558 * I
%! 0.8 + I * 1.8 ,  2.318944084 + 1.88491973 * I
%! 0.8 + I * 2. ,  2.869716809 + 2.225506523 * I
%! 1. + I * 0. ,  0.8392965923 + 0. * I
%! 1. + I * 0.2 ,  0.8559363407 + 0.108250955 * I
%! 1. + I * 0.4 ,  0.906529758 + 0.2204040232 * I
%! 1. + I * 0.6 ,  0.9931306727 + 0.3403783409 * I
%! 1. + I * 0.8 ,  1.119268095 + 0.4720784944 * I
%! 1. + I * 1. ,  1.29010951 + 0.6192468708 * I
%! 1. + I * 1.2 ,  1.512691987 + 0.7850890595 * I
%! 1. + I * 1.4 ,  1.796200374 + 0.9714821804 * I
%! 1. + I * 1.6 ,  2.152201882 + 1.177446413 * I
%! 1. + I * 1.8 ,  2.594547417 + 1.396378892 * I
%! 1. + I * 2. ,  3.138145339 + 1.611394819 * I
%! ];
%! CN = [
%! -1. + I * 0. , 0.5436738271 + 0. * I
%! -1. + I * 0.2 , 0.5541219664 + 0.1672121517 * I
%! -1. + I * 0.4 , 0.5857703552 + 0.3410940893 * I
%! -1. + I * 0.6 , 0.6395034233 + 0.5285979063 * I
%! -1. + I * 0.8 , 0.716688504 + 0.7372552987 * I
%! -1. + I * 1. , 0.8189576795 + 0.9755037374 * I
%! -1. + I * 1.2 , 0.9477661951 + 1.253049471 * I
%! -1. + I * 1.4 , 1.103540657 + 1.581252712 * I
%! -1. + I * 1.6 , 1.284098214 + 1.973449038 * I
%! -1. + I * 1.8 , 1.481835651 + 2.4449211 * I
%! -1. + I * 2. , 1.679032464 + 3.011729224 * I
%! -0.8 + I * 0. , 0.6982891589 + 0. * I
%! -0.8 + I * 0.2 , 0.71187169 + 0.1430549855 * I
%! -0.8 + I * 0.4 , 0.7530744458 + 0.2920273465 * I
%! -0.8 + I * 0.6 , 0.8232501212 + 0.4531616768 * I
%! -0.8 + I * 0.8 , 0.9245978896 + 0.6334016187 * I
%! -0.8 + I * 1. , 1.060030206 + 0.8408616109 * I
%! -0.8 + I * 1.2 , 1.232861756 + 1.085475913 * I
%! -0.8 + I * 1.4 , 1.446126965 + 1.379933558 * I
%! -0.8 + I * 1.6 , 1.701139468 + 1.741030588 * I
%! -0.8 + I * 1.8 , 1.994526268 + 2.191509596 * I
%! -0.8 + I * 2. , 2.312257188 + 2.762051518 * I
%! -0.6 + I * 0. , 0.8258917445 + 0. * I
%! -0.6 + I * 0.2 , 0.842151698 + 0.1130337928 * I
%! -0.6 + I * 0.4 , 0.8915487431 + 0.2309124769 * I
%! -0.6 + I * 0.6 , 0.975948103 + 0.3588102098 * I
%! -0.6 + I * 0.8 , 1.098499209 + 0.5026234141 * I
%! -0.6 + I * 1. , 1.263676101 + 0.6695125973 * I
%! -0.6 + I * 1.2 , 1.477275851 + 0.8687285705 * I
%! -0.6 + I * 1.4 , 1.746262523 + 1.112955966 * I
%! -0.6 + I * 1.6 , 2.078179075 + 1.420581466 * I
%! -0.6 + I * 1.8 , 2.479425208 + 1.819580713 * I
%! -0.6 + I * 2. , 2.950586798 + 2.354077344 * I
%! -0.4 + I * 0. , 0.9211793498 + 0. * I
%! -0.4 + I * 0.2 , 0.9395019377 + 0.07822091534 * I
%! -0.4 + I * 0.4 , 0.9952345231 + 0.1598950363 * I
%! -0.4 + I * 0.6 , 1.090715991 + 0.2487465067 * I
%! -0.4 + I * 0.8 , 1.229998843 + 0.34910407 * I
%! -0.4 + I * 1. , 1.419103868 + 0.4663848201 * I
%! -0.4 + I * 1.2 , 1.666426377 + 0.607877235 * I
%! -0.4 + I * 1.4 , 1.983347336 + 0.7841054404 * I
%! -0.4 + I * 1.6 , 2.385101684 + 1.01134031 * I
%! -0.4 + I * 1.8 , 2.89185416 + 1.316448705 * I
%! -0.4 + I * 2. , 3.529393374 + 1.74670531 * I
%! -0.2 + I * 0. , 0.9800743122 + 0. * I
%! -0.2 + I * 0.2 , 0.9997019476 + 0.03999835809 * I
%! -0.2 + I * 0.4 , 1.059453907 + 0.08179712295 * I
%! -0.2 + I * 0.6 , 1.16200643 + 0.1273503824 * I
%! -0.2 + I * 0.8 , 1.312066413 + 0.1789585449 * I
%! -0.2 + I * 1. , 1.516804331 + 0.2395555269 * I
%! -0.2 + I * 1.2 , 1.786613221 + 0.313189147 * I
%! -0.2 + I * 1.4 , 2.136422971 + 0.405890925 * I
%! -0.2 + I * 1.6 , 2.588021972 + 0.527357091 * I
%! -0.2 + I * 1.8 , 3.174302819 + 0.6944201617 * I
%! -0.2 + I * 2. , 3.947361147 + 0.9387994989 * I
%! 0. + I * 0. , 1. + 0. * I
%! 0. + I * 0.2 , 1.020074723 + 0. * I
%! 0. + I * 0.4 , 1.08120563 + 0. * I
%! 0. + I * 0.6 , 1.18619146 + 0. * I
%! 0. + I * 0.8 , 1.339978715 + 0. * I
%! 0. + I * 1. , 1.550164037 + 0. * I
%! 0. + I * 1.2 , 1.827893279 + 0. * I
%! 0. + I * 1.4 , 2.189462954 + 0. * I
%! 0. + I * 1.6 , 2.659259752 + 0. * I
%! 0. + I * 1.8 , 3.275434266 + 0. * I
%! 0. + I * 2. , 4.101632484 + 0. * I
%! 0.2 + I * 0. , 0.9800743122 + 0. * I
%! 0.2 + I * 0.2 , 0.9997019476 - 0.03999835809 * I
%! 0.2 + I * 0.4 , 1.059453907 - 0.08179712295 * I
%! 0.2 + I * 0.6 , 1.16200643 - 0.1273503824 * I
%! 0.2 + I * 0.8 , 1.312066413 - 0.1789585449 * I
%! 0.2 + I * 1. , 1.516804331 - 0.2395555269 * I
%! 0.2 + I * 1.2 , 1.786613221 - 0.313189147 * I
%! 0.2 + I * 1.4 , 2.136422971 - 0.405890925 * I
%! 0.2 + I * 1.6 , 2.588021972 - 0.527357091 * I
%! 0.2 + I * 1.8 , 3.174302819 - 0.6944201617 * I
%! 0.2 + I * 2. , 3.947361147 - 0.9387994989 * I
%! 0.4 + I * 0. , 0.9211793498 + 0. * I
%! 0.4 + I * 0.2 , 0.9395019377 - 0.07822091534 * I
%! 0.4 + I * 0.4 , 0.9952345231 - 0.1598950363 * I
%! 0.4 + I * 0.6 , 1.090715991 - 0.2487465067 * I
%! 0.4 + I * 0.8 , 1.229998843 - 0.34910407 * I
%! 0.4 + I * 1. , 1.419103868 - 0.4663848201 * I
%! 0.4 + I * 1.2 , 1.666426377 - 0.607877235 * I
%! 0.4 + I * 1.4 , 1.983347336 - 0.7841054404 * I
%! 0.4 + I * 1.6 , 2.385101684 - 1.01134031 * I
%! 0.4 + I * 1.8 , 2.89185416 - 1.316448705 * I
%! 0.4 + I * 2. , 3.529393374 - 1.74670531 * I
%! 0.6 + I * 0. , 0.8258917445 + 0. * I
%! 0.6 + I * 0.2 , 0.842151698 - 0.1130337928 * I
%! 0.6 + I * 0.4 , 0.8915487431 - 0.2309124769 * I
%! 0.6 + I * 0.6 , 0.975948103 - 0.3588102098 * I
%! 0.6 + I * 0.8 , 1.098499209 - 0.5026234141 * I
%! 0.6 + I * 1. , 1.263676101 - 0.6695125973 * I
%! 0.6 + I * 1.2 , 1.477275851 - 0.8687285705 * I
%! 0.6 + I * 1.4 , 1.746262523 - 1.112955966 * I
%! 0.6 + I * 1.6 , 2.078179075 - 1.420581466 * I
%! 0.6 + I * 1.8 , 2.479425208 - 1.819580713 * I
%! 0.6 + I * 2. , 2.950586798 - 2.354077344 * I
%! 0.8 + I * 0. , 0.6982891589 + 0. * I
%! 0.8 + I * 0.2 , 0.71187169 - 0.1430549855 * I
%! 0.8 + I * 0.4 , 0.7530744458 - 0.2920273465 * I
%! 0.8 + I * 0.6 , 0.8232501212 - 0.4531616768 * I
%! 0.8 + I * 0.8 , 0.9245978896 - 0.6334016187 * I
%! 0.8 + I * 1. , 1.060030206 - 0.8408616109 * I
%! 0.8 + I * 1.2 , 1.232861756 - 1.085475913 * I
%! 0.8 + I * 1.4 , 1.446126965 - 1.379933558 * I
%! 0.8 + I * 1.6 , 1.701139468 - 1.741030588 * I
%! 0.8 + I * 1.8 , 1.994526268 - 2.191509596 * I
%! 0.8 + I * 2. , 2.312257188 - 2.762051518 * I
%! 1. + I * 0. , 0.5436738271 + 0. * I
%! 1. + I * 0.2 , 0.5541219664 - 0.1672121517 * I
%! 1. + I * 0.4 , 0.5857703552 - 0.3410940893 * I
%! 1. + I * 0.6 , 0.6395034233 - 0.5285979063 * I
%! 1. + I * 0.8 , 0.716688504 - 0.7372552987 * I
%! 1. + I * 1. , 0.8189576795 - 0.9755037374 * I
%! 1. + I * 1.2 , 0.9477661951 - 1.253049471 * I
%! 1. + I * 1.4 , 1.103540657 - 1.581252712 * I
%! 1. + I * 1.6 , 1.284098214 - 1.973449038 * I
%! 1. + I * 1.8 , 1.481835651 - 2.4449211 * I
%! 1. + I * 2. , 1.679032464 - 3.011729224 * I
%! ];
%! DN = [
%! -1. + I * 0. , 0.9895776106 + 0. * I
%! -1. + I * 0.2 , 0.9893361555 + 0.002756935338 * I
%! -1. + I * 0.4 , 0.9885716856 + 0.005949639805 * I
%! -1. + I * 0.6 , 0.9871564855 + 0.01008044183 * I
%! -1. + I * 0.8 , 0.9848512162 + 0.01579337596 * I
%! -1. + I * 1. , 0.9812582484 + 0.02396648455 * I
%! -1. + I * 1.2 , 0.9757399152 + 0.0358288294 * I
%! -1. + I * 1.4 , 0.9672786056 + 0.0531049859 * I
%! -1. + I * 1.6 , 0.954237868 + 0.0781744383 * I
%! -1. + I * 1.8 , 0.933957524 + 0.1141918269 * I
%! -1. + I * 2. , 0.9020917489 + 0.1650142936 * I
%! -0.8 + I * 0. , 0.992429635 + 0. * I
%! -0.8 + I * 0.2 , 0.9924147861 + 0.003020708044 * I
%! -0.8 + I * 0.4 , 0.99236555 + 0.00652359532 * I
%! -0.8 + I * 0.6 , 0.9922655715 + 0.0110676219 * I
%! -0.8 + I * 0.8 , 0.9920785856 + 0.01737733806 * I
%! -0.8 + I * 1. , 0.9917291795 + 0.02645738598 * I
%! -0.8 + I * 1.2 , 0.9910606387 + 0.03974949378 * I
%! -0.8 + I * 1.4 , 0.9897435004 + 0.05935252515 * I
%! -0.8 + I * 1.6 , 0.987077644 + 0.08832675281 * I
%! -0.8 + I * 1.8 , 0.9815667458 + 0.1310872821 * I
%! -0.8 + I * 2. , 0.970020127 + 0.1938136793 * I
%! -0.6 + I * 0. , 0.9953099088 + 0. * I
%! -0.6 + I * 0.2 , 0.995526009 + 0.002814772354 * I
%! -0.6 + I * 0.4 , 0.9962071136 + 0.006083312292 * I
%! -0.6 + I * 0.6 , 0.9974557125 + 0.01033463525 * I
%! -0.6 + I * 0.8 , 0.9994560563 + 0.01626207722 * I
%! -0.6 + I * 1. , 1.00249312 + 0.02484336286 * I
%! -0.6 + I * 1.2 , 1.006973922 + 0.0375167093 * I
%! -0.6 + I * 1.4 , 1.013436509 + 0.05645315628 * I
%! -0.6 + I * 1.6 , 1.022504295 + 0.08499262247 * I
%! -0.6 + I * 1.8 , 1.034670023 + 0.1283564595 * I
%! -0.6 + I * 2. , 1.049599899 + 0.194806122 * I
%! -0.4 + I * 0. , 0.9977686897 + 0. * I
%! -0.4 + I * 0.2 , 0.9981836165 + 0.002167241934 * I
%! -0.4 + I * 0.4 , 0.9994946045 + 0.004686808612 * I
%! -0.4 + I * 0.6 , 1.001910789 + 0.00797144174 * I
%! -0.4 + I * 0.8 , 1.005817375 + 0.01256717724 * I
%! -0.4 + I * 1. , 1.011836374 + 0.01925509038 * I
%! -0.4 + I * 1.2 , 1.020923572 + 0.02920828367 * I
%! -0.4 + I * 1.4 , 1.034513743 + 0.04425213602 * I
%! -0.4 + I * 1.6 , 1.054725746 + 0.06732276244 * I
%! -0.4 + I * 1.8 , 1.08462027 + 0.1033236812 * I
%! -0.4 + I * 2. , 1.128407402 + 0.1608240664 * I
%! -0.2 + I * 0. , 0.9994191176 + 0. * I
%! -0.2 + I * 0.2 , 0.9999683719 + 0.001177128019 * I
%! -0.2 + I * 0.4 , 1.001705496 + 0.00254669712 * I
%! -0.2 + I * 0.6 , 1.004913944 + 0.004334880912 * I
%! -0.2 + I * 0.8 , 1.010120575 + 0.006842775622 * I
%! -0.2 + I * 1. , 1.018189543 + 0.01050520136 * I
%! -0.2 + I * 1.2 , 1.030482479 + 0.01598431001 * I
%! -0.2 + I * 1.4 , 1.049126108 + 0.02433134655 * I
%! -0.2 + I * 1.6 , 1.077466003 + 0.0372877718 * I
%! -0.2 + I * 1.8 , 1.120863308 + 0.05789156398 * I
%! -0.2 + I * 2. , 1.188162088 + 0.09181238708 * I
%! 0. + I * 0. , 1. + 0. * I
%! 0. + I * 0.2 , 1.000596698 + 0. * I
%! 0. + I * 0.4 , 1.002484444 + 0. * I
%! 0. + I * 0.6 , 1.005973379 + 0. * I
%! 0. + I * 0.8 , 1.011641536 + 0. * I
%! 0. + I * 1. , 1.020441432 + 0. * I
%! 0. + I * 1.2 , 1.033885057 + 0. * I
%! 0. + I * 1.4 , 1.054361188 + 0. * I
%! 0. + I * 1.6 , 1.085694733 + 0. * I
%! 0. + I * 1.8 , 1.134186672 + 0. * I
%! 0. + I * 2. , 1.210701071 + 0. * I
%! 0.2 + I * 0. , 0.9994191176 + 0. * I
%! 0.2 + I * 0.2 , 0.9999683719 - 0.001177128019 * I
%! 0.2 + I * 0.4 , 1.001705496 - 0.00254669712 * I
%! 0.2 + I * 0.6 , 1.004913944 - 0.004334880912 * I
%! 0.2 + I * 0.8 , 1.010120575 - 0.006842775622 * I
%! 0.2 + I * 1. , 1.018189543 - 0.01050520136 * I
%! 0.2 + I * 1.2 , 1.030482479 - 0.01598431001 * I
%! 0.2 + I * 1.4 , 1.049126108 - 0.02433134655 * I
%! 0.2 + I * 1.6 , 1.077466003 - 0.0372877718 * I
%! 0.2 + I * 1.8 , 1.120863308 - 0.05789156398 * I
%! 0.2 + I * 2. , 1.188162088 - 0.09181238708 * I
%! 0.4 + I * 0. , 0.9977686897 + 0. * I
%! 0.4 + I * 0.2 , 0.9981836165 - 0.002167241934 * I
%! 0.4 + I * 0.4 , 0.9994946045 - 0.004686808612 * I
%! 0.4 + I * 0.6 , 1.001910789 - 0.00797144174 * I
%! 0.4 + I * 0.8 , 1.005817375 - 0.01256717724 * I
%! 0.4 + I * 1. , 1.011836374 - 0.01925509038 * I
%! 0.4 + I * 1.2 , 1.020923572 - 0.02920828367 * I
%! 0.4 + I * 1.4 , 1.034513743 - 0.04425213602 * I
%! 0.4 + I * 1.6 , 1.054725746 - 0.06732276244 * I
%! 0.4 + I * 1.8 , 1.08462027 - 0.1033236812 * I
%! 0.4 + I * 2. , 1.128407402 - 0.1608240664 * I
%! 0.6 + I * 0. , 0.9953099088 + 0. * I
%! 0.6 + I * 0.2 , 0.995526009 - 0.002814772354 * I
%! 0.6 + I * 0.4 , 0.9962071136 - 0.006083312292 * I
%! 0.6 + I * 0.6 , 0.9974557125 - 0.01033463525 * I
%! 0.6 + I * 0.8 , 0.9994560563 - 0.01626207722 * I
%! 0.6 + I * 1. , 1.00249312 - 0.02484336286 * I
%! 0.6 + I * 1.2 , 1.006973922 - 0.0375167093 * I
%! 0.6 + I * 1.4 , 1.013436509 - 0.05645315628 * I
%! 0.6 + I * 1.6 , 1.022504295 - 0.08499262247 * I
%! 0.6 + I * 1.8 , 1.034670023 - 0.1283564595 * I
%! 0.6 + I * 2. , 1.049599899 - 0.194806122 * I
%! 0.8 + I * 0. , 0.992429635 + 0. * I
%! 0.8 + I * 0.2 , 0.9924147861 - 0.003020708044 * I
%! 0.8 + I * 0.4 , 0.99236555 - 0.00652359532 * I
%! 0.8 + I * 0.6 , 0.9922655715 - 0.0110676219 * I
%! 0.8 + I * 0.8 , 0.9920785856 - 0.01737733806 * I
%! 0.8 + I * 1. , 0.9917291795 - 0.02645738598 * I
%! 0.8 + I * 1.2 , 0.9910606387 - 0.03974949378 * I
%! 0.8 + I * 1.4 , 0.9897435004 - 0.05935252515 * I
%! 0.8 + I * 1.6 , 0.987077644 - 0.08832675281 * I
%! 0.8 + I * 1.8 , 0.9815667458 - 0.1310872821 * I
%! 0.8 + I * 2. , 0.970020127 - 0.1938136793 * I
%! 1. + I * 0. , 0.9895776106 + 0. * I
%! 1. + I * 0.2 , 0.9893361555 - 0.002756935338 * I
%! 1. + I * 0.4 , 0.9885716856 - 0.005949639805 * I
%! 1. + I * 0.6 , 0.9871564855 - 0.01008044183 * I
%! 1. + I * 0.8 , 0.9848512162 - 0.01579337596 * I
%! 1. + I * 1. , 0.9812582484 - 0.02396648455 * I
%! 1. + I * 1.2 , 0.9757399152 - 0.0358288294 * I
%! 1. + I * 1.4 , 0.9672786056 - 0.0531049859 * I
%! 1. + I * 1.6 , 0.954237868 - 0.0781744383 * I
%! 1. + I * 1.8 , 0.933957524 - 0.1141918269 * I
%! 1. + I * 2. , 0.9020917489 - 0.1650142936 * I
%! ];
%! tol = 1e-9;
%! for x = 0:10
%!   for y = 0:10
%!     ur = -1 + x * 0.2;
%!     ui =  y * 0.2;
%!     ii = 1 + y + x*11;
%!     [sn, cn, dn] = ellipj (ur + I * ui, m);
%!     assert (SN (ii, 2), sn, tol);
%!     assert (CN (ii, 2), cn, tol);
%!     assert (DN (ii, 2), dn, tol);
%!   endfor
%! endfor

## tests taken from test_ellipj.m
%!test
%! u1 = pi/3; m1 = 0;
%! res1 = [sin(pi/3), cos(pi/3), 1];
%! [sn,cn,dn]=ellipj(u1,m1);
%! assert([sn,cn,dn], res1, 10*eps);

%!test
%! u2 = log(2); m2 = 1;
%! res2 = [ 3/5, 4/5, 4/5 ];
%! [sn,cn,dn]=ellipj(u2,m2);
%! assert([sn,cn,dn], res2, 10*eps);

%!test
%! u3 = log(2)*1i; m3 = 0;
%! res3 = [3i/4,5/4,1];
%! [sn,cn,dn]=ellipj(u3,m3);
%! assert([sn,cn,dn], res3, 10*eps);

%!test
%! u4 = -1; m4 = tan(pi/8)^4;
%! res4 = [-0.8392965923,0.5436738271,0.9895776106];
%! [sn,cn,dn]=ellipj(u4, m4);
%! assert([sn,cn,dn], res4, 1e-10);

%!test
%! u5 = -0.2 + 0.4i; m5 = tan(pi/8)^4;
%! res5 = [ -0.2152524522 + 0.402598347i, ...
%!           1.059453907  + 0.08179712295i, ...
%!           1.001705496  + 0.00254669712i ];
%! [sn,cn,dn]=ellipj(u5,m5);
%! assert([sn,cn,dn], res5, 1e-9);

%!test
%! u6 = 0.2 + 0.6i; m6 = tan(pi/8)^4;
%! res6 = [ 0.2369100139 + 0.624633635i, ...
%!          1.16200643   - 0.1273503824i, ...
%!          1.004913944 - 0.004334880912i ];
%! [sn,cn,dn]=ellipj(u6,m6);
%! assert([sn,cn,dn], res6, 1e-8);

%!test
%! u7 = 0.8 + 0.8i; m7 = tan(pi/8)^4;
%! res7 = [0.9588386397 + 0.6107824358i, ...
%!         0.9245978896 - 0.6334016187i, ...
%!         0.9920785856 - 0.01737733806i ];
%! [sn,cn,dn]=ellipj(u7,m7);
%! assert([sn,cn,dn], res7, 1e-10);

%!test
%! u=[0,pi/6,pi/4,pi/2]; m=0;
%! res = [0,1/2,1/sqrt(2),1;1,cos(pi/6),1/sqrt(2),0;1,1,1,1];
%! [sn,cn,dn]=ellipj(u,m);
%! assert([sn;cn;dn],res, 100*eps);
%! [sn,cn,dn]=ellipj(u',0);
%! assert([sn,cn,dn],res', 100*eps);

## XXX FIXME XXX
## need to check [real,complex]x[scalar,rowvec,colvec,matrix]x[u,m]

%!test
%! ## Test Jacobi elliptic functions
%! ## against "exact" solution from Mathematica 3.0
%! ## David Billinghurst <David.Billinghurst@riotinto.com>
%! ## 1 February 2001
%! u = [ 0.25; 0.25; 0.20; 0.20; 0.672; 0.5];
%! m = [ 0.0;  1.0;  0.19; 0.81; 0.36;  0.9999999999];
%! S = [ sin(0.25); tanh(0.25);
%!  0.19842311013970879516;
%!  0.19762082367187648571;
%!  0.6095196917919021945;
%!  0.4621171572617320908 ];
%! C = [ cos(0.25); sech(0.25);
%!  0.9801164570409401062;
%!  0.9802785369736752032;
%!  0.7927709286533560550;
%!  0.8868188839691764094 ];
%! D = [ 1.0;  sech(0.25);
%!  0.9962526643271134302;
%!  0.9840560289645665155;
%!  0.9307281387786906491;
%!  0.8868188839812167635 ];
%! [sn,cn,dn] = ellipj(u,m);
%! assert(sn,S,8*eps);
%! assert(cn,C,8*eps);
%! assert(dn,D,8*eps);

%!error ellipj ()
%!error ellipj (1)
%!error ellipj (1,2,3,4)
*/