/* $Id: ellik.c,v 1.1 2003/04/15 03:20:17 cher Exp $ */

/* This file is taken from Cephes math library */

/*
  Cephes Math Library Release 2.8:  June, 2000
  Copyright 1984, 1987, 2000 by Stephen L. Moshier
*/

#include "cephes.h"
#include "mconf.h"
#include <math.h>

/*							ellik.c
 *
 *	Incomplete elliptic integral of the first kind
 *
 *
 *
 * SYNOPSIS:
 *
 * double phi, m, y, ellik();
 *
 * y = ellik( phi, m );
 *
 *
 *
 * DESCRIPTION:
 *
 * Approximates the integral
 *
 *
 *
 *                phi
 *                 -
 *                | |
 *                |           dt
 * F(phi_\m)  =    |    ------------------
 *                |                   2
 *              | |    sqrt( 1 - m sin t )
 *               -
 *                0
 *
 * of amplitude phi and modulus m, using the arithmetic -
 * geometric mean algorithm.
 *
 *
 *
 *
 * ACCURACY:
 *
 * Tested at random points with m in [0, 1] and phi as indicated.
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE     -10,10       200000      7.4e-16     1.0e-16
 *
 *
 */

extern double PI, PIO2, MACHEP, MAXNUM;

double
cephes_ellik(double phi, double m)
{
  double a, b, c, e, temp, t, K;
  int d, mod, sign, npio2;

  if( m == 0.0 )
    return( phi );
  a = 1.0 - m;
  if( a == 0.0 ) {
    if( fabs(phi) >= PIO2 ) {
      cephes_mtherr( "ellik", SING );
      return( MAXNUM );
    }
    return(  log(  tan( (PIO2 + phi)/2.0 )  )   );
  }

  npio2 = floor( phi/PIO2 );
  if( npio2 & 1 )
    npio2 += 1;
  if( npio2 ) {
    K = cephes_ellpk( a );
    phi = phi - npio2 * PIO2;
  } else
    K = 0.0;

  if( phi < 0.0 ) {
    phi = -phi;
    sign = -1;
  } else
    sign = 0;

  b = sqrt(a);
  t = tan( phi );
  if( fabs(t) > 10.0 ) {
    /* Transform the amplitude */
    e = 1.0/(b*t);
    /* ... but avoid multiple recursions.  */
    if( fabs(e) < 10.0 ) {
      e = atan(e);
      if( npio2 == 0 )
        K = cephes_ellpk( a );
      temp = K - cephes_ellik( e, m );
      goto done;
    }
  }
  a = 1.0;
  c = sqrt(m);
  d = 1;
  mod = 0;

  while( fabs(c/a) > MACHEP ) {
    temp = b/a;
    phi = phi + atan(t*temp) + mod * PI;
    mod = (phi + PIO2)/PI;
    t = t * ( 1.0 + temp )/( 1.0 - temp * t * t );
    c = ( a - b )/2.0;
    temp = sqrt( a * b );
    a = ( a + b )/2.0;
    b = temp;
    d += d;
  }

  temp = (atan(t) + mod * PI)/(d * a);

done:
  if( sign < 0 )
    temp = -temp;
  temp += npio2 * K;
  return( temp );
}

/*
 * Local variables:
 *  compile-command: "make -C .."
 * End:
 */