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SHLOMIF / Math-Cephes-0.5307 / libmd / ellik.c 

            

              
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/*                                                      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 md_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
 *
 *
 */

/*
Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*/
/*      Incomplete elliptic integral of first kind      */
#include "mconf.h"
#ifdef ANSIPROT
extern double sqrt ( double );
extern double md_fabs ( double );
extern double md_log ( double );
extern double md_tan ( double );
extern double md_atan ( double );
extern double md_floor ( double );
extern double ellpk ( double );
double ellik ( double, double );
#else
double sqrt(), md_fabs(), md_log(), md_tan(), md_atan(), md_floor(), ellpk();
double ellik();
#endif
extern double PI, PIO2, MACHEP, MAXNUM;
double ellik( phi, m )
double phi, 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( md_fabs(phi) >= PIO2 )
                {
                mtherr( "ellik", SING );
                return( MAXNUM );
                }
        return(  md_log(  md_tan( (PIO2 + phi)/2.0 )  )   );
        }
npio2 = md_floor( phi/PIO2 );
if( npio2 & 1 )
        npio2 += 1;
if( npio2 )
        {
        K = 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 = md_tan( phi );
if( md_fabs(t) > 10.0 )
        {
        /* Transform the amplitude */
        e = 1.0/(b*t);
        /* ... but avoid multiple recursions.  */
        if( md_fabs(e) < 10.0 )
                {
                e = md_atan(e);
                if( npio2 == 0 )
                        K = ellpk( a );
                temp = K - ellik( e, m );
                goto done;
                }
        }
a = 1.0;
c = sqrt(m);
d = 1;
mod = 0;
while( md_fabs(c/a) > MACHEP )
        {
        temp = b/a;
        phi = phi + md_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 = (md_atan(t) + mod * PI)/(d * a);
done:
if( sign < 0 )
        temp = -temp;
temp += npio2 * K;
return( temp );
}