/*							ellpkf.c
 *
 *	Complete elliptic integral of the first kind
 *
 *
 *
 * SYNOPSIS:
 *
 * float m1, y, ellpkf();
 *
 * y = ellpkf( m1 );
 *
 *
 *
 * DESCRIPTION:
 *
 * Approximates the integral
 *
 *
 *
 *            pi/2
 *             -
 *            | |
 *            |           dt
 * K(m)  =    |    ------------------
 *            |                   2
 *          | |    sqrt( 1 - m sin t )
 *           -
 *            0
 *
 * where m = 1 - m1, using the approximation
 *
 *     P(x)  -  log x Q(x).
 *
 * The argument m1 is used rather than m so that the logarithmic
 * singularity at m = 1 will be shifted to the origin; this
 * preserves maximum accuracy.
 *
 * K(0) = pi/2.
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE       0,1        30000       1.3e-7      3.4e-8
 *
 * ERROR MESSAGES:
 *
 *   message         condition      value returned
 * ellpkf domain      x<0, x>1           0.0
 *
 */

/*							ellpk.c */


/*
Cephes Math Library, Release 2.0:  April, 1987
Copyright 1984, 1987 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/

#include <math.h>

static float P[] =
{
 1.37982864606273237150E-4,
 2.28025724005875567385E-3,
 7.97404013220415179367E-3,
 9.85821379021226008714E-3,
 6.87489687449949877925E-3,
 6.18901033637687613229E-3,
 8.79078273952743772254E-3,
 1.49380448916805252718E-2,
 3.08851465246711995998E-2,
 9.65735902811690126535E-2,
 1.38629436111989062502E0
};

static float Q[] =
{
 2.94078955048598507511E-5,
 9.14184723865917226571E-4,
 5.94058303753167793257E-3,
 1.54850516649762399335E-2,
 2.39089602715924892727E-2,
 3.01204715227604046988E-2,
 3.73774314173823228969E-2,
 4.88280347570998239232E-2,
 7.03124996963957469739E-2,
 1.24999999999870820058E-1,
 4.99999999999999999821E-1
};
static float C1 = 1.3862943611198906188E0; /* log(4) */

extern float MACHEPF, MAXNUMF;

float polevlf(float, float *, int);
float p1evlf(float, float *, int);
float logf(float);
float ellpkf(float xx)
{
float x;

x = xx;
if( (x < 0.0) || (x > 1.0) )
	{
	mtherr( "ellpkf", DOMAIN );
	return( 0.0 );
	}

if( x > MACHEPF )
	{
	return( polevlf(x,P,10) - logf(x) * polevlf(x,Q,10) );
	}
else
	{
	if( x == 0.0 )
		{
		mtherr( "ellpkf", SING );
		return( MAXNUMF );
		}
	else
		{
		return( C1 - 0.5 * logf(x) );
		}
	}
}