ellpk.c

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/*                                                     ellpk.c
 *
 *     Complete elliptic integral of the first kind
 *
 *
 *
 * SYNOPSIS:
 *
 * double m1, y, ellpk();
 *
 * y = ellpk( 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 internally 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       2.5e-16     6.8e-17
 *
 * ERROR MESSAGES:
 *
 *   message         condition      value returned
 * ellpk 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 "mconf.h"
 
static double 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 double 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 double C1 = 1.3862943611198906188E0;     /* log(4) */
 
extern double MACHEP;
 
double ellpk(double x)
{
 
    if (x < 0.0) {
        sf_error("ellpk", SF_ERROR_DOMAIN, NULL);
        return (NAN);
    }
 
    if (x > 1.0) {
        if (cephes_isinf(x)) {
            return 0.0;
        }
        return ellpk(1/x)/sqrt(x);
    }
 
    if (x > MACHEP) {
        return (polevl(x, P, 10) - log(x) * polevl(x, Q, 10));
    }
    else {
        if (x == 0.0) {
            sf_error("ellpk", SF_ERROR_SINGULAR, NULL);
            return (INFINITY);
        }
        else {
            return (C1 - 0.5 * log(x));
        }
    }
}