ellpe.c

Go to the documentation of this file.

/*                                                     ellpe.c
 *
 *     Complete elliptic integral of the second kind
 *
 *
 *
 * SYNOPSIS:
 *
 * double m, y, ellpe();
 *
 * y = ellpe( m );
 *
 *
 *
 * DESCRIPTION:
 *
 * Approximates the integral
 *
 *
 *            pi/2
 *             -
 *            | |                 2
 * E(m)  =    |    sqrt( 1 - m sin t ) dt
 *          | |
 *           -
 *            0
 *
 * Where m = 1 - m1, using the approximation
 *
 *      P(x)  -  x log x Q(x).
 *
 * Though there are no singularities, the argument m1 is used
 * internally rather than m for compatibility with ellpk().
 *
 * E(1) = 1; E(0) = pi/2.
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE       0, 1       10000       2.1e-16     7.3e-17
 *
 *
 * ERROR MESSAGES:
 *
 *   message         condition      value returned
 * ellpe domain      x<0, x>1            0.0
 *
 */
 
/*                                                     ellpe.c         */
 
/* Elliptic integral of second kind */
 
/*
 * Cephes Math Library, Release 2.1:  February, 1989
 * Copyright 1984, 1987, 1989 by Stephen L. Moshier
 * Direct inquiries to 30 Frost Street, Cambridge, MA 02140
 *
 * Feb, 2002:  altered by Travis Oliphant
 * so that it is called with argument m
 * (which gets immediately converted to m1 = 1-m)
 */
 
#include "mconf.h"
 
static double P[] = {
    1.53552577301013293365E-4,
    2.50888492163602060990E-3,
    8.68786816565889628429E-3,
    1.07350949056076193403E-2,
    7.77395492516787092951E-3,
    7.58395289413514708519E-3,
    1.15688436810574127319E-2,
    2.18317996015557253103E-2,
    5.68051945617860553470E-2,
    4.43147180560990850618E-1,
    1.00000000000000000299E0
};
 
static double Q[] = {
    3.27954898576485872656E-5,
    1.00962792679356715133E-3,
    6.50609489976927491433E-3,
    1.68862163993311317300E-2,
    2.61769742454493659583E-2,
    3.34833904888224918614E-2,
    4.27180926518931511717E-2,
    5.85936634471101055642E-2,
    9.37499997197644278445E-2,
    2.49999999999888314361E-1
};
 
double ellpe(double x)
{
    x = 1.0 - x;
    if (x <= 0.0) {
        if (x == 0.0)
            return (1.0);
        sf_error("ellpe", SF_ERROR_DOMAIN, NULL);
        return (NAN);
    }
    if (x > 1.0) {
        return ellpe(1.0 - 1/x) * sqrt(x);
    }
    return (polevl(x, P, 10) - log(x) * (x * polevl(x, Q, 9)));
}