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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(x)
double x;
{
if (x < 0.0) {
mtherr("ellpk", DOMAIN);
return (NPY_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) {
mtherr("ellpk", SING);
return (NPY_INFINITY);
}
else {
return (C1 - 0.5 * log(x));
}
}
}
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