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/* ellpe.c
*
* Complete elliptic integral of the second kind
*
*
*
* SYNOPSIS:
*
* double m1, y, ellpe();
*
* y = ellpe( m1 );
*
*
*
* 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
* rather than m for compatibility with ellpk().
*
* E(1) = 1; E(0) = pi/2.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0, 1 13000 3.1e-17 9.4e-18
* 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
*/
#include "mconf.h"
#include "cephes.h"
#ifdef UNK
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
};
#endif
#ifdef DEC
static unsigned short P[] = {
0035041,0001364,0141572,0117555,
0036044,0066032,0130027,0033404,
0036416,0053617,0064456,0102632,
0036457,0161100,0061177,0122612,
0036376,0136251,0012403,0124162,
0036370,0101316,0151715,0131613,
0036475,0105477,0050317,0133272,
0036662,0154232,0024645,0171552,
0037150,0126220,0047054,0030064,
0037742,0162057,0167645,0165612,
0040200,0000000,0000000,0000000
};
static unsigned short Q[] = {
0034411,0106743,0115771,0055462,
0035604,0052575,0155171,0045540,
0036325,0030424,0064332,0167756,
0036612,0052366,0063006,0115175,
0036726,0070430,0004533,0124654,
0037011,0022741,0030675,0030711,
0037056,0174452,0127062,0132122,
0037157,0177750,0142041,0072523,
0037277,0177777,0173137,0002627,
0037577,0177777,0177777,0101101
};
#endif
#ifdef IBMPC
static unsigned short P[] = {
0x53ee,0x986f,0x205e,0x3f24,
0xe6e0,0x5602,0x8d83,0x3f64,
0xd0b3,0xed25,0xcaf1,0x3f81,
0xf4b1,0x0c4f,0xfc48,0x3f85,
0x750e,0x22a0,0xd795,0x3f7f,
0xb671,0xda79,0x1059,0x3f7f,
0xf6d7,0xea19,0xb167,0x3f87,
0xbe6d,0x4534,0x5b13,0x3f96,
0x8607,0x09c5,0x1592,0x3fad,
0xbd71,0xfdf4,0x5c85,0x3fdc,
0x0000,0x0000,0x0000,0x3ff0
};
static unsigned short Q[] = {
0x2b66,0x737f,0x31bc,0x3f01,
0x296c,0xbb4f,0x8aaf,0x3f50,
0x5dfe,0x8d1b,0xa622,0x3f7a,
0xd350,0xccc0,0x4a9e,0x3f91,
0x7535,0x012b,0xce23,0x3f9a,
0xa639,0x2637,0x24bc,0x3fa1,
0x568a,0x55c6,0xdf25,0x3fa5,
0x2eaa,0x1884,0xfffd,0x3fad,
0xe0b3,0xfecb,0xffff,0x3fb7,
0xf048,0xffff,0xffff,0x3fcf
};
#endif
#ifdef MIEEE
static unsigned short P[] = {
0x3f24,0x205e,0x986f,0x53ee,
0x3f64,0x8d83,0x5602,0xe6e0,
0x3f81,0xcaf1,0xed25,0xd0b3,
0x3f85,0xfc48,0x0c4f,0xf4b1,
0x3f7f,0xd795,0x22a0,0x750e,
0x3f7f,0x1059,0xda79,0xb671,
0x3f87,0xb167,0xea19,0xf6d7,
0x3f96,0x5b13,0x4534,0xbe6d,
0x3fad,0x1592,0x09c5,0x8607,
0x3fdc,0x5c85,0xfdf4,0xbd71,
0x3ff0,0x0000,0x0000,0x0000
};
static unsigned short Q[] = {
0x3f01,0x31bc,0x737f,0x2b66,
0x3f50,0x8aaf,0xbb4f,0x296c,
0x3f7a,0xa622,0x8d1b,0x5dfe,
0x3f91,0x4a9e,0xccc0,0xd350,
0x3f9a,0xce23,0x012b,0x7535,
0x3fa1,0x24bc,0x2637,0xa639,
0x3fa5,0xdf25,0x55c6,0x568a,
0x3fad,0xfffd,0x1884,0x2eaa,
0x3fb7,0xffff,0xfecb,0xe0b3,
0x3fcf,0xffff,0xffff,0xf048
};
#endif
double ellpe(x)
double x;
{
if( (x <= 0.0) || (x > 1.0) )
{
if( x == 0.0 )
return( 1.0 );
mtherr( "ellpe", DOMAIN );
return( 0.0 );
}
return( polevl(x,P,10) - log(x) * (x * polevl(x,Q,9)) );
}
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