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/* ellik.c
*
* Incomplete elliptic integral of the first kind
*
*
*
* SYNOPSIS:
*
* double phi, m, y, ellik();
*
* y = ellik( phi, m );
*
*
*
* DESCRIPTION:
*
* Approximates the integral
*
*
*
* phi
* -
* | |
* | dt
* F(phi | m) = | ------------------
* | 2
* | | sqrt( 1 - m sin t )
* -
* 0
*
* of amplitude phi and modulus m, using the arithmetic -
* geometric mean algorithm.
*
*
*
*
* ACCURACY:
*
* Tested at random points with m in [0, 1] and phi as indicated.
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -10,10 200000 7.4e-16 1.0e-16
*
*
*/
/*
Cephes Math Library Release 2.0: April, 1987
Copyright 1984, 1987 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
/* Incomplete elliptic integral of first kind */
#include "mconf.h"
extern double PI, PIO2, MACHEP, MAXNUM;
double ellik( phi, m )
double phi, m;
{
double a, b, c, e, temp, t, K;
int d, mod, sign, npio2;
if( m == 0.0 )
return( phi );
a = 1.0 - m;
if( a == 0.0 )
{
if( fabs(phi) >= PIO2 )
{
mtherr( "ellik", SING );
return( MAXNUM );
}
return( log( tan( (PIO2 + phi)/2.0 ) ) );
}
npio2 = floor( phi/PIO2 );
if( npio2 & 1 )
npio2 += 1;
if( npio2 )
{
K = ellpk( a );
phi = phi - npio2 * PIO2;
}
else
K = 0.0;
if( phi < 0.0 )
{
phi = -phi;
sign = -1;
}
else
sign = 0;
b = sqrt(a);
t = tan( phi );
if( fabs(t) > 10.0 )
{
/* Transform the amplitude */
e = 1.0/(b*t);
/* ... but avoid multiple recursions. */
if( fabs(e) < 10.0 )
{
e = atan(e);
if( npio2 == 0 )
K = ellpk( a );
temp = K - ellik( e, m );
goto done;
}
}
a = 1.0;
c = sqrt(m);
d = 1;
mod = 0;
while( fabs(c/a) > MACHEP )
{
temp = b/a;
phi = phi + atan(t*temp) + mod * PI;
mod = (phi + PIO2)/PI;
t = t * ( 1.0 + temp )/( 1.0 - temp * t * t );
c = ( a - b )/2.0;
temp = sqrt( a * b );
a = ( a + b )/2.0;
b = temp;
d += d;
}
temp = (atan(t) + mod * PI)/(d * a);
done:
if( sign < 0 )
temp = -temp;
temp += npio2 * K;
return( temp );
}
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