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/* ellikl.c
*
* Incomplete elliptic integral of the first kind
*
*
*
* SYNOPSIS:
*
* long double phi, m, y, ellikl();
*
* y = ellikl( 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 30000 3.6e-18 4.1e-19
*
*
*/
/*
Cephes Math Library Release 2.3: November, 1995
Copyright 1984, 1987, 1995 by Stephen L. Moshier
*/
/* Incomplete elliptic integral of first kind */
#include <math.h>
#ifdef ANSIPROT
extern long double sqrtl ( long double );
extern long double fabsl ( long double );
extern long double logl ( long double );
extern long double tanl ( long double );
extern long double atanl ( long double );
extern long double floorl ( long double );
extern long double ellpkl ( long double );
long double ellikl ( long double, long double );
#else
long double sqrtl(), fabsl(), logl(), tanl(), atanl(), floorl(), ellpkl();
long double ellikl();
#endif
extern long double PIL, PIO2L, MACHEPL, MAXNUML;
long double ellikl( phi, m )
long double phi, m;
{
long double a, b, c, e, temp, t, K;
int d, mod, sign, npio2;
if( m == 0.0L )
return( phi );
a = 1.0L - m;
if( a == 0.0L )
{
if( fabsl(phi) >= PIO2L )
{
mtherr( "ellikl", SING );
return( MAXNUML );
}
return( logl( tanl( 0.5L*(PIO2L + phi) ) ) );
}
npio2 = floorl( phi/PIO2L );
if( npio2 & 1 )
npio2 += 1;
if( npio2 )
{
K = ellpkl( a );
phi = phi - npio2 * PIO2L;
}
else
K = 0.0L;
if( phi < 0.0L )
{
phi = -phi;
sign = -1;
}
else
sign = 0;
b = sqrtl(a);
t = tanl( phi );
if( fabsl(t) > 10.0L )
{
/* Transform the amplitude */
e = 1.0L/(b*t);
/* ... but avoid multiple recursions. */
if( fabsl(e) < 10.0L )
{
e = atanl(e);
if( npio2 == 0 )
K = ellpkl( a );
temp = K - ellikl( e, m );
goto done;
}
}
a = 1.0L;
c = sqrtl(m);
d = 1;
mod = 0;
while( fabsl(c/a) > MACHEPL )
{
temp = b/a;
phi = phi + atanl(t*temp) + mod * PIL;
mod = (phi + PIO2L)/PIL;
t = t * ( 1.0L + temp )/( 1.0L - temp * t * t );
c = 0.5L * ( a - b );
temp = sqrtl( a * b );
a = 0.5L * ( a + b );
b = temp;
d += d;
}
temp = (atanl(t) + mod * PIL)/(d * a);
done:
if( sign < 0 )
temp = -temp;
temp += npio2 * K;
return( temp );
}
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