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/* elliel.c
*
* Incomplete elliptic integral of the second kind
*
*
*
* SYNOPSIS:
*
* long double phi, m, y, elliel();
*
* y = elliel( phi, m );
*
*
*
* DESCRIPTION:
*
* Approximates the integral
*
*
* phi
* -
* | |
* | 2
* E(phi_\m) = | sqrt( 1 - m sin t ) dt
* |
* | |
* -
* 0
*
* of amplitude phi and modulus m, using the arithmetic -
* geometric mean algorithm.
*
*
*
* ACCURACY:
*
* Tested at random arguments with phi in [-10, 10] and m in
* [0, 1].
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -10,10 50000 2.7e-18 2.3e-19
*
*
*/
/*
Cephes Math Library Release 2.3: November, 1995
Copyright 1984, 1987, 1993, 1995 by Stephen L. Moshier
*/
/* Incomplete elliptic integral of second 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 sinl ( long double );
extern long double tanl ( long double );
extern long double atanl ( long double );
extern long double floorl ( long double );
extern long double ellpel ( long double );
extern long double ellpkl ( long double );
long double elliel ( long double, long double );
#else
long double sqrtl(), fabsl(), logl(), sinl(), tanl(), atanl(), floorl();
long double ellpel(), ellpkl(), elliel();
#endif
extern long double PIL, PIO2L, MACHEPL;
long double elliel( phi, m )
long double phi, m;
{
long double a, b, c, e, temp, lphi, t, E;
int d, mod, npio2, sign;
if( m == 0.0L )
return( phi );
lphi = phi;
npio2 = floorl( lphi/PIO2L );
if( npio2 & 1 )
npio2 += 1;
lphi = lphi - npio2 * PIO2L;
if( lphi < 0.0L )
{
lphi = -lphi;
sign = -1;
}
else
{
sign = 1;
}
a = 1.0L - m;
E = ellpel( a );
if( a == 0.0L )
{
temp = sinl( lphi );
goto done;
}
t = tanl( lphi );
b = sqrtl(a);
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);
temp = E + m * sinl( lphi ) * sinl( e ) - elliel( e, m );
goto done;
}
}
c = sqrtl(m);
a = 1.0L;
d = 1;
e = 0.0L;
mod = 0;
while( fabsl(c/a) > MACHEPL )
{
temp = b/a;
lphi = lphi + atanl(t*temp) + mod * PIL;
mod = (lphi + 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;
e += c * sinl(lphi);
}
temp = E / ellpkl( 1.0L - m );
temp *= (atanl(t) + mod * PIL)/(d * a);
temp += e;
done:
if( sign < 0 )
temp = -temp;
temp += npio2 * E;
return( temp );
}
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