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/* ellie.c
*
* Incomplete elliptic integral of the second kind
*
*
*
* SYNOPSIS:
*
* double phi, m, y, ellie();
*
* y = ellie( 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
* DEC 0,2 2000 1.9e-16 3.4e-17
* IEEE -10,10 150000 3.3e-15 1.4e-16
*
*
*/
/*
Cephes Math Library Release 2.0: April, 1987
Copyright 1984, 1987, 1993 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
/* Incomplete elliptic integral of second kind */
#include "mconf.h"
extern double PI, PIO2, MACHEP;
double ellie( phi, m )
double phi, m;
{
double a, b, c, e, temp;
double lphi, t, E;
int d, mod, npio2, sign;
if( m == 0.0 )
return( phi );
lphi = phi;
npio2 = floor( lphi/PIO2 );
if( npio2 & 1 )
npio2 += 1;
lphi = lphi - npio2 * PIO2;
if( lphi < 0.0 )
{
lphi = -lphi;
sign = -1;
}
else
{
sign = 1;
}
a = 1.0 - m;
E = ellpe( m );
if( a == 0.0 )
{
temp = sin( lphi );
goto done;
}
t = tan( lphi );
b = sqrt(a);
/* Thanks to Brian Fitzgerald <fitzgb@mml0.meche.rpi.edu>
for pointing out an instability near odd multiples of pi/2. */
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);
temp = E + m * sin( lphi ) * sin( e ) - ellie( e, m );
goto done;
}
}
c = sqrt(m);
a = 1.0;
d = 1;
e = 0.0;
mod = 0;
while( fabs(c/a) > MACHEP )
{
temp = b/a;
lphi = lphi + atan(t*temp) + mod * PI;
mod = (lphi + 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;
e += c * sin(lphi);
}
temp = E / ellpk( 1.0 - m );
temp *= (atan(t) + mod * PI)/(d * a);
temp += e;
done:
if( sign < 0 )
temp = -temp;
temp += npio2 * E;
return( temp );
}
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