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-rw-r--r--libm/double/ellie.c148
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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 2.8: June, 2000
-Copyright 1984, 1987, 1993, 2000 by Stephen L. Moshier
-*/
-
-/* Incomplete elliptic integral of second kind */
-#include <math.h>
-extern double PI, PIO2, MACHEP;
-#ifdef ANSIPROT
-extern double sqrt ( double );
-extern double fabs ( double );
-extern double log ( double );
-extern double sin ( double x );
-extern double tan ( double x );
-extern double atan ( double );
-extern double floor ( double );
-extern double ellpe ( double );
-extern double ellpk ( double );
-double ellie ( double, double );
-#else
-double sqrt(), fabs(), log(), sin(), tan(), atan(), floor();
-double ellpe(), ellpk(), ellie();
-#endif
-
-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( a );
-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 );
-}