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/* ellpjl.c
*
* Jacobian Elliptic Functions
*
*
*
* SYNOPSIS:
*
* long double u, m, sn, cn, dn, phi;
* int ellpjl();
*
* ellpjl( u, m, _&sn, _&cn, _&dn, _&phi );
*
*
*
* DESCRIPTION:
*
*
* Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m),
* and dn(u|m) of parameter m between 0 and 1, and real
* argument u.
*
* These functions are periodic, with quarter-period on the
* real axis equal to the complete elliptic integral
* ellpk(1.0-m).
*
* Relation to incomplete elliptic integral:
* If u = ellik(phi,m), then sn(u|m) = sin(phi),
* and cn(u|m) = cos(phi). Phi is called the amplitude of u.
*
* Computation is by means of the arithmetic-geometric mean
* algorithm, except when m is within 1e-12 of 0 or 1. In the
* latter case with m close to 1, the approximation applies
* only for phi < pi/2.
*
* ACCURACY:
*
* Tested at random points with u between 0 and 10, m between
* 0 and 1.
*
* Absolute error (* = relative error):
* arithmetic function # trials peak rms
* IEEE sn 10000 1.7e-18 2.3e-19
* IEEE cn 20000 1.6e-18 2.2e-19
* IEEE dn 10000 4.7e-15 2.7e-17
* IEEE phi 10000 4.0e-19* 6.6e-20*
*
* Accuracy deteriorates when u is large.
*
*/
/*
Cephes Math Library Release 2.3: November, 1995
Copyright 1984, 1987, 1995 by Stephen L. Moshier
*/
#include <math.h>
#ifdef ANSIPROT
extern long double sqrtl ( long double );
extern long double fabsl ( long double );
extern long double sinl ( long double );
extern long double cosl ( long double );
extern long double asinl ( long double );
extern long double tanhl ( long double );
extern long double sinhl ( long double );
extern long double coshl ( long double );
extern long double atanl ( long double );
extern long double expl ( long double );
#else
long double sqrtl(), fabsl(), sinl(), cosl(), asinl(), tanhl();
long double sinhl(), coshl(), atanl(), expl();
#endif
extern long double PIO2L, MACHEPL;
int ellpjl( u, m, sn, cn, dn, ph )
long double u, m;
long double *sn, *cn, *dn, *ph;
{
long double ai, b, phi, t, twon;
long double a[9], c[9];
int i;
/* Check for special cases */
if( m < 0.0L || m > 1.0L )
{
mtherr( "ellpjl", DOMAIN );
*sn = 0.0L;
*cn = 0.0L;
*ph = 0.0L;
*dn = 0.0L;
return(-1);
}
if( m < 1.0e-12L )
{
t = sinl(u);
b = cosl(u);
ai = 0.25L * m * (u - t*b);
*sn = t - ai*b;
*cn = b + ai*t;
*ph = u - ai;
*dn = 1.0L - 0.5L*m*t*t;
return(0);
}
if( m >= 0.999999999999L )
{
ai = 0.25L * (1.0L-m);
b = coshl(u);
t = tanhl(u);
phi = 1.0L/b;
twon = b * sinhl(u);
*sn = t + ai * (twon - u)/(b*b);
*ph = 2.0L*atanl(expl(u)) - PIO2L + ai*(twon - u)/b;
ai *= t * phi;
*cn = phi - ai * (twon - u);
*dn = phi + ai * (twon + u);
return(0);
}
/* A. G. M. scale */
a[0] = 1.0L;
b = sqrtl(1.0L - m);
c[0] = sqrtl(m);
twon = 1.0L;
i = 0;
while( fabsl(c[i]/a[i]) > MACHEPL )
{
if( i > 7 )
{
mtherr( "ellpjl", OVERFLOW );
goto done;
}
ai = a[i];
++i;
c[i] = 0.5L * ( ai - b );
t = sqrtl( ai * b );
a[i] = 0.5L * ( ai + b );
b = t;
twon *= 2.0L;
}
done:
/* backward recurrence */
phi = twon * a[i] * u;
do
{
t = c[i] * sinl(phi) / a[i];
b = phi;
phi = 0.5L * (asinl(t) + phi);
}
while( --i );
*sn = sinl(phi);
t = cosl(phi);
*cn = t;
*dn = t/cosl(phi-b);
*ph = phi;
return(0);
}
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