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/* ellpj.c
*
* Jacobian Elliptic Functions
*
*
*
* SYNOPSIS:
*
* double u, m, sn, cn, dn, phi;
* int ellpj();
*
* ellpj( 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-9 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
* DEC sn 1800 4.5e-16 8.7e-17
* IEEE phi 10000 9.2e-16* 1.4e-16*
* IEEE sn 50000 4.1e-15 4.6e-16
* IEEE cn 40000 3.6e-15 4.4e-16
* IEEE dn 10000 1.3e-12 1.8e-14
*
* Peak error observed in consistency check using addition
* theorem for sn(u+v) was 4e-16 (absolute). Also tested by
* the above relation to the incomplete elliptic integral.
* Accuracy deteriorates when u is large.
*
*/
/* ellpj.c */
/*
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*/
#include <math.h>
#ifdef ANSIPROT
extern double sqrt ( double );
extern double fabs ( double );
extern double sin ( double );
extern double cos ( double );
extern double asin ( double );
extern double tanh ( double );
extern double sinh ( double );
extern double cosh ( double );
extern double atan ( double );
extern double exp ( double );
#else
double sqrt(), fabs(), sin(), cos(), asin(), tanh();
double sinh(), cosh(), atan(), exp();
#endif
extern double PIO2, MACHEP;
int ellpj( u, m, sn, cn, dn, ph )
double u, m;
double *sn, *cn, *dn, *ph;
{
double ai, b, phi, t, twon;
double a[9], c[9];
int i;
/* Check for special cases */
if( m < 0.0 || m > 1.0 )
{
mtherr( "ellpj", DOMAIN );
*sn = 0.0;
*cn = 0.0;
*ph = 0.0;
*dn = 0.0;
return(-1);
}
if( m < 1.0e-9 )
{
t = sin(u);
b = cos(u);
ai = 0.25 * m * (u - t*b);
*sn = t - ai*b;
*cn = b + ai*t;
*ph = u - ai;
*dn = 1.0 - 0.5*m*t*t;
return(0);
}
if( m >= 0.9999999999 )
{
ai = 0.25 * (1.0-m);
b = cosh(u);
t = tanh(u);
phi = 1.0/b;
twon = b * sinh(u);
*sn = t + ai * (twon - u)/(b*b);
*ph = 2.0*atan(exp(u)) - PIO2 + 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] = 1.0;
b = sqrt(1.0 - m);
c[0] = sqrt(m);
twon = 1.0;
i = 0;
while( fabs(c[i]/a[i]) > MACHEP )
{
if( i > 7 )
{
mtherr( "ellpj", OVERFLOW );
goto done;
}
ai = a[i];
++i;
c[i] = ( ai - b )/2.0;
t = sqrt( ai * b );
a[i] = ( ai + b )/2.0;
b = t;
twon *= 2.0;
}
done:
/* backward recurrence */
phi = twon * a[i] * u;
do
{
t = c[i] * sin(phi) / a[i];
b = phi;
phi = (asin(t) + phi)/2.0;
}
while( --i );
*sn = sin(phi);
t = cos(phi);
*cn = t;
*dn = t/cos(phi-b);
*ph = phi;
return(0);
}
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