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/* ellpjf.c
*
* Jacobian Elliptic Functions
*
*
*
* SYNOPSIS:
*
* float 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
* IEEE sn 10000 1.7e-6 2.2e-7
* IEEE cn 10000 1.6e-6 2.2e-7
* IEEE dn 10000 1.4e-3 1.9e-5
* IEEE phi 10000 3.9e-7* 6.7e-8*
*
* 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 2.2: July, 1992
Copyright 1984, 1987, 1992 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#include <math.h>
extern float PIO2F, MACHEPF;
#define fabsf(x) ( (x) < 0 ? -(x) : (x) )
#ifdef ANSIC
float sqrtf(float), sinf(float), cosf(float), asinf(float), tanhf(float);
float sinhf(float), coshf(float), atanf(float), expf(float);
#else
float sqrtf(), sinf(), cosf(), asinf(), tanhf();
float sinhf(), coshf(), atanf(), expf();
#endif
int ellpjf( float uu, float mm,
float *sn, float *cn, float *dn, float *ph )
{
float u, m, ai, b, phi, t, twon;
float a[10], c[10];
int i;
u = uu;
m = mm;
/* Check for special cases */
if( m < 0.0 || m > 1.0 )
{
mtherr( "ellpjf", DOMAIN );
return(-1);
}
if( m < 1.0e-5 )
{
t = sinf(u);
b = cosf(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 >= 0.99999 )
{
ai = 0.25 * (1.0-m);
b = coshf(u);
t = tanhf(u);
phi = 1.0/b;
twon = b * sinhf(u);
*sn = t + ai * (twon - u)/(b*b);
*ph = 2.0*atanf(expf(u)) - PIO2F + 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 = sqrtf(1.0 - m);
c[0] = sqrtf(m);
twon = 1.0;
i = 0;
while( fabsf( (c[i]/a[i]) ) > MACHEPF )
{
if( i > 8 )
{
/* mtherr( "ellpjf", OVERFLOW );*/
break;
}
ai = a[i];
++i;
c[i] = 0.5 * ( ai - b );
t = sqrtf( ai * b );
a[i] = 0.5 * ( ai + b );
b = t;
twon += twon;
}
/* backward recurrence */
phi = twon * a[i] * u;
do
{
t = c[i] * sinf(phi) / a[i];
b = phi;
phi = 0.5 * (asinf(t) + phi);
}
while( --i );
*sn = sinf(phi);
t = cosf(phi);
*cn = t;
*dn = t/cosf(phi-b);
*ph = phi;
return(0);
}
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