source: sasview/sansmodels/src/cephes/struve.c @ 476977b

/*                                                      struve.c
 *
 *      Struve function
 *
 *
 *
 * SYNOPSIS:
 *
 * double v, x, y, struve();
 *
 * y = struve( v, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Computes the Struve function Hv(x) of order v, argument x.
 * Negative x is rejected unless v is an integer.
 *
 * This module also contains the hypergeometric functions 1F2
 * and 3F0 and a routine for the Bessel function Yv(x) with
 * noninteger v.
 *
 *
 *
 * ACCURACY:
 *
 * Not accurately characterized, but spot checked against tables.
 *
 */


/*
Cephes Math Library Release 2.81:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*/
#include "mconf.h"
#define DEBUG 0
#ifdef ANSIPROT
extern double gamma ( double );
extern double pow ( double, double );
extern double sqrt ( double );
extern double yn ( int, double );
extern double jv ( double, double );
extern double fabs ( double );
extern double floor ( double );
extern double sin ( double );
extern double cos ( double );
double yv ( double, double );
double onef2 (double, double, double, double, double * );
double threef0 (double, double, double, double, double * );
#else
double gamma(), pow(), sqrt(), yn(), yv(), jv(), fabs(), floor();
double sin(), cos();
double onef2(), threef0();
#endif
static double stop = 1.37e-17;
extern double MACHEP;

double onef2( a, b, c, x, err )
double a, b, c, x;
double *err;
{
double n, a0, sum, t;
double an, bn, cn, max, z;

an = a;
bn = b;
cn = c;
a0 = 1.0;
sum = 1.0;
n = 1.0;
t = 1.0;
max = 0.0;

do
        {
        if( an == 0 )
                goto done;
        if( bn == 0 )
                goto error;
        if( cn == 0 )
                goto error;
        if( (a0 > 1.0e34) || (n > 200) )
                goto error;
        a0 *= (an * x) / (bn * cn * n);
        sum += a0;
        an += 1.0;
        bn += 1.0;
        cn += 1.0;
        n += 1.0;
        z = fabs( a0 );
        if( z > max )
                max = z;
        if( sum != 0 )
                t = fabs( a0 / sum );
        else
                t = z;
        }
while( t > stop );

done:

*err = fabs( MACHEP*max /sum );

#if DEBUG
        printf(" onef2 cancellation error %.5E\n", *err );
#endif

goto xit;

error:
#if DEBUG
printf("onef2 does not converge\n");
#endif
*err = 1.0e38;

xit:

#if DEBUG
printf("onef2( %.2E %.2E %.2E %.5E ) =  %.3E  %.6E\n", a, b, c, x, n, sum);
#endif
return(sum);
}




double threef0( a, b, c, x, err )
double a, b, c, x;
double *err;
{
double n, a0, sum, t, conv, conv1;
double an, bn, cn, max, z;

an = a;
bn = b;
cn = c;
a0 = 1.0;
sum = 1.0;
n = 1.0;
t = 1.0;
max = 0.0;
conv = 1.0e38;
conv1 = conv;

do
        {
        if( an == 0.0 )
                goto done;
        if( bn == 0.0 )
                goto done;
        if( cn == 0.0 )
                goto done;
        if( (a0 > 1.0e34) || (n > 200) )
                goto error;
        a0 *= (an * bn * cn * x) / n;
        an += 1.0;
        bn += 1.0;
        cn += 1.0;
        n += 1.0;
        z = fabs( a0 );
        if( z > max )
                max = z;
        if( z >= conv )
                {
                if( (z < max) && (z > conv1) )
                        goto done;
                }
        conv1 = conv;
        conv = z;
        sum += a0;
        if( sum != 0 )
                t = fabs( a0 / sum );
        else
                t = z;
        }
while( t > stop );

done:

t = fabs( MACHEP*max/sum );
#if DEBUG
        printf(" threef0 cancellation error %.5E\n", t );
#endif

max = fabs( conv/sum );
if( max > t )
        t = max;
#if DEBUG
        printf(" threef0 convergence %.5E\n", max );
#endif

goto xit;

error:
#if DEBUG
printf("threef0 does not converge\n");
#endif
t = 1.0e38;

xit:

#if DEBUG
printf("threef0( %.2E %.2E %.2E %.5E ) =  %.3E  %.6E\n", a, b, c, x, n, sum);
#endif

*err = t;
return(sum);
}




extern double PI;

double struve( v, x )
double v, x;
{
double y, ya, f, g, h, t;
double onef2err, threef0err;

f = floor(v);
if( (v < 0) && ( v-f == 0.5 ) )
        {
        y = jv( -v, x );
        f = 1.0 - f;
        g =  2.0 * floor(f/2.0);
        if( g != f )
                y = -y;
        return(y);
        }
t = 0.25*x*x;
f = fabs(x);
g = 1.5 * fabs(v);
if( (f > 30.0) && (f > g) )
        {
        onef2err = 1.0e38;
        y = 0.0;
        }
else
        {
        y = onef2( 1.0, 1.5, 1.5+v, -t, &onef2err );
        }

if( (f < 18.0) || (x < 0.0) )
        {
        threef0err = 1.0e38;
        ya = 0.0;
        }
else
        {
        ya = threef0( 1.0, 0.5, 0.5-v, -1.0/t, &threef0err );
        }

f = sqrt( PI );
h = pow( 0.5*x, v-1.0 );

if( onef2err <= threef0err )
        {
        g = gamma( v + 1.5 );
        y = y * h * t / ( 0.5 * f * g );
        return(y);
        }
else
        {
        g = gamma( v + 0.5 );
        ya = ya * h / ( f * g );
        ya = ya + yv( v, x );
        return(ya);
        }
}




/* Bessel function of noninteger order
 */

double yv( v, x )
double v, x;
{
double y, t;
int n;

y = floor( v );
if( y == v )
        {
        n = v;
        y = yn( n, x );
        return( y );
        }
t = PI * v;
y = (cos(t) * jv( v, x ) - jv( -v, x ))/sin(t);
return( y );
}

/* Crossover points between ascending series and asymptotic series
 * for Struve function
 *
 *       v       x
 * 
 *       0      19.2
 *       1      18.95
 *       2      19.15
 *       3      19.3
 *       5      19.7
 *      10      21.35
 *      20      26.35
 *      30      32.31
 *      40      40.0
 */