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/* ivf.c
*
* Modified Bessel function of noninteger order
*
*
*
* SYNOPSIS:
*
* float v, x, y, ivf();
*
* y = ivf( v, x );
*
*
*
* DESCRIPTION:
*
* Returns modified Bessel function of order v of the
* argument. If x is negative, v must be integer valued.
*
* The function is defined as Iv(x) = Jv( ix ). It is
* here computed in terms of the confluent hypergeometric
* function, according to the formula
*
* v -x
* Iv(x) = (x/2) e hyperg( v+0.5, 2v+1, 2x ) / gamma(v+1)
*
* If v is a negative integer, then v is replaced by -v.
*
*
* ACCURACY:
*
* Tested at random points (v, x), with v between 0 and
* 30, x between 0 and 28.
* arithmetic domain # trials peak rms
* Relative error:
* IEEE 0,15 3000 4.7e-6 5.4e-7
* Absolute error (relative when function > 1)
* IEEE 0,30 5000 8.5e-6 1.3e-6
*
* Accuracy is diminished if v is near a negative integer.
* The useful domain for relative error is limited by overflow
* of the single precision exponential function.
*
* See also hyperg.c.
*
*/
/* iv.c */
/* Modified Bessel function of noninteger order */
/* If x < 0, then v must be an integer. */
/*
Cephes Math Library Release 2.2: June, 1992
Copyright 1984, 1987, 1988, 1992 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#include <math.h>
extern float MAXNUMF;
#define fabsf(x) ( (x) < 0 ? -(x) : (x) )
float hypergf(float, float, float);
float expf(float), gammaf(float), logf(float), floorf(float);
float ivf( float v, float x )
{
int sign;
float t, ax;
/* If v is a negative integer, invoke symmetry */
t = floorf(v);
if( v < 0.0 )
{
if( t == v )
{
v = -v; /* symmetry */
t = -t;
}
}
/* If x is negative, require v to be an integer */
sign = 1;
if( x < 0.0 )
{
if( t != v )
{
mtherr( "ivf", DOMAIN );
return( 0.0 );
}
if( v != 2.0 * floorf(v/2.0) )
sign = -1;
}
/* Avoid logarithm singularity */
if( x == 0.0 )
{
if( v == 0.0 )
return( 1.0 );
if( v < 0.0 )
{
mtherr( "ivf", OVERFLOW );
return( MAXNUMF );
}
else
return( 0.0 );
}
ax = fabsf(x);
t = v * logf( 0.5 * ax ) - x;
t = sign * expf(t) / gammaf( v + 1.0 );
ax = v + 0.5;
return( t * hypergf( ax, 2.0 * ax, 2.0 * x ) );
}
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