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/* k0f.c
*
* Modified Bessel function, third kind, order zero
*
*
*
* SYNOPSIS:
*
* float x, y, k0f();
*
* y = k0f( x );
*
*
*
* DESCRIPTION:
*
* Returns modified Bessel function of the third kind
* of order zero of the argument.
*
* The range is partitioned into the two intervals [0,8] and
* (8, infinity). Chebyshev polynomial expansions are employed
* in each interval.
*
*
*
* ACCURACY:
*
* Tested at 2000 random points between 0 and 8. Peak absolute
* error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15.
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0, 30 30000 7.8e-7 8.5e-8
*
* ERROR MESSAGES:
*
* message condition value returned
* K0 domain x <= 0 MAXNUM
*
*/
/* k0ef()
*
* Modified Bessel function, third kind, order zero,
* exponentially scaled
*
*
*
* SYNOPSIS:
*
* float x, y, k0ef();
*
* y = k0ef( x );
*
*
*
* DESCRIPTION:
*
* Returns exponentially scaled modified Bessel function
* of the third kind of order zero of the argument.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0, 30 30000 8.1e-7 7.8e-8
* See k0().
*
*/
/*
Cephes Math Library Release 2.0: April, 1987
Copyright 1984, 1987 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#include <math.h>
/* Chebyshev coefficients for K0(x) + log(x/2) I0(x)
* in the interval [0,2]. The odd order coefficients are all
* zero; only the even order coefficients are listed.
*
* lim(x->0){ K0(x) + log(x/2) I0(x) } = -EUL.
*/
static float A[] =
{
1.90451637722020886025E-9f,
2.53479107902614945675E-7f,
2.28621210311945178607E-5f,
1.26461541144692592338E-3f,
3.59799365153615016266E-2f,
3.44289899924628486886E-1f,
-5.35327393233902768720E-1f
};
/* Chebyshev coefficients for exp(x) sqrt(x) K0(x)
* in the inverted interval [2,infinity].
*
* lim(x->inf){ exp(x) sqrt(x) K0(x) } = sqrt(pi/2).
*/
static float B[] = {
-1.69753450938905987466E-9f,
8.57403401741422608519E-9f,
-4.66048989768794782956E-8f,
2.76681363944501510342E-7f,
-1.83175552271911948767E-6f,
1.39498137188764993662E-5f,
-1.28495495816278026384E-4f,
1.56988388573005337491E-3f,
-3.14481013119645005427E-2f,
2.44030308206595545468E0f
};
/* k0.c */
extern float MAXNUMF;
#ifdef ANSIC
float chbevlf(float, float *, int);
float expf(float), i0f(float), logf(float), sqrtf(float);
#else
float chbevlf(), expf(), i0f(), logf(), sqrtf();
#endif
float k0f( float xx )
{
float x, y, z;
x = xx;
if( x <= 0.0f )
{
mtherr( "k0f", DOMAIN );
return( MAXNUMF );
}
if( x <= 2.0f )
{
y = x * x - 2.0f;
y = chbevlf( y, A, 7 ) - logf( 0.5f * x ) * i0f(x);
return( y );
}
z = 8.0f/x - 2.0f;
y = expf(-x) * chbevlf( z, B, 10 ) / sqrtf(x);
return(y);
}
float k0ef( float xx )
{
float x, y;
x = xx;
if( x <= 0.0f )
{
mtherr( "k0ef", DOMAIN );
return( MAXNUMF );
}
if( x <= 2.0f )
{
y = x * x - 2.0f;
y = chbevlf( y, A, 7 ) - logf( 0.5f * x ) * i0f(x);
return( y * expf(x) );
}
y = chbevlf( 8.0f/x - 2.0f, B, 10 ) / sqrtf(x);
return(y);
}
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