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/* j0f.c
*
* Bessel function of order zero
*
*
*
* SYNOPSIS:
*
* float x, y, j0f();
*
* y = j0f( x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of order zero of the argument.
*
* The domain is divided into the intervals [0, 2] and
* (2, infinity). In the first interval the following polynomial
* approximation is used:
*
*
* 2 2 2
* (w - r ) (w - r ) (w - r ) P(w)
* 1 2 3
*
* 2
* where w = x and the three r's are zeros of the function.
*
* In the second interval, the modulus and phase are approximated
* by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x)
* and Phase(x) = x + 1/x R(1/x^2) - pi/4. The function is
*
* j0(x) = Modulus(x) cos( Phase(x) ).
*
*
*
* ACCURACY:
*
* Absolute error:
* arithmetic domain # trials peak rms
* IEEE 0, 2 100000 1.3e-7 3.6e-8
* IEEE 2, 32 100000 1.9e-7 5.4e-8
*
*/
/* y0f.c
*
* Bessel function of the second kind, order zero
*
*
*
* SYNOPSIS:
*
* float x, y, y0f();
*
* y = y0f( x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of the second kind, of order
* zero, of the argument.
*
* The domain is divided into the intervals [0, 2] and
* (2, infinity). In the first interval a rational approximation
* R(x) is employed to compute
*
* 2 2 2
* y0(x) = (w - r ) (w - r ) (w - r ) R(x) + 2/pi ln(x) j0(x).
* 1 2 3
*
* Thus a call to j0() is required. The three zeros are removed
* from R(x) to improve its numerical stability.
*
* In the second interval, the modulus and phase are approximated
* by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x)
* and Phase(x) = x + 1/x S(1/x^2) - pi/4. Then the function is
*
* y0(x) = Modulus(x) sin( Phase(x) ).
*
*
*
*
* ACCURACY:
*
* Absolute error, when y0(x) < 1; else relative error:
*
* arithmetic domain # trials peak rms
* IEEE 0, 2 100000 2.4e-7 3.4e-8
* IEEE 2, 32 100000 1.8e-7 5.3e-8
*
*/
/*
Cephes Math Library Release 2.2: June, 1992
Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#include <math.h>
static float MO[8] = {
-6.838999669318810E-002f,
1.864949361379502E-001f,
-2.145007480346739E-001f,
1.197549369473540E-001f,
-3.560281861530129E-003f,
-4.969382655296620E-002f,
-3.355424622293709E-006f,
7.978845717621440E-001f
};
static float PH[8] = {
3.242077816988247E+001f,
-3.630592630518434E+001f,
1.756221482109099E+001f,
-4.974978466280903E+000f,
1.001973420681837E+000f,
-1.939906941791308E-001f,
6.490598792654666E-002f,
-1.249992184872738E-001f
};
static float YP[5] = {
9.454583683980369E-008f,
-9.413212653797057E-006f,
5.344486707214273E-004f,
-1.584289289821316E-002f,
1.707584643733568E-001f
};
float YZ1 = 0.43221455686510834878f;
float YZ2 = 22.401876406482861405f;
float YZ3 = 64.130620282338755553f;
static float DR1 = 5.78318596294678452118f;
/*
static float DR2 = 30.4712623436620863991;
static float DR3 = 74.887006790695183444889;
*/
static float JP[5] = {
-6.068350350393235E-008f,
6.388945720783375E-006f,
-3.969646342510940E-004f,
1.332913422519003E-002f,
-1.729150680240724E-001f
};
extern float PIO4F;
float polevlf(float, float *, int);
float logf(float), sinf(float), cosf(float), sqrtf(float);
float j0f( float xx )
{
float x, w, z, p, q, xn;
if( xx < 0 )
x = -xx;
else
x = xx;
if( x <= 2.0f )
{
z = x * x;
if( x < 1.0e-3f )
return( 1.0f - 0.25f*z );
p = (z-DR1) * polevlf( z, JP, 4);
return( p );
}
q = 1.0f/x;
w = sqrtf(q);
p = w * polevlf( q, MO, 7);
w = q*q;
xn = q * polevlf( w, PH, 7) - PIO4F;
p = p * cosf(xn + x);
return(p);
}
/* y0() 2 */
/* Bessel function of second kind, order zero */
/* Rational approximation coefficients YP[] are used for x < 6.5.
* The function computed is y0(x) - 2 ln(x) j0(x) / pi,
* whose value at x = 0 is 2 * ( log(0.5) + EUL ) / pi
* = 0.073804295108687225 , EUL is Euler's constant.
*/
static float TWOOPI = 0.636619772367581343075535f; /* 2/pi */
extern float MAXNUMF;
float y0f( float xx )
{
float x, w, z, p, q, xn;
x = xx;
if( x <= 2.0f )
{
if( x <= 0.0f )
{
mtherr( "y0f", DOMAIN );
return( -MAXNUMF );
}
z = x * x;
/* w = (z-YZ1)*(z-YZ2)*(z-YZ3) * polevlf( z, YP, 4);*/
w = (z-YZ1) * polevlf( z, YP, 4);
w += TWOOPI * logf(x) * j0f(x);
return( w );
}
q = 1.0f/x;
w = sqrtf(q);
p = w * polevlf( q, MO, 7);
w = q*q;
xn = q * polevlf( w, PH, 7) - PIO4F;
p = p * sinf(xn + x);
return( p );
}
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