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/* j1f.c
*
* Bessel function of order one
*
*
*
* SYNOPSIS:
*
* float x, y, j1f();
*
* y = j1f( x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of order one of the argument.
*
* The domain is divided into the intervals [0, 2] and
* (2, infinity). In the first interval a polynomial approximation
* 2
* (w - r ) x P(w)
* 1
* 2
* is used, where w = x and r is the first zero 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) - 3pi/4. The function is
*
* j0(x) = Modulus(x) cos( Phase(x) ).
*
*
*
* ACCURACY:
*
* Absolute error:
* arithmetic domain # trials peak rms
* IEEE 0, 2 100000 1.2e-7 2.5e-8
* IEEE 2, 32 100000 2.0e-7 5.3e-8
*
*
*/
/* y1.c
*
* Bessel function of second kind of order one
*
*
*
* SYNOPSIS:
*
* double x, y, y1();
*
* y = y1( x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of the second kind of order one
* 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
* y0(x) = (w - r ) x R(x^2) + 2/pi (ln(x) j1(x) - 1/x) .
* 1
*
* Thus a call to j1() is required.
*
* 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) - 3pi/4. Then the function is
*
* y0(x) = Modulus(x) sin( Phase(x) ).
*
*
*
*
* ACCURACY:
*
* Absolute error:
* arithmetic domain # trials peak rms
* IEEE 0, 2 100000 2.2e-7 4.6e-8
* IEEE 2, 32 100000 1.9e-7 5.3e-8
*
* (error criterion relative when |y1| > 1).
*
*/
/*
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 JP[5] = {
-4.878788132172128E-009f,
6.009061827883699E-007f,
-4.541343896997497E-005f,
1.937383947804541E-003f,
-3.405537384615824E-002f
};
static float YP[5] = {
8.061978323326852E-009f,
-9.496460629917016E-007f,
6.719543806674249E-005f,
-2.641785726447862E-003f,
4.202369946500099E-002f
};
static float MO1[8] = {
6.913942741265801E-002f,
-2.284801500053359E-001f,
3.138238455499697E-001f,
-2.102302420403875E-001f,
5.435364690523026E-003f,
1.493389585089498E-001f,
4.976029650847191E-006f,
7.978845453073848E-001f
};
static float PH1[8] = {
-4.497014141919556E+001f,
5.073465654089319E+001f,
-2.485774108720340E+001f,
7.222973196770240E+000f,
-1.544842782180211E+000f,
3.503787691653334E-001f,
-1.637986776941202E-001f,
3.749989509080821E-001f
};
static float YO1 = 4.66539330185668857532f;
static float Z1 = 1.46819706421238932572E1f;
static float THPIO4F = 2.35619449019234492885f; /* 3*pi/4 */
static float TWOOPI = 0.636619772367581343075535f; /* 2/pi */
extern float PIO4;
float polevlf(float, float *, int);
float logf(float), sinf(float), cosf(float), sqrtf(float);
float j1f( float xx )
{
float x, w, z, p, q, xn;
x = xx;
if( x < 0 )
x = -xx;
if( x <= 2.0f )
{
z = x * x;
p = (z-Z1) * x * polevlf( z, JP, 4 );
return( p );
}
q = 1.0f/x;
w = sqrtf(q);
p = w * polevlf( q, MO1, 7);
w = q*q;
xn = q * polevlf( w, PH1, 7) - THPIO4F;
p = p * cosf(xn + x);
return(p);
}
extern float MAXNUMF;
float y1f( float xx )
{
float x, w, z, p, q, xn;
x = xx;
if( x <= 2.0f )
{
if( x <= 0.0f )
{
mtherr( "y1f", DOMAIN );
return( -MAXNUMF );
}
z = x * x;
w = (z - YO1) * x * polevlf( z, YP, 4 );
w += TWOOPI * ( j1f(x) * logf(x) - 1.0f/x );
return( w );
}
q = 1.0f/x;
w = sqrtf(q);
p = w * polevlf( q, MO1, 7);
w = q*q;
xn = q * polevlf( w, PH1, 7) - THPIO4F;
p = p * sinf(xn + x);
return(p);
}
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