V7/usr/src/libm/j1.c

/*
	floating point Bessel's function
	of the first and second kinds
	of order one

	j1(x) returns the value of J1(x)
	for all real values of x.

	There are no error returns.
	Calls sin, cos, sqrt.

	There is a niggling bug in J1 which
	causes errors up to 2e-16 for x in the
	interval [-8,8].
	The bug is caused by an inappropriate order
	of summation of the series.  rhm will fix it
	someday.

	Coefficients are from Hart & Cheney.
	#6050 (20.98D)
	#6750 (19.19D)
	#7150 (19.35D)

	y1(x) returns the value of Y1(x)
	for positive real values of x.
	For x<=0, error number EDOM is set and a
	large negative value is returned.

	Calls sin, cos, sqrt, log, j1.

	The values of Y1 have not been checked
	to more than ten places.

	Coefficients are from Hart & Cheney.
	#6447 (22.18D)
	#6750 (19.19D)
	#7150 (19.35D)
*/

#include <math.h>
#include <errno.h>

int	errno;
static double pzero, qzero;
static double tpi	= .6366197723675813430755350535e0;
static double pio4	= .7853981633974483096156608458e0;
static double p1[] = {
	0.581199354001606143928050809e21,
	-.6672106568924916298020941484e20,
	0.2316433580634002297931815435e19,
	-.3588817569910106050743641413e17,
	0.2908795263834775409737601689e15,
	-.1322983480332126453125473247e13,
	0.3413234182301700539091292655e10,
	-.4695753530642995859767162166e7,
	0.2701122710892323414856790990e4,
};
static double q1[] = {
	0.1162398708003212287858529400e22,
	0.1185770712190320999837113348e20,
	0.6092061398917521746105196863e17,
	0.2081661221307607351240184229e15,
	0.5243710262167649715406728642e12,
	0.1013863514358673989967045588e10,
	0.1501793594998585505921097578e7,
	0.1606931573481487801970916749e4,
	1.0,
};
static double p2[] = {
	-.4435757816794127857114720794e7,
	-.9942246505077641195658377899e7,
	-.6603373248364939109255245434e7,
	-.1523529351181137383255105722e7,
	-.1098240554345934672737413139e6,
	-.1611616644324610116477412898e4,
	0.0,
};
static double q2[] = {
	-.4435757816794127856828016962e7,
	-.9934124389934585658967556309e7,
	-.6585339479723087072826915069e7,
	-.1511809506634160881644546358e7,
	-.1072638599110382011903063867e6,
	-.1455009440190496182453565068e4,
	1.0,
};
static double p3[] = {
	0.3322091340985722351859704442e5,
	0.8514516067533570196555001171e5,
	0.6617883658127083517939992166e5,
	0.1849426287322386679652009819e5,
	0.1706375429020768002061283546e4,
	0.3526513384663603218592175580e2,
	0.0,
};
static double q3[] = {
	0.7087128194102874357377502472e6,
	0.1819458042243997298924553839e7,
	0.1419460669603720892855755253e7,
	0.4002944358226697511708610813e6,
	0.3789022974577220264142952256e5,
	0.8638367769604990967475517183e3,
	1.0,
};
static double p4[] = {
	-.9963753424306922225996744354e23,
	0.2655473831434854326894248968e23,
	-.1212297555414509577913561535e22,
	0.2193107339917797592111427556e20,
	-.1965887462722140658820322248e18,
	0.9569930239921683481121552788e15,
	-.2580681702194450950541426399e13,
	0.3639488548124002058278999428e10,
	-.2108847540133123652824139923e7,
	0.0,
};
static double q4[] = {
	0.5082067366941243245314424152e24,
	0.5435310377188854170800653097e22,
	0.2954987935897148674290758119e20,
	0.1082258259408819552553850180e18,
	0.2976632125647276729292742282e15,
	0.6465340881265275571961681500e12,
	0.1128686837169442121732366891e10,
	0.1563282754899580604737366452e7,
	0.1612361029677000859332072312e4,
	1.0,
};

double
j1(arg) double arg;{
	double xsq, n, d, x;
	double sin(), cos(), sqrt();
	int i;

	x = arg;
	if(x < 0.) x = -x;
	if(x > 8.){
		asympt(x);
		n = x - 3.*pio4;
		n = sqrt(tpi/x)*(pzero*cos(n) - qzero*sin(n));
		if(arg <0.) n = -n;
		return(n);
	}
	xsq = x*x;
	for(n=0,d=0,i=8;i>=0;i--){
		n = n*xsq + p1[i];
		d = d*xsq + q1[i];
	}
	return(arg*n/d);
}

double
y1(arg) double arg;{
	double xsq, n, d, x;
	double sin(), cos(), sqrt(), log(), j1();
	int i;

	errno = 0;
	x = arg;
	if(x <= 0.){
		errno = EDOM;
		return(-HUGE);
	}
	if(x > 8.){
		asympt(x);
		n = x - 3*pio4;
		return(sqrt(tpi/x)*(pzero*sin(n) + qzero*cos(n)));
	}
	xsq = x*x;
	for(n=0,d=0,i=9;i>=0;i--){
		n = n*xsq + p4[i];
		d = d*xsq + q4[i];
	}
	return(x*n/d + tpi*(j1(x)*log(x)-1./x));
}

static
asympt(arg) double arg;{
	double zsq, n, d;
	int i;
	zsq = 64./(arg*arg);
	for(n=0,d=0,i=6;i>=0;i--){
		n = n*zsq + p2[i];
		d = d*zsq + q2[i];
	}
	pzero = n/d;
	for(n=0,d=0,i=6;i>=0;i--){
		n = n*zsq + p3[i];
		d = d*zsq + q3[i];
	}
	qzero = (8./arg)*(n/d);
}