e_jn.c

/* @(#)e_jn.c 1.4 95/01/18 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunSoft, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

/*
 * __ieee754_jn(n, x)
 * floating point Bessel's function of the 1st and 2nd kind
 * of order n
 *
 * Special cases:
 *	y0(0)=y1(0)=yn(n,0) = -inf with overflow signal;
 *	y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
 * Note 2. About jn(n,x), yn(n,x)
 *	For n=0, j0(x) is called,
 *	for n=1, j1(x) is called,
 *	for n<x, forward recursion us used starting
 *	from values of j0(x) and j1(x).
 *	for n>x, a continued fraction approximation to
 *	j(n,x)/j(n-1,x) is evaluated and then backward
 *	recursion is used starting from a supposed value
 *	for j(n,x). The resulting value of j(0,x) is
 *	compared with the actual value to correct the
 *	supposed value of j(n,x).
 *
 *	yn(n,x) is similar in all respects, except
 *	that forward recursion is used for all
 *	values of n>1.
 */

#ifndef __FDLIBM_H__
#include "fdlibm.h"
#endif

double __ieee754_jn(int n, double x)
{
	int32_t i, hx, ix, lx;
	int sgn;
	double a, b, temp, di;
	double z, w;

	static const double invsqrtpi = 5.64189583547756279280e-01;	/* 0x3FE20DD7, 0x50429B6D */
	static const double two = 2.00000000000000000000e+00;	/* 0x40000000, 0x00000000 */
	static const double one = 1.00000000000000000000e+00;	/* 0x3FF00000, 0x00000000 */
	static const double zero = 0.00000000000000000000e+00;

	/* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
	 * Thus, J(-n,x) = J(n,-x)
	 */
	GET_DOUBLE_WORDS(hx, lx, x);
	ix = IC(0x7fffffff) & hx;
	/* if J(n,NaN) is NaN */
	if ((ix | (((uint32_t) (lx | -lx)) >> 31)) > IC(0x7ff00000))
		return x + x;
	if (n < 0)
	{
		n = -n;
		x = -x;
		hx ^= IC(0x80000000);
	}
	if (n == 0)
		return (__ieee754_j0(x));
	if (n == 1)
		return (__ieee754_j1(x));
	sgn = (int)((n & 1) & (hx >> 31));			/* even n -- 0, odd n -- sign(x) */
	x = __ieee754_fabs(x);
	if ((ix | lx) == 0 || ix >= IC(0x7ff00000))	/* if x is 0 or inf */
		b = zero;
	else if ((double) n <= x)
	{
		/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
		if (ix >= IC(0x52D00000))
		{								/* x > 2**302 */
			/* (x >> n**2)
			 *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
			 *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
			 *      Let s=sin(x), c=cos(x),
			 *      xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
			 *
			 *         n    sin(xn)*sqt2    cos(xn)*sqt2
			 *      ----------------------------------
			 *         0     s-c         c+s
			 *         1    -s-c        -c+s
			 *         2    -s+c        -c-s
			 *         3     s+c         c-s
			 */
			double s;
			double c;

			__ieee754_sincos(x, &s, &c);
			switch (n & 3)
			{
			case 0:
				temp = c + s;
				break;
			case 1:
				temp = -c + s;
				break;
			case 2:
				temp = -c - s;
				break;
			case 3:
				temp = c - s;
				break;
			}
			b = invsqrtpi * temp / __ieee754_sqrt(x);
		} else
		{
			a = __ieee754_j0(x);
			b = __ieee754_j1(x);
			for (i = 1; i < n; i++)
			{
				temp = b;
				b = b * ((double) (i + i) / x) - a;	/* avoid underflow */
				a = temp;
			}
		}
	} else
	{
		if (ix < IC(0x3e100000))
		{								/* x < 2**-29 */
			/* x is tiny, return the first Taylor expansion of J(n,x)
			 * J(n,x) = 1/n!*(x/2)^n  - ...
			 */
			if (n > 33)					/* underflow */
				b = zero;
			else
			{
				temp = x * 0.5;
				b = temp;
				for (a = one, i = 2; i <= n; i++)
				{
					a *= (double) i;	/* a = n! */
					b *= temp;			/* b = (x/2)^n */
				}
				b = b / a;
			}
		} else
		{
			/* use backward recurrence */
			/*          x      x^2      x^2
			 *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
			 *          2n  - 2(n+1) - 2(n+2)
			 *
			 *          1      1        1
			 *  (for large x)   =  ----  ------   ------   .....
			 *          2n   2(n+1)   2(n+2)
			 *          -- - ------ - ------ -
			 *           x     x         x
			 *
			 * Let w = 2n/x and h=2/x, then the above quotient
			 * is equal to the continued fraction:
			 *          1
			 *  = -----------------------
			 *             1
			 *     w - -----------------
			 *            1
			 *          w+h - ---------
			 *             w+2h - ...
			 *
			 * To determine how many terms needed, let
			 * Q(0) = w, Q(1) = w(w+h) - 1,
			 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
			 * When Q(k) > 1e4  good for single
			 * When Q(k) > 1e9  good for double
			 * When Q(k) > 1e17 good for quadruple
			 */
			/* determine k */
			double t, v;
			double q0, q1, h, tmp;
			int32_t k, m;

			w = (n + n) / (double) x;
			h = 2.0 / (double) x;
			q0 = w;
			z = w + h;
			q1 = w * z - 1.0;
			k = 1;
			while (q1 < 1.0e9)
			{
				k += 1;
				z += h;
				tmp = z * q1 - q0;
				q0 = q1;
				q1 = tmp;
			}
			m = n + n;
			for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
				t = one / (i / x - t);
			a = t;
			b = one;
			/*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
			 *  Hence, if n*(log(2n/x)) > ...
			 *  single 8.8722839355e+01
			 *  double 7.09782712893383973096e+02
			 *  long double 1.1356523406294143949491931077970765006170e+04
			 *  then recurrent value may overflow and the result is
			 *  likely underflow to zero
			 */
			tmp = n;
			v = two / x;
			tmp = tmp * __ieee754_log(__ieee754_fabs(v * tmp));
			if (tmp < 7.09782712893383973096e+02)
			{
				for (i = n - 1, di = (double) (i + i); i > 0; i--)
				{
					temp = b;
					b *= di;
					b = b / x - a;
					a = temp;
					di -= two;
				}
			} else
			{
				for (i = n - 1, di = (double) (i + i); i > 0; i--)
				{
					temp = b;
					b *= di;
					b = b / x - a;
					a = temp;
					di -= two;
					/* scale b to avoid spurious overflow */
					if (b > 1e100)
					{
						a /= b;
						t /= b;
						b = one;
					}
				}
			}
			/* j0() and j1() suffer enormous loss of precision at and
			 * near zero; however, we know that their zero points never
			 * coincide, so just choose the one further away from zero.
			 */
			z = __ieee754_j0(x);
			w = __ieee754_j1(x);
			if (__ieee754_fabs(z) >= __ieee754_fabs(w))
				b = (t * z / b);
			else
				b = (t * w / a);
		}
	}
	if (sgn)
		return -b;
	return b;
}

/* wrapper jn */
double __jn(int n, double x)
{
	if (isgreater(__ieee754_fabs(x), X_TLOSS) && _LIB_VERSION != _IEEE_ && _LIB_VERSION != _POSIX_)
		/* jn(n,|x|>X_TLOSS) */
		return __kernel_standard(n, x, 0.0, KMATHERR_JN_TLOSS);

	return __ieee754_jn(n, x);
}

__typeof(__jn) jn __attribute__((weak, alias("__jn")));
#ifdef __NO_LONG_DOUBLE_MATH
__typeof(__jnl) __jnl __attribute__((alias("__jn")));
__typeof(__jnl) jnl __attribute__((weak, alias("__jnl")));
#endif