354 lines
8.8 KiB
C
354 lines
8.8 KiB
C
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/* @(#)e_jn.c 5.1 93/09/24 */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/*
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* __ieee754_jn(n, x), __ieee754_yn(n, x)
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* floating point Bessel's function of the 1st and 2nd kind
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* of order n
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*
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* Special cases:
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* y0(0)=y1(0)=yn(n,0) = -inf with overflow signal;
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* y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
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* Note 2. About jn(n,x), yn(n,x)
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* For n=0, j0(x) is called,
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* for n=1, j1(x) is called,
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* for n<x, forward recursion us used starting
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* from values of j0(x) and j1(x).
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* for n>x, a continued fraction approximation to
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* j(n,x)/j(n-1,x) is evaluated and then backward
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* recursion is used starting from a supposed value
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* for j(n,x). The resulting value of j(0,x) is
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* compared with the actual value to correct the
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* supposed value of j(n,x).
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*
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* yn(n,x) is similar in all respects, except
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* that forward recursion is used for all
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* values of n>1.
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*
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*/
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#include <errno.h>
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#include <float.h>
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#include <math.h>
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#include <math-narrow-eval.h>
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#include <math_private.h>
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#include <fenv_private.h>
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#include <math-underflow.h>
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#include <libm-alias-finite.h>
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static const double
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invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
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two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
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one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
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static const double zero = 0.00000000000000000000e+00;
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double
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__ieee754_jn (int n, double x)
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{
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int32_t i, hx, ix, lx, sgn;
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double a, b, temp, di, ret;
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double z, w;
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/* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
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* Thus, J(-n,x) = J(n,-x)
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*/
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EXTRACT_WORDS (hx, lx, x);
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ix = 0x7fffffff & hx;
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/* if J(n,NaN) is NaN */
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if (__glibc_unlikely ((ix | ((uint32_t) (lx | -lx)) >> 31) > 0x7ff00000))
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return x + x;
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if (n < 0)
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{
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n = -n;
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x = -x;
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hx ^= 0x80000000;
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}
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if (n == 0)
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return (__ieee754_j0 (x));
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if (n == 1)
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return (__ieee754_j1 (x));
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sgn = (n & 1) & (hx >> 31); /* even n -- 0, odd n -- sign(x) */
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x = fabs (x);
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{
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SET_RESTORE_ROUND (FE_TONEAREST);
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if (__glibc_unlikely ((ix | lx) == 0 || ix >= 0x7ff00000))
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/* if x is 0 or inf */
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return sgn == 1 ? -zero : zero;
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else if ((double) n <= x)
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{
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/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
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if (ix >= 0x52D00000) /* x > 2**302 */
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{ /* (x >> n**2)
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* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Let s=sin(x), c=cos(x),
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* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
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*
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* n sin(xn)*sqt2 cos(xn)*sqt2
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* ----------------------------------
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* 0 s-c c+s
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* 1 -s-c -c+s
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* 2 -s+c -c-s
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* 3 s+c c-s
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*/
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double s;
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double c;
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__sincos (x, &s, &c);
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switch (n & 3)
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{
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case 0: temp = c + s; break;
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case 1: temp = -c + s; break;
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case 2: temp = -c - s; break;
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case 3: temp = c - s; break;
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default: __builtin_unreachable ();
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}
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b = invsqrtpi * temp / sqrt (x);
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}
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else
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{
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a = __ieee754_j0 (x);
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b = __ieee754_j1 (x);
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for (i = 1; i < n; i++)
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{
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temp = b;
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b = b * ((double) (i + i) / x) - a; /* avoid underflow */
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a = temp;
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}
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}
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}
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else
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{
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if (ix < 0x3e100000) /* x < 2**-29 */
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{ /* x is tiny, return the first Taylor expansion of J(n,x)
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* J(n,x) = 1/n!*(x/2)^n - ...
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*/
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if (n > 33) /* underflow */
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b = zero;
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else
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{
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temp = x * 0.5; b = temp;
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for (a = one, i = 2; i <= n; i++)
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{
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a *= (double) i; /* a = n! */
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b *= temp; /* b = (x/2)^n */
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}
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b = b / a;
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}
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}
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else
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{
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/* use backward recurrence */
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/* x x^2 x^2
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* J(n,x)/J(n-1,x) = ---- ------ ------ .....
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* 2n - 2(n+1) - 2(n+2)
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*
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* 1 1 1
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* (for large x) = ---- ------ ------ .....
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* 2n 2(n+1) 2(n+2)
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* -- - ------ - ------ -
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* x x x
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*
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* Let w = 2n/x and h=2/x, then the above quotient
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* is equal to the continued fraction:
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* 1
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* = -----------------------
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* 1
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* w - -----------------
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* 1
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* w+h - ---------
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* w+2h - ...
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*
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* To determine how many terms needed, let
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* Q(0) = w, Q(1) = w(w+h) - 1,
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* Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
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* When Q(k) > 1e4 good for single
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* When Q(k) > 1e9 good for double
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* When Q(k) > 1e17 good for quadruple
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*/
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/* determine k */
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double t, v;
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double q0, q1, h, tmp; int32_t k, m;
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w = (n + n) / (double) x; h = 2.0 / (double) x;
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q0 = w; z = w + h; q1 = w * z - 1.0; k = 1;
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while (q1 < 1.0e9)
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{
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k += 1; z += h;
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tmp = z * q1 - q0;
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q0 = q1;
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q1 = tmp;
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}
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m = n + n;
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for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
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t = one / (i / x - t);
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a = t;
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b = one;
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/* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
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* Hence, if n*(log(2n/x)) > ...
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* single 8.8722839355e+01
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* double 7.09782712893383973096e+02
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* long double 1.1356523406294143949491931077970765006170e+04
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* then recurrent value may overflow and the result is
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* likely underflow to zero
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*/
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tmp = n;
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v = two / x;
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tmp = tmp * __ieee754_log (fabs (v * tmp));
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if (tmp < 7.09782712893383973096e+02)
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{
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for (i = n - 1, di = (double) (i + i); i > 0; i--)
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{
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temp = b;
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b *= di;
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b = b / x - a;
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a = temp;
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di -= two;
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}
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}
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else
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{
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for (i = n - 1, di = (double) (i + i); i > 0; i--)
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{
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temp = b;
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b *= di;
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b = b / x - a;
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a = temp;
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di -= two;
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/* scale b to avoid spurious overflow */
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if (b > 1e100)
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{
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a /= b;
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t /= b;
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b = one;
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}
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}
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}
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/* j0() and j1() suffer enormous loss of precision at and
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* near zero; however, we know that their zero points never
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* coincide, so just choose the one further away from zero.
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*/
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z = __ieee754_j0 (x);
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w = __ieee754_j1 (x);
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if (fabs (z) >= fabs (w))
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b = (t * z / b);
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else
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b = (t * w / a);
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}
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}
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if (sgn == 1)
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ret = -b;
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else
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ret = b;
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ret = math_narrow_eval (ret);
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}
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if (ret == 0)
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{
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ret = math_narrow_eval (copysign (DBL_MIN, ret) * DBL_MIN);
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__set_errno (ERANGE);
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}
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else
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math_check_force_underflow (ret);
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return ret;
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}
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libm_alias_finite (__ieee754_jn, __jn)
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double
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__ieee754_yn (int n, double x)
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{
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int32_t i, hx, ix, lx;
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int32_t sign;
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double a, b, temp, ret;
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EXTRACT_WORDS (hx, lx, x);
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ix = 0x7fffffff & hx;
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/* if Y(n,NaN) is NaN */
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if (__glibc_unlikely ((ix | ((uint32_t) (lx | -lx)) >> 31) > 0x7ff00000))
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return x + x;
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sign = 1;
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if (n < 0)
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{
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n = -n;
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sign = 1 - ((n & 1) << 1);
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}
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if (n == 0)
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return (__ieee754_y0 (x));
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if (__glibc_unlikely ((ix | lx) == 0))
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return -sign / zero;
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/* -inf and overflow exception. */;
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if (__glibc_unlikely (hx < 0))
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return zero / (zero * x);
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{
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SET_RESTORE_ROUND (FE_TONEAREST);
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if (n == 1)
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{
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ret = sign * __ieee754_y1 (x);
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goto out;
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}
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if (__glibc_unlikely (ix == 0x7ff00000))
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return zero;
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if (ix >= 0x52D00000) /* x > 2**302 */
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{ /* (x >> n**2)
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* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Let s=sin(x), c=cos(x),
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* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
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*
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* n sin(xn)*sqt2 cos(xn)*sqt2
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* ----------------------------------
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* 0 s-c c+s
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* 1 -s-c -c+s
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* 2 -s+c -c-s
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* 3 s+c c-s
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*/
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double c;
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double s;
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__sincos (x, &s, &c);
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switch (n & 3)
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{
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case 0: temp = s - c; break;
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case 1: temp = -s - c; break;
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case 2: temp = -s + c; break;
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case 3: temp = s + c; break;
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default: __builtin_unreachable ();
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}
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b = invsqrtpi * temp / sqrt (x);
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}
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else
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{
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uint32_t high;
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a = __ieee754_y0 (x);
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b = __ieee754_y1 (x);
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/* quit if b is -inf */
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GET_HIGH_WORD (high, b);
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for (i = 1; i < n && high != 0xfff00000; i++)
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{
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temp = b;
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b = ((double) (i + i) / x) * b - a;
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GET_HIGH_WORD (high, b);
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a = temp;
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}
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/* If B is +-Inf, set up errno accordingly. */
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if (!isfinite (b))
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__set_errno (ERANGE);
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}
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if (sign > 0)
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ret = b;
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else
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ret = -b;
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}
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out:
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if (isinf (ret))
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ret = copysign (DBL_MAX, ret) * DBL_MAX;
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return ret;
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}
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libm_alias_finite (__ieee754_yn, __yn)
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