lib/libm/noieee_src/n_jn.c

   1 /*      $NetBSD: n_jn.c,v 1.7 2011/11/02 02:34:56 christos Exp $        */
   2 /*-
   3  * Copyright (c) 1992, 1993
   4  *      The Regents of the University of California.  All rights reserved.
   5  *
   6  * Redistribution and use in source and binary forms, with or without
   7  * modification, are permitted provided that the following conditions
   8  * are met:
   9  * 1. Redistributions of source code must retain the above copyright
  10  *    notice, this list of conditions and the following disclaimer.
  11  * 2. Redistributions in binary form must reproduce the above copyright
  12  *    notice, this list of conditions and the following disclaimer in the
  13  *    documentation and/or other materials provided with the distribution.
  14  * 3. Neither the name of the University nor the names of its contributors
  15  *    may be used to endorse or promote products derived from this software
  16  *    without specific prior written permission.
  17  *
  18  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
  19  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  20  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  21  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
  22  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  23  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  24  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  25  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  26  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  27  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  28  * SUCH DAMAGE.
  29  */
  30
  31 #ifndef lint
  32 #if 0
  33 static char sccsid[] = "@(#)jn.c        8.2 (Berkeley) 11/30/93";
  34 #endif
  35 #endif /* not lint */
  36
  37 /*
  38  * 16 December 1992
  39  * Minor modifications by Peter McIlroy to adapt non-IEEE architecture.
  40  */
  41
  42 /*
  43  * ====================================================
  44  * Copyright (C) 1992 by Sun Microsystems, Inc.
  45  *
  46  * Developed at SunPro, a Sun Microsystems, Inc. business.
  47  * Permission to use, copy, modify, and distribute this
  48  * software is freely granted, provided that this notice
  49  * is preserved.
  50  * ====================================================
  51  *
  52  * ******************* WARNING ********************
  53  * This is an alpha version of SunPro's FDLIBM (Freely
  54  * Distributable Math Library) for IEEE double precision
  55  * arithmetic. FDLIBM is a basic math library written
  56  * in C that runs on machines that conform to IEEE
  57  * Standard 754/854. This alpha version is distributed
  58  * for testing purpose. Those who use this software
  59  * should report any bugs to
  60  *
  61  *              fdlibm-comments@sunpro.eng.sun.com
  62  *
  63  * -- K.C. Ng, Oct 12, 1992
  64  * ************************************************
  65  */
  66
  67 /*
  68  * jn(int n, double x), yn(int n, double x)
  69  * floating point Bessel's function of the 1st and 2nd kind
  70  * of order n
  71  *
  72  * Special cases:
  73  *      y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
  74  *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
  75  * Note 2. About jn(n,x), yn(n,x)
  76  *      For n=0, j0(x) is called,
  77  *      for n=1, j1(x) is called,
  78  *      for n<x, forward recursion us used starting
  79  *      from values of j0(x) and j1(x).
  80  *      for n>x, a continued fraction approximation to
  81  *      j(n,x)/j(n-1,x) is evaluated and then backward
  82  *      recursion is used starting from a supposed value
  83  *      for j(n,x). The resulting value of j(0,x) is
  84  *      compared with the actual value to correct the
  85  *      supposed value of j(n,x).
  86  *
  87  *      yn(n,x) is similar in all respects, except
  88  *      that forward recursion is used for all
  89  *      values of n>1.
  90  *
  91  */
  92
  93 #include "mathimpl.h"
  94 #include <float.h>
  95 #include <errno.h>
  96
  97 #if defined(__vax__) || defined(tahoe)
  98 #define _IEEE   0
  99 #else
 100 #define _IEEE   1
 101 #define infnan(x) (0.0)
 102 #endif
 103
 104 static const double
 105 invsqrtpi= 5.641895835477562869480794515607725858441e-0001,
 106 two  = 2.0,
 107 zero = 0.0,
 108 one  = 1.0;
 109
 110 double
 111 jn(int n, double x)
 112 {
 113         int i, sgn;
 114         double a, b, temp;
 115         double z, w;
 116
 117     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
 118      * Thus, J(-n,x) = J(n,-x)
 119      */
 120     /* if J(n,NaN) is NaN */
 121 #if _IEEE
 122         if (snan(x)) return x+x;
 123 #endif
 124         if (n<0){
 125                 n = -n;
 126                 x = -x;
 127         }
 128         if (n==0) return(j0(x));
 129         if (n==1) return(j1(x));
 130         sgn = (n&1)&(x < zero);         /* even n -- 0, odd n -- sign(x) */
 131         x = fabs(x);
 132         if (x == 0 || !finite (x))      /* if x is 0 or inf */
 133             b = zero;
 134         else if ((double) n <= x) {
 135                         /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
 136 #if _IEEE
 137             if (x >= 8.148143905337944345e+090) {
 138                                         /* x >= 2**302 */
 139     /* (x >> n**2)
 140      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 141      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 142      *      Let s=sin(x), c=cos(x),
 143      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 144      *
 145      *             n    sin(xn)*sqt2    cos(xn)*sqt2
 146      *          ----------------------------------
 147      *             0     s-c             c+s
 148      *             1    -s-c            -c+s
 149      *             2    -s+c            -c-s
 150      *             3     s+c             c-s
 151      */
 152                 switch(n&3) {
 153                     case 0: temp =  cos(x)+sin(x); break;
 154                     case 1: temp = -cos(x)+sin(x); break;
 155                     case 2: temp = -cos(x)-sin(x); break;
 156                     case 3: temp =  cos(x)-sin(x); break;
 157                 }
 158                 b = invsqrtpi*temp/sqrt(x);
 159             } else
 160 #endif
 161             {
 162                 a = j0(x);
 163                 b = j1(x);
 164                 for(i=1;i<n;i++){
 165                     temp = b;
 166                     b = b*((double)(i+i)/x) - a; /* avoid underflow */
 167                     a = temp;
 168                 }
 169             }
 170         } else {
 171             if (x < 1.86264514923095703125e-009) { /* x < 2**-29 */
 172     /* x is tiny, return the first Taylor expansion of J(n,x)
 173      * J(n,x) = 1/n!*(x/2)^n  - ...
 174      */
 175                 if (n > 33)     /* underflow */
 176                     b = zero;
 177                 else {
 178                     temp = x*0.5; b = temp;
 179                     for (a=one,i=2;i<=n;i++) {
 180                         a *= (double)i;         /* a = n! */
 181                         b *= temp;              /* b = (x/2)^n */
 182                     }
 183                     b = b/a;
 184                 }
 185             } else {
 186                 /* use backward recurrence */
 187                 /*                      x      x^2      x^2
 188                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 189                  *                      2n  - 2(n+1) - 2(n+2)
 190                  *
 191                  *                      1      1        1
 192                  *  (for large x)   =  ----  ------   ------   .....
 193                  *                      2n   2(n+1)   2(n+2)
 194                  *                      -- - ------ - ------ -
 195                  *                       x     x         x
 196                  *
 197                  * Let w = 2n/x and h=2/x, then the above quotient
 198                  * is equal to the continued fraction:
 199                  *                  1
 200                  *      = -----------------------
 201                  *                     1
 202                  *         w - -----------------
 203                  *                        1
 204                  *              w+h - ---------
 205                  *                     w+2h - ...
 206                  *
 207                  * To determine how many terms needed, let
 208                  * Q(0) = w, Q(1) = w(w+h) - 1,
 209                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 210                  * When Q(k) > 1e4      good for single
 211                  * When Q(k) > 1e9      good for double
 212                  * When Q(k) > 1e17     good for quadruple
 213                  */
 214             /* determine k */
 215                 double t,v;
 216                 double q0,q1,h,tmp; int k,m;
 217                 w  = (n+n)/(double)x; h = 2.0/(double)x;
 218                 q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
 219                 while (q1<1.0e9) {
 220                         k += 1; z += h;
 221                         tmp = z*q1 - q0;
 222                         q0 = q1;
 223                         q1 = tmp;
 224                 }
 225                 m = n+n;
 226                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 227                 a = t;
 228                 b = one;
 229                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 230                  *  Hence, if n*(log(2n/x)) > ...
 231                  *  single 8.8722839355e+01
 232                  *  double 7.09782712893383973096e+02
 233                  *  long double 1.1356523406294143949491931077970765006170e+04
 234                  *  then recurrent value may overflow and the result will
 235                  *  likely underflow to zero
 236                  */
 237                 tmp = n;
 238                 v = two/x;
 239                 tmp = tmp*log(fabs(v*tmp));
 240                 for (i=n-1;i>0;i--){
 241                         temp = b;
 242                         b = ((i+i)/x)*b - a;
 243                         a = temp;
 244                     /* scale b to avoid spurious overflow */
 245 #                       if defined(__vax__) || defined(tahoe)
 246 #                               define BMAX 1e13
 247 #                       else
 248 #                               define BMAX 1e100
 249 #                       endif /* defined(__vax__) || defined(tahoe) */
 250                         if (b > BMAX) {
 251                                 a /= b;
 252                                 t /= b;
 253                                 b = one;
 254                         }
 255                 }
 256                 b = (t*j0(x)/b);
 257             }
 258         }
 259         return ((sgn == 1) ? -b : b);
 260 }
 261
 262 double
 263 yn(int n, double x)
 264 {
 265         int i, sign;
 266         double a, b, temp;
 267
 268     /* Y(n,NaN), Y(n, x < 0) is NaN */
 269         if (x <= 0 || (_IEEE && x != x))
 270                 if (_IEEE && x < 0) return zero/zero;
 271                 else if (x < 0)     return (infnan(EDOM));
 272                 else if (_IEEE)     return -one/zero;
 273                 else                return(infnan(-ERANGE));
 274         else if (!finite(x)) return(0);
 275         sign = 1;
 276         if (n<0){
 277                 n = -n;
 278                 sign = 1 - ((n&1)<<2);
 279         }
 280         if (n == 0) return(y0(x));
 281         if (n == 1) return(sign*y1(x));
 282 #if _IEEE
 283         if(x >= 8.148143905337944345e+090) { /* x > 2**302 */
 284     /* (x >> n**2)
 285      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 286      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 287      *      Let s=sin(x), c=cos(x),
 288      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 289      *
 290      *             n    sin(xn)*sqt2    cos(xn)*sqt2
 291      *          ----------------------------------
 292      *             0     s-c             c+s
 293      *             1    -s-c            -c+s
 294      *             2    -s+c            -c-s
 295      *             3     s+c             c-s
 296      */
 297                 switch (n&3) {
 298                     case 0: temp =  sin(x)-cos(x); break;
 299                     case 1: temp = -sin(x)-cos(x); break;
 300                     case 2: temp = -sin(x)+cos(x); break;
 301                     case 3: temp =  sin(x)+cos(x); break;
 302                 }
 303                 b = invsqrtpi*temp/sqrt(x);
 304         } else
 305 #endif
 306         {
 307             a = y0(x);
 308             b = y1(x);
 309         /* quit if b is -inf */
 310             for (i = 1; i < n && !finite(b); i++){
 311                 temp = b;
 312                 b = ((double)(i+i)/x)*b - a;
 313                 a = temp;
 314             }
 315         }
 316         if (!_IEEE && !finite(b))
 317                 return (infnan(-sign * ERANGE));
 318         return ((sign > 0) ? b : -b);
 319 }