usr/src/lib/libm/common/C/jn.c

   1 /*
   2  * CDDL HEADER START
   3  *
   4  * The contents of this file are subject to the terms of the
   5  * Common Development and Distribution License (the "License").
   6  * You may not use this file except in compliance with the License.
   7  *
   8  * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
   9  * or http://www.opensolaris.org/os/licensing.
  10  * See the License for the specific language governing permissions
  11  * and limitations under the License.
  12  *
  13  * When distributing Covered Code, include this CDDL HEADER in each
  14  * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
  15  * If applicable, add the following below this CDDL HEADER, with the
  16  * fields enclosed by brackets "[]" replaced with your own identifying
  17  * information: Portions Copyright [yyyy] [name of copyright owner]
  18  *
  19  * CDDL HEADER END
  20  */
  21
  22 /*
  23  * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
  24  */
  25 /*
  26  * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
  27  * Use is subject to license terms.
  28  */
  29
  30 #pragma weak __jn = jn
  31 #pragma weak __yn = yn
  32
  33 /*
  34  * floating point Bessel's function of the 1st and 2nd kind
  35  * of order n: jn(n,x),yn(n,x);
  36  *
  37  * Special cases:
  38  *      y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
  39  *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
  40  * Note 2. About jn(n,x), yn(n,x)
  41  *      For n=0, j0(x) is called,
  42  *      for n=1, j1(x) is called,
  43  *      for n<x, forward recursion us used starting
  44  *      from values of j0(x) and j1(x).
  45  *      for n>x, a continued fraction approximation to
  46  *      j(n,x)/j(n-1,x) is evaluated and then backward
  47  *      recursion is used starting from a supposed value
  48  *      for j(n,x). The resulting value of j(0,x) is
  49  *      compared with the actual value to correct the
  50  *      supposed value of j(n,x).
  51  *
  52  *      yn(n,x) is similar in all respects, except
  53  *      that forward recursion is used for all
  54  *      values of n>1.
  55  *
  56  */
  57
  58 #include "libm.h"
  59 #include <float.h>      /* DBL_MIN */
  60 #include <values.h>     /* X_TLOSS */
  61 #include "xpg6.h"       /* __xpg6 */
  62
  63 #define GENERIC double
  64
  65 static const GENERIC
  66         invsqrtpi = 5.641895835477562869480794515607725858441e-0001,
  67         two     = 2.0,
  68         zero    = 0.0,
  69         one     = 1.0;
  70
  71 GENERIC
  72 jn(int n, GENERIC x) {
  73         int i, sgn;
  74         GENERIC a, b, temp = 0;
  75         GENERIC z, w, ox, on;
  76
  77         /*
  78          * J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  79          * Thus, J(-n,x) = J(n,-x)
  80          */
  81         ox = x; on = (GENERIC)n;
  82         if (n < 0) {
  83                 n = -n;
  84                 x = -x;
  85         }
  86         if (isnan(x))
  87                 return (x*x);   /* + -> * for Cheetah */
  88         if (!((int) _lib_version == libm_ieee ||
  89                 (__xpg6 & _C99SUSv3_math_errexcept) != 0)) {
  90             if (fabs(x) > X_TLOSS)
  91                         return (_SVID_libm_err(on, ox, 38));
  92         }
  93         if (n == 0)
  94                 return (j0(x));
  95         if (n == 1)
  96                 return (j1(x));
  97         if ((n&1) == 0)
  98                 sgn = 0;                        /* even n */
  99         else
 100                 sgn = signbit(x);       /* old n  */
 101         x = fabs(x);
 102         if (x == zero||!finite(x)) b = zero;
 103         else if ((GENERIC)n <= x) {
 104                                         /*
 105                                          * Safe to use
 106                                          *  J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
 107                                          */
 108             if (x > 1.0e91) {
 109                                 /*
 110                                  * x >> n**2
 111                                  *    Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 112                                  *   Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 113                                  *   Let s=sin(x), c=cos(x),
 114                                  *      xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 115                                  *
 116                                  *         n    sin(xn)*sqt2    cos(xn)*sqt2
 117                                  *      ----------------------------------
 118                                  *         0     s-c             c+s
 119                                  *         1    -s-c            -c+s
 120                                  *         2    -s+c            -c-s
 121                                  *         3     s+c             c-s
 122                                  */
 123                 switch (n&3) {
 124                     case 0: temp =  cos(x)+sin(x); break;
 125                     case 1: temp = -cos(x)+sin(x); break;
 126                     case 2: temp = -cos(x)-sin(x); break;
 127                     case 3: temp =  cos(x)-sin(x); break;
 128                 }
 129                 b = invsqrtpi*temp/sqrt(x);
 130             } else {
 131                         a = j0(x);
 132                         b = j1(x);
 133                         for (i = 1; i < n; i++) {
 134                     temp = b;
 135                     b = b*((GENERIC)(i+i)/x) - a; /* avoid underflow */
 136                     a = temp;
 137                         }
 138             }
 139         } else {
 140             if (x < 1e-9) {     /* use J(n,x) = 1/n!*(x/2)^n */
 141                 b = pow(0.5*x, (GENERIC) n);
 142                 if (b != zero) {
 143                     for (a = one, i = 1; i <= n; i++) a *= (GENERIC)i;
 144                     b = b/a;
 145                 }
 146             } else {
 147                 /*
 148                  * use backward recurrence
 149                  *                      x         x^2     x^2
 150                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 151                  *                      2n  - 2(n+1) - 2(n+2)
 152                  *
 153                  *                      1         1         1
 154                  *  (for large x)   =  ----  ------   ------   .....
 155                  *                      2n   2(n+1)   2(n+2)
 156                  *                      -- - ------ - ------ -
 157                  *                       x       x               x
 158                  *
 159                  * Let w = 2n/x and h = 2/x, then the above quotient
 160                  * is equal to the continued fraction:
 161                  *                  1
 162                  *      = -----------------------
 163                  *                         1
 164                  *         w - -----------------
 165                  *                        1
 166                  *                      w+h - ---------
 167                  *                         w+2h - ...
 168                  *
 169                  * To determine how many terms needed, let
 170                  * Q(0) = w, Q(1) = w(w+h) - 1,
 171                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 172                  * When Q(k) > 1e4      good for single
 173                  * When Q(k) > 1e9      good for double
 174                  * When Q(k) > 1e17     good for quaduple
 175                  */
 176             /* determin k */
 177                 GENERIC t, v;
 178                 double q0, q1, h, tmp; int k, m;
 179                 w  = (n+n)/(double)x; h = 2.0/(double)x;
 180                 q0 = w;  z = w + h; q1 = w*z - 1.0; k = 1;
 181                 while (q1 < 1.0e9) {
 182                         k += 1; z += h;
 183                         tmp = z*q1 - q0;
 184                         q0 = q1;
 185                         q1 = tmp;
 186                 }
 187                 m = n+n;
 188                 for (t = zero, i = 2*(n+k); i >= m; i -= 2) t = one/(i/x-t);
 189                 a = t;
 190                 b = one;
 191                 /*
 192                  * estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 193                  *  hence, if n*(log(2n/x)) > ...
 194                  *  single 8.8722839355e+01
 195                  *  double 7.09782712893383973096e+02
 196                  *  long double 1.1356523406294143949491931077970765006170e+04
 197                  *  then recurrent value may overflow and the result is
 198                  *  likely underflow to zero
 199                  */
 200                 tmp = n;
 201                 v = two/x;
 202                 tmp = tmp*log(fabs(v*tmp));
 203                 if (tmp < 7.09782712893383973096e+02) {
 204                             for (i = n-1; i > 0; i--) {
 205                                 temp = b;
 206                                 b = ((i+i)/x)*b - a;
 207                             a = temp;
 208                                 }
 209                 } else {
 210                                 for (i = n-1; i > 0; i--) {
 211                                     temp = b;
 212                                     b = ((i+i)/x)*b - a;
 213                                     a = temp;
 214                                         if (b > 1e100) {
 215                                                 a /= b;
 216                                                 t /= b;
 217                                                 b  = 1.0;
 218                                         }
 219                                 }
 220                 }
 221                         b = (t*j0(x)/b);
 222             }
 223         }
 224         if (sgn == 1)
 225                 return (-b);
 226         else
 227                 return (b);
 228 }
 229
 230 GENERIC
 231 yn(int n, GENERIC x) {
 232         int i;
 233         int sign;
 234         GENERIC a, b, temp = 0, ox, on;
 235
 236         ox = x; on = (GENERIC)n;
 237         if (isnan(x))
 238                 return (x*x);   /* + -> * for Cheetah */
 239         if (x <= zero) {
 240                 if (x == zero) {
 241                         /* return -one/zero; */
 242                         return (_SVID_libm_err((GENERIC)n, x, 12));
 243                 } else {
 244                         /* return zero/zero; */
 245                         return (_SVID_libm_err((GENERIC)n, x, 13));
 246                 }
 247         }
 248         if (!((int) _lib_version == libm_ieee ||
 249                 (__xpg6 & _C99SUSv3_math_errexcept) != 0)) {
 250             if (x > X_TLOSS)
 251                         return (_SVID_libm_err(on, ox, 39));
 252         }
 253         sign = 1;
 254         if (n < 0) {
 255                 n = -n;
 256                 if ((n&1) == 1) sign = -1;
 257         }
 258         if (n == 0)
 259                 return (y0(x));
 260         if (n == 1)
 261                 return (sign*y1(x));
 262         if (!finite(x))
 263                 return (zero);
 264
 265         if (x > 1.0e91) {
 266                                 /*
 267                                  * x >> n**2
 268                                  *  Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 269                                  *  Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 270                                  *  Let s = sin(x), c = cos(x),
 271                                  *  xn = x-(2n+1)*pi/4, sqt2 = sqrt(2), then
 272                                  *
 273                                  *    n sin(xn)*sqt2    cos(xn)*sqt2
 274                                  *      ----------------------------------
 275                                  *       0       s-c             c+s
 276                                  *       1      -s-c            -c+s
 277                                  *       2      -s+c            -c-s
 278                                  *       3       s+c             c-s
 279                                  */
 280                 switch (n&3) {
 281                     case 0: temp =  sin(x)-cos(x); break;
 282                     case 1: temp = -sin(x)-cos(x); break;
 283                     case 2: temp = -sin(x)+cos(x); break;
 284                     case 3: temp =  sin(x)+cos(x); break;
 285                 }
 286                 b = invsqrtpi*temp/sqrt(x);
 287         } else {
 288                 a = y0(x);
 289                 b = y1(x);
 290                 /*
 291                  * fix 1262058 and take care of non-default rounding
 292                  */
 293                 for (i = 1; i < n; i++) {
 294                         temp = b;
 295                         b *= (GENERIC) (i + i) / x;
 296                         if (b <= -DBL_MAX)
 297                                 break;
 298                         b -= a;
 299                         a = temp;
 300                 }
 301         }
 302         if (sign > 0)
 303                 return (b);
 304         else
 305                 return (-b);
 306 }