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 {
  74         int i, sgn;
  75         GENERIC a, b, temp = 0;
  76         GENERIC z, w, ox, on;
  77 
  78         /*
  79          * J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  80          * Thus, J(-n,x) = J(n,-x)
  81          */
  82         ox = x;
  83         on = (GENERIC)n;
  84 
  85         if (n < 0) {
  86                 n = -n;
  87                 x = -x;
  88         }
  89         if (isnan(x))
  90                 return (x*x);   /* + -> * for Cheetah */
  91         if (!((int)_lib_version == libm_ieee ||
  92             (__xpg6 & _C99SUSv3_math_errexcept) != 0)) {
  93                 if (fabs(x) > X_TLOSS)
  94                         return (_SVID_libm_err(on, ox, 38));
  95         }
  96         if (n == 0)
  97                 return (j0(x));
  98         if (n == 1)
  99                 return (j1(x));
 100         if ((n&1) == 0)
 101                 sgn = 0;                        /* even n */
 102         else
 103                 sgn = signbit(x);       /* old n  */
 104         x = fabs(x);
 105         if (x == zero||!finite(x)) b = zero;
 106         else if ((GENERIC)n <= x) {
 107                                         /*
 108                                          * Safe to use
 109                                          *  J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
 110                                          */
 111                 if (x > 1.0e91) {
 112                                 /*
 113                                  * x >> n**2
 114                                  *    Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 115                                  *   Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 116                                  *   Let s=sin(x), c=cos(x),
 117                                  *      xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 118                                  *
 119                                  *         n    sin(xn)*sqt2    cos(xn)*sqt2
 120                                  *      ----------------------------------
 121                                  *         0     s-c             c+s
 122                                  *         1    -s-c            -c+s
 123                                  *         2    -s+c            -c-s
 124                                  *         3     s+c             c-s
 125                                  */
 126                         switch (n&3) {
 127                         case 0:
 128                                 temp =  cos(x)+sin(x);
 129                                 break;
 130                         case 1:
 131                                 temp = -cos(x)+sin(x);
 132                                 break;
 133                         case 2:
 134                                 temp = -cos(x)-sin(x);
 135                                 break;
 136                         case 3:
 137                                 temp =  cos(x)-sin(x);
 138                                 break;
 139                         }
 140                         b = invsqrtpi*temp/sqrt(x);
 141                 } else {
 142                         a = j0(x);
 143                         b = j1(x);
 144                         for (i = 1; i < n; i++) {
 145                                 temp = b;
 146                                 /* avoid underflow */
 147                                 b = b*((GENERIC)(i+i)/x) - a;
 148                                 a = temp;
 149                         }
 150                 }
 151         } else {
 152                 if (x < 1e-9) {      /* use J(n,x) = 1/n!*(x/2)^n */
 153                         b = pow(0.5*x, (GENERIC) n);
 154                         if (b != zero) {
 155                                 for (a = one, i = 1; i <= n; i++)
 156                                         a *= (GENERIC)i;
 157                                 b = b/a;
 158                         }
 159                 } else {
 160                         /*
 161                          * use backward recurrence
 162                          *                      x         x^2     x^2
 163                          *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 164                          *                      2n  - 2(n+1) - 2(n+2)
 165                          *
 166                          *                      1         1         1
 167                          *  (for large x)   =  ----  ------   ------   .....
 168                          *                      2n   2(n+1)   2(n+2)
 169                          *                      -- - ------ - ------ -
 170                          *                       x       x               x
 171                          *
 172                          * Let w = 2n/x and h = 2/x, then the above quotient
 173                          * is equal to the continued fraction:
 174                          *                  1
 175                          *      = -----------------------
 176                          *                         1
 177                          *         w - -----------------
 178                          *                        1
 179                          *                      w+h - ---------
 180                          *                         w+2h - ...
 181                          *
 182                          * To determine how many terms needed, let
 183                          * Q(0) = w, Q(1) = w(w+h) - 1,
 184                          * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 185                          * When Q(k) > 1e4   good for single
 186                          * When Q(k) > 1e9   good for double
 187                          * When Q(k) > 1e17  good for quaduple
 188                          */
 189                         /* determine k */
 190                         GENERIC t, v;
 191                         double q0, q1, h, tmp;
 192                         int k, m;
 193                         w  = (n+n)/(double)x;
 194                         h = 2.0/(double)x;
 195                         q0 = w;
 196                         z = w + h;
 197                         q1 = w*z - 1.0;
 198                         k = 1;
 199 
 200                         while (q1 < 1.0e9) {
 201                                 k += 1;
 202                                 z += h;
 203                                 tmp = z*q1 - q0;
 204                                 q0 = q1;
 205                                 q1 = tmp;
 206                         }
 207                         m = n+n;
 208                         for (t = zero, i = 2*(n+k); i >= m; i -= 2)
 209                                 t = one/(i/x-t);
 210                         a = t;
 211                         b = one;
 212                         /*
 213                          * estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 214                          * hence, if n*(log(2n/x)) > ...
 215                          *  single:
 216                          *    8.8722839355e+01
 217                          *  double:
 218                          *    7.09782712893383973096e+02
 219                          *  long double:
 220                          *    1.1356523406294143949491931077970765006170e+04
 221                          * then recurrent value may overflow and the result is
 222                          * likely underflow to zero
 223                          */
 224                         tmp = n;
 225                         v = two/x;
 226                         tmp = tmp*log(fabs(v*tmp));
 227                         if (tmp < 7.09782712893383973096e+02) {
 228                                 for (i = n-1; i > 0; i--) {
 229                                         temp = b;
 230                                         b = ((i+i)/x)*b - a;
 231                                         a = temp;
 232                                 }
 233                         } else {
 234                                 for (i = n-1; i > 0; i--) {
 235                                         temp = b;
 236                                         b = ((i+i)/x)*b - a;
 237                                         a = temp;
 238                                         if (b > 1e100) {
 239                                                 a /= b;
 240                                                 t /= b;
 241                                                 b  = 1.0;
 242                                         }
 243                                 }
 244                         }
 245                         b = (t*j0(x)/b);
 246                 }
 247         }
 248         if (sgn != 0)
 249                 return (-b);
 250         else
 251                 return (b);
 252 }
 253 
 254 GENERIC
 255 yn(int n, GENERIC x)
 256 {
 257         int i;
 258         int sign;
 259         GENERIC a, b, temp = 0, ox, on;
 260 
 261         ox = x;
 262         on = (GENERIC)n;
 263         if (isnan(x))
 264                 return (x*x);   /* + -> * for Cheetah */
 265         if (x <= zero) {
 266                 if (x == zero) {
 267                         /* return -one/zero; */
 268                         return (_SVID_libm_err((GENERIC)n, x, 12));
 269                 } else {
 270                         /* return zero/zero; */
 271                         return (_SVID_libm_err((GENERIC)n, x, 13));
 272                 }
 273         }
 274         if (!((int)_lib_version == libm_ieee ||
 275             (__xpg6 & _C99SUSv3_math_errexcept) != 0)) {
 276                 if (x > X_TLOSS)
 277                         return (_SVID_libm_err(on, ox, 39));
 278         }
 279         sign = 1;
 280         if (n < 0) {
 281                 n = -n;
 282                 if ((n&1) == 1) sign = -1;
 283         }
 284         if (n == 0)
 285                 return (y0(x));
 286         if (n == 1)
 287                 return (sign*y1(x));
 288         if (!finite(x))
 289                 return (zero);
 290 
 291         if (x > 1.0e91) {
 292                                 /*
 293                                  * x >> n**2
 294                                  *  Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 295                                  *  Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 296                                  *  Let s = sin(x), c = cos(x),
 297                                  *  xn = x-(2n+1)*pi/4, sqt2 = sqrt(2), then
 298                                  *
 299                                  *    n sin(xn)*sqt2    cos(xn)*sqt2
 300                                  *      ----------------------------------
 301                                  *       0       s-c             c+s
 302                                  *       1      -s-c            -c+s
 303                                  *       2      -s+c            -c-s
 304                                  *       3       s+c             c-s
 305                                  */
 306                 switch (n&3) {
 307                 case 0:
 308                         temp =  sin(x)-cos(x);
 309                         break;
 310                 case 1:
 311                         temp = -sin(x)-cos(x);
 312                         break;
 313                 case 2:
 314                         temp = -sin(x)+cos(x);
 315                         break;
 316                 case 3:
 317                         temp =  sin(x)+cos(x);
 318                         break;
 319                 }
 320                 b = invsqrtpi*temp/sqrt(x);
 321         } else {
 322                 a = y0(x);
 323                 b = y1(x);
 324                 /*
 325                  * fix 1262058 and take care of non-default rounding
 326                  */
 327                 for (i = 1; i < n; i++) {
 328                         temp = b;
 329                         b *= (GENERIC) (i + i) / x;
 330                         if (b <= -DBL_MAX)
 331                                 break;
 332                         b -= a;
 333                         a = temp;
 334                 }
 335         }
 336         if (sign > 0)
 337                 return (b);
 338         else
 339                 return (-b);
 340 }