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 __jnl = jnl
  31 #pragma weak __ynl = ynl
  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 "longdouble.h"
  60 #include <float.h>        /* LDBL_MAX */
  61 
  62 #define GENERIC long double
  63 
  64 static const GENERIC
  65 invsqrtpi = 5.641895835477562869480794515607725858441e-0001L,
  66 two  = 2.0L,
  67 zero = 0.0L,
  68 one  = 1.0L;
  69 
  70 GENERIC
  71 jnl(int n, GENERIC x)
  72 {
  73         int i, sgn;
  74         GENERIC a, b, temp, z, w;
  75 
  76         /*
  77          * J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  78          * Thus, J(-n,x) = J(n,-x)
  79          */
  80         if (n < 0) {
  81                 n = -n;
  82                 x = -x;
  83         }
  84         if (n == 0)
  85                 return (j0l(x));
  86         if (n == 1)
  87                 return (j1l(x));
  88         if (x != x)
  89                 return (x+x);
  90         if ((n&1) == 0)
  91                 sgn = 0;                        /* even n */
  92         else
  93                 sgn = signbitl(x);      /* old n  */
  94         x = fabsl(x);
  95         if (x == zero || !finitel(x)) b = zero;
  96         else if ((GENERIC)n <= x) {
  97                                         /*
  98                                          * Safe to use
  99                                          * J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
 100                                          */
 101                 if (x > 1.0e91L) {
 102                                 /*
 103                                  * x >> n**2
 104                                  *  Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 105                                  *   Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 106                                  *   Let s=sin(x), c=cos(x),
 107                                  *      xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 108                                  *
 109                                  *         n    sin(xn)*sqt2    cos(xn)*sqt2
 110                                  *      ----------------------------------
 111                                  *         0     s-c             c+s
 112                                  *         1    -s-c            -c+s
 113                                  *         2    -s+c            -c-s
 114                                  *         3     s+c             c-s
 115                                  */
 116                         switch (n&3) {
 117                         case 0:
 118                                 temp =  cosl(x)+sinl(x);
 119                                 break;
 120                         case 1:
 121                                 temp = -cosl(x)+sinl(x);
 122                                 break;
 123                         case 2:
 124                                 temp = -cosl(x)-sinl(x);
 125                                 break;
 126                         case 3:
 127                                 temp =  cosl(x)-sinl(x);
 128                                 break;
 129                         }
 130                         b = invsqrtpi*temp/sqrtl(x);
 131                 } else {
 132                         a = j0l(x);
 133                         b = j1l(x);
 134                         for (i = 1; i < n; i++) {
 135                                 temp = b;
 136                                 /* avoid underflow */
 137                                 b = b*((GENERIC)(i+i)/x) - a;
 138                                 a = temp;
 139                         }
 140                 }
 141         } else {
 142                 if (x < 1e-17L) {    /* use J(n,x) = 1/n!*(x/2)^n */
 143                         b = powl(0.5L*x, (GENERIC)n);
 144                         if (b != zero) {
 145                                 for (a = one, i = 1; i <= n; i++)
 146                                         a *= (GENERIC)i;
 147                                 b = b/a;
 148                         }
 149                 } else {
 150                         /* use backward recurrence */
 151                         /*
 152                          *                      x      x^2      x^2
 153                          *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 154                          *                      2n  - 2(n+1) - 2(n+2)
 155                          *
 156                          *                      1      1        1
 157                          *  (for large x)   =  ----  ------   ------   .....
 158                          *                      2n   2(n+1)   2(n+2)
 159                          *                      -- - ------ - ------ -
 160                          *                       x     x         x
 161                          *
 162                          * Let w = 2n/x and h=2/x, then the above quotient
 163                          * is equal to the continued fraction:
 164                          *                  1
 165                          *      = -----------------------
 166                          *                     1
 167                          *         w - -----------------
 168                          *                        1
 169                          *              w+h - ---------
 170                          *                     w+2h - ...
 171                          *
 172                          * To determine how many terms needed, let
 173                          * Q(0) = w, Q(1) = w(w+h) - 1,
 174                          * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 175                          * When Q(k) > 1e4   good for single
 176                          * When Q(k) > 1e9   good for double
 177                          * When Q(k) > 1e17  good for quaduple
 178                          */
 179                         /* determine k */
 180                         GENERIC t, v;
 181                         double q0, q1, h, tmp;
 182                         int k, m;
 183                         w  = (n+n)/(double)x;
 184                         h = 2.0/(double)x;
 185                         q0 = w;
 186                         z = w+h;
 187                         q1 = w*z - 1.0;
 188                         k = 1;
 189                         while (q1 < 1.0e17) {
 190                                 k += 1;
 191                                 z += h;
 192                                 tmp = z*q1 - q0;
 193                                 q0 = q1;
 194                                 q1 = tmp;
 195                         }
 196                         m = n+n;
 197                         for (t = zero, i = 2*(n+k); i >= m; i -= 2)
 198                                 t = one/(i/x-t);
 199                         a = t;
 200                         b = one;
 201                         /*
 202                          * estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 203                          * hence, if n*(log(2n/x)) > ...
 204                          *  single 8.8722839355e+01
 205                          *  double 7.09782712893383973096e+02
 206                          *  long double 1.1356523406294143949491931077970765006170e+04
 207                          *  then recurrent value may overflow and the result is
 208                          *  likely underflow to zero
 209                          */
 210                         tmp = n;
 211                         v = two/x;
 212                         tmp = tmp*logl(fabsl(v*tmp));
 213                         if (tmp < 1.1356523406294143949491931077970765e+04L) {
 214                                 for (i = n-1; i > 0; i--) {
 215                                         temp = b;
 216                                         b = ((i+i)/x)*b - a;
 217                                         a = temp;
 218                                 }
 219                         } else {
 220                                 for (i = n-1; i > 0; i--) {
 221                                         temp = b;
 222                                         b = ((i+i)/x)*b - a;
 223                                         a = temp;
 224                                         if (b > 1e1000L) {
 225                                                 a /= b;
 226                                                 t /= b;
 227                                                 b  = 1.0;
 228                                         }
 229                                 }
 230                         }
 231                         b = (t*j0l(x)/b);
 232                 }
 233         }
 234         if (sgn != 0)
 235                 return (-b);
 236         else
 237                 return (b);
 238 }
 239 
 240 GENERIC
 241 ynl(int n, GENERIC x)
 242 {
 243         int i;
 244         int sign;
 245         GENERIC a, b, temp;
 246 
 247         if (x != x)
 248                 return (x+x);
 249         if (x <= zero) {
 250                 if (x == zero)
 251                         return (-one/zero);
 252                 else
 253                         return (zero/zero);
 254         }
 255         sign = 1;
 256         if (n < 0) {
 257                 n = -n;
 258                 if ((n&1) == 1) sign = -1;
 259         }
 260         if (n == 0)
 261                 return (y0l(x));
 262         if (n == 1)
 263                 return (sign*y1l(x));
 264         if (!finitel(x))
 265                 return (zero);
 266 
 267         if (x > 1.0e91L) {
 268                 /*
 269                  * x >> n**2
 270                  *   Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 271                  *   Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 272                  *   Let s = sin(x), c = cos(x),
 273                  *      xn = x-(2n+1)*pi/4, sqt2 = sqrt(2), then
 274                  *
 275                  *         n    sin(xn)*sqt2    cos(xn)*sqt2
 276                  *      ----------------------------------
 277                  *         0     s-c             c+s
 278                  *         1    -s-c            -c+s
 279                  *         2    -s+c            -c-s
 280                  *         3     s+c             c-s
 281                  */
 282                 switch (n&3) {
 283                 case 0:
 284                         temp =  sinl(x)-cosl(x);
 285                         break;
 286                 case 1:
 287                         temp = -sinl(x)-cosl(x);
 288                         break;
 289                 case 2:
 290                         temp = -sinl(x)+cosl(x);
 291                         break;
 292                 case 3:
 293                         temp =  sinl(x)+cosl(x);
 294                         break;
 295                 }
 296                 b = invsqrtpi*temp/sqrtl(x);
 297         } else {
 298                 a = y0l(x);
 299                 b = y1l(x);
 300                 /*
 301                  * fix 1262058 and take care of non-default rounding
 302                  */
 303                 for (i = 1; i < n; i++) {
 304                         temp = b;
 305                         b *= (GENERIC) (i + i) / x;
 306                         if (b <= -LDBL_MAX)
 307                                 break;
 308                         b -= a;
 309                         a = temp;
 310                 }
 311         }
 312         if (sign > 0)
 313                 return (b);
 314         else
 315                 return (-b);
 316 }