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 #if defined(ELFOBJ)
  31 #pragma weak jnl = __jnl
  32 #pragma weak ynl = __ynl
  33 #endif
  34 
  35 /*
  36  * floating point Bessel's function of the 1st and 2nd kind
  37  * of order n: jn(n,x),yn(n,x);
  38  *          
  39  * Special cases:
  40  *      y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
  41  *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
  42  * Note 2. About jn(n,x), yn(n,x)
  43  *      For n=0, j0(x) is called,
  44  *      for n=1, j1(x) is called,
  45  *      for n<x, forward recursion us used starting
  46  *      from values of j0(x) and j1(x).
  47  *      for n>x, a continued fraction approximation to
  48  *      j(n,x)/j(n-1,x) is evaluated and then backward
  49  *      recursion is used starting from a supposed value
  50  *      for j(n,x). The resulting value of j(0,x) is
  51  *      compared with the actual value to correct the
  52  *      supposed value of j(n,x).
  53  *
  54  *      yn(n,x) is similar in all respects, except
  55  *      that forward recursion is used for all
  56  *      values of n>1.
  57  *      
  58  */
  59 
  60 #include "libm.h"
  61 #include "longdouble.h"
  62 #include <float.h>        /* LDBL_MAX */
  63 
  64 #define GENERIC long double
  65 
  66 static const GENERIC
  67 invsqrtpi= 5.641895835477562869480794515607725858441e-0001L,
  68 two  = 2.0L,
  69 zero = 0.0L,
  70 one  = 1.0L;
  71 
  72 GENERIC
  73 jnl(n,x) int n; GENERIC x;{
  74         int i, sgn;
  75         GENERIC a, b, temp = 0, z, w;
  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) return(j0l(x));
  85         if(n==1) return(j1l(x));
  86         if(x!=x) return x+x;
  87         if((n&1)==0)        
  88                 sgn=0;                  /* even n */
  89         else 
  90                 sgn = signbitl(x);      /* old n  */
  91         x = fabsl(x);
  92         if(x == zero||!finitel(x)) b = zero;
  93         else if((GENERIC)n<=x) {     /* Safe to use 
  94                                            J(n+1,x)=2n/x *J(n,x)-J(n-1,x) 
  95                                          */
  96             if(x>1.0e91L) {  /* x >> n**2 
  97                                     Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  98                                     Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  99                                     Let s=sin(x), c=cos(x), 
 100                                         xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 101 
 102                                            n    sin(xn)*sqt2    cos(xn)*sqt2
 103                                         ----------------------------------
 104                                            0     s-c             c+s
 105                                            1    -s-c            -c+s
 106                                            2    -s+c            -c-s
 107                                            3     s+c             c-s
 108                                  */
 109                 switch(n&3) {
 110                     case 0: temp =  cosl(x)+sinl(x); break;
 111                     case 1: temp = -cosl(x)+sinl(x); break;
 112                     case 2: temp = -cosl(x)-sinl(x); break;
 113                     case 3: temp =  cosl(x)-sinl(x); break;
 114                 }
 115                 b = invsqrtpi*temp/sqrtl(x);
 116             } else {    
 117                 a = j0l(x);
 118                 b = j1l(x);
 119                 for(i=1;i<n;i++){
 120                     temp = b;
 121                     b = b*((GENERIC)(i+i)/x) - a; /* avoid underflow */
 122                     a = temp;
 123                 }
 124             }
 125         } else {
 126             if(x<1e-17L) {   /* use J(n,x) = 1/n!*(x/2)^n */
 127                 b = powl(0.5L*x,(GENERIC) n);
 128                 if (b!=zero) {
 129                     for(a=one,i=1;i<=n;i++) a *= (GENERIC)i;
 130                     b = b/a;
 131                 }
 132             } else {
 133                 /* use backward recurrence */
 134                 /*                      x      x^2      x^2       
 135                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 136                  *                      2n  - 2(n+1) - 2(n+2)
 137                  *
 138                  *                      1      1        1       
 139                  *  (for large x)   =  ----  ------   ------   .....
 140                  *                      2n   2(n+1)   2(n+2)
 141                  *                      -- - ------ - ------ - 
 142                  *                       x     x         x
 143                  *
 144                  * Let w = 2n/x and h=2/x, then the above quotient
 145                  * is equal to the continued fraction:
 146                  *                  1
 147                  *      = -----------------------
 148                  *                     1
 149                  *         w - -----------------
 150                  *                        1
 151                  *              w+h - ---------
 152                  *                     w+2h - ...
 153                  *
 154                  * To determine how many terms needed, let
 155                  * Q(0) = w, Q(1) = w(w+h) - 1,
 156                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 157                  * When Q(k) > 1e4   good for single 
 158                  * When Q(k) > 1e9   good for double 
 159                  * When Q(k) > 1e17  good for quaduple 
 160                  */
 161             /* determin k */
 162                 GENERIC t,v;
 163                 double q0,q1,h,tmp; int k,m;
 164                 w  = (n+n)/(double)x; h = 2.0/(double)x;
 165                 q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
 166                 while(q1<1.0e17) {
 167                         k += 1; z += h;
 168                         tmp = z*q1 - q0;
 169                         q0 = q1;
 170                         q1 = tmp;
 171                 }
 172                 m = n+n;
 173                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 174                 a = t;
 175                 b = one;
 176                 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 177                    hence, if n*(log(2n/x)) > ...
 178                     single 8.8722839355e+01
 179                     double 7.09782712893383973096e+02
 180                     long double 1.1356523406294143949491931077970765006170e+04
 181                     then recurrent value may overflow and the result is
 182                     likely underflow to zero
 183                  */
 184                 tmp = n;
 185                 v = two/x;
 186                 tmp = tmp*logl(fabsl(v*tmp));
 187                 if(tmp<1.1356523406294143949491931077970765e+04L) {
 188                     for(i=n-1;i>0;i--){
 189                         temp = b;
 190                         b = ((i+i)/x)*b - a;
 191                         a = temp;
 192                     }
 193                 } else {
 194                     for(i=n-1;i>0;i--){
 195                         temp = b;
 196                         b = ((i+i)/x)*b - a;
 197                         a = temp;
 198                         if(b>1e1000L) {
 199                             a /= b;
 200                             t /= b;
 201                             b  = 1.0;
 202                         }
 203                     }
 204                 }
 205                 b = (t*j0l(x)/b);
 206             }
 207         }
 208         if(sgn==1) return -b; else return b;
 209 }
 210 
 211 GENERIC ynl(n,x) 
 212 int n; GENERIC x;{
 213         int i;
 214         int sign;
 215         GENERIC a, b, temp = 0;
 216 
 217         if(x!=x)
 218                 return x+x;
 219         if (x <= zero) {
 220                 if(x==zero) 
 221                         return -one/zero;
 222                 else 
 223                         return zero/zero;
 224         }
 225         sign = 1;
 226         if(n<0){
 227                 n = -n;
 228                 if((n&1) == 1) sign = -1;
 229         }
 230         if(n==0) return(y0l(x));
 231         if(n==1) return(sign*y1l(x));
 232         if(!finitel(x)) return zero;
 233 
 234         if(x>1.0e91L) {      /* x >> n**2 
 235                                     Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 236                                     Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 237                                     Let s=sin(x), c=cos(x), 
 238                                         xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 239 
 240                                            n    sin(xn)*sqt2    cos(xn)*sqt2
 241                                         ----------------------------------
 242                                            0     s-c             c+s
 243                                            1    -s-c            -c+s
 244                                            2    -s+c            -c-s
 245                                            3     s+c             c-s
 246                                  */
 247                 switch(n&3) {
 248                     case 0: temp =  sinl(x)-cosl(x); break;
 249                     case 1: temp = -sinl(x)-cosl(x); break;
 250                     case 2: temp = -sinl(x)+cosl(x); break;
 251                     case 3: temp =  sinl(x)+cosl(x); break;
 252                 }
 253                 b = invsqrtpi*temp/sqrtl(x);
 254         } else {
 255                 a = y0l(x);
 256                 b = y1l(x);
 257                 /*
 258                  * fix 1262058 and take care of non-default rounding
 259                  */
 260                 for (i = 1; i < n; i++) {
 261                         temp = b;
 262                         b *= (GENERIC) (i + i) / x;
 263                         if (b <= -LDBL_MAX)
 264                                 break;
 265                         b -= a;
 266                         a = temp;
 267                 }
 268         }
 269         if(sign>0) return b; else return -b;
 270 }