gonzui

    1: /* @(#)e_jn.c 5.1 93/09/24 */
    2: /*
    3:  * ====================================================
    4:  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
    5:  *
    6:  * Developed at SunPro, a Sun Microsystems, Inc. business.
    7:  * Permission to use, copy, modify, and distribute this
    8:  * software is freely granted, provided that this notice 
    9:  * is preserved.
   10:  * ====================================================
   11:  */
   12: 
   13: /*
   14:  * jn(n, x), yn(n, x)
   15:  * floating point Bessel's function of the 1st and 2nd kind
   16:  * of order n
   17:  *          
   18:  * Special cases:
   19:  *      y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
   20:  *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
   21:  * Note 2. About jn(n,x), yn(n,x)
   22:  *      For n=0, j0(x) is called,
   23:  *      for n=1, j1(x) is called,
   24:  *      for n<x, forward recursion us used starting
   25:  *      from values of j0(x) and j1(x).
   26:  *      for n>x, a continued fraction approximation to
   27:  *      j(n,x)/j(n-1,x) is evaluated and then backward
   28:  *      recursion is used starting from a supposed value
   29:  *      for j(n,x). The resulting value of j(0,x) is
   30:  *      compared with the actual value to correct the
   31:  *      supposed value of j(n,x).
   32:  *
   33:  *      yn(n,x) is similar in all respects, except
   34:  *      that forward recursion is used for all
   35:  *      values of n>1.
   36:  *      
   37:  */
   38: 
   39: #include "math.h"
   40: #include "math_private.h"
   41: 
   42: static const double
   43: invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
   44: two   =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
   45: one   =  1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
   46: 
   47: static const double zero  =  0.00000000000000000000e+00;
   48: 
   49: double
   50: jn(int n, double x)
   51: {
   52:         int32_t i,hx,ix,lx, sgn;
   53:         double a, b, temp, di;
   54:         double z, w;
   55: 
   56:     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
   57:      * Thus, J(-n,x) = J(n,-x)
   58:      */
   59:         EXTRACT_WORDS(hx,lx,x);
   60:         ix = 0x7fffffff&hx;
   61:     /* if J(n,NaN) is NaN */
   62:         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
   63:         if(n<0){               
   64:                 n = -n;
   65:                 x = -x;
   66:                 hx ^= 0x80000000;
   67:         }
   68:         if(n==0) return(j0(x));
   69:         if(n==1) return(j1(x));
   70:         sgn = (n&1)&(hx>>31);  /* even n -- 0, odd n -- sign(x) */
   71:         x = fabs(x);
   72:         if((ix|lx)==0||ix>=0x7ff00000)         /* if x is 0 or inf */
   73:             b = zero;
   74:         else if((double)n<=x) {   
   75:                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
   76:             if(ix>=0x52D00000) { /* x > 2**302 */
   77:     /* (x >> n**2) 
   78:      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
   79:      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
   80:      *      Let s=sin(x), c=cos(x), 
   81:      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
   82:      *
   83:      *             n    sin(xn)*sqt2       cos(xn)*sqt2
   84:      *          ----------------------------------
   85:      *             0     s-c                c+s
   86:      *             1    -s-c               -c+s
   87:      *             2    -s+c               -c-s
   88:      *             3     s+c                c-s
   89:      */
   90:                 switch(n&3) {
   91:                     case 0: temp =  cos(x)+sin(x); break;
   92:                     case 1: temp = -cos(x)+sin(x); break;
   93:                     case 2: temp = -cos(x)-sin(x); break;
   94:                     case 3: temp =  cos(x)-sin(x); break;
   95:                 }
   96:                 b = invsqrtpi*temp/sqrt(x);
   97:             } else {   
   98:                 a = j0(x);
   99:                 b = j1(x);
  100:                 for(i=1;i<n;i++){
  101:                     temp = b;
  102:                     b = b*((double)(i+i)/x) - a; /* avoid underflow */
  103:                     a = temp;
  104:                 }
  105:             }
  106:         } else {
  107:             if(ix<0x3e100000) {        /* x < 2**-29 */
  108:     /* x is tiny, return the first Taylor expansion of J(n,x) 
  109:      * J(n,x) = 1/n!*(x/2)^n  - ...
  110:      */
  111:                 if(n>33)      /* underflow */
  112:                     b = zero;
  113:                 else {
  114:                     temp = x*0.5; b = temp;
  115:                     for (a=one,i=2;i<=n;i++) {
  116:                         a *= (double)i;              /* a = n! */
  117:                         b *= temp;           /* b = (x/2)^n */
  118:                     }
  119:                     b = b/a;
  120:                 }
  121:             } else {
  122:                 /* use backward recurrence */
  123:                 /*                    x      x^2      x^2       
  124:                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
  125:                  *                    2n  - 2(n+1) - 2(n+2)
  126:                  *
  127:                  *                    1      1        1       
  128:                  *  (for large x)   =  ----  ------   ------   .....
  129:                  *                    2n   2(n+1)   2(n+2)
  130:                  *                    -- - ------ - ------ - 
  131:                  *                     x     x         x
  132:                  *
  133:                  * Let w = 2n/x and h=2/x, then the above quotient
  134:                  * is equal to the continued fraction:
  135:                  *                1
  136:                  *    = -----------------------
  137:                  *                   1
  138:                  *       w - -----------------
  139:                  *                      1
  140:                  *            w+h - ---------
  141:                  *                   w+2h - ...
  142:                  *
  143:                  * To determine how many terms needed, let
  144:                  * Q(0) = w, Q(1) = w(w+h) - 1,
  145:                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
  146:                  * When Q(k) > 1e4    good for single 
  147:                  * When Q(k) > 1e9    good for double 
  148:                  * When Q(k) > 1e17   good for quadruple 
  149:                  */
  150:             /* determine k */
  151:                 double t,v;
  152:                 double q0,q1,h,tmp; int32_t k,m;
  153:                 w  = (n+n)/(double)x; h = 2.0/(double)x;
  154:                 q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
  155:                 while(q1<1.0e9) {
  156:                         k += 1; z += h;
  157:                         tmp = z*q1 - q0;
  158:                         q0 = q1;
  159:                         q1 = tmp;
  160:                 }
  161:                 m = n+n;
  162:                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
  163:                 a = t;
  164:                 b = one;
  165:                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
  166:                  *  Hence, if n*(log(2n/x)) > ...
  167:                  *  single 8.8722839355e+01
  168:                  *  double 7.09782712893383973096e+02
  169:                  *  long double 1.1356523406294143949491931077970765006170e+04
  170:                  *  then recurrent value may overflow and the result is 
  171:                  *  likely underflow to zero
  172:                  */
  173:                 tmp = n;
  174:                 v = two/x;
  175:                 tmp = tmp*log(fabs(v*tmp));
  176:                 if(tmp<7.09782712893383973096e+02) {
  177:                    for(i=n-1,di=(double)(i+i);i>0;i--){
  178:                         temp = b;
  179:                         b *= di;
  180:                         b  = b/x - a;
  181:                         a = temp;
  182:                         di -= two;
  183:                    }
  184:                 } else {
  185:                    for(i=n-1,di=(double)(i+i);i>0;i--){
  186:                         temp = b;
  187:                         b *= di;
  188:                         b  = b/x - a;
  189:                         a = temp;
  190:                         di -= two;
  191:                     /* scale b to avoid spurious overflow */
  192:                         if(b>1e100) {
  193:                             a /= b;
  194:                             t /= b;
  195:                             b  = one;
  196:                         }
  197:                    }
  198:                 }
  199:                b = (t*j0(x)/b);
  200:             }
  201:         }
  202:         if(sgn==1) return -b; else return b;
  203: }
  204: 
  205: double
  206: yn(int n, double x) 
  207: {
  208:         int32_t i,hx,ix,lx;
  209:         int32_t sign;
  210:         double a, b, temp;
  211: 
  212:         EXTRACT_WORDS(hx,lx,x);
  213:         ix = 0x7fffffff&hx;
  214:     /* if Y(n,NaN) is NaN */
  215:         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
  216:         if((ix|lx)==0) return -one/zero;
  217:         if(hx<0) return zero/zero;
  218:         sign = 1;
  219:         if(n<0){
  220:                 n = -n;
  221:                 sign = 1 - ((n&1)<<1);
  222:         }
  223:         if(n==0) return(y0(x));
  224:         if(n==1) return(sign*y1(x));
  225:         if(ix==0x7ff00000) return zero;
  226:         if(ix>=0x52D00000) { /* x > 2**302 */
  227:     /* (x >> n**2) 
  228:      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  229:      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  230:      *      Let s=sin(x), c=cos(x), 
  231:      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
  232:      *
  233:      *             n    sin(xn)*sqt2       cos(xn)*sqt2
  234:      *          ----------------------------------
  235:      *             0     s-c                c+s
  236:      *             1    -s-c               -c+s
  237:      *             2    -s+c               -c-s
  238:      *             3     s+c                c-s
  239:      */
  240:                 switch(n&3) {
  241:                     case 0: temp =  sin(x)-cos(x); break;
  242:                     case 1: temp = -sin(x)-cos(x); break;
  243:                     case 2: temp = -sin(x)+cos(x); break;
  244:                     case 3: temp =  sin(x)+cos(x); break;
  245:                 }
  246:                 b = invsqrtpi*temp/sqrt(x);
  247:         } else {
  248:             u_int32_t high;
  249:             a = y0(x);
  250:             b = y1(x);
  251:         /* quit if b is -inf */
  252:             GET_HIGH_WORD(high,b);
  253:             for(i=1;i<n&&high!=0xfff00000;i++){ 
  254:                 temp = b;
  255:                 b = ((double)(i+i)/x)*b - a;
  256:                 GET_HIGH_WORD(high,b);
  257:                 a = temp;
  258:             }
  259:         }
  260:         if(sign>0) return b; else return -b;
  261: }