sysdeps/ieee754/dbl-64/e_jn.c

   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  * __ieee754_jn(n, x), __ieee754_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 overflow 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 <errno.h>
  40 #include <float.h>
  41 #include <math.h>
  42 #include <math_private.h>
  43
  44 static const double
  45   invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  46   two = 2.00000000000000000000e+00,  /* 0x40000000, 0x00000000 */
  47   one = 1.00000000000000000000e+00;  /* 0x3FF00000, 0x00000000 */
  48
  49 static const double zero = 0.00000000000000000000e+00;
  50
  51 double
  52 __ieee754_jn (int n, double x)
  53 {
  54   int32_t i, hx, ix, lx, sgn;
  55   double a, b, temp, di;
  56   double z, w;
  57
  58   /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  59    * Thus, J(-n,x) = J(n,-x)
  60    */
  61   EXTRACT_WORDS (hx, lx, x);
  62   ix = 0x7fffffff & hx;
  63   /* if J(n,NaN) is NaN */
  64   if (__glibc_unlikely ((ix | ((u_int32_t) (lx | -lx)) >> 31) > 0x7ff00000))
  65     return x + x;
  66   if (n < 0)
  67     {
  68       n = -n;
  69       x = -x;
  70       hx ^= 0x80000000;
  71     }
  72   if (n == 0)
  73     return (__ieee754_j0 (x));
  74   if (n == 1)
  75     return (__ieee754_j1 (x));
  76   sgn = (n & 1) & (hx >> 31);   /* even n -- 0, odd n -- sign(x) */
  77   x = fabs (x);
  78   if (__glibc_unlikely ((ix | lx) == 0 || ix >= 0x7ff00000))
  79     /* if x is 0 or inf */
  80     b = zero;
  81   else if ((double) n <= x)
  82     {
  83       /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  84       if (ix >= 0x52D00000)      /* x > 2**302 */
  85         { /* (x >> n**2)
  86                          *          Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  87                          *          Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  88                          *          Let s=sin(x), c=cos(x),
  89                          *              xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
  90                          *
  91                          *                 n    sin(xn)*sqt2    cos(xn)*sqt2
  92                          *              ----------------------------------
  93                          *                 0     s-c             c+s
  94                          *                 1    -s-c            -c+s
  95                          *                 2    -s+c            -c-s
  96                          *                 3     s+c             c-s
  97                          */
  98           double s;
  99           double c;
 100           __sincos (x, &s, &c);
 101           switch (n & 3)
 102             {
 103             case 0: temp = c + s; break;
 104             case 1: temp = -c + s; break;
 105             case 2: temp = -c - s; break;
 106             case 3: temp = c - s; break;
 107             }
 108           b = invsqrtpi * temp / __ieee754_sqrt (x);
 109         }
 110       else
 111         {
 112           a = __ieee754_j0 (x);
 113           b = __ieee754_j1 (x);
 114           for (i = 1; i < n; i++)
 115             {
 116               temp = b;
 117               b = b * ((double) (i + i) / x) - a; /* avoid underflow */
 118               a = temp;
 119             }
 120         }
 121     }
 122   else
 123     {
 124       if (ix < 0x3e100000)      /* x < 2**-29 */
 125         { /* x is tiny, return the first Taylor expansion of J(n,x)
 126                          * J(n,x) = 1/n!*(x/2)^n  - ...
 127                          */
 128           if (n > 33)           /* underflow */
 129             b = zero;
 130           else
 131             {
 132               temp = x * 0.5; b = temp;
 133               for (a = one, i = 2; i <= n; i++)
 134                 {
 135                   a *= (double) i;              /* a = n! */
 136                   b *= temp;                    /* b = (x/2)^n */
 137                 }
 138               b = b / a;
 139             }
 140         }
 141       else
 142         {
 143           /* use backward recurrence */
 144           /*                    x      x^2      x^2
 145            *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 146            *                    2n  - 2(n+1) - 2(n+2)
 147            *
 148            *                    1      1        1
 149            *  (for large x)   =  ----  ------   ------   .....
 150            *                    2n   2(n+1)   2(n+2)
 151            *                    -- - ------ - ------ -
 152            *                     x     x         x
 153            *
 154            * Let w = 2n/x and h=2/x, then the above quotient
 155            * is equal to the continued fraction:
 156            *                1
 157            *    = -----------------------
 158            *                   1
 159            *       w - -----------------
 160            *                      1
 161            *            w+h - ---------
 162            *                   w+2h - ...
 163            *
 164            * To determine how many terms needed, let
 165            * Q(0) = w, Q(1) = w(w+h) - 1,
 166            * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 167            * When Q(k) > 1e4    good for single
 168            * When Q(k) > 1e9    good for double
 169            * When Q(k) > 1e17   good for quadruple
 170            */
 171           /* determine k */
 172           double t, v;
 173           double q0, q1, h, tmp; int32_t k, m;
 174           w = (n + n) / (double) x; h = 2.0 / (double) x;
 175           q0 = w;  z = w + h; q1 = w * z - 1.0; k = 1;
 176           while (q1 < 1.0e9)
 177             {
 178               k += 1; z += h;
 179               tmp = z * q1 - q0;
 180               q0 = q1;
 181               q1 = tmp;
 182             }
 183           m = n + n;
 184           for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
 185             t = one / (i / x - t);
 186           a = t;
 187           b = one;
 188           /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 189            *  Hence, if n*(log(2n/x)) > ...
 190            *  single 8.8722839355e+01
 191            *  double 7.09782712893383973096e+02
 192            *  long double 1.1356523406294143949491931077970765006170e+04
 193            *  then recurrent value may overflow and the result is
 194            *  likely underflow to zero
 195            */
 196           tmp = n;
 197           v = two / x;
 198           tmp = tmp * __ieee754_log (fabs (v * tmp));
 199           if (tmp < 7.09782712893383973096e+02)
 200             {
 201               for (i = n - 1, di = (double) (i + i); i > 0; i--)
 202                 {
 203                   temp = b;
 204                   b *= di;
 205                   b = b / x - a;
 206                   a = temp;
 207                   di -= two;
 208                 }
 209             }
 210           else
 211             {
 212               for (i = n - 1, di = (double) (i + i); i > 0; i--)
 213                 {
 214                   temp = b;
 215                   b *= di;
 216                   b = b / x - a;
 217                   a = temp;
 218                   di -= two;
 219                   /* scale b to avoid spurious overflow */
 220                   if (b > 1e100)
 221                     {
 222                       a /= b;
 223                       t /= b;
 224                       b = one;
 225                     }
 226                 }
 227             }
 228           /* j0() and j1() suffer enormous loss of precision at and
 229            * near zero; however, we know that their zero points never
 230            * coincide, so just choose the one further away from zero.
 231            */
 232           z = __ieee754_j0 (x);
 233           w = __ieee754_j1 (x);
 234           if (fabs (z) >= fabs (w))
 235             b = (t * z / b);
 236           else
 237             b = (t * w / a);
 238         }
 239     }
 240   if (sgn == 1)
 241     return -b;
 242   else
 243     return b;
 244 }
 245 strong_alias (__ieee754_jn, __jn_finite)
 246
 247 double
 248 __ieee754_yn (int n, double x)
 249 {
 250   int32_t i, hx, ix, lx;
 251   int32_t sign;
 252   double a, b, temp, ret;
 253
 254   EXTRACT_WORDS (hx, lx, x);
 255   ix = 0x7fffffff & hx;
 256   /* if Y(n,NaN) is NaN */
 257   if (__glibc_unlikely ((ix | ((u_int32_t) (lx | -lx)) >> 31) > 0x7ff00000))
 258     return x + x;
 259   if (__glibc_unlikely ((ix | lx) == 0))
 260     return -HUGE_VAL + x;
 261   /* -inf and overflow exception.  */;
 262   if (__glibc_unlikely (hx < 0))
 263     return zero / (zero * x);
 264   sign = 1;
 265   if (n < 0)
 266     {
 267       n = -n;
 268       sign = 1 - ((n & 1) << 1);
 269     }
 270   if (n == 0)
 271     return (__ieee754_y0 (x));
 272   {
 273     SET_RESTORE_ROUND (FE_TONEAREST);
 274     if (n == 1)
 275       {
 276         ret = sign * __ieee754_y1 (x);
 277         goto out;
 278       }
 279     if (__glibc_unlikely (ix == 0x7ff00000))
 280       return zero;
 281     if (ix >= 0x52D00000)      /* x > 2**302 */
 282       { /* (x >> n**2)
 283          *          Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 284          *          Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 285          *          Let s=sin(x), c=cos(x),
 286          *              xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 287          *
 288          *                 n    sin(xn)*sqt2    cos(xn)*sqt2
 289          *              ----------------------------------
 290          *                 0     s-c             c+s
 291          *                 1    -s-c            -c+s
 292          *                 2    -s+c            -c-s
 293          *                 3     s+c             c-s
 294          */
 295         double c;
 296         double s;
 297         __sincos (x, &s, &c);
 298         switch (n & 3)
 299           {
 300           case 0: temp = s - c; break;
 301           case 1: temp = -s - c; break;
 302           case 2: temp = -s + c; break;
 303           case 3: temp = s + c; break;
 304           }
 305         b = invsqrtpi * temp / __ieee754_sqrt (x);
 306       }
 307     else
 308       {
 309         u_int32_t high;
 310         a = __ieee754_y0 (x);
 311         b = __ieee754_y1 (x);
 312         /* quit if b is -inf */
 313         GET_HIGH_WORD (high, b);
 314         for (i = 1; i < n && high != 0xfff00000; i++)
 315           {
 316             temp = b;
 317             b = ((double) (i + i) / x) * b - a;
 318             GET_HIGH_WORD (high, b);
 319             a = temp;
 320           }
 321         /* If B is +-Inf, set up errno accordingly.  */
 322         if (!isfinite (b))
 323           __set_errno (ERANGE);
 324       }
 325     if (sign > 0)
 326       ret = b;
 327     else
 328       ret = -b;
 329   }
 330  out:
 331   if (isinf (ret))
 332     ret = __copysign (DBL_MAX, ret) * DBL_MAX;
 333   return ret;
 334 }
 335 strong_alias (__ieee754_yn, __yn_finite)