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glibc  2.9
Functions | Variables
e_jn.c File Reference
#include "math.h"
#include "math_private.h"

Go to the source code of this file.

Functions

double __ieee754_jn (int n, double x)
double __ieee754_yn (int n, double x)

Variables

static double invsqrtpi = 5.64189583547756279280e-01
static double two = 2.00000000000000000000e+00
static double one = 1.00000000000000000000e+00
static double zero = 0.00000000000000000000e+00

Function Documentation

double __ieee754_jn ( int  n,
double  x 
)

Definition at line 64 of file e_jn.c.

{
       int32_t i,hx,ix,lx, sgn;
       double a, b, temp, di;
       double z, w;

    /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
     * Thus, J(-n,x) = J(n,-x)
     */
       EXTRACT_WORDS(hx,lx,x);
       ix = 0x7fffffff&hx;
    /* if J(n,NaN) is NaN */
       if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
       if(n<0){
              n = -n;
              x = -x;
              hx ^= 0x80000000;
       }
       if(n==0) return(__ieee754_j0(x));
       if(n==1) return(__ieee754_j1(x));
       sgn = (n&1)&(hx>>31);       /* even n -- 0, odd n -- sign(x) */
       x = fabs(x);
       if((ix|lx)==0||ix>=0x7ff00000)     /* if x is 0 or inf */
           b = zero;
       else if((double)n<=x) {
              /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
           if(ix>=0x52D00000) { /* x > 2**302 */
    /* (x >> n**2)
     *     Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
     *     Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
     *     Let s=sin(x), c=cos(x),
     *        xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
     *
     *           n   sin(xn)*sqt2  cos(xn)*sqt2
     *        ----------------------------------
     *           0    s-c           c+s
     *           1   -s-c          -c+s
     *           2   -s+c          -c-s
     *           3    s+c           c-s
     */
              double s;
              double c;
              __sincos (x, &s, &c);
              switch(n&3) {
                  case 0: temp =  c + s; break;
                  case 1: temp = -c + s; break;
                  case 2: temp = -c - s; break;
                  case 3: temp =  c - s; break;
              }
              b = invsqrtpi*temp/__ieee754_sqrt(x);
           } else {
               a = __ieee754_j0(x);
               b = __ieee754_j1(x);
               for(i=1;i<n;i++){
                  temp = b;
                  b = b*((double)(i+i)/x) - a; /* avoid underflow */
                  a = temp;
               }
           }
       } else {
           if(ix<0x3e100000) {     /* x < 2**-29 */
    /* x is tiny, return the first Taylor expansion of J(n,x)
     * J(n,x) = 1/n!*(x/2)^n  - ...
     */
              if(n>33)      /* underflow */
                  b = zero;
              else {
                  temp = x*0.5; b = temp;
                  for (a=one,i=2;i<=n;i++) {
                     a *= (double)i;             /* a = n! */
                     b *= temp;           /* b = (x/2)^n */
                  }
                  b = b/a;
              }
           } else {
              /* use backward recurrence */
              /*                   x      x^2      x^2
               *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
               *                   2n  - 2(n+1) - 2(n+2)
               *
               *                   1      1        1
               *  (for large x)   =  ----  ------   ------   .....
               *                   2n   2(n+1)   2(n+2)
               *                   -- - ------ - ------ -
               *                    x     x         x
               *
               * Let w = 2n/x and h=2/x, then the above quotient
               * is equal to the continued fraction:
               *                1
               *     = -----------------------
               *                   1
               *        w - -----------------
               *                     1
               *             w+h - ---------
               *                   w+2h - ...
               *
               * To determine how many terms needed, let
               * Q(0) = w, Q(1) = w(w+h) - 1,
               * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
               * When Q(k) > 1e4   good for single
               * When Q(k) > 1e9   good for double
               * When Q(k) > 1e17  good for quadruple
               */
           /* determine k */
              double t,v;
              double q0,q1,h,tmp; int32_t k,m;
              w  = (n+n)/(double)x; h = 2.0/(double)x;
              q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
              while(q1<1.0e9) {
                     k += 1; z += h;
                     tmp = z*q1 - q0;
                     q0 = q1;
                     q1 = tmp;
              }
              m = n+n;
              for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
              a = t;
              b = one;
              /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
               *  Hence, if n*(log(2n/x)) > ...
               *  single 8.8722839355e+01
               *  double 7.09782712893383973096e+02
               *  long double 1.1356523406294143949491931077970765006170e+04
               *  then recurrent value may overflow and the result is
               *  likely underflow to zero
               */
              tmp = n;
              v = two/x;
              tmp = tmp*__ieee754_log(fabs(v*tmp));
              if(tmp<7.09782712893383973096e+02) {
                  for(i=n-1,di=(double)(i+i);i>0;i--){
                      temp = b;
                     b *= di;
                     b  = b/x - a;
                      a = temp;
                     di -= two;
                  }
              } else {
                  for(i=n-1,di=(double)(i+i);i>0;i--){
                      temp = b;
                     b *= di;
                     b  = b/x - a;
                      a = temp;
                     di -= two;
                  /* scale b to avoid spurious overflow */
                     if(b>1e100) {
                         a /= b;
                         t /= b;
                         b  = one;
                     }
                  }
              }
              b = (t*__ieee754_j0(x)/b);
           }
       }
       if(sgn==1) return -b; else return b;
}

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double __ieee754_yn ( int  n,
double  x 
)

Definition at line 227 of file e_jn.c.

{
       int32_t i,hx,ix,lx;
       int32_t sign;
       double a, b, temp;

       EXTRACT_WORDS(hx,lx,x);
       ix = 0x7fffffff&hx;
    /* if Y(n,NaN) is NaN */
       if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
       if((ix|lx)==0) return -HUGE_VAL+x; /* -inf and overflow exception.  */;
       if(hx<0) return zero/(zero*x);
       sign = 1;
       if(n<0){
              n = -n;
              sign = 1 - ((n&1)<<1);
       }
       if(n==0) return(__ieee754_y0(x));
       if(n==1) return(sign*__ieee754_y1(x));
       if(ix==0x7ff00000) return zero;
       if(ix>=0x52D00000) { /* x > 2**302 */
    /* (x >> n**2)
     *     Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
     *     Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
     *     Let s=sin(x), c=cos(x),
     *        xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
     *
     *           n   sin(xn)*sqt2  cos(xn)*sqt2
     *        ----------------------------------
     *           0    s-c           c+s
     *           1   -s-c          -c+s
     *           2   -s+c          -c-s
     *           3    s+c           c-s
     */
              double c;
              double s;
              __sincos (x, &s, &c);
              switch(n&3) {
                  case 0: temp =  s - c; break;
                  case 1: temp = -s - c; break;
                  case 2: temp = -s + c; break;
                  case 3: temp =  s + c; break;
              }
              b = invsqrtpi*temp/__ieee754_sqrt(x);
       } else {
           u_int32_t high;
           a = __ieee754_y0(x);
           b = __ieee754_y1(x);
       /* quit if b is -inf */
           GET_HIGH_WORD(high,b);
           for(i=1;i<n&&high!=0xfff00000;i++){
              temp = b;
              b = ((double)(i+i)/x)*b - a;
              GET_HIGH_WORD(high,b);
              a = temp;
           }
       }
       if(sign>0) return b; else return -b;
}

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Variable Documentation

double invsqrtpi = 5.64189583547756279280e-01 [static]

Definition at line 51 of file e_jn.c.

double one = 1.00000000000000000000e+00 [static]

Definition at line 53 of file e_jn.c.

double two = 2.00000000000000000000e+00 [static]

Definition at line 52 of file e_jn.c.

double zero = 0.00000000000000000000e+00 [static]

Definition at line 58 of file e_jn.c.