| /* origin: FreeBSD /usr/src/lib/msun/src/e_jn.c */ |
| /* |
| * ==================================================== |
| * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
| * |
| * Developed at SunSoft, a Sun Microsystems, Inc. business. |
| * Permission to use, copy, modify, and distribute this |
| * software is freely granted, provided that this notice |
| * is preserved. |
| * ==================================================== |
| */ |
| /* |
| * jn(n, x), yn(n, x) |
| * floating point Bessel's function of the 1st and 2nd kind |
| * of order n |
| * |
| * Special cases: |
| * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; |
| * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. |
| * Note 2. About jn(n,x), yn(n,x) |
| * For n=0, j0(x) is called, |
| * for n=1, j1(x) is called, |
| * for n<=x, forward recursion is used starting |
| * from values of j0(x) and j1(x). |
| * for n>x, a continued fraction approximation to |
| * j(n,x)/j(n-1,x) is evaluated and then backward |
| * recursion is used starting from a supposed value |
| * for j(n,x). The resulting value of j(0,x) is |
| * compared with the actual value to correct the |
| * supposed value of j(n,x). |
| * |
| * yn(n,x) is similar in all respects, except |
| * that forward recursion is used for all |
| * values of n>1. |
| */ |
| |
| #include "libm.h" |
| |
| static const double invsqrtpi = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */ |
| |
| double jn(int n, double x) |
| { |
| uint32_t ix, lx; |
| int nm1, i, sign; |
| double a, b, temp; |
| |
| EXTRACT_WORDS(ix, lx, x); |
| sign = ix>>31; |
| ix &= 0x7fffffff; |
| |
| if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */ |
| return x; |
| |
| /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) |
| * Thus, J(-n,x) = J(n,-x) |
| */ |
| /* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */ |
| if (n == 0) |
| return j0(x); |
| if (n < 0) { |
| nm1 = -(n+1); |
| x = -x; |
| sign ^= 1; |
| } else |
| nm1 = n-1; |
| if (nm1 == 0) |
| return j1(x); |
| |
| sign &= n; /* even n: 0, odd n: signbit(x) */ |
| x = fabs(x); |
| if ((ix|lx) == 0 || ix == 0x7ff00000) /* if x is 0 or inf */ |
| b = 0.0; |
| else if (nm1 < 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 |
| */ |
| switch(nm1&3) { |
| case 0: temp = -cos(x)+sin(x); break; |
| case 1: temp = -cos(x)-sin(x); break; |
| case 2: temp = cos(x)-sin(x); break; |
| default: |
| case 3: temp = cos(x)+sin(x); break; |
| } |
| b = invsqrtpi*temp/sqrt(x); |
| } else { |
| a = j0(x); |
| b = j1(x); |
| for (i=0; i<nm1; ) { |
| i++; |
| temp = b; |
| b = b*(2.0*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 (nm1 > 32) /* underflow */ |
| b = 0.0; |
| else { |
| temp = x*0.5; |
| b = temp; |
| a = 1.0; |
| for (i=2; i<=nm1+1; 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,q0,q1,w,h,z,tmp,nf; |
| int k; |
| |
| nf = nm1 + 1.0; |
| w = 2*nf/x; |
| h = 2/x; |
| z = w+h; |
| q0 = w; |
| q1 = w*z - 1.0; |
| k = 1; |
| while (q1 < 1.0e9) { |
| k += 1; |
| z += h; |
| tmp = z*q1 - q0; |
| q0 = q1; |
| q1 = tmp; |
| } |
| for (t=0.0, i=k; i>=0; i--) |
| t = 1/(2*(i+nf)/x - t); |
| a = t; |
| b = 1.0; |
| /* 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 = nf*log(fabs(w)); |
| if (tmp < 7.09782712893383973096e+02) { |
| for (i=nm1; i>0; i--) { |
| temp = b; |
| b = b*(2.0*i)/x - a; |
| a = temp; |
| } |
| } else { |
| for (i=nm1; i>0; i--) { |
| temp = b; |
| b = b*(2.0*i)/x - a; |
| a = temp; |
| /* scale b to avoid spurious overflow */ |
| if (b > 0x1p500) { |
| a /= b; |
| t /= b; |
| b = 1.0; |
| } |
| } |
| } |
| z = j0(x); |
| w = j1(x); |
| if (fabs(z) >= fabs(w)) |
| b = t*z/b; |
| else |
| b = t*w/a; |
| } |
| } |
| return sign ? -b : b; |
| } |
| |
| |
| double yn(int n, double x) |
| { |
| uint32_t ix, lx, ib; |
| int nm1, sign, i; |
| double a, b, temp; |
| |
| EXTRACT_WORDS(ix, lx, x); |
| sign = ix>>31; |
| ix &= 0x7fffffff; |
| |
| if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */ |
| return x; |
| if (sign && (ix|lx)!=0) /* x < 0 */ |
| return 0/0.0; |
| if (ix == 0x7ff00000) |
| return 0.0; |
| |
| if (n == 0) |
| return y0(x); |
| if (n < 0) { |
| nm1 = -(n+1); |
| sign = n&1; |
| } else { |
| nm1 = n-1; |
| sign = 0; |
| } |
| if (nm1 == 0) |
| return sign ? -y1(x) : y1(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 |
| */ |
| switch(nm1&3) { |
| case 0: temp = -sin(x)-cos(x); break; |
| case 1: temp = -sin(x)+cos(x); break; |
| case 2: temp = sin(x)+cos(x); break; |
| default: |
| case 3: temp = sin(x)-cos(x); break; |
| } |
| b = invsqrtpi*temp/sqrt(x); |
| } else { |
| a = y0(x); |
| b = y1(x); |
| /* quit if b is -inf */ |
| GET_HIGH_WORD(ib, b); |
| for (i=0; i<nm1 && ib!=0xfff00000; ){ |
| i++; |
| temp = b; |
| b = (2.0*i/x)*b - a; |
| GET_HIGH_WORD(ib, b); |
| a = temp; |
| } |
| } |
| return sign ? -b : b; |
| } |
