| /* @(#)e_jn.c 5.1 93/09/24 */ |
| /* |
| * ==================================================== |
| * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
| * |
| * Developed at SunPro, a Sun Microsystems, Inc. business. |
| * Permission to use, copy, modify, and distribute this |
| * software is freely granted, provided that this notice |
| * is preserved. |
| * ==================================================== |
| */ |
| |
| /* |
| * __ieee754_jn(n, x), __ieee754_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 overflow 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 us 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 <errno.h> |
| #include <math.h> |
| #include <math_private.h> |
| |
| static const double |
| invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ |
| two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ |
| one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ |
| |
| static const double zero = 0.00000000000000000000e+00; |
| |
| double |
| __ieee754_jn (int n, double x) |
| { |
| 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 (__builtin_expect ((ix | ((u_int32_t) (lx | -lx)) >> 31) > 0x7ff00000, 0)) |
| 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 (__builtin_expect ((ix | lx) == 0 || ix >= 0x7ff00000, 0)) |
| /* 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; |
| } |
| } |
| } |
| /* j0() and j1() suffer enormous loss of precision at and |
| * near zero; however, we know that their zero points never |
| * coincide, so just choose the one further away from zero. |
| */ |
| z = __ieee754_j0 (x); |
| w = __ieee754_j1 (x); |
| if (fabs (z) >= fabs (w)) |
| b = (t * z / b); |
| else |
| b = (t * w / a); |
| } |
| } |
| if (sgn == 1) |
| return -b; |
| else |
| return b; |
| } |
| strong_alias (__ieee754_jn, __jn_finite) |
| |
| double |
| __ieee754_yn (int n, double x) |
| { |
| 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 (__builtin_expect ((ix | ((u_int32_t) (lx | -lx)) >> 31) > 0x7ff00000, 0)) |
| return x + x; |
| if (__builtin_expect ((ix | lx) == 0, 0)) |
| return -HUGE_VAL + x; |
| /* -inf and overflow exception. */; |
| if (__builtin_expect (hx < 0, 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 (__builtin_expect (ix == 0x7ff00000, 0)) |
| 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 B is +-Inf, set up errno accordingly. */ |
| if (!__finite (b)) |
| __set_errno (ERANGE); |
| } |
| if (sign > 0) |
| return b; |
| else |
| return -b; |
| } |
| strong_alias (__ieee754_yn, __yn_finite) |
