/* @(#)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 nx, 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 #include #include #include 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, ret; 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 (__glibc_unlikely ((ix | ((uint32_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); { SET_RESTORE_ROUND (FE_TONEAREST); if (__glibc_unlikely ((ix | lx) == 0 || ix >= 0x7ff00000)) /* if x is 0 or inf */ return sgn == 1 ? -zero : 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 / 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) ret = -b; else ret = b; ret = math_narrow_eval (ret); } if (ret == 0) { ret = math_narrow_eval (__copysign (DBL_MIN, ret) * DBL_MIN); __set_errno (ERANGE); } else math_check_force_underflow (ret); return ret; } 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, ret; EXTRACT_WORDS (hx, lx, x); ix = 0x7fffffff & hx; /* if Y(n,NaN) is NaN */ if (__glibc_unlikely ((ix | ((uint32_t) (lx | -lx)) >> 31) > 0x7ff00000)) return x + x; sign = 1; if (n < 0) { n = -n; sign = 1 - ((n & 1) << 1); } if (n == 0) return (__ieee754_y0 (x)); if (__glibc_unlikely ((ix | lx) == 0)) return -sign / zero; /* -inf and overflow exception. */; if (__glibc_unlikely (hx < 0)) return zero / (zero * x); { SET_RESTORE_ROUND (FE_TONEAREST); if (n == 1) { ret = sign * __ieee754_y1 (x); goto out; } if (__glibc_unlikely (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 / sqrt (x); } else { uint32_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 (!isfinite (b)) __set_errno (ERANGE); } if (sign > 0) ret = b; else ret = -b; } out: if (isinf (ret)) ret = __copysign (DBL_MAX, ret) * DBL_MAX; return ret; } strong_alias (__ieee754_yn, __yn_finite)