/* * ==================================================== * 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 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 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 "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 attribute_hidden __ieee754_jn(int n, double x) { int32_t i,hx,ix,lx, sgn; double a, b, temp=0, 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 */ switch(n&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; case 3: temp = cos(x)-sin(x); break; } b = invsqrtpi*temp/sqrt(x); } else { a = __ieee754_j0(x); b = __ieee754_j1(x); for(i=1;i33) /* 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; } /* * wrapper jn(int n, double x) */ #ifndef _IEEE_LIBM double jn(int n, double x) { double z = __ieee754_jn(n, x); if (_LIB_VERSION == _IEEE_ || isnan(x)) return z; if (fabs(x) > X_TLOSS) return __kernel_standard((double)n, x, 38); /* jn(|x|>X_TLOSS,n) */ return z; } #else strong_alias(__ieee754_jn, jn) #endif double attribute_hidden __ieee754_yn(int n, double x) { int32_t i,hx,ix,lx; int32_t sign; double a, b, temp=0; 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 -one/zero; if(hx<0) return zero/zero; 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 */ switch(n&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; case 3: temp = sin(x)+cos(x); break; } b = invsqrtpi*temp/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;i0) return b; else return -b; } /* * wrapper yn(int n, double x) */ #ifndef _IEEE_LIBM double yn(int n, double x) /* wrapper yn */ { double z = __ieee754_yn(n, x); if (_LIB_VERSION == _IEEE_ || isnan(x)) return z; if (x <= 0.0) { if(x == 0.0) /* d= -one/(x-x); */ return __kernel_standard((double)n, x, 12); /* d = zero/(x-x); */ return __kernel_standard((double)n, x, 13); } if (x > X_TLOSS) return __kernel_standard((double)n, x, 39); /* yn(x>X_TLOSS,n) */ return z; } #else strong_alias(__ieee754_yn, yn) #endif