RTAI API: base/math/e_jn.c Source File

00001 /* @(#)e_jn.c 5.1 93/09/24 */ 00002 /* 00003 * ==================================================== 00004 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 00005 * 00006 * Developed at SunPro, a Sun Microsystems, Inc. business. 00007 * Permission to use, copy, modify, and distribute this 00008 * software is freely granted, provided that this notice 00009 * is preserved. 00010 * ==================================================== 00011 */ 00012 00013 #if defined(LIBM_SCCS) && !defined(lint) 00014 static char rcsid[] = "$NetBSD: e_jn.c,v 1.9 1995/05/10 20:45:34 jtc Exp $"; 00015 #endif 00016 00017 /* 00018 * __ieee754_jn(n, x), __ieee754_yn(n, x) 00019 * floating point Bessel's function of the 1st and 2nd kind 00020 * of order n 00021 * 00022 * Special cases: 00023 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 00024 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 00025 * Note 2. About jn(n,x), yn(n,x) 00026 * For n=0, j0(x) is called, 00027 * for n=1, j1(x) is called, 00028 * for n<x, forward recursion us used starting 00029 * from values of j0(x) and j1(x). 00030 * for n>x, a continued fraction approximation to 00031 * j(n,x)/j(n-1,x) is evaluated and then backward 00032 * recursion is used starting from a supposed value 00033 * for j(n,x). The resulting value of j(0,x) is 00034 * compared with the actual value to correct the 00035 * supposed value of j(n,x). 00036 * 00037 * yn(n,x) is similar in all respects, except 00038 * that forward recursion is used for all 00039 * values of n>1. 00040 * 00041 */ 00042 00043 #include "math.h" 00044 #include "mathP.h" 00045 00046 #ifdef __STDC__ 00047 static const double 00048 #else 00049 static double 00050 #endif 00051 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ 00052 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 00053 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ 00054 00055 #ifdef __STDC__ 00056 static const double zero = 0.00000000000000000000e+00; 00057 #else 00058 static double zero = 0.00000000000000000000e+00; 00059 #endif 00060 00061 #ifdef __STDC__ 00062 double __ieee754_jn(int n, double x) 00063 #else 00064 double __ieee754_jn(n,x) 00065 int n; double x; 00066 #endif 00067 { 00068 int32_t i,hx,ix,lx, sgn; 00069 double a, b, temp, di; 00070 double z, w; 00071 00072 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 00073 * Thus, J(-n,x) = J(n,-x) 00074 */ 00075 EXTRACT_WORDS(hx,lx,x); 00076 ix = 0x7fffffff&hx; 00077 /* if J(n,NaN) is NaN */ 00078 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 00079 if(n<0){ 00080 n = -n; 00081 x = -x; 00082 hx ^= 0x80000000; 00083 } 00084 if(n==0) return(__ieee754_j0(x)); 00085 if(n==1) return(__ieee754_j1(x)); 00086 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 00087 x = fabs(x); 00088 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ 00089 b = zero; 00090 else if((double)n<=x) { 00091 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 00092 if(ix>=0x52D00000) { /* x > 2**302 */ 00093 /* (x >> n**2) 00094 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 00095 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 00096 * Let s=sin(x), c=cos(x), 00097 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 00098 * 00099 * n sin(xn)*sqt2 cos(xn)*sqt2 00100 * ---------------------------------- 00101 * 0 s-c c+s 00102 * 1 -s-c -c+s 00103 * 2 -s+c -c-s 00104 * 3 s+c c-s 00105 */ 00106 switch(n&3) { 00107 case 0: temp = cos(x)+sin(x); break; 00108 case 1: temp = -cos(x)+sin(x); break; 00109 case 2: temp = -cos(x)-sin(x); break; 00110 case 3: temp = cos(x)-sin(x); break; 00111 default: temp = 0.0; 00112 } 00113 b = invsqrtpi*temp/sqrt(x); 00114 } else { 00115 a = __ieee754_j0(x); 00116 b = __ieee754_j1(x); 00117 for(i=1;i<n;i++){ 00118 temp = b; 00119 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 00120 a = temp; 00121 } 00122 } 00123 } else { 00124 if(ix<0x3e100000) { /* x < 2**-29 */ 00125 /* x is tiny, return the first Taylor expansion of J(n,x) 00126 * J(n,x) = 1/n!*(x/2)^n - ... 00127 */ 00128 if(n>33) /* underflow */ 00129 b = zero; 00130 else { 00131 temp = x*0.5; b = temp; 00132 for (a=one,i=2;i<=n;i++) { 00133 a *= (double)i; /* a = n! */ 00134 b *= temp; /* b = (x/2)^n */ 00135 } 00136 b = b/a; 00137 } 00138 } else { 00139 /* use backward recurrence */ 00140 /* x x^2 x^2 00141 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 00142 * 2n - 2(n+1) - 2(n+2) 00143 * 00144 * 1 1 1 00145 * (for large x) = ---- ------ ------ ..... 00146 * 2n 2(n+1) 2(n+2) 00147 * -- - ------ - ------ - 00148 * x x x 00149 * 00150 * Let w = 2n/x and h=2/x, then the above quotient 00151 * is equal to the continued fraction: 00152 * 1 00153 * = ----------------------- 00154 * 1 00155 * w - ----------------- 00156 * 1 00157 * w+h - --------- 00158 * w+2h - ... 00159 * 00160 * To determine how many terms needed, let 00161 * Q(0) = w, Q(1) = w(w+h) - 1, 00162 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 00163 * When Q(k) > 1e4 good for single 00164 * When Q(k) > 1e9 good for double 00165 * When Q(k) > 1e17 good for quadruple 00166 */ 00167 /* determine k */ 00168 double t,v; 00169 double q0,q1,h,tmp; int32_t k,m; 00170 w = (n+n)/(double)x; h = 2.0/(double)x; 00171 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 00172 while(q1<1.0e9) { 00173 k += 1; z += h; 00174 tmp = z*q1 - q0; 00175 q0 = q1; 00176 q1 = tmp; 00177 } 00178 m = n+n; 00179 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 00180 a = t; 00181 b = one; 00182 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 00183 * Hence, if n*(log(2n/x)) > ... 00184 * single 8.8722839355e+01 00185 * double 7.09782712893383973096e+02 00186 * long double 1.1356523406294143949491931077970765006170e+04 00187 * then recurrent value may overflow and the result is 00188 * likely underflow to zero 00189 */ 00190 tmp = n; 00191 v = two/x; 00192 tmp = tmp*__ieee754_log(fabs(v*tmp)); 00193 if(tmp<7.09782712893383973096e+02) { 00194 for(i=n-1,di=(double)(i+i);i>0;i--){ 00195 temp = b; 00196 b *= di; 00197 b = b/x - a; 00198 a = temp; 00199 di -= two; 00200 } 00201 } else { 00202 for(i=n-1,di=(double)(i+i);i>0;i--){ 00203 temp = b; 00204 b *= di; 00205 b = b/x - a; 00206 a = temp; 00207 di -= two; 00208 /* scale b to avoid spurious overflow */ 00209 if(b>1e100) { 00210 a /= b; 00211 t /= b; 00212 b = one; 00213 } 00214 } 00215 } 00216 b = (t*__ieee754_j0(x)/b); 00217 } 00218 } 00219 if(sgn==1) return -b; else return b; 00220 } 00221 00222 #ifdef __STDC__ 00223 double __ieee754_yn(int n, double x) 00224 #else 00225 double __ieee754_yn(n,x) 00226 int n; double x; 00227 #endif 00228 { 00229 int32_t i,hx,ix,lx; 00230 int32_t sign; 00231 double a, b, temp; 00232 00233 EXTRACT_WORDS(hx,lx,x); 00234 ix = 0x7fffffff&hx; 00235 /* if Y(n,NaN) is NaN */ 00236 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; 00237 if((ix|lx)==0) return -one/zero; 00238 if(hx<0) return zero/zero; 00239 sign = 1; 00240 if(n<0){ 00241 n = -n; 00242 sign = 1 - ((n&1)<<1); 00243 } 00244 if(n==0) return(__ieee754_y0(x)); 00245 if(n==1) return(sign*__ieee754_y1(x)); 00246 if(ix==0x7ff00000) return zero; 00247 if(ix>=0x52D00000) { /* x > 2**302 */ 00248 /* (x >> n**2) 00249 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 00250 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) 00251 * Let s=sin(x), c=cos(x), 00252 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then 00253 * 00254 * n sin(xn)*sqt2 cos(xn)*sqt2 00255 * ---------------------------------- 00256 * 0 s-c c+s 00257 * 1 -s-c -c+s 00258 * 2 -s+c -c-s 00259 * 3 s+c c-s 00260 */ 00261 switch(n&3) { 00262 case 0: temp = sin(x)-cos(x); break; 00263 case 1: temp = -sin(x)-cos(x); break; 00264 case 2: temp = -sin(x)+cos(x); break; 00265 case 3: temp = sin(x)+cos(x); break; 00266 default: temp = 0.0; 00267 } 00268 b = invsqrtpi*temp/sqrt(x); 00269 } else { 00270 u_int32_t high; 00271 a = __ieee754_y0(x); 00272 b = __ieee754_y1(x); 00273 /* quit if b is -inf */ 00274 GET_HIGH_WORD(high,b); 00275 for(i=1;i<n&&high!=0xfff00000;i++){ 00276 temp = b; 00277 b = ((double)(i+i)/x)*b - a; 00278 GET_HIGH_WORD(high,b); 00279 a = temp; 00280 } 00281 } 00282 if(sign>0) return b; else return -b; 00283 }