| e_jn.c (302408) | e_jn.c (336196) |
|---|---|
| 1 | |
| 2/* @(#)e_jn.c 1.4 95/01/18 */ 3/* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * 7 * Developed at SunSoft, a Sun Microsystems, Inc. business. 8 * Permission to use, copy, modify, and distribute this | 1/* @(#)e_jn.c 1.4 95/01/18 */ 2/* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunSoft, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this |
| 9 * software is freely granted, provided that this notice | 8 * software is freely granted, provided that this notice |
| 10 * is preserved. 11 * ==================================================== 12 */ 13 14#include <sys/cdefs.h> | 9 * is preserved. 10 * ==================================================== 11 */ 12 13#include <sys/cdefs.h> |
| 15__FBSDID("$FreeBSD: stable/11/lib/msun/src/e_jn.c 279856 2015-03-10 17:10:54Z kargl $"); | 14__FBSDID("$FreeBSD: stable/11/lib/msun/src/e_jn.c 336196 2018-07-11 12:12:49Z markj $"); |
| 16 17/* 18 * __ieee754_jn(n, x), __ieee754_yn(n, x) 19 * floating point Bessel's function of the 1st and 2nd kind 20 * of order n | 15 16/* 17 * __ieee754_jn(n, x), __ieee754_yn(n, x) 18 * floating point Bessel's function of the 1st and 2nd kind 19 * of order n |
| 21 * | 20 * |
| 22 * Special cases: 23 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 24 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 25 * Note 2. About jn(n,x), yn(n,x) 26 * For n=0, j0(x) is called, 27 * for n=1, j1(x) is called, 28 * for n<x, forward recursion us used starting 29 * from values of j0(x) and j1(x). 30 * for n>x, a continued fraction approximation to 31 * j(n,x)/j(n-1,x) is evaluated and then backward 32 * recursion is used starting from a supposed value 33 * for j(n,x). The resulting value of j(0,x) is 34 * compared with the actual value to correct the 35 * supposed value of j(n,x). 36 * 37 * yn(n,x) is similar in all respects, except 38 * that forward recursion is used for all 39 * values of n>1. | 21 * Special cases: 22 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 23 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 24 * Note 2. About jn(n,x), yn(n,x) 25 * For n=0, j0(x) is called, 26 * for n=1, j1(x) is called, 27 * for n<x, forward recursion us used starting 28 * from values of j0(x) and j1(x). 29 * for n>x, a continued fraction approximation to 30 * j(n,x)/j(n-1,x) is evaluated and then backward 31 * recursion is used starting from a supposed value 32 * for j(n,x). The resulting value of j(0,x) is 33 * compared with the actual value to correct the 34 * supposed value of j(n,x). 35 * 36 * yn(n,x) is similar in all respects, except 37 * that forward recursion is used for all 38 * values of n>1. |
| 40 * | |
| 41 */ 42 43#include "math.h" 44#include "math_private.h" 45 46static const volatile double vone = 1, vzero = 0; 47 48static const double --- 12 unchanged lines hidden (view full) --- 61 62 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 63 * Thus, J(-n,x) = J(n,-x) 64 */ 65 EXTRACT_WORDS(hx,lx,x); 66 ix = 0x7fffffff&hx; 67 /* if J(n,NaN) is NaN */ 68 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; | 39 */ 40 41#include "math.h" 42#include "math_private.h" 43 44static const volatile double vone = 1, vzero = 0; 45 46static const double --- 12 unchanged lines hidden (view full) --- 59 60 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 61 * Thus, J(-n,x) = J(n,-x) 62 */ 63 EXTRACT_WORDS(hx,lx,x); 64 ix = 0x7fffffff&hx; 65 /* if J(n,NaN) is NaN */ 66 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; |
| 69 if(n<0){ | 67 if(n<0){ |
| 70 n = -n; 71 x = -x; 72 hx ^= 0x80000000; 73 } 74 if(n==0) return(__ieee754_j0(x)); 75 if(n==1) return(__ieee754_j1(x)); 76 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 77 x = fabs(x); 78 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ 79 b = zero; | 68 n = -n; 69 x = -x; 70 hx ^= 0x80000000; 71 } 72 if(n==0) return(__ieee754_j0(x)); 73 if(n==1) return(__ieee754_j1(x)); 74 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 75 x = fabs(x); 76 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ 77 b = zero; |
| 80 else if((double)n<=x) { | 78 else if((double)n<=x) { |
| 81 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 82 if(ix>=0x52D00000) { /* x > 2**302 */ | 79 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 80 if(ix>=0x52D00000) { /* x > 2**302 */ |
| 83 /* (x >> n**2) | 81 /* (x >> n**2) |
| 84 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 85 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) | 82 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 83 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) |
| 86 * Let s=sin(x), c=cos(x), 87 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then | 84 * Let s=sin(x), c=cos(x), 85 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then |
| 88 * 89 * n sin(xn)*sqt2 cos(xn)*sqt2 90 * ---------------------------------- 91 * 0 s-c c+s 92 * 1 -s-c -c+s 93 * 2 -s+c -c-s 94 * 3 s+c c-s 95 */ 96 switch(n&3) { 97 case 0: temp = cos(x)+sin(x); break; 98 case 1: temp = -cos(x)+sin(x); break; 99 case 2: temp = -cos(x)-sin(x); break; 100 case 3: temp = cos(x)-sin(x); break; 101 } 102 b = invsqrtpi*temp/sqrt(x); | 86 * 87 * n sin(xn)*sqt2 cos(xn)*sqt2 88 * ---------------------------------- 89 * 0 s-c c+s 90 * 1 -s-c -c+s 91 * 2 -s+c -c-s 92 * 3 s+c c-s 93 */ 94 switch(n&3) { 95 case 0: temp = cos(x)+sin(x); break; 96 case 1: temp = -cos(x)+sin(x); break; 97 case 2: temp = -cos(x)-sin(x); break; 98 case 3: temp = cos(x)-sin(x); break; 99 } 100 b = invsqrtpi*temp/sqrt(x); |
| 103 } else { | 101 } else { |
| 104 a = __ieee754_j0(x); 105 b = __ieee754_j1(x); 106 for(i=1;i<n;i++){ 107 temp = b; 108 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 109 a = temp; 110 } 111 } 112 } else { 113 if(ix<0x3e100000) { /* x < 2**-29 */ | 102 a = __ieee754_j0(x); 103 b = __ieee754_j1(x); 104 for(i=1;i<n;i++){ 105 temp = b; 106 b = b*((double)(i+i)/x) - a; /* avoid underflow */ 107 a = temp; 108 } 109 } 110 } else { 111 if(ix<0x3e100000) { /* x < 2**-29 */ |
| 114 /* x is tiny, return the first Taylor expansion of J(n,x) | 112 /* x is tiny, return the first Taylor expansion of J(n,x) |
| 115 * J(n,x) = 1/n!*(x/2)^n - ... 116 */ 117 if(n>33) /* underflow */ 118 b = zero; 119 else { 120 temp = x*0.5; b = temp; 121 for (a=one,i=2;i<=n;i++) { 122 a *= (double)i; /* a = n! */ 123 b *= temp; /* b = (x/2)^n */ 124 } 125 b = b/a; 126 } 127 } else { 128 /* use backward recurrence */ | 113 * J(n,x) = 1/n!*(x/2)^n - ... 114 */ 115 if(n>33) /* underflow */ 116 b = zero; 117 else { 118 temp = x*0.5; b = temp; 119 for (a=one,i=2;i<=n;i++) { 120 a *= (double)i; /* a = n! */ 121 b *= temp; /* b = (x/2)^n */ 122 } 123 b = b/a; 124 } 125 } else { 126 /* use backward recurrence */ |
| 129 /* x x^2 x^2 | 127 /* x x^2 x^2 |
| 130 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 131 * 2n - 2(n+1) - 2(n+2) 132 * | 128 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 129 * 2n - 2(n+1) - 2(n+2) 130 * |
| 133 * 1 1 1 | 131 * 1 1 1 |
| 134 * (for large x) = ---- ------ ------ ..... 135 * 2n 2(n+1) 2(n+2) | 132 * (for large x) = ---- ------ ------ ..... 133 * 2n 2(n+1) 2(n+2) |
| 136 * -- - ------ - ------ - | 134 * -- - ------ - ------ - |
| 137 * x x x 138 * 139 * Let w = 2n/x and h=2/x, then the above quotient 140 * is equal to the continued fraction: 141 * 1 142 * = ----------------------- 143 * 1 144 * w - ----------------- 145 * 1 146 * w+h - --------- 147 * w+2h - ... 148 * 149 * To determine how many terms needed, let 150 * Q(0) = w, Q(1) = w(w+h) - 1, 151 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), | 135 * x x x 136 * 137 * Let w = 2n/x and h=2/x, then the above quotient 138 * is equal to the continued fraction: 139 * 1 140 * = ----------------------- 141 * 1 142 * w - ----------------- 143 * 1 144 * w+h - --------- 145 * w+2h - ... 146 * 147 * To determine how many terms needed, let 148 * Q(0) = w, Q(1) = w(w+h) - 1, 149 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), |
| 152 * When Q(k) > 1e4 good for single 153 * When Q(k) > 1e9 good for double 154 * When Q(k) > 1e17 good for quadruple | 150 * When Q(k) > 1e4 good for single 151 * When Q(k) > 1e9 good for double 152 * When Q(k) > 1e17 good for quadruple |
| 155 */ 156 /* determine k */ 157 double t,v; 158 double q0,q1,h,tmp; int32_t k,m; 159 w = (n+n)/(double)x; h = 2.0/(double)x; 160 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 161 while(q1<1.0e9) { 162 k += 1; z += h; --- 69 unchanged lines hidden (view full) --- 232 if(n<0){ 233 n = -n; 234 sign = 1 - ((n&1)<<1); 235 } 236 if(n==0) return(__ieee754_y0(x)); 237 if(n==1) return(sign*__ieee754_y1(x)); 238 if(ix==0x7ff00000) return zero; 239 if(ix>=0x52D00000) { /* x > 2**302 */ | 153 */ 154 /* determine k */ 155 double t,v; 156 double q0,q1,h,tmp; int32_t k,m; 157 w = (n+n)/(double)x; h = 2.0/(double)x; 158 q0 = w; z = w+h; q1 = w*z - 1.0; k=1; 159 while(q1<1.0e9) { 160 k += 1; z += h; --- 69 unchanged lines hidden (view full) --- 230 if(n<0){ 231 n = -n; 232 sign = 1 - ((n&1)<<1); 233 } 234 if(n==0) return(__ieee754_y0(x)); 235 if(n==1) return(sign*__ieee754_y1(x)); 236 if(ix==0x7ff00000) return zero; 237 if(ix>=0x52D00000) { /* x > 2**302 */ |
| 240 /* (x >> n**2) | 238 /* (x >> n**2) |
| 241 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 242 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) | 239 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) 240 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) |
| 243 * Let s=sin(x), c=cos(x), 244 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then | 241 * Let s=sin(x), c=cos(x), 242 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then |
| 245 * 246 * n sin(xn)*sqt2 cos(xn)*sqt2 247 * ---------------------------------- 248 * 0 s-c c+s 249 * 1 -s-c -c+s 250 * 2 -s+c -c-s 251 * 3 s+c c-s 252 */ --- 22 unchanged lines hidden --- | 243 * 244 * n sin(xn)*sqt2 cos(xn)*sqt2 245 * ---------------------------------- 246 * 0 s-c c+s 247 * 1 -s-c -c+s 248 * 2 -s+c -c-s 249 * 3 s+c c-s 250 */ --- 22 unchanged lines hidden --- |