1 // Copyright 2010 The Go Authors. All rights reserved.
2 // Use of this source code is governed by a BSD-style
3 // license that can be found in the LICENSE file.
4 5 package math
6 7 /*
8 Bessel function of the first and second kinds of order n.
9 */
10 11 // The original C code and the long comment below are
12 // from FreeBSD's /usr/src/lib/msun/src/e_jn.c and
13 // came with this notice. The go code is a simplified
14 // version of the original C.
15 //
16 // ====================================================
17 // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
18 //
19 // Developed at SunPro, a Sun Microsystems, Inc. business.
20 // Permission to use, copy, modify, and distribute this
21 // software is freely granted, provided that this notice
22 // is preserved.
23 // ====================================================
24 //
25 // __ieee754_jn(n, x), __ieee754_yn(n, x)
26 // floating point Bessel's function of the 1st and 2nd kind
27 // of order n
28 //
29 // Special cases:
30 // y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
31 // y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
32 // Note 2. About jn(n,x), yn(n,x)
33 // For n=0, j0(x) is called,
34 // for n=1, j1(x) is called,
35 // for n<x, forward recursion is used starting
36 // from values of j0(x) and j1(x).
37 // for n>x, a continued fraction approximation to
38 // j(n,x)/j(n-1,x) is evaluated and then backward
39 // recursion is used starting from a supposed value
40 // for j(n,x). The resulting value of j(0,x) is
41 // compared with the actual value to correct the
42 // supposed value of j(n,x).
43 //
44 // yn(n,x) is similar in all respects, except
45 // that forward recursion is used for all
46 // values of n>1.
47 48 // Jn returns the order-n Bessel function of the first kind.
49 //
50 // Special cases are:
51 //
52 // Jn(n, ±Inf) = 0
53 // Jn(n, NaN) = NaN
54 func Jn(n int, x float64) float64 {
55 const (
56 TwoM29 = 1.0 / (1 << 29) // 2**-29 0x3e10000000000000
57 Two302 = 1 << 302 // 2**302 0x52D0000000000000
58 )
59 // special cases
60 switch {
61 case IsNaN(x):
62 return x
63 case IsInf(x, 0):
64 return 0
65 }
66 // J(-n, x) = (-1)**n * J(n, x), J(n, -x) = (-1)**n * J(n, x)
67 // Thus, J(-n, x) = J(n, -x)
68 69 if n == 0 {
70 return J0(x)
71 }
72 if x == 0 {
73 return 0
74 }
75 if n < 0 {
76 n, x = -n, -x
77 }
78 if n == 1 {
79 return J1(x)
80 }
81 sign := false
82 if x < 0 {
83 x = -x
84 if n&1 == 1 {
85 sign = true // odd n and negative x
86 }
87 }
88 var b float64
89 if float64(n) <= x {
90 // Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
91 if x >= Two302 { // x > 2**302
92 93 // (x >> n**2)
94 // Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
95 // Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
96 // Let s=sin(x), c=cos(x),
97 // xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
98 //
99 // n sin(xn)*sqt2 cos(xn)*sqt2
100 // ----------------------------------
101 // 0 s-c c+s
102 // 1 -s-c -c+s
103 // 2 -s+c -c-s
104 // 3 s+c c-s
105 106 var temp float64
107 switch s, c := Sincos(x); n & 3 {
108 case 0:
109 temp = c + s
110 case 1:
111 temp = -c + s
112 case 2:
113 temp = -c - s
114 case 3:
115 temp = c - s
116 }
117 b = (1 / SqrtPi) * temp / Sqrt(x)
118 } else {
119 b = J1(x)
120 for i, a := 1, J0(x); i < n; i++ {
121 a, b = b, b*(float64(i+i)/x)-a // avoid underflow
122 }
123 }
124 } else {
125 if x < TwoM29 { // x < 2**-29
126 // x is tiny, return the first Taylor expansion of J(n,x)
127 // J(n,x) = 1/n!*(x/2)**n - ...
128 129 if n > 33 { // underflow
130 b = 0
131 } else {
132 temp := x * 0.5
133 b = temp
134 a := 1.0
135 for i := 2; i <= n; i++ {
136 a *= float64(i) // a = n!
137 b *= temp // b = (x/2)**n
138 }
139 b /= a
140 }
141 } else {
142 // use backward recurrence
143 // x x**2 x**2
144 // J(n,x)/J(n-1,x) = ---- ------ ------ .....
145 // 2n - 2(n+1) - 2(n+2)
146 //
147 // 1 1 1
148 // (for large x) = ---- ------ ------ .....
149 // 2n 2(n+1) 2(n+2)
150 // -- - ------ - ------ -
151 // x x x
152 //
153 // Let w = 2n/x and h=2/x, then the above quotient
154 // is equal to the continued fraction:
155 // 1
156 // = -----------------------
157 // 1
158 // w - -----------------
159 // 1
160 // w+h - ---------
161 // w+2h - ...
162 //
163 // To determine how many terms needed, let
164 // Q(0) = w, Q(1) = w(w+h) - 1,
165 // Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
166 // When Q(k) > 1e4 good for single
167 // When Q(k) > 1e9 good for double
168 // When Q(k) > 1e17 good for quadruple
169 170 // determine k
171 w := float64(n+n) / x
172 h := 2 / x
173 q0 := w
174 z := w + h
175 q1 := w*z - 1
176 k := 1
177 for q1 < 1e9 {
178 k++
179 z += h
180 q0, q1 = q1, z*q1-q0
181 }
182 m := n + n
183 t := 0.0
184 for i := 2 * (n + k); i >= m; i -= 2 {
185 t = 1 / (float64(i)/x - t)
186 }
187 a := t
188 b = 1
189 // estimate log((2/x)**n*n!) = n*log(2/x)+n*ln(n)
190 // Hence, if n*(log(2n/x)) > ...
191 // single 8.8722839355e+01
192 // double 7.09782712893383973096e+02
193 // long double 1.1356523406294143949491931077970765006170e+04
194 // then recurrent value may overflow and the result is
195 // likely underflow to zero
196 197 tmp := float64(n)
198 v := 2 / x
199 tmp = tmp * Log(Abs(v*tmp))
200 if tmp < 7.09782712893383973096e+02 {
201 for i := n - 1; i > 0; i-- {
202 di := float64(i + i)
203 a, b = b, b*di/x-a
204 }
205 } else {
206 for i := n - 1; i > 0; i-- {
207 di := float64(i + i)
208 a, b = b, b*di/x-a
209 // scale b to avoid spurious overflow
210 if b > 1e100 {
211 a /= b
212 t /= b
213 b = 1
214 }
215 }
216 }
217 b = t * J0(x) / b
218 }
219 }
220 if sign {
221 return -b
222 }
223 return b
224 }
225 226 // Yn returns the order-n Bessel function of the second kind.
227 //
228 // Special cases are:
229 //
230 // Yn(n, +Inf) = 0
231 // Yn(n ≥ 0, 0) = -Inf
232 // Yn(n < 0, 0) = +Inf if n is odd, -Inf if n is even
233 // Yn(n, x < 0) = NaN
234 // Yn(n, NaN) = NaN
235 func Yn(n int, x float64) float64 {
236 const Two302 = 1 << 302 // 2**302 0x52D0000000000000
237 // special cases
238 switch {
239 case x < 0 || IsNaN(x):
240 return NaN()
241 case IsInf(x, 1):
242 return 0
243 }
244 245 if n == 0 {
246 return Y0(x)
247 }
248 if x == 0 {
249 if n < 0 && n&1 == 1 {
250 return Inf(1)
251 }
252 return Inf(-1)
253 }
254 sign := false
255 if n < 0 {
256 n = -n
257 if n&1 == 1 {
258 sign = true // sign true if n < 0 && |n| odd
259 }
260 }
261 if n == 1 {
262 if sign {
263 return -Y1(x)
264 }
265 return Y1(x)
266 }
267 var b float64
268 if x >= Two302 { // x > 2**302
269 // (x >> n**2)
270 // Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
271 // Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
272 // Let s=sin(x), c=cos(x),
273 // xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
274 //
275 // n sin(xn)*sqt2 cos(xn)*sqt2
276 // ----------------------------------
277 // 0 s-c c+s
278 // 1 -s-c -c+s
279 // 2 -s+c -c-s
280 // 3 s+c c-s
281 282 var temp float64
283 switch s, c := Sincos(x); n & 3 {
284 case 0:
285 temp = s - c
286 case 1:
287 temp = -s - c
288 case 2:
289 temp = -s + c
290 case 3:
291 temp = s + c
292 }
293 b = (1 / SqrtPi) * temp / Sqrt(x)
294 } else {
295 a := Y0(x)
296 b = Y1(x)
297 // quit if b is -inf
298 for i := 1; i < n && !IsInf(b, -1); i++ {
299 a, b = b, (float64(i+i)/x)*b-a
300 }
301 }
302 if sign {
303 return -b
304 }
305 return b
306 }
307