Source file src/pkg/math/jn.go
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 // Jn(n, ±Inf) = 0
52 // Jn(n, NaN) = NaN
53 func Jn(n int, x float64) float64 {
54 const (
55 TwoM29 = 1.0 / (1 << 29) // 2**-29 0x3e10000000000000
56 Two302 = 1 << 302 // 2**302 0x52D0000000000000
57 )
58 // special cases
59 switch {
60 case IsNaN(x):
61 return x
62 case IsInf(x, 0):
63 return 0
64 }
65 // J(-n, x) = (-1)**n * J(n, x), J(n, -x) = (-1)**n * J(n, x)
66 // Thus, J(-n, x) = J(n, -x)
67
68 if n == 0 {
69 return J0(x)
70 }
71 if x == 0 {
72 return 0
73 }
74 if n < 0 {
75 n, x = -n, -x
76 }
77 if n == 1 {
78 return J1(x)
79 }
80 sign := false
81 if x < 0 {
82 x = -x
83 if n&1 == 1 {
84 sign = true // odd n and negative x
85 }
86 }
87 var b float64
88 if float64(n) <= x {
89 // Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
90 if x >= Two302 { // x > 2**302
91
92 // (x >> n**2)
93 // Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
94 // Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
95 // Let s=sin(x), c=cos(x),
96 // xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
97 //
98 // n sin(xn)*sqt2 cos(xn)*sqt2
99 // ----------------------------------
100 // 0 s-c c+s
101 // 1 -s-c -c+s
102 // 2 -s+c -c-s
103 // 3 s+c c-s
104
105 var temp float64
106 switch n & 3 {
107 case 0:
108 temp = Cos(x) + Sin(x)
109 case 1:
110 temp = -Cos(x) + Sin(x)
111 case 2:
112 temp = -Cos(x) - Sin(x)
113 case 3:
114 temp = Cos(x) - Sin(x)
115 }
116 b = (1 / SqrtPi) * temp / Sqrt(x)
117 } else {
118 b = J1(x)
119 for i, a := 1, J0(x); i < n; i++ {
120 a, b = b, b*(float64(i+i)/x)-a // avoid underflow
121 }
122 }
123 } else {
124 if x < TwoM29 { // x < 2**-29
125 // x is tiny, return the first Taylor expansion of J(n,x)
126 // J(n,x) = 1/n!*(x/2)**n - ...
127
128 if n > 33 { // underflow
129 b = 0
130 } else {
131 temp := x * 0.5
132 b = temp
133 a := 1.0
134 for i := 2; i <= n; i++ {
135 a *= float64(i) // a = n!
136 b *= temp // b = (x/2)**n
137 }
138 b /= a
139 }
140 } else {
141 // use backward recurrence
142 // x x**2 x**2
143 // J(n,x)/J(n-1,x) = ---- ------ ------ .....
144 // 2n - 2(n+1) - 2(n+2)
145 //
146 // 1 1 1
147 // (for large x) = ---- ------ ------ .....
148 // 2n 2(n+1) 2(n+2)
149 // -- - ------ - ------ -
150 // x x x
151 //
152 // Let w = 2n/x and h=2/x, then the above quotient
153 // is equal to the continued fraction:
154 // 1
155 // = -----------------------
156 // 1
157 // w - -----------------
158 // 1
159 // w+h - ---------
160 // w+2h - ...
161 //
162 // To determine how many terms needed, let
163 // Q(0) = w, Q(1) = w(w+h) - 1,
164 // Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
165 // When Q(k) > 1e4 good for single
166 // When Q(k) > 1e9 good for double
167 // When Q(k) > 1e17 good for quadruple
168
169 // determine k
170 w := float64(n+n) / x
171 h := 2 / x
172 q0 := w
173 z := w + h
174 q1 := w*z - 1
175 k := 1
176 for q1 < 1e9 {
177 k += 1
178 z += h
179 q0, q1 = q1, z*q1-q0
180 }
181 m := n + n
182 t := 0.0
183 for i := 2 * (n + k); i >= m; i -= 2 {
184 t = 1 / (float64(i)/x - t)
185 }
186 a := t
187 b = 1
188 // estimate log((2/x)**n*n!) = n*log(2/x)+n*ln(n)
189 // Hence, if n*(log(2n/x)) > ...
190 // single 8.8722839355e+01
191 // double 7.09782712893383973096e+02
192 // long double 1.1356523406294143949491931077970765006170e+04
193 // then recurrent value may overflow and the result is
194 // likely underflow to zero
195
196 tmp := float64(n)
197 v := 2 / x
198 tmp = tmp * Log(Abs(v*tmp))
199 if tmp < 7.09782712893383973096e+02 {
200 for i := n - 1; i > 0; i-- {
201 di := float64(i + i)
202 a, b = b, b*di/x-a
203 di -= 2
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 di -= 2
210 // scale b to avoid spurious overflow
211 if b > 1e100 {
212 a /= b
213 t /= b
214 b = 1
215 }
216 }
217 }
218 b = t * J0(x) / b
219 }
220 }
221 if sign {
222 return -b
223 }
224 return b
225 }
226
227 // Yn returns the order-n Bessel function of the second kind.
228 //
229 // Special cases are:
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 // Y1(n, x < 0) = NaN
234 // Y1(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 n & 3 {
284 case 0:
285 temp = Sin(x) - Cos(x)
286 case 1:
287 temp = -Sin(x) - Cos(x)
288 case 2:
289 temp = -Sin(x) + Cos(x)
290 case 3:
291 temp = Sin(x) + Cos(x)
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 }