summaryrefslogtreecommitdiffstats
path: root/src/math/jn.go
blob: b1aca8ff6be74722ea1e76349f6f2470626c307f (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.

package math

/*
	Bessel function of the first and second kinds of order n.
*/

// The original C code and the long comment below are
// from FreeBSD's /usr/src/lib/msun/src/e_jn.c and
// came with this notice. The go code is a simplified
// version of the original C.
//
// ====================================================
// 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 n<x, forward recursion is used starting
//      from values of j0(x) and j1(x).
//      for n>x, 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.

// Jn returns the order-n Bessel function of the first kind.
//
// Special cases are:
//	Jn(n, ±Inf) = 0
//	Jn(n, NaN) = NaN
func Jn(n int, x float64) float64 {
	const (
		TwoM29 = 1.0 / (1 << 29) // 2**-29 0x3e10000000000000
		Two302 = 1 << 302        // 2**302 0x52D0000000000000
	)
	// special cases
	switch {
	case IsNaN(x):
		return x
	case IsInf(x, 0):
		return 0
	}
	// J(-n, x) = (-1)**n * J(n, x), J(n, -x) = (-1)**n * J(n, x)
	// Thus, J(-n, x) = J(n, -x)

	if n == 0 {
		return J0(x)
	}
	if x == 0 {
		return 0
	}
	if n < 0 {
		n, x = -n, -x
	}
	if n == 1 {
		return J1(x)
	}
	sign := false
	if x < 0 {
		x = -x
		if n&1 == 1 {
			sign = true // odd n and negative x
		}
	}
	var b float64
	if float64(n) <= x {
		// Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
		if x >= Two302 { // 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

			var temp float64
			switch s, c := Sincos(x); n & 3 {
			case 0:
				temp = c + s
			case 1:
				temp = -c + s
			case 2:
				temp = -c - s
			case 3:
				temp = c - s
			}
			b = (1 / SqrtPi) * temp / Sqrt(x)
		} else {
			b = J1(x)
			for i, a := 1, J0(x); i < n; i++ {
				a, b = b, b*(float64(i+i)/x)-a // avoid underflow
			}
		}
	} else {
		if x < TwoM29 { // x < 2**-29
			// x is tiny, return the first Taylor expansion of J(n,x)
			// J(n,x) = 1/n!*(x/2)**n  - ...

			if n > 33 { // underflow
				b = 0
			} else {
				temp := x * 0.5
				b = temp
				a := 1.0
				for i := 2; i <= n; i++ {
					a *= float64(i) // a = n!
					b *= temp       // b = (x/2)**n
				}
				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
			w := float64(n+n) / x
			h := 2 / x
			q0 := w
			z := w + h
			q1 := w*z - 1
			k := 1
			for q1 < 1e9 {
				k++
				z += h
				q0, q1 = q1, z*q1-q0
			}
			m := n + n
			t := 0.0
			for i := 2 * (n + k); i >= m; i -= 2 {
				t = 1 / (float64(i)/x - t)
			}
			a := t
			b = 1
			//  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 := float64(n)
			v := 2 / x
			tmp = tmp * Log(Abs(v*tmp))
			if tmp < 7.09782712893383973096e+02 {
				for i := n - 1; i > 0; i-- {
					di := float64(i + i)
					a, b = b, b*di/x-a
				}
			} else {
				for i := n - 1; i > 0; i-- {
					di := float64(i + i)
					a, b = b, b*di/x-a
					// scale b to avoid spurious overflow
					if b > 1e100 {
						a /= b
						t /= b
						b = 1
					}
				}
			}
			b = t * J0(x) / b
		}
	}
	if sign {
		return -b
	}
	return b
}

// Yn returns the order-n Bessel function of the second kind.
//
// Special cases are:
//	Yn(n, +Inf) = 0
//	Yn(n ≥ 0, 0) = -Inf
//	Yn(n < 0, 0) = +Inf if n is odd, -Inf if n is even
//	Yn(n, x < 0) = NaN
//	Yn(n, NaN) = NaN
func Yn(n int, x float64) float64 {
	const Two302 = 1 << 302 // 2**302 0x52D0000000000000
	// special cases
	switch {
	case x < 0 || IsNaN(x):
		return NaN()
	case IsInf(x, 1):
		return 0
	}

	if n == 0 {
		return Y0(x)
	}
	if x == 0 {
		if n < 0 && n&1 == 1 {
			return Inf(1)
		}
		return Inf(-1)
	}
	sign := false
	if n < 0 {
		n = -n
		if n&1 == 1 {
			sign = true // sign true if n < 0 && |n| odd
		}
	}
	if n == 1 {
		if sign {
			return -Y1(x)
		}
		return Y1(x)
	}
	var b float64
	if x >= Two302 { // 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

		var temp float64
		switch s, c := Sincos(x); n & 3 {
		case 0:
			temp = s - c
		case 1:
			temp = -s - c
		case 2:
			temp = -s + c
		case 3:
			temp = s + c
		}
		b = (1 / SqrtPi) * temp / Sqrt(x)
	} else {
		a := Y0(x)
		b = Y1(x)
		// quit if b is -inf
		for i := 1; i < n && !IsInf(b, -1); i++ {
			a, b = b, (float64(i+i)/x)*b-a
		}
	}
	if sign {
		return -b
	}
	return b
}