Black Lives Matter. Support the Equal Justice Initiative.

# Source file src/math/j0.go

## Documentation: math

```     1  // Copyright 2010 The Go Authors. All rights reserved.
2  // Use of this source code is governed by a BSD-style
4
5  package math
6
7  /*
8  	Bessel function of the first and second kinds of order zero.
9  */
10
11  // The original C code and the long comment below are
12  // from FreeBSD's /usr/src/lib/msun/src/e_j0.c and
13  // came with this notice. The go code is a simplified
14  // version of the original C.
15  //
16  // ====================================================
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_j0(x), __ieee754_y0(x)
26  // Bessel function of the first and second kinds of order zero.
27  // Method -- j0(x):
28  //      1. For tiny x, we use j0(x) = 1 - x**2/4 + x**4/64 - ...
29  //      2. Reduce x to |x| since j0(x)=j0(-x),  and
30  //         for x in (0,2)
31  //              j0(x) = 1-z/4+ z**2*R0/S0,  where z = x*x;
32  //         (precision:  |j0-1+z/4-z**2R0/S0 |<2**-63.67 )
33  //         for x in (2,inf)
34  //              j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
35  //         where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
36  //         as follow:
37  //              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
38  //                      = 1/sqrt(2) * (cos(x) + sin(x))
39  //              sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
40  //                      = 1/sqrt(2) * (sin(x) - cos(x))
41  //         (To avoid cancellation, use
42  //              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
43  //         to compute the worse one.)
44  //
45  //      3 Special cases
46  //              j0(nan)= nan
47  //              j0(0) = 1
48  //              j0(inf) = 0
49  //
50  // Method -- y0(x):
51  //      1. For x<2.
52  //         Since
53  //              y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x**2/4 - ...)
54  //         therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
55  //         We use the following function to approximate y0,
56  //              y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x**2
57  //         where
58  //              U(z) = u00 + u01*z + ... + u06*z**6
59  //              V(z) = 1  + v01*z + ... + v04*z**4
60  //         with absolute approximation error bounded by 2**-72.
61  //         Note: For tiny x, U/V = u0 and j0(x)~1, hence
62  //              y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
63  //      2. For x>=2.
64  //              y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
65  //         where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
66  //         by the method mentioned above.
67  //      3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
68  //
69
70  // J0 returns the order-zero Bessel function of the first kind.
71  //
72  // Special cases are:
73  //	J0(±Inf) = 0
74  //	J0(0) = 1
75  //	J0(NaN) = NaN
76  func J0(x float64) float64 {
77  	const (
78  		Huge   = 1e300
79  		TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000
80  		TwoM13 = 1.0 / (1 << 13) // 2**-13 0x3f20000000000000
81  		Two129 = 1 << 129        // 2**129 0x4800000000000000
82  		// R0/S0 on [0, 2]
83  		R02 = 1.56249999999999947958e-02  // 0x3F8FFFFFFFFFFFFD
84  		R03 = -1.89979294238854721751e-04 // 0xBF28E6A5B61AC6E9
85  		R04 = 1.82954049532700665670e-06  // 0x3EBEB1D10C503919
86  		R05 = -4.61832688532103189199e-09 // 0xBE33D5E773D63FCE
87  		S01 = 1.56191029464890010492e-02  // 0x3F8FFCE882C8C2A4
88  		S02 = 1.16926784663337450260e-04  // 0x3F1EA6D2DD57DBF4
89  		S03 = 5.13546550207318111446e-07  // 0x3EA13B54CE84D5A9
90  		S04 = 1.16614003333790000205e-09  // 0x3E1408BCF4745D8F
91  	)
92  	// special cases
93  	switch {
94  	case IsNaN(x):
95  		return x
96  	case IsInf(x, 0):
97  		return 0
98  	case x == 0:
99  		return 1
100  	}
101
102  	x = Abs(x)
103  	if x >= 2 {
104  		s, c := Sincos(x)
105  		ss := s - c
106  		cc := s + c
107
108  		// make sure x+x does not overflow
109  		if x < MaxFloat64/2 {
110  			z := -Cos(x + x)
111  			if s*c < 0 {
112  				cc = z / ss
113  			} else {
114  				ss = z / cc
115  			}
116  		}
117
118  		// j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
119  		// y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
120
121  		var z float64
122  		if x > Two129 { // |x| > ~6.8056e+38
123  			z = (1 / SqrtPi) * cc / Sqrt(x)
124  		} else {
125  			u := pzero(x)
126  			v := qzero(x)
127  			z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x)
128  		}
129  		return z // |x| >= 2.0
130  	}
131  	if x < TwoM13 { // |x| < ~1.2207e-4
132  		if x < TwoM27 {
133  			return 1 // |x| < ~7.4506e-9
134  		}
135  		return 1 - 0.25*x*x // ~7.4506e-9 < |x| < ~1.2207e-4
136  	}
137  	z := x * x
138  	r := z * (R02 + z*(R03+z*(R04+z*R05)))
139  	s := 1 + z*(S01+z*(S02+z*(S03+z*S04)))
140  	if x < 1 {
141  		return 1 + z*(-0.25+(r/s)) // |x| < 1.00
142  	}
143  	u := 0.5 * x
144  	return (1+u)*(1-u) + z*(r/s) // 1.0 < |x| < 2.0
145  }
146
147  // Y0 returns the order-zero Bessel function of the second kind.
148  //
149  // Special cases are:
150  //	Y0(+Inf) = 0
151  //	Y0(0) = -Inf
152  //	Y0(x < 0) = NaN
153  //	Y0(NaN) = NaN
154  func Y0(x float64) float64 {
155  	const (
156  		TwoM27 = 1.0 / (1 << 27)             // 2**-27 0x3e40000000000000
157  		Two129 = 1 << 129                    // 2**129 0x4800000000000000
158  		U00    = -7.38042951086872317523e-02 // 0xBFB2E4D699CBD01F
159  		U01    = 1.76666452509181115538e-01  // 0x3FC69D019DE9E3FC
160  		U02    = -1.38185671945596898896e-02 // 0xBF8C4CE8B16CFA97
161  		U03    = 3.47453432093683650238e-04  // 0x3F36C54D20B29B6B
162  		U04    = -3.81407053724364161125e-06 // 0xBECFFEA773D25CAD
163  		U05    = 1.95590137035022920206e-08  // 0x3E5500573B4EABD4
164  		U06    = -3.98205194132103398453e-11 // 0xBDC5E43D693FB3C8
165  		V01    = 1.27304834834123699328e-02  // 0x3F8A127091C9C71A
166  		V02    = 7.60068627350353253702e-05  // 0x3F13ECBBF578C6C1
167  		V03    = 2.59150851840457805467e-07  // 0x3E91642D7FF202FD
168  		V04    = 4.41110311332675467403e-10  // 0x3DFE50183BD6D9EF
169  	)
170  	// special cases
171  	switch {
172  	case x < 0 || IsNaN(x):
173  		return NaN()
174  	case IsInf(x, 1):
175  		return 0
176  	case x == 0:
177  		return Inf(-1)
178  	}
179
180  	if x >= 2 { // |x| >= 2.0
181
182  		// y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
183  		//     where x0 = x-pi/4
184  		// Better formula:
185  		//     cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
186  		//             =  1/sqrt(2) * (sin(x) + cos(x))
187  		//     sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
188  		//             =  1/sqrt(2) * (sin(x) - cos(x))
189  		// To avoid cancellation, use
190  		//     sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
191  		// to compute the worse one.
192
193  		s, c := Sincos(x)
194  		ss := s - c
195  		cc := s + c
196
197  		// j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
198  		// y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
199
200  		// make sure x+x does not overflow
201  		if x < MaxFloat64/2 {
202  			z := -Cos(x + x)
203  			if s*c < 0 {
204  				cc = z / ss
205  			} else {
206  				ss = z / cc
207  			}
208  		}
209  		var z float64
210  		if x > Two129 { // |x| > ~6.8056e+38
211  			z = (1 / SqrtPi) * ss / Sqrt(x)
212  		} else {
213  			u := pzero(x)
214  			v := qzero(x)
215  			z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x)
216  		}
217  		return z // |x| >= 2.0
218  	}
219  	if x <= TwoM27 {
220  		return U00 + (2/Pi)*Log(x) // |x| < ~7.4506e-9
221  	}
222  	z := x * x
223  	u := U00 + z*(U01+z*(U02+z*(U03+z*(U04+z*(U05+z*U06)))))
224  	v := 1 + z*(V01+z*(V02+z*(V03+z*V04)))
225  	return u/v + (2/Pi)*J0(x)*Log(x) // ~7.4506e-9 < |x| < 2.0
226  }
227
228  // The asymptotic expansions of pzero is
229  //      1 - 9/128 s**2 + 11025/98304 s**4 - ..., where s = 1/x.
230  // For x >= 2, We approximate pzero by
231  // 	pzero(x) = 1 + (R/S)
232  // where  R = pR0 + pR1*s**2 + pR2*s**4 + ... + pR5*s**10
233  // 	  S = 1 + pS0*s**2 + ... + pS4*s**10
234  // and
235  //      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
236
237  // for x in [inf, 8]=1/[0,0.125]
238  var p0R8 = [6]float64{
239  	0.00000000000000000000e+00,  // 0x0000000000000000
240  	-7.03124999999900357484e-02, // 0xBFB1FFFFFFFFFD32
241  	-8.08167041275349795626e+00, // 0xC02029D0B44FA779
242  	-2.57063105679704847262e+02, // 0xC07011027B19E863
243  	-2.48521641009428822144e+03, // 0xC0A36A6ECD4DCAFC
244  	-5.25304380490729545272e+03, // 0xC0B4850B36CC643D
245  }
246  var p0S8 = [5]float64{
247  	1.16534364619668181717e+02, // 0x405D223307A96751
249  	4.05978572648472545552e+04, // 0x40E3D2BB6EB6B05F
250  	1.16752972564375915681e+05, // 0x40FC810F8F9FA9BD
251  	4.76277284146730962675e+04, // 0x40E741774F2C49DC
252  }
253
254  // for x in [8,4.5454]=1/[0.125,0.22001]
255  var p0R5 = [6]float64{
256  	-1.14125464691894502584e-11, // 0xBDA918B147E495CC
257  	-7.03124940873599280078e-02, // 0xBFB1FFFFE69AFBC6
258  	-4.15961064470587782438e+00, // 0xC010A370F90C6BBF
259  	-6.76747652265167261021e+01, // 0xC050EB2F5A7D1783
260  	-3.31231299649172967747e+02, // 0xC074B3B36742CC63
261  	-3.46433388365604912451e+02, // 0xC075A6EF28A38BD7
262  }
263  var p0S5 = [5]float64{
264  	6.07539382692300335975e+01, // 0x404E60810C98C5DE
265  	1.05125230595704579173e+03, // 0x40906D025C7E2864
266  	5.97897094333855784498e+03, // 0x40B75AF88FBE1D60
267  	9.62544514357774460223e+03, // 0x40C2CCB8FA76FA38
268  	2.40605815922939109441e+03, // 0x40A2CC1DC70BE864
269  }
270
271  // for x in [4.547,2.8571]=1/[0.2199,0.35001]
272  var p0R3 = [6]float64{
273  	-2.54704601771951915620e-09, // 0xBE25E1036FE1AA86
274  	-7.03119616381481654654e-02, // 0xBFB1FFF6F7C0E24B
275  	-2.40903221549529611423e+00, // 0xC00345B2AEA48074
276  	-2.19659774734883086467e+01, // 0xC035F74A4CB94E14
277  	-5.80791704701737572236e+01, // 0xC04D0A22420A1A45
278  	-3.14479470594888503854e+01, // 0xC03F72ACA892D80F
279  }
280  var p0S3 = [5]float64{
281  	3.58560338055209726349e+01, // 0x4041ED9284077DD3
282  	3.61513983050303863820e+02, // 0x40769839464A7C0E
283  	1.19360783792111533330e+03, // 0x4092A66E6D1061D6
284  	1.12799679856907414432e+03, // 0x40919FFCB8C39B7E
285  	1.73580930813335754692e+02, // 0x4065B296FC379081
286  }
287
288  // for x in [2.8570,2]=1/[0.3499,0.5]
289  var p0R2 = [6]float64{
290  	-8.87534333032526411254e-08, // 0xBE77D316E927026D
291  	-7.03030995483624743247e-02, // 0xBFB1FF62495E1E42
292  	-1.45073846780952986357e+00, // 0xBFF736398A24A843
293  	-7.63569613823527770791e+00, // 0xC01E8AF3EDAFA7F3
294  	-1.11931668860356747786e+01, // 0xC02662E6C5246303
295  	-3.23364579351335335033e+00, // 0xC009DE81AF8FE70F
296  }
297  var p0S2 = [5]float64{
298  	2.22202997532088808441e+01, // 0x40363865908B5959
299  	1.36206794218215208048e+02, // 0x4061069E0EE8878F
300  	2.70470278658083486789e+02, // 0x4070E78642EA079B
301  	1.53875394208320329881e+02, // 0x40633C033AB6FAFF
302  	1.46576176948256193810e+01, // 0x402D50B344391809
303  }
304
305  func pzero(x float64) float64 {
306  	var p *[6]float64
307  	var q *[5]float64
308  	if x >= 8 {
309  		p = &p0R8
310  		q = &p0S8
311  	} else if x >= 4.5454 {
312  		p = &p0R5
313  		q = &p0S5
314  	} else if x >= 2.8571 {
315  		p = &p0R3
316  		q = &p0S3
317  	} else if x >= 2 {
318  		p = &p0R2
319  		q = &p0S2
320  	}
321  	z := 1 / (x * x)
322  	r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
323  	s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))))
324  	return 1 + r/s
325  }
326
327  // For x >= 8, the asymptotic expansions of qzero is
328  //      -1/8 s + 75/1024 s**3 - ..., where s = 1/x.
329  // We approximate pzero by
330  //      qzero(x) = s*(-1.25 + (R/S))
331  // where R = qR0 + qR1*s**2 + qR2*s**4 + ... + qR5*s**10
332  //       S = 1 + qS0*s**2 + ... + qS5*s**12
333  // and
334  //      | qzero(x)/s +1.25-R/S | <= 2**(-61.22)
335
336  // for x in [inf, 8]=1/[0,0.125]
337  var q0R8 = [6]float64{
338  	0.00000000000000000000e+00, // 0x0000000000000000
339  	7.32421874999935051953e-02, // 0x3FB2BFFFFFFFFE2C
340  	1.17682064682252693899e+01, // 0x402789525BB334D6
341  	5.57673380256401856059e+02, // 0x40816D6315301825
342  	8.85919720756468632317e+03, // 0x40C14D993E18F46D
343  	3.70146267776887834771e+04, // 0x40E212D40E901566
344  }
345  var q0S8 = [6]float64{
346  	1.63776026895689824414e+02,  // 0x406478D5365B39BC
347  	8.09834494656449805916e+03,  // 0x40BFA2584E6B0563
348  	1.42538291419120476348e+05,  // 0x4101665254D38C3F
349  	8.03309257119514397345e+05,  // 0x412883DA83A52B43
350  	8.40501579819060512818e+05,  // 0x4129A66B28DE0B3D
351  	-3.43899293537866615225e+05, // 0xC114FD6D2C9530C5
352  }
353
354  // for x in [8,4.5454]=1/[0.125,0.22001]
355  var q0R5 = [6]float64{
356  	1.84085963594515531381e-11, // 0x3DB43D8F29CC8CD9
357  	7.32421766612684765896e-02, // 0x3FB2BFFFD172B04C
358  	5.83563508962056953777e+00, // 0x401757B0B9953DD3
359  	1.35111577286449829671e+02, // 0x4060E3920A8788E9
360  	1.02724376596164097464e+03, // 0x40900CF99DC8C481
361  	1.98997785864605384631e+03, // 0x409F17E953C6E3A6
362  }
363  var q0S5 = [6]float64{
364  	8.27766102236537761883e+01,  // 0x4054B1B3FB5E1543
365  	2.07781416421392987104e+03,  // 0x40A03BA0DA21C0CE
366  	1.88472887785718085070e+04,  // 0x40D267D27B591E6D
367  	5.67511122894947329769e+04,  // 0x40EBB5E397E02372
368  	3.59767538425114471465e+04,  // 0x40E191181F7A54A0
369  	-5.35434275601944773371e+03, // 0xC0B4EA57BEDBC609
370  }
371
372  // for x in [4.547,2.8571]=1/[0.2199,0.35001]
373  var q0R3 = [6]float64{
375  	7.32411180042911447163e-02, // 0x3FB2BFEE0E8D0842
376  	3.34423137516170720929e+00, // 0x400AC0FC61149CF5
377  	4.26218440745412650017e+01, // 0x40454F98962DAEDD
378  	1.70808091340565596283e+02, // 0x406559DBE25EFD1F
379  	1.66733948696651168575e+02, // 0x4064D77C81FA21E0
380  }
381  var q0S3 = [6]float64{
382  	4.87588729724587182091e+01,  // 0x40486122BFE343A6
383  	7.09689221056606015736e+02,  // 0x40862D8386544EB3
384  	3.70414822620111362994e+03,  // 0x40ACF04BE44DFC63
385  	6.46042516752568917582e+03,  // 0x40B93C6CD7C76A28
387  	-1.49247451836156386662e+02, // 0xC062A7EB201CF40F
388  }
389
390  // for x in [2.8570,2]=1/[0.3499,0.5]
391  var q0R2 = [6]float64{
392  	1.50444444886983272379e-07, // 0x3E84313B54F76BDB
393  	7.32234265963079278272e-02, // 0x3FB2BEC53E883E34
394  	1.99819174093815998816e+00, // 0x3FFFF897E727779C
395  	1.44956029347885735348e+01, // 0x402CFDBFAAF96FE5
396  	3.16662317504781540833e+01, // 0x403FAA8E29FBDC4A
397  	1.62527075710929267416e+01, // 0x403040B171814BB4
398  }
399  var q0S2 = [6]float64{
400  	3.03655848355219184498e+01,  // 0x403E5D96F7C07AED
401  	2.69348118608049844624e+02,  // 0x4070D591E4D14B40
402  	8.44783757595320139444e+02,  // 0x408A664522B3BF22
403  	8.82935845112488550512e+02,  // 0x408B977C9C5CC214
404  	2.12666388511798828631e+02,  // 0x406A95530E001365
405  	-5.31095493882666946917e+00, // 0xC0153E6AF8B32931
406  }
407
408  func qzero(x float64) float64 {
409  	var p, q *[6]float64
410  	if x >= 8 {
411  		p = &q0R8
412  		q = &q0S8
413  	} else if x >= 4.5454 {
414  		p = &q0R5
415  		q = &q0S5
416  	} else if x >= 2.8571 {
417  		p = &q0R3
418  		q = &q0S3
419  	} else if x >= 2 {
420  		p = &q0R2
421  		q = &q0S2
422  	}
423  	z := 1 / (x * x)
424  	r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
425  	s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))))
426  	return (-0.125 + r/s) / x
427  }
428
```

View as plain text