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 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 // ====================================================
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_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 //
74 // J0(±Inf) = 0
75 // J0(0) = 1
76 // J0(NaN) = NaN
77 func J0(x float64) float64 {
78 const (
79 Huge = 1e300
80 TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000
81 TwoM13 = 1.0 / (1 << 13) // 2**-13 0x3f20000000000000
82 Two129 = 1 << 129 // 2**129 0x4800000000000000
83 // R0/S0 on [0, 2]
84 R02 = 1.56249999999999947958e-02 // 0x3F8FFFFFFFFFFFFD
85 R03 = -1.89979294238854721751e-04 // 0xBF28E6A5B61AC6E9
86 R04 = 1.82954049532700665670e-06 // 0x3EBEB1D10C503919
87 R05 = -4.61832688532103189199e-09 // 0xBE33D5E773D63FCE
88 S01 = 1.56191029464890010492e-02 // 0x3F8FFCE882C8C2A4
89 S02 = 1.16926784663337450260e-04 // 0x3F1EA6D2DD57DBF4
90 S03 = 5.13546550207318111446e-07 // 0x3EA13B54CE84D5A9
91 S04 = 1.16614003333790000205e-09 // 0x3E1408BCF4745D8F
92 )
93 // special cases
94 switch {
95 case IsNaN(x):
96 return x
97 case IsInf(x, 0):
98 return 0
99 case x == 0:
100 return 1
101 }
102
103 x = Abs(x)
104 if x >= 2 {
105 s, c := Sincos(x)
106 ss := s - c
107 cc := s + c
108
109 // make sure x+x does not overflow
110 if x < MaxFloat64/2 {
111 z := -Cos(x + x)
112 if s*c < 0 {
113 cc = z / ss
114 } else {
115 ss = z / cc
116 }
117 }
118
119 // j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
120 // y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
121
122 var z float64
123 if x > Two129 { // |x| > ~6.8056e+38
124 z = (1 / SqrtPi) * cc / Sqrt(x)
125 } else {
126 u := pzero(x)
127 v := qzero(x)
128 z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x)
129 }
130 return z // |x| >= 2.0
131 }
132 if x < TwoM13 { // |x| < ~1.2207e-4
133 if x < TwoM27 {
134 return 1 // |x| < ~7.4506e-9
135 }
136 return 1 - 0.25*x*x // ~7.4506e-9 < |x| < ~1.2207e-4
137 }
138 z := x * x
139 r := z * (R02 + z*(R03+z*(R04+z*R05)))
140 s := 1 + z*(S01+z*(S02+z*(S03+z*S04)))
141 if x < 1 {
142 return 1 + z*(-0.25+(r/s)) // |x| < 1.00
143 }
144 u := 0.5 * x
145 return (1+u)*(1-u) + z*(r/s) // 1.0 < |x| < 2.0
146 }
147
148 // Y0 returns the order-zero Bessel function of the second kind.
149 //
150 // Special cases are:
151 //
152 // Y0(+Inf) = 0
153 // Y0(0) = -Inf
154 // Y0(x < 0) = NaN
155 // Y0(NaN) = NaN
156 func Y0(x float64) float64 {
157 const (
158 TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000
159 Two129 = 1 << 129 // 2**129 0x4800000000000000
160 U00 = -7.38042951086872317523e-02 // 0xBFB2E4D699CBD01F
161 U01 = 1.76666452509181115538e-01 // 0x3FC69D019DE9E3FC
162 U02 = -1.38185671945596898896e-02 // 0xBF8C4CE8B16CFA97
163 U03 = 3.47453432093683650238e-04 // 0x3F36C54D20B29B6B
164 U04 = -3.81407053724364161125e-06 // 0xBECFFEA773D25CAD
165 U05 = 1.95590137035022920206e-08 // 0x3E5500573B4EABD4
166 U06 = -3.98205194132103398453e-11 // 0xBDC5E43D693FB3C8
167 V01 = 1.27304834834123699328e-02 // 0x3F8A127091C9C71A
168 V02 = 7.60068627350353253702e-05 // 0x3F13ECBBF578C6C1
169 V03 = 2.59150851840457805467e-07 // 0x3E91642D7FF202FD
170 V04 = 4.41110311332675467403e-10 // 0x3DFE50183BD6D9EF
171 )
172 // special cases
173 switch {
174 case x < 0 || IsNaN(x):
175 return NaN()
176 case IsInf(x, 1):
177 return 0
178 case x == 0:
179 return Inf(-1)
180 }
181
182 if x >= 2 { // |x| >= 2.0
183
184 // y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
185 // where x0 = x-pi/4
186 // Better formula:
187 // cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
188 // = 1/sqrt(2) * (sin(x) + cos(x))
189 // sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
190 // = 1/sqrt(2) * (sin(x) - cos(x))
191 // To avoid cancellation, use
192 // sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
193 // to compute the worse one.
194
195 s, c := Sincos(x)
196 ss := s - c
197 cc := s + c
198
199 // j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
200 // y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
201
202 // make sure x+x does not overflow
203 if x < MaxFloat64/2 {
204 z := -Cos(x + x)
205 if s*c < 0 {
206 cc = z / ss
207 } else {
208 ss = z / cc
209 }
210 }
211 var z float64
212 if x > Two129 { // |x| > ~6.8056e+38
213 z = (1 / SqrtPi) * ss / Sqrt(x)
214 } else {
215 u := pzero(x)
216 v := qzero(x)
217 z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x)
218 }
219 return z // |x| >= 2.0
220 }
221 if x <= TwoM27 {
222 return U00 + (2/Pi)*Log(x) // |x| < ~7.4506e-9
223 }
224 z := x * x
225 u := U00 + z*(U01+z*(U02+z*(U03+z*(U04+z*(U05+z*U06)))))
226 v := 1 + z*(V01+z*(V02+z*(V03+z*V04)))
227 return u/v + (2/Pi)*J0(x)*Log(x) // ~7.4506e-9 < |x| < 2.0
228 }
229
230 // The asymptotic expansions of pzero is
231 // 1 - 9/128 s**2 + 11025/98304 s**4 - ..., where s = 1/x.
232 // For x >= 2, We approximate pzero by
233 // pzero(x) = 1 + (R/S)
234 // where R = pR0 + pR1*s**2 + pR2*s**4 + ... + pR5*s**10
235 // S = 1 + pS0*s**2 + ... + pS4*s**10
236 // and
237 // | pzero(x)-1-R/S | <= 2 ** ( -60.26)
238
239 // for x in [inf, 8]=1/[0,0.125]
240 var p0R8 = [6]float64{
241 0.00000000000000000000e+00, // 0x0000000000000000
242 -7.03124999999900357484e-02, // 0xBFB1FFFFFFFFFD32
243 -8.08167041275349795626e+00, // 0xC02029D0B44FA779
244 -2.57063105679704847262e+02, // 0xC07011027B19E863
245 -2.48521641009428822144e+03, // 0xC0A36A6ECD4DCAFC
246 -5.25304380490729545272e+03, // 0xC0B4850B36CC643D
247 }
248 var p0S8 = [5]float64{
249 1.16534364619668181717e+02, // 0x405D223307A96751
250 3.83374475364121826715e+03, // 0x40ADF37D50596938
251 4.05978572648472545552e+04, // 0x40E3D2BB6EB6B05F
252 1.16752972564375915681e+05, // 0x40FC810F8F9FA9BD
253 4.76277284146730962675e+04, // 0x40E741774F2C49DC
254 }
255
256 // for x in [8,4.5454]=1/[0.125,0.22001]
257 var p0R5 = [6]float64{
258 -1.14125464691894502584e-11, // 0xBDA918B147E495CC
259 -7.03124940873599280078e-02, // 0xBFB1FFFFE69AFBC6
260 -4.15961064470587782438e+00, // 0xC010A370F90C6BBF
261 -6.76747652265167261021e+01, // 0xC050EB2F5A7D1783
262 -3.31231299649172967747e+02, // 0xC074B3B36742CC63
263 -3.46433388365604912451e+02, // 0xC075A6EF28A38BD7
264 }
265 var p0S5 = [5]float64{
266 6.07539382692300335975e+01, // 0x404E60810C98C5DE
267 1.05125230595704579173e+03, // 0x40906D025C7E2864
268 5.97897094333855784498e+03, // 0x40B75AF88FBE1D60
269 9.62544514357774460223e+03, // 0x40C2CCB8FA76FA38
270 2.40605815922939109441e+03, // 0x40A2CC1DC70BE864
271 }
272
273 // for x in [4.547,2.8571]=1/[0.2199,0.35001]
274 var p0R3 = [6]float64{
275 -2.54704601771951915620e-09, // 0xBE25E1036FE1AA86
276 -7.03119616381481654654e-02, // 0xBFB1FFF6F7C0E24B
277 -2.40903221549529611423e+00, // 0xC00345B2AEA48074
278 -2.19659774734883086467e+01, // 0xC035F74A4CB94E14
279 -5.80791704701737572236e+01, // 0xC04D0A22420A1A45
280 -3.14479470594888503854e+01, // 0xC03F72ACA892D80F
281 }
282 var p0S3 = [5]float64{
283 3.58560338055209726349e+01, // 0x4041ED9284077DD3
284 3.61513983050303863820e+02, // 0x40769839464A7C0E
285 1.19360783792111533330e+03, // 0x4092A66E6D1061D6
286 1.12799679856907414432e+03, // 0x40919FFCB8C39B7E
287 1.73580930813335754692e+02, // 0x4065B296FC379081
288 }
289
290 // for x in [2.8570,2]=1/[0.3499,0.5]
291 var p0R2 = [6]float64{
292 -8.87534333032526411254e-08, // 0xBE77D316E927026D
293 -7.03030995483624743247e-02, // 0xBFB1FF62495E1E42
294 -1.45073846780952986357e+00, // 0xBFF736398A24A843
295 -7.63569613823527770791e+00, // 0xC01E8AF3EDAFA7F3
296 -1.11931668860356747786e+01, // 0xC02662E6C5246303
297 -3.23364579351335335033e+00, // 0xC009DE81AF8FE70F
298 }
299 var p0S2 = [5]float64{
300 2.22202997532088808441e+01, // 0x40363865908B5959
301 1.36206794218215208048e+02, // 0x4061069E0EE8878F
302 2.70470278658083486789e+02, // 0x4070E78642EA079B
303 1.53875394208320329881e+02, // 0x40633C033AB6FAFF
304 1.46576176948256193810e+01, // 0x402D50B344391809
305 }
306
307 func pzero(x float64) float64 {
308 var p *[6]float64
309 var q *[5]float64
310 if x >= 8 {
311 p = &p0R8
312 q = &p0S8
313 } else if x >= 4.5454 {
314 p = &p0R5
315 q = &p0S5
316 } else if x >= 2.8571 {
317 p = &p0R3
318 q = &p0S3
319 } else if x >= 2 {
320 p = &p0R2
321 q = &p0S2
322 }
323 z := 1 / (x * x)
324 r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
325 s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))))
326 return 1 + r/s
327 }
328
329 // For x >= 8, the asymptotic expansions of qzero is
330 // -1/8 s + 75/1024 s**3 - ..., where s = 1/x.
331 // We approximate pzero by
332 // qzero(x) = s*(-1.25 + (R/S))
333 // where R = qR0 + qR1*s**2 + qR2*s**4 + ... + qR5*s**10
334 // S = 1 + qS0*s**2 + ... + qS5*s**12
335 // and
336 // | qzero(x)/s +1.25-R/S | <= 2**(-61.22)
337
338 // for x in [inf, 8]=1/[0,0.125]
339 var q0R8 = [6]float64{
340 0.00000000000000000000e+00, // 0x0000000000000000
341 7.32421874999935051953e-02, // 0x3FB2BFFFFFFFFE2C
342 1.17682064682252693899e+01, // 0x402789525BB334D6
343 5.57673380256401856059e+02, // 0x40816D6315301825
344 8.85919720756468632317e+03, // 0x40C14D993E18F46D
345 3.70146267776887834771e+04, // 0x40E212D40E901566
346 }
347 var q0S8 = [6]float64{
348 1.63776026895689824414e+02, // 0x406478D5365B39BC
349 8.09834494656449805916e+03, // 0x40BFA2584E6B0563
350 1.42538291419120476348e+05, // 0x4101665254D38C3F
351 8.03309257119514397345e+05, // 0x412883DA83A52B43
352 8.40501579819060512818e+05, // 0x4129A66B28DE0B3D
353 -3.43899293537866615225e+05, // 0xC114FD6D2C9530C5
354 }
355
356 // for x in [8,4.5454]=1/[0.125,0.22001]
357 var q0R5 = [6]float64{
358 1.84085963594515531381e-11, // 0x3DB43D8F29CC8CD9
359 7.32421766612684765896e-02, // 0x3FB2BFFFD172B04C
360 5.83563508962056953777e+00, // 0x401757B0B9953DD3
361 1.35111577286449829671e+02, // 0x4060E3920A8788E9
362 1.02724376596164097464e+03, // 0x40900CF99DC8C481
363 1.98997785864605384631e+03, // 0x409F17E953C6E3A6
364 }
365 var q0S5 = [6]float64{
366 8.27766102236537761883e+01, // 0x4054B1B3FB5E1543
367 2.07781416421392987104e+03, // 0x40A03BA0DA21C0CE
368 1.88472887785718085070e+04, // 0x40D267D27B591E6D
369 5.67511122894947329769e+04, // 0x40EBB5E397E02372
370 3.59767538425114471465e+04, // 0x40E191181F7A54A0
371 -5.35434275601944773371e+03, // 0xC0B4EA57BEDBC609
372 }
373
374 // for x in [4.547,2.8571]=1/[0.2199,0.35001]
375 var q0R3 = [6]float64{
376 4.37741014089738620906e-09, // 0x3E32CD036ADECB82
377 7.32411180042911447163e-02, // 0x3FB2BFEE0E8D0842
378 3.34423137516170720929e+00, // 0x400AC0FC61149CF5
379 4.26218440745412650017e+01, // 0x40454F98962DAEDD
380 1.70808091340565596283e+02, // 0x406559DBE25EFD1F
381 1.66733948696651168575e+02, // 0x4064D77C81FA21E0
382 }
383 var q0S3 = [6]float64{
384 4.87588729724587182091e+01, // 0x40486122BFE343A6
385 7.09689221056606015736e+02, // 0x40862D8386544EB3
386 3.70414822620111362994e+03, // 0x40ACF04BE44DFC63
387 6.46042516752568917582e+03, // 0x40B93C6CD7C76A28
388 2.51633368920368957333e+03, // 0x40A3A8AAD94FB1C0
389 -1.49247451836156386662e+02, // 0xC062A7EB201CF40F
390 }
391
392 // for x in [2.8570,2]=1/[0.3499,0.5]
393 var q0R2 = [6]float64{
394 1.50444444886983272379e-07, // 0x3E84313B54F76BDB
395 7.32234265963079278272e-02, // 0x3FB2BEC53E883E34
396 1.99819174093815998816e+00, // 0x3FFFF897E727779C
397 1.44956029347885735348e+01, // 0x402CFDBFAAF96FE5
398 3.16662317504781540833e+01, // 0x403FAA8E29FBDC4A
399 1.62527075710929267416e+01, // 0x403040B171814BB4
400 }
401 var q0S2 = [6]float64{
402 3.03655848355219184498e+01, // 0x403E5D96F7C07AED
403 2.69348118608049844624e+02, // 0x4070D591E4D14B40
404 8.44783757595320139444e+02, // 0x408A664522B3BF22
405 8.82935845112488550512e+02, // 0x408B977C9C5CC214
406 2.12666388511798828631e+02, // 0x406A95530E001365
407 -5.31095493882666946917e+00, // 0xC0153E6AF8B32931
408 }
409
410 func qzero(x float64) float64 {
411 var p, q *[6]float64
412 if x >= 8 {
413 p = &q0R8
414 q = &q0S8
415 } else if x >= 4.5454 {
416 p = &q0R5
417 q = &q0S5
418 } else if x >= 2.8571 {
419 p = &q0R3
420 q = &q0S3
421 } else if x >= 2 {
422 p = &q0R2
423 q = &q0S2
424 }
425 z := 1 / (x * x)
426 r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
427 s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))))
428 return (-0.125 + r/s) / x
429 }
430