libgo/go/math/j0.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 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 cancelation, 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 cancelation, 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
 248         3.83374475364121826715e+03, // 0x40ADF37D50596938
 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{
 374         4.37741014089738620906e-09, // 0x3E32CD036ADECB82
 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
 386         2.51633368920368957333e+03,  // 0x40A3A8AAD94FB1C0
 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 }