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 // 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 if x < 0 { 103 x = -x 104 } 105 if x >= 2 { 106 s, c := Sincos(x) 107 ss := s - c 108 cc := s + c 109 110 // make sure x+x does not overflow 111 if x < MaxFloat64/2 { 112 z := -Cos(x + x) 113 if s*c < 0 { 114 cc = z / ss 115 } else { 116 ss = z / cc 117 } 118 } 119 120 // j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) 121 // y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) 122 123 var z float64 124 if x > Two129 { // |x| > ~6.8056e+38 125 z = (1 / SqrtPi) * cc / Sqrt(x) 126 } else { 127 u := pzero(x) 128 v := qzero(x) 129 z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x) 130 } 131 return z // |x| >= 2.0 132 } 133 if x < TwoM13 { // |x| < ~1.2207e-4 134 if x < TwoM27 { 135 return 1 // |x| < ~7.4506e-9 136 } 137 return 1 - 0.25*x*x // ~7.4506e-9 < |x| < ~1.2207e-4 138 } 139 z := x * x 140 r := z * (R02 + z*(R03+z*(R04+z*R05))) 141 s := 1 + z*(S01+z*(S02+z*(S03+z*S04))) 142 if x < 1 { 143 return 1 + z*(-0.25+(r/s)) // |x| < 1.00 144 } 145 u := 0.5 * x 146 return (1+u)*(1-u) + z*(r/s) // 1.0 < |x| < 2.0 147 } 148 149 // Y0 returns the order-zero Bessel function of the second kind. 150 // 151 // Special cases are: 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
