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 one. 9 */ 10 11 // The original C code and the long comment below are 12 // from FreeBSD's /usr/src/lib/msun/src/e_j1.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_j1(x), __ieee754_y1(x) 26 // Bessel function of the first and second kinds of order one. 27 // Method -- j1(x): 28 // 1. For tiny x, we use j1(x) = x/2 - x**3/16 + x**5/384 - ... 29 // 2. Reduce x to |x| since j1(x)=-j1(-x), and 30 // for x in (0,2) 31 // j1(x) = x/2 + x*z*R0/S0, where z = x*x; 32 // (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 ) 33 // for x in (2,inf) 34 // j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1)) 35 // y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) 36 // where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) 37 // as follow: 38 // cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 39 // = 1/sqrt(2) * (sin(x) - cos(x)) 40 // sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 41 // = -1/sqrt(2) * (sin(x) + cos(x)) 42 // (To avoid cancelation, use 43 // sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 44 // to compute the worse one.) 45 // 46 // 3 Special cases 47 // j1(nan)= nan 48 // j1(0) = 0 49 // j1(inf) = 0 50 // 51 // Method -- y1(x): 52 // 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN 53 // 2. For x<2. 54 // Since 55 // y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x**3-...) 56 // therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function. 57 // We use the following function to approximate y1, 58 // y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x**2 59 // where for x in [0,2] (abs err less than 2**-65.89) 60 // U(z) = U0[0] + U0[1]*z + ... + U0[4]*z**4 61 // V(z) = 1 + v0[0]*z + ... + v0[4]*z**5 62 // Note: For tiny x, 1/x dominate y1 and hence 63 // y1(tiny) = -2/pi/tiny, (choose tiny<2**-54) 64 // 3. For x>=2. 65 // y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) 66 // where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) 67 // by method mentioned above. 68 69 // J1 returns the order-one Bessel function of the first kind. 70 // 71 // Special cases are: 72 // J1(Inf) = 0 73 // J1(NaN) = NaN 74 func J1(x float64) float64 { 75 const ( 76 TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000 77 Two129 = 1 << 129 // 2**129 0x4800000000000000 78 // R0/S0 on [0, 2] 79 R00 = -6.25000000000000000000e-02 // 0xBFB0000000000000 80 R01 = 1.40705666955189706048e-03 // 0x3F570D9F98472C61 81 R02 = -1.59955631084035597520e-05 // 0xBEF0C5C6BA169668 82 R03 = 4.96727999609584448412e-08 // 0x3E6AAAFA46CA0BD9 83 S01 = 1.91537599538363460805e-02 // 0x3F939D0B12637E53 84 S02 = 1.85946785588630915560e-04 // 0x3F285F56B9CDF664 85 S03 = 1.17718464042623683263e-06 // 0x3EB3BFF8333F8498 86 S04 = 5.04636257076217042715e-09 // 0x3E35AC88C97DFF2C 87 S05 = 1.23542274426137913908e-11 // 0x3DAB2ACFCFB97ED8 88 ) 89 // special cases 90 switch { 91 case IsNaN(x): 92 return x 93 case IsInf(x, 0) || x == 0: 94 return 0 95 } 96 97 sign := false 98 if x < 0 { 99 x = -x 100 sign = true 101 } 102 if x >= 2 { 103 s, c := Sincos(x) 104 ss := -s - c 105 cc := s - c 106 107 // make sure x+x does not overflow 108 if x < MaxFloat64/2 { 109 z := Cos(x + x) 110 if s*c > 0 { 111 cc = z / ss 112 } else { 113 ss = z / cc 114 } 115 } 116 117 // j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) 118 // y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) 119 120 var z float64 121 if x > Two129 { 122 z = (1 / SqrtPi) * cc / Sqrt(x) 123 } else { 124 u := pone(x) 125 v := qone(x) 126 z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x) 127 } 128 if sign { 129 return -z 130 } 131 return z 132 } 133 if x < TwoM27 { // |x|<2**-27 134 return 0.5 * x // inexact if x!=0 necessary 135 } 136 z := x * x 137 r := z * (R00 + z*(R01+z*(R02+z*R03))) 138 s := 1.0 + z*(S01+z*(S02+z*(S03+z*(S04+z*S05)))) 139 r *= x 140 z = 0.5*x + r/s 141 if sign { 142 return -z 143 } 144 return z 145 } 146 147 // Y1 returns the order-one Bessel function of the second kind. 148 // 149 // Special cases are: 150 // Y1(+Inf) = 0 151 // Y1(0) = -Inf 152 // Y1(x < 0) = NaN 153 // Y1(NaN) = NaN 154 func Y1(x float64) float64 { 155 const ( 156 TwoM54 = 1.0 / (1 << 54) // 2**-54 0x3c90000000000000 157 Two129 = 1 << 129 // 2**129 0x4800000000000000 158 U00 = -1.96057090646238940668e-01 // 0xBFC91866143CBC8A 159 U01 = 5.04438716639811282616e-02 // 0x3FA9D3C776292CD1 160 U02 = -1.91256895875763547298e-03 // 0xBF5F55E54844F50F 161 U03 = 2.35252600561610495928e-05 // 0x3EF8AB038FA6B88E 162 U04 = -9.19099158039878874504e-08 // 0xBE78AC00569105B8 163 V00 = 1.99167318236649903973e-02 // 0x3F94650D3F4DA9F0 164 V01 = 2.02552581025135171496e-04 // 0x3F2A8C896C257764 165 V02 = 1.35608801097516229404e-06 // 0x3EB6C05A894E8CA6 166 V03 = 6.22741452364621501295e-09 // 0x3E3ABF1D5BA69A86 167 V04 = 1.66559246207992079114e-11 // 0x3DB25039DACA772A 168 ) 169 // special cases 170 switch { 171 case x < 0 || IsNaN(x): 172 return NaN() 173 case IsInf(x, 1): 174 return 0 175 case x == 0: 176 return Inf(-1) 177 } 178 179 if x >= 2 { 180 s, c := Sincos(x) 181 ss := -s - c 182 cc := s - c 183 184 // make sure x+x does not overflow 185 if x < MaxFloat64/2 { 186 z := Cos(x + x) 187 if s*c > 0 { 188 cc = z / ss 189 } else { 190 ss = z / cc 191 } 192 } 193 // y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) 194 // where x0 = x-3pi/4 195 // Better formula: 196 // cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 197 // = 1/sqrt(2) * (sin(x) - cos(x)) 198 // sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 199 // = -1/sqrt(2) * (cos(x) + sin(x)) 200 // To avoid cancelation, use 201 // sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 202 // to compute the worse one. 203 204 var z float64 205 if x > Two129 { 206 z = (1 / SqrtPi) * ss / Sqrt(x) 207 } else { 208 u := pone(x) 209 v := qone(x) 210 z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x) 211 } 212 return z 213 } 214 if x <= TwoM54 { // x < 2**-54 215 return -(2 / Pi) / x 216 } 217 z := x * x 218 u := U00 + z*(U01+z*(U02+z*(U03+z*U04))) 219 v := 1 + z*(V00+z*(V01+z*(V02+z*(V03+z*V04)))) 220 return x*(u/v) + (2/Pi)*(J1(x)*Log(x)-1/x) 221 } 222 223 // For x >= 8, the asymptotic expansions of pone is 224 // 1 + 15/128 s**2 - 4725/2**15 s**4 - ..., where s = 1/x. 225 // We approximate pone by 226 // pone(x) = 1 + (R/S) 227 // where R = pr0 + pr1*s**2 + pr2*s**4 + ... + pr5*s**10 228 // S = 1 + ps0*s**2 + ... + ps4*s**10 229 // and 230 // | pone(x)-1-R/S | <= 2**(-60.06) 231 232 // for x in [inf, 8]=1/[0,0.125] 233 var p1R8 = [6]float64{ 234 0.00000000000000000000e+00, // 0x0000000000000000 235 1.17187499999988647970e-01, // 0x3FBDFFFFFFFFFCCE 236 1.32394806593073575129e+01, // 0x402A7A9D357F7FCE 237 4.12051854307378562225e+02, // 0x4079C0D4652EA590 238 3.87474538913960532227e+03, // 0x40AE457DA3A532CC 239 7.91447954031891731574e+03, // 0x40BEEA7AC32782DD 240 } 241 var p1S8 = [5]float64{ 242 1.14207370375678408436e+02, // 0x405C8D458E656CAC 243 3.65093083420853463394e+03, // 0x40AC85DC964D274F 244 3.69562060269033463555e+04, // 0x40E20B8697C5BB7F 245 9.76027935934950801311e+04, // 0x40F7D42CB28F17BB 246 3.08042720627888811578e+04, // 0x40DE1511697A0B2D 247 } 248 249 // for x in [8,4.5454] = 1/[0.125,0.22001] 250 var p1R5 = [6]float64{ 251 1.31990519556243522749e-11, // 0x3DAD0667DAE1CA7D 252 1.17187493190614097638e-01, // 0x3FBDFFFFE2C10043 253 6.80275127868432871736e+00, // 0x401B36046E6315E3 254 1.08308182990189109773e+02, // 0x405B13B9452602ED 255 5.17636139533199752805e+02, // 0x40802D16D052D649 256 5.28715201363337541807e+02, // 0x408085B8BB7E0CB7 257 } 258 var p1S5 = [5]float64{ 259 5.92805987221131331921e+01, // 0x404DA3EAA8AF633D 260 9.91401418733614377743e+02, // 0x408EFB361B066701 261 5.35326695291487976647e+03, // 0x40B4E9445706B6FB 262 7.84469031749551231769e+03, // 0x40BEA4B0B8A5BB15 263 1.50404688810361062679e+03, // 0x40978030036F5E51 264 } 265 266 // for x in[4.5453,2.8571] = 1/[0.2199,0.35001] 267 var p1R3 = [6]float64{ 268 3.02503916137373618024e-09, // 0x3E29FC21A7AD9EDD 269 1.17186865567253592491e-01, // 0x3FBDFFF55B21D17B 270 3.93297750033315640650e+00, // 0x400F76BCE85EAD8A 271 3.51194035591636932736e+01, // 0x40418F489DA6D129 272 9.10550110750781271918e+01, // 0x4056C3854D2C1837 273 4.85590685197364919645e+01, // 0x4048478F8EA83EE5 274 } 275 var p1S3 = [5]float64{ 276 3.47913095001251519989e+01, // 0x40416549A134069C 277 3.36762458747825746741e+02, // 0x40750C3307F1A75F 278 1.04687139975775130551e+03, // 0x40905B7C5037D523 279 8.90811346398256432622e+02, // 0x408BD67DA32E31E9 280 1.03787932439639277504e+02, // 0x4059F26D7C2EED53 281 } 282 283 // for x in [2.8570,2] = 1/[0.3499,0.5] 284 var p1R2 = [6]float64{ 285 1.07710830106873743082e-07, // 0x3E7CE9D4F65544F4 286 1.17176219462683348094e-01, // 0x3FBDFF42BE760D83 287 2.36851496667608785174e+00, // 0x4002F2B7F98FAEC0 288 1.22426109148261232917e+01, // 0x40287C377F71A964 289 1.76939711271687727390e+01, // 0x4031B1A8177F8EE2 290 5.07352312588818499250e+00, // 0x40144B49A574C1FE 291 } 292 var p1S2 = [5]float64{ 293 2.14364859363821409488e+01, // 0x40356FBD8AD5ECDC 294 1.25290227168402751090e+02, // 0x405F529314F92CD5 295 2.32276469057162813669e+02, // 0x406D08D8D5A2DBD9 296 1.17679373287147100768e+02, // 0x405D6B7ADA1884A9 297 8.36463893371618283368e+00, // 0x4020BAB1F44E5192 298 } 299 300 func pone(x float64) float64 { 301 var p *[6]float64 302 var q *[5]float64 303 if x >= 8 { 304 p = &p1R8 305 q = &p1S8 306 } else if x >= 4.5454 { 307 p = &p1R5 308 q = &p1S5 309 } else if x >= 2.8571 { 310 p = &p1R3 311 q = &p1S3 312 } else if x >= 2 { 313 p = &p1R2 314 q = &p1S2 315 } 316 z := 1 / (x * x) 317 r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))) 318 s := 1.0 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))) 319 return 1 + r/s 320 } 321 322 // For x >= 8, the asymptotic expansions of qone is 323 // 3/8 s - 105/1024 s**3 - ..., where s = 1/x. 324 // We approximate qone by 325 // qone(x) = s*(0.375 + (R/S)) 326 // where R = qr1*s**2 + qr2*s**4 + ... + qr5*s**10 327 // S = 1 + qs1*s**2 + ... + qs6*s**12 328 // and 329 // | qone(x)/s -0.375-R/S | <= 2**(-61.13) 330 331 // for x in [inf, 8] = 1/[0,0.125] 332 var q1R8 = [6]float64{ 333 0.00000000000000000000e+00, // 0x0000000000000000 334 -1.02539062499992714161e-01, // 0xBFBA3FFFFFFFFDF3 335 -1.62717534544589987888e+01, // 0xC0304591A26779F7 336 -7.59601722513950107896e+02, // 0xC087BCD053E4B576 337 -1.18498066702429587167e+04, // 0xC0C724E740F87415 338 -4.84385124285750353010e+04, // 0xC0E7A6D065D09C6A 339 } 340 var q1S8 = [6]float64{ 341 1.61395369700722909556e+02, // 0x40642CA6DE5BCDE5 342 7.82538599923348465381e+03, // 0x40BE9162D0D88419 343 1.33875336287249578163e+05, // 0x4100579AB0B75E98 344 7.19657723683240939863e+05, // 0x4125F65372869C19 345 6.66601232617776375264e+05, // 0x412457D27719AD5C 346 -2.94490264303834643215e+05, // 0xC111F9690EA5AA18 347 } 348 349 // for x in [8,4.5454] = 1/[0.125,0.22001] 350 var q1R5 = [6]float64{ 351 -2.08979931141764104297e-11, // 0xBDB6FA431AA1A098 352 -1.02539050241375426231e-01, // 0xBFBA3FFFCB597FEF 353 -8.05644828123936029840e+00, // 0xC0201CE6CA03AD4B 354 -1.83669607474888380239e+02, // 0xC066F56D6CA7B9B0 355 -1.37319376065508163265e+03, // 0xC09574C66931734F 356 -2.61244440453215656817e+03, // 0xC0A468E388FDA79D 357 } 358 var q1S5 = [6]float64{ 359 8.12765501384335777857e+01, // 0x405451B2FF5A11B2 360 1.99179873460485964642e+03, // 0x409F1F31E77BF839 361 1.74684851924908907677e+04, // 0x40D10F1F0D64CE29 362 4.98514270910352279316e+04, // 0x40E8576DAABAD197 363 2.79480751638918118260e+04, // 0x40DB4B04CF7C364B 364 -4.71918354795128470869e+03, // 0xC0B26F2EFCFFA004 365 } 366 367 // for x in [4.5454,2.8571] = 1/[0.2199,0.35001] ??? 368 var q1R3 = [6]float64{ 369 -5.07831226461766561369e-09, // 0xBE35CFA9D38FC84F 370 -1.02537829820837089745e-01, // 0xBFBA3FEB51AEED54 371 -4.61011581139473403113e+00, // 0xC01270C23302D9FF 372 -5.78472216562783643212e+01, // 0xC04CEC71C25D16DA 373 -2.28244540737631695038e+02, // 0xC06C87D34718D55F 374 -2.19210128478909325622e+02, // 0xC06B66B95F5C1BF6 375 } 376 var q1S3 = [6]float64{ 377 4.76651550323729509273e+01, // 0x4047D523CCD367E4 378 6.73865112676699709482e+02, // 0x40850EEBC031EE3E 379 3.38015286679526343505e+03, // 0x40AA684E448E7C9A 380 5.54772909720722782367e+03, // 0x40B5ABBAA61D54A6 381 1.90311919338810798763e+03, // 0x409DBC7A0DD4DF4B 382 -1.35201191444307340817e+02, // 0xC060E670290A311F 383 } 384 385 // for x in [2.8570,2] = 1/[0.3499,0.5] 386 var q1R2 = [6]float64{ 387 -1.78381727510958865572e-07, // 0xBE87F12644C626D2 388 -1.02517042607985553460e-01, // 0xBFBA3E8E9148B010 389 -2.75220568278187460720e+00, // 0xC006048469BB4EDA 390 -1.96636162643703720221e+01, // 0xC033A9E2C168907F 391 -4.23253133372830490089e+01, // 0xC04529A3DE104AAA 392 -2.13719211703704061733e+01, // 0xC0355F3639CF6E52 393 } 394 var q1S2 = [6]float64{ 395 2.95333629060523854548e+01, // 0x403D888A78AE64FF 396 2.52981549982190529136e+02, // 0x406F9F68DB821CBA 397 7.57502834868645436472e+02, // 0x4087AC05CE49A0F7 398 7.39393205320467245656e+02, // 0x40871B2548D4C029 399 1.55949003336666123687e+02, // 0x40637E5E3C3ED8D4 400 -4.95949898822628210127e+00, // 0xC013D686E71BE86B 401 } 402 403 func qone(x float64) float64 { 404 var p, q *[6]float64 405 if x >= 8 { 406 p = &q1R8 407 q = &q1S8 408 } else if x >= 4.5454 { 409 p = &q1R5 410 q = &q1S5 411 } else if x >= 2.8571 { 412 p = &q1R3 413 q = &q1S3 414 } else if x >= 2 { 415 p = &q1R2 416 q = &q1S2 417 } 418 z := 1 / (x * x) 419 r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))) 420 s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))) 421 return (0.375 + r/s) / x 422 } 423
