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 // Copyright 2010 The Go Authors. All rights reserved. // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. package math /* Bessel function of the first and second kinds of order one. */ // The original C code and the long comment below are // from FreeBSD's /usr/src/lib/msun/src/e_j1.c and // came with this notice. The go code is a simplified // version of the original C. // // ==================================================== // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. // // Developed at SunPro, a Sun Microsystems, Inc. business. // Permission to use, copy, modify, and distribute this // software is freely granted, provided that this notice // is preserved. // ==================================================== // // __ieee754_j1(x), __ieee754_y1(x) // Bessel function of the first and second kinds of order one. // Method -- j1(x): // 1. For tiny x, we use j1(x) = x/2 - x**3/16 + x**5/384 - ... // 2. Reduce x to |x| since j1(x)=-j1(-x), and // for x in (0,2) // j1(x) = x/2 + x*z*R0/S0, where z = x*x; // (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 ) // for x in (2,inf) // j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1)) // y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) // where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) // as follow: // cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) // = 1/sqrt(2) * (sin(x) - cos(x)) // sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) // = -1/sqrt(2) * (sin(x) + cos(x)) // (To avoid cancelation, use // sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) // to compute the worse one.) // // 3 Special cases // j1(nan)= nan // j1(0) = 0 // j1(inf) = 0 // // Method -- y1(x): // 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN // 2. For x<2. // Since // y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x**3-...) // therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function. // We use the following function to approximate y1, // y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x**2 // where for x in [0,2] (abs err less than 2**-65.89) // U(z) = U0[0] + U0[1]*z + ... + U0[4]*z**4 // V(z) = 1 + v0[0]*z + ... + v0[4]*z**5 // Note: For tiny x, 1/x dominate y1 and hence // y1(tiny) = -2/pi/tiny, (choose tiny<2**-54) // 3. For x>=2. // y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1)) // where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1) // by method mentioned above. // J1 returns the order-one Bessel function of the first kind. // // Special cases are: // J1(±Inf) = 0 // J1(NaN) = NaN func J1(x float64) float64 { const ( TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000 Two129 = 1 << 129 // 2**129 0x4800000000000000 // R0/S0 on [0, 2] R00 = -6.25000000000000000000e-02 // 0xBFB0000000000000 R01 = 1.40705666955189706048e-03 // 0x3F570D9F98472C61 R02 = -1.59955631084035597520e-05 // 0xBEF0C5C6BA169668 R03 = 4.96727999609584448412e-08 // 0x3E6AAAFA46CA0BD9 S01 = 1.91537599538363460805e-02 // 0x3F939D0B12637E53 S02 = 1.85946785588630915560e-04 // 0x3F285F56B9CDF664 S03 = 1.17718464042623683263e-06 // 0x3EB3BFF8333F8498 S04 = 5.04636257076217042715e-09 // 0x3E35AC88C97DFF2C S05 = 1.23542274426137913908e-11 // 0x3DAB2ACFCFB97ED8 ) // special cases switch { case IsNaN(x): return x case IsInf(x, 0) || x == 0: return 0 } sign := false if x < 0 { x = -x sign = true } if x >= 2 { s, c := Sincos(x) ss := -s - c cc := s - c // make sure x+x does not overflow if x < MaxFloat64/2 { z := Cos(x + x) if s*c > 0 { cc = z / ss } else { ss = z / cc } } // j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) // y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) var z float64 if x > Two129 { z = (1 / SqrtPi) * cc / Sqrt(x) } else { u := pone(x) v := qone(x) z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x) } if sign { return -z } return z } if x < TwoM27 { // |x|<2**-27 return 0.5 * x // inexact if x!=0 necessary } z := x * x r := z * (R00 + z*(R01+z*(R02+z*R03))) s := 1.0 + z*(S01+z*(S02+z*(S03+z*(S04+z*S05)))) r *= x z = 0.5*x + r/s if sign { return -z } return z } // Y1 returns the order-one Bessel function of the second kind. // // Special cases are: // Y1(+Inf) = 0 // Y1(0) = -Inf // Y1(x < 0) = NaN // Y1(NaN) = NaN func Y1(x float64) float64 { const ( TwoM54 = 1.0 / (1 << 54) // 2**-54 0x3c90000000000000 Two129 = 1 << 129 // 2**129 0x4800000000000000 U00 = -1.96057090646238940668e-01 // 0xBFC91866143CBC8A U01 = 5.04438716639811282616e-02 // 0x3FA9D3C776292CD1 U02 = -1.91256895875763547298e-03 // 0xBF5F55E54844F50F U03 = 2.35252600561610495928e-05 // 0x3EF8AB038FA6B88E U04 = -9.19099158039878874504e-08 // 0xBE78AC00569105B8 V00 = 1.99167318236649903973e-02 // 0x3F94650D3F4DA9F0 V01 = 2.02552581025135171496e-04 // 0x3F2A8C896C257764 V02 = 1.35608801097516229404e-06 // 0x3EB6C05A894E8CA6 V03 = 6.22741452364621501295e-09 // 0x3E3ABF1D5BA69A86 V04 = 1.66559246207992079114e-11 // 0x3DB25039DACA772A ) // special cases switch { case x < 0 || IsNaN(x): return NaN() case IsInf(x, 1): return 0 case x == 0: return Inf(-1) } if x >= 2 { s, c := Sincos(x) ss := -s - c cc := s - c // make sure x+x does not overflow if x < MaxFloat64/2 { z := Cos(x + x) if s*c > 0 { cc = z / ss } else { ss = z / cc } } // y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) // where x0 = x-3pi/4 // Better formula: // cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) // = 1/sqrt(2) * (sin(x) - cos(x)) // sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) // = -1/sqrt(2) * (cos(x) + sin(x)) // To avoid cancelation, use // sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) // to compute the worse one. var z float64 if x > Two129 { z = (1 / SqrtPi) * ss / Sqrt(x) } else { u := pone(x) v := qone(x) z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x) } return z } if x <= TwoM54 { // x < 2**-54 return -(2 / Pi) / x } z := x * x u := U00 + z*(U01+z*(U02+z*(U03+z*U04))) v := 1 + z*(V00+z*(V01+z*(V02+z*(V03+z*V04)))) return x*(u/v) + (2/Pi)*(J1(x)*Log(x)-1/x) } // For x >= 8, the asymptotic expansions of pone is // 1 + 15/128 s**2 - 4725/2**15 s**4 - ..., where s = 1/x. // We approximate pone by // pone(x) = 1 + (R/S) // where R = pr0 + pr1*s**2 + pr2*s**4 + ... + pr5*s**10 // S = 1 + ps0*s**2 + ... + ps4*s**10 // and // | pone(x)-1-R/S | <= 2**(-60.06) // for x in [inf, 8]=1/[0,0.125] var p1R8 = [6]float64{ 0.00000000000000000000e+00, // 0x0000000000000000 1.17187499999988647970e-01, // 0x3FBDFFFFFFFFFCCE 1.32394806593073575129e+01, // 0x402A7A9D357F7FCE 4.12051854307378562225e+02, // 0x4079C0D4652EA590 3.87474538913960532227e+03, // 0x40AE457DA3A532CC 7.91447954031891731574e+03, // 0x40BEEA7AC32782DD } var p1S8 = [5]float64{ 1.14207370375678408436e+02, // 0x405C8D458E656CAC 3.65093083420853463394e+03, // 0x40AC85DC964D274F 3.69562060269033463555e+04, // 0x40E20B8697C5BB7F 9.76027935934950801311e+04, // 0x40F7D42CB28F17BB 3.08042720627888811578e+04, // 0x40DE1511697A0B2D } // for x in [8,4.5454] = 1/[0.125,0.22001] var p1R5 = [6]float64{ 1.31990519556243522749e-11, // 0x3DAD0667DAE1CA7D 1.17187493190614097638e-01, // 0x3FBDFFFFE2C10043 6.80275127868432871736e+00, // 0x401B36046E6315E3 1.08308182990189109773e+02, // 0x405B13B9452602ED 5.17636139533199752805e+02, // 0x40802D16D052D649 5.28715201363337541807e+02, // 0x408085B8BB7E0CB7 } var p1S5 = [5]float64{ 5.92805987221131331921e+01, // 0x404DA3EAA8AF633D 9.91401418733614377743e+02, // 0x408EFB361B066701 5.35326695291487976647e+03, // 0x40B4E9445706B6FB 7.84469031749551231769e+03, // 0x40BEA4B0B8A5BB15 1.50404688810361062679e+03, // 0x40978030036F5E51 } // for x in[4.5453,2.8571] = 1/[0.2199,0.35001] var p1R3 = [6]float64{ 3.02503916137373618024e-09, // 0x3E29FC21A7AD9EDD 1.17186865567253592491e-01, // 0x3FBDFFF55B21D17B 3.93297750033315640650e+00, // 0x400F76BCE85EAD8A 3.51194035591636932736e+01, // 0x40418F489DA6D129 9.10550110750781271918e+01, // 0x4056C3854D2C1837 4.85590685197364919645e+01, // 0x4048478F8EA83EE5 } var p1S3 = [5]float64{ 3.47913095001251519989e+01, // 0x40416549A134069C 3.36762458747825746741e+02, // 0x40750C3307F1A75F 1.04687139975775130551e+03, // 0x40905B7C5037D523 8.90811346398256432622e+02, // 0x408BD67DA32E31E9 1.03787932439639277504e+02, // 0x4059F26D7C2EED53 } // for x in [2.8570,2] = 1/[0.3499,0.5] var p1R2 = [6]float64{ 1.07710830106873743082e-07, // 0x3E7CE9D4F65544F4 1.17176219462683348094e-01, // 0x3FBDFF42BE760D83 2.36851496667608785174e+00, // 0x4002F2B7F98FAEC0 1.22426109148261232917e+01, // 0x40287C377F71A964 1.76939711271687727390e+01, // 0x4031B1A8177F8EE2 5.07352312588818499250e+00, // 0x40144B49A574C1FE } var p1S2 = [5]float64{ 2.14364859363821409488e+01, // 0x40356FBD8AD5ECDC 1.25290227168402751090e+02, // 0x405F529314F92CD5 2.32276469057162813669e+02, // 0x406D08D8D5A2DBD9 1.17679373287147100768e+02, // 0x405D6B7ADA1884A9 8.36463893371618283368e+00, // 0x4020BAB1F44E5192 } func pone(x float64) float64 { var p [6]float64 var q [5]float64 if x >= 8 { p = p1R8 q = p1S8 } else if x >= 4.5454 { p = p1R5 q = p1S5 } else if x >= 2.8571 { p = p1R3 q = p1S3 } else if x >= 2 { p = p1R2 q = p1S2 } z := 1 / (x * x) r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))) s := 1.0 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))) return 1 + r/s } // For x >= 8, the asymptotic expansions of qone is // 3/8 s - 105/1024 s**3 - ..., where s = 1/x. // We approximate qone by // qone(x) = s*(0.375 + (R/S)) // where R = qr1*s**2 + qr2*s**4 + ... + qr5*s**10 // S = 1 + qs1*s**2 + ... + qs6*s**12 // and // | qone(x)/s -0.375-R/S | <= 2**(-61.13) // for x in [inf, 8] = 1/[0,0.125] var q1R8 = [6]float64{ 0.00000000000000000000e+00, // 0x0000000000000000 -1.02539062499992714161e-01, // 0xBFBA3FFFFFFFFDF3 -1.62717534544589987888e+01, // 0xC0304591A26779F7 -7.59601722513950107896e+02, // 0xC087BCD053E4B576 -1.18498066702429587167e+04, // 0xC0C724E740F87415 -4.84385124285750353010e+04, // 0xC0E7A6D065D09C6A } var q1S8 = [6]float64{ 1.61395369700722909556e+02, // 0x40642CA6DE5BCDE5 7.82538599923348465381e+03, // 0x40BE9162D0D88419 1.33875336287249578163e+05, // 0x4100579AB0B75E98 7.19657723683240939863e+05, // 0x4125F65372869C19 6.66601232617776375264e+05, // 0x412457D27719AD5C -2.94490264303834643215e+05, // 0xC111F9690EA5AA18 } // for x in [8,4.5454] = 1/[0.125,0.22001] var q1R5 = [6]float64{ -2.08979931141764104297e-11, // 0xBDB6FA431AA1A098 -1.02539050241375426231e-01, // 0xBFBA3FFFCB597FEF -8.05644828123936029840e+00, // 0xC0201CE6CA03AD4B -1.83669607474888380239e+02, // 0xC066F56D6CA7B9B0 -1.37319376065508163265e+03, // 0xC09574C66931734F -2.61244440453215656817e+03, // 0xC0A468E388FDA79D } var q1S5 = [6]float64{ 8.12765501384335777857e+01, // 0x405451B2FF5A11B2 1.99179873460485964642e+03, // 0x409F1F31E77BF839 1.74684851924908907677e+04, // 0x40D10F1F0D64CE29 4.98514270910352279316e+04, // 0x40E8576DAABAD197 2.79480751638918118260e+04, // 0x40DB4B04CF7C364B -4.71918354795128470869e+03, // 0xC0B26F2EFCFFA004 } // for x in [4.5454,2.8571] = 1/[0.2199,0.35001] ??? var q1R3 = [6]float64{ -5.07831226461766561369e-09, // 0xBE35CFA9D38FC84F -1.02537829820837089745e-01, // 0xBFBA3FEB51AEED54 -4.61011581139473403113e+00, // 0xC01270C23302D9FF -5.78472216562783643212e+01, // 0xC04CEC71C25D16DA -2.28244540737631695038e+02, // 0xC06C87D34718D55F -2.19210128478909325622e+02, // 0xC06B66B95F5C1BF6 } var q1S3 = [6]float64{ 4.76651550323729509273e+01, // 0x4047D523CCD367E4 6.73865112676699709482e+02, // 0x40850EEBC031EE3E 3.38015286679526343505e+03, // 0x40AA684E448E7C9A 5.54772909720722782367e+03, // 0x40B5ABBAA61D54A6 1.90311919338810798763e+03, // 0x409DBC7A0DD4DF4B -1.35201191444307340817e+02, // 0xC060E670290A311F } // for x in [2.8570,2] = 1/[0.3499,0.5] var q1R2 = [6]float64{ -1.78381727510958865572e-07, // 0xBE87F12644C626D2 -1.02517042607985553460e-01, // 0xBFBA3E8E9148B010 -2.75220568278187460720e+00, // 0xC006048469BB4EDA -1.96636162643703720221e+01, // 0xC033A9E2C168907F -4.23253133372830490089e+01, // 0xC04529A3DE104AAA -2.13719211703704061733e+01, // 0xC0355F3639CF6E52 } var q1S2 = [6]float64{ 2.95333629060523854548e+01, // 0x403D888A78AE64FF 2.52981549982190529136e+02, // 0x406F9F68DB821CBA 7.57502834868645436472e+02, // 0x4087AC05CE49A0F7 7.39393205320467245656e+02, // 0x40871B2548D4C029 1.55949003336666123687e+02, // 0x40637E5E3C3ED8D4 -4.95949898822628210127e+00, // 0xC013D686E71BE86B } func qone(x float64) float64 { var p, q [6]float64 if x >= 8 { p = q1R8 q = q1S8 } else if x >= 4.5454 { p = q1R5 q = q1S5 } else if x >= 2.8571 { p = q1R3 q = q1S3 } else if x >= 2 { p = q1R2 q = q1S2 } z := 1 / (x * x) r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))) s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))) return (0.375 + r/s) / x }