diff options
Diffstat (limited to '')
| -rw-r--r-- | src/math/j0.go | 427 |
1 files changed, 427 insertions, 0 deletions
diff --git a/src/math/j0.go b/src/math/j0.go new file mode 100644 index 0000000..cb5f07b --- /dev/null +++ b/src/math/j0.go @@ -0,0 +1,427 @@ +// 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 zero. +*/ + +// The original C code and the long comment below are +// from FreeBSD's /usr/src/lib/msun/src/e_j0.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_j0(x), __ieee754_y0(x) +// Bessel function of the first and second kinds of order zero. +// Method -- j0(x): +// 1. For tiny x, we use j0(x) = 1 - x**2/4 + x**4/64 - ... +// 2. Reduce x to |x| since j0(x)=j0(-x), and +// for x in (0,2) +// j0(x) = 1-z/4+ z**2*R0/S0, where z = x*x; +// (precision: |j0-1+z/4-z**2R0/S0 |<2**-63.67 ) +// for x in (2,inf) +// j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0)) +// where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) +// as follow: +// cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) +// = 1/sqrt(2) * (cos(x) + sin(x)) +// sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4) +// = 1/sqrt(2) * (sin(x) - cos(x)) +// (To avoid cancellation, use +// sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) +// to compute the worse one.) +// +// 3 Special cases +// j0(nan)= nan +// j0(0) = 1 +// j0(inf) = 0 +// +// Method -- y0(x): +// 1. For x<2. +// Since +// y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x**2/4 - ...) +// therefore y0(x)-2/pi*j0(x)*ln(x) is an even function. +// We use the following function to approximate y0, +// y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x**2 +// where +// U(z) = u00 + u01*z + ... + u06*z**6 +// V(z) = 1 + v01*z + ... + v04*z**4 +// with absolute approximation error bounded by 2**-72. +// Note: For tiny x, U/V = u0 and j0(x)~1, hence +// y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27) +// 2. For x>=2. +// y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0)) +// where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) +// by the method mentioned above. +// 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0. +// + +// J0 returns the order-zero Bessel function of the first kind. +// +// Special cases are: +// J0(±Inf) = 0 +// J0(0) = 1 +// J0(NaN) = NaN +func J0(x float64) float64 { + const ( + Huge = 1e300 + TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000 + TwoM13 = 1.0 / (1 << 13) // 2**-13 0x3f20000000000000 + Two129 = 1 << 129 // 2**129 0x4800000000000000 + // R0/S0 on [0, 2] + R02 = 1.56249999999999947958e-02 // 0x3F8FFFFFFFFFFFFD + R03 = -1.89979294238854721751e-04 // 0xBF28E6A5B61AC6E9 + R04 = 1.82954049532700665670e-06 // 0x3EBEB1D10C503919 + R05 = -4.61832688532103189199e-09 // 0xBE33D5E773D63FCE + S01 = 1.56191029464890010492e-02 // 0x3F8FFCE882C8C2A4 + S02 = 1.16926784663337450260e-04 // 0x3F1EA6D2DD57DBF4 + S03 = 5.13546550207318111446e-07 // 0x3EA13B54CE84D5A9 + S04 = 1.16614003333790000205e-09 // 0x3E1408BCF4745D8F + ) + // special cases + switch { + case IsNaN(x): + return x + case IsInf(x, 0): + return 0 + case x == 0: + return 1 + } + + x = Abs(x) + 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 + } + } + + // j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) + // y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) + + var z float64 + if x > Two129 { // |x| > ~6.8056e+38 + z = (1 / SqrtPi) * cc / Sqrt(x) + } else { + u := pzero(x) + v := qzero(x) + z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x) + } + return z // |x| >= 2.0 + } + if x < TwoM13 { // |x| < ~1.2207e-4 + if x < TwoM27 { + return 1 // |x| < ~7.4506e-9 + } + return 1 - 0.25*x*x // ~7.4506e-9 < |x| < ~1.2207e-4 + } + z := x * x + r := z * (R02 + z*(R03+z*(R04+z*R05))) + s := 1 + z*(S01+z*(S02+z*(S03+z*S04))) + if x < 1 { + return 1 + z*(-0.25+(r/s)) // |x| < 1.00 + } + u := 0.5 * x + return (1+u)*(1-u) + z*(r/s) // 1.0 < |x| < 2.0 +} + +// Y0 returns the order-zero Bessel function of the second kind. +// +// Special cases are: +// Y0(+Inf) = 0 +// Y0(0) = -Inf +// Y0(x < 0) = NaN +// Y0(NaN) = NaN +func Y0(x float64) float64 { + const ( + TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000 + Two129 = 1 << 129 // 2**129 0x4800000000000000 + U00 = -7.38042951086872317523e-02 // 0xBFB2E4D699CBD01F + U01 = 1.76666452509181115538e-01 // 0x3FC69D019DE9E3FC + U02 = -1.38185671945596898896e-02 // 0xBF8C4CE8B16CFA97 + U03 = 3.47453432093683650238e-04 // 0x3F36C54D20B29B6B + U04 = -3.81407053724364161125e-06 // 0xBECFFEA773D25CAD + U05 = 1.95590137035022920206e-08 // 0x3E5500573B4EABD4 + U06 = -3.98205194132103398453e-11 // 0xBDC5E43D693FB3C8 + V01 = 1.27304834834123699328e-02 // 0x3F8A127091C9C71A + V02 = 7.60068627350353253702e-05 // 0x3F13ECBBF578C6C1 + V03 = 2.59150851840457805467e-07 // 0x3E91642D7FF202FD + V04 = 4.41110311332675467403e-10 // 0x3DFE50183BD6D9EF + ) + // 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 { // |x| >= 2.0 + + // y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) + // where x0 = x-pi/4 + // Better formula: + // cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) + // = 1/sqrt(2) * (sin(x) + cos(x)) + // sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) + // = 1/sqrt(2) * (sin(x) - cos(x)) + // To avoid cancellation, use + // sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) + // to compute the worse one. + + s, c := Sincos(x) + ss := s - c + cc := s + c + + // j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) + // y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) + + // 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 + } + } + var z float64 + if x > Two129 { // |x| > ~6.8056e+38 + z = (1 / SqrtPi) * ss / Sqrt(x) + } else { + u := pzero(x) + v := qzero(x) + z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x) + } + return z // |x| >= 2.0 + } + if x <= TwoM27 { + return U00 + (2/Pi)*Log(x) // |x| < ~7.4506e-9 + } + z := x * x + u := U00 + z*(U01+z*(U02+z*(U03+z*(U04+z*(U05+z*U06))))) + v := 1 + z*(V01+z*(V02+z*(V03+z*V04))) + return u/v + (2/Pi)*J0(x)*Log(x) // ~7.4506e-9 < |x| < 2.0 +} + +// The asymptotic expansions of pzero is +// 1 - 9/128 s**2 + 11025/98304 s**4 - ..., where s = 1/x. +// For x >= 2, We approximate pzero by +// pzero(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 +// | pzero(x)-1-R/S | <= 2 ** ( -60.26) + +// for x in [inf, 8]=1/[0,0.125] +var p0R8 = [6]float64{ + 0.00000000000000000000e+00, // 0x0000000000000000 + -7.03124999999900357484e-02, // 0xBFB1FFFFFFFFFD32 + -8.08167041275349795626e+00, // 0xC02029D0B44FA779 + -2.57063105679704847262e+02, // 0xC07011027B19E863 + -2.48521641009428822144e+03, // 0xC0A36A6ECD4DCAFC + -5.25304380490729545272e+03, // 0xC0B4850B36CC643D +} +var p0S8 = [5]float64{ + 1.16534364619668181717e+02, // 0x405D223307A96751 + 3.83374475364121826715e+03, // 0x40ADF37D50596938 + 4.05978572648472545552e+04, // 0x40E3D2BB6EB6B05F + 1.16752972564375915681e+05, // 0x40FC810F8F9FA9BD + 4.76277284146730962675e+04, // 0x40E741774F2C49DC +} + +// for x in [8,4.5454]=1/[0.125,0.22001] +var p0R5 = [6]float64{ + -1.14125464691894502584e-11, // 0xBDA918B147E495CC + -7.03124940873599280078e-02, // 0xBFB1FFFFE69AFBC6 + -4.15961064470587782438e+00, // 0xC010A370F90C6BBF + -6.76747652265167261021e+01, // 0xC050EB2F5A7D1783 + -3.31231299649172967747e+02, // 0xC074B3B36742CC63 + -3.46433388365604912451e+02, // 0xC075A6EF28A38BD7 +} +var p0S5 = [5]float64{ + 6.07539382692300335975e+01, // 0x404E60810C98C5DE + 1.05125230595704579173e+03, // 0x40906D025C7E2864 + 5.97897094333855784498e+03, // 0x40B75AF88FBE1D60 + 9.62544514357774460223e+03, // 0x40C2CCB8FA76FA38 + 2.40605815922939109441e+03, // 0x40A2CC1DC70BE864 +} + +// for x in [4.547,2.8571]=1/[0.2199,0.35001] +var p0R3 = [6]float64{ + -2.54704601771951915620e-09, // 0xBE25E1036FE1AA86 + -7.03119616381481654654e-02, // 0xBFB1FFF6F7C0E24B + -2.40903221549529611423e+00, // 0xC00345B2AEA48074 + -2.19659774734883086467e+01, // 0xC035F74A4CB94E14 + -5.80791704701737572236e+01, // 0xC04D0A22420A1A45 + -3.14479470594888503854e+01, // 0xC03F72ACA892D80F +} +var p0S3 = [5]float64{ + 3.58560338055209726349e+01, // 0x4041ED9284077DD3 + 3.61513983050303863820e+02, // 0x40769839464A7C0E + 1.19360783792111533330e+03, // 0x4092A66E6D1061D6 + 1.12799679856907414432e+03, // 0x40919FFCB8C39B7E + 1.73580930813335754692e+02, // 0x4065B296FC379081 +} + +// for x in [2.8570,2]=1/[0.3499,0.5] +var p0R2 = [6]float64{ + -8.87534333032526411254e-08, // 0xBE77D316E927026D + -7.03030995483624743247e-02, // 0xBFB1FF62495E1E42 + -1.45073846780952986357e+00, // 0xBFF736398A24A843 + -7.63569613823527770791e+00, // 0xC01E8AF3EDAFA7F3 + -1.11931668860356747786e+01, // 0xC02662E6C5246303 + -3.23364579351335335033e+00, // 0xC009DE81AF8FE70F +} +var p0S2 = [5]float64{ + 2.22202997532088808441e+01, // 0x40363865908B5959 + 1.36206794218215208048e+02, // 0x4061069E0EE8878F + 2.70470278658083486789e+02, // 0x4070E78642EA079B + 1.53875394208320329881e+02, // 0x40633C033AB6FAFF + 1.46576176948256193810e+01, // 0x402D50B344391809 +} + +func pzero(x float64) float64 { + var p *[6]float64 + var q *[5]float64 + if x >= 8 { + p = &p0R8 + q = &p0S8 + } else if x >= 4.5454 { + p = &p0R5 + q = &p0S5 + } else if x >= 2.8571 { + p = &p0R3 + q = &p0S3 + } else if x >= 2 { + p = &p0R2 + q = &p0S2 + } + 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])))) + return 1 + r/s +} + +// For x >= 8, the asymptotic expansions of qzero is +// -1/8 s + 75/1024 s**3 - ..., where s = 1/x. +// We approximate pzero by +// qzero(x) = s*(-1.25 + (R/S)) +// where R = qR0 + qR1*s**2 + qR2*s**4 + ... + qR5*s**10 +// S = 1 + qS0*s**2 + ... + qS5*s**12 +// and +// | qzero(x)/s +1.25-R/S | <= 2**(-61.22) + +// for x in [inf, 8]=1/[0,0.125] +var q0R8 = [6]float64{ + 0.00000000000000000000e+00, // 0x0000000000000000 + 7.32421874999935051953e-02, // 0x3FB2BFFFFFFFFE2C + 1.17682064682252693899e+01, // 0x402789525BB334D6 + 5.57673380256401856059e+02, // 0x40816D6315301825 + 8.85919720756468632317e+03, // 0x40C14D993E18F46D + 3.70146267776887834771e+04, // 0x40E212D40E901566 +} +var q0S8 = [6]float64{ + 1.63776026895689824414e+02, // 0x406478D5365B39BC + 8.09834494656449805916e+03, // 0x40BFA2584E6B0563 + 1.42538291419120476348e+05, // 0x4101665254D38C3F + 8.03309257119514397345e+05, // 0x412883DA83A52B43 + 8.40501579819060512818e+05, // 0x4129A66B28DE0B3D + -3.43899293537866615225e+05, // 0xC114FD6D2C9530C5 +} + +// for x in [8,4.5454]=1/[0.125,0.22001] +var q0R5 = [6]float64{ + 1.84085963594515531381e-11, // 0x3DB43D8F29CC8CD9 + 7.32421766612684765896e-02, // 0x3FB2BFFFD172B04C + 5.83563508962056953777e+00, // 0x401757B0B9953DD3 + 1.35111577286449829671e+02, // 0x4060E3920A8788E9 + 1.02724376596164097464e+03, // 0x40900CF99DC8C481 + 1.98997785864605384631e+03, // 0x409F17E953C6E3A6 +} +var q0S5 = [6]float64{ + 8.27766102236537761883e+01, // 0x4054B1B3FB5E1543 + 2.07781416421392987104e+03, // 0x40A03BA0DA21C0CE + 1.88472887785718085070e+04, // 0x40D267D27B591E6D + 5.67511122894947329769e+04, // 0x40EBB5E397E02372 + 3.59767538425114471465e+04, // 0x40E191181F7A54A0 + -5.35434275601944773371e+03, // 0xC0B4EA57BEDBC609 +} + +// for x in [4.547,2.8571]=1/[0.2199,0.35001] +var q0R3 = [6]float64{ + 4.37741014089738620906e-09, // 0x3E32CD036ADECB82 + 7.32411180042911447163e-02, // 0x3FB2BFEE0E8D0842 + 3.34423137516170720929e+00, // 0x400AC0FC61149CF5 + 4.26218440745412650017e+01, // 0x40454F98962DAEDD + 1.70808091340565596283e+02, // 0x406559DBE25EFD1F + 1.66733948696651168575e+02, // 0x4064D77C81FA21E0 +} +var q0S3 = [6]float64{ + 4.87588729724587182091e+01, // 0x40486122BFE343A6 + 7.09689221056606015736e+02, // 0x40862D8386544EB3 + 3.70414822620111362994e+03, // 0x40ACF04BE44DFC63 + 6.46042516752568917582e+03, // 0x40B93C6CD7C76A28 + 2.51633368920368957333e+03, // 0x40A3A8AAD94FB1C0 + -1.49247451836156386662e+02, // 0xC062A7EB201CF40F +} + +// for x in [2.8570,2]=1/[0.3499,0.5] +var q0R2 = [6]float64{ + 1.50444444886983272379e-07, // 0x3E84313B54F76BDB + 7.32234265963079278272e-02, // 0x3FB2BEC53E883E34 + 1.99819174093815998816e+00, // 0x3FFFF897E727779C + 1.44956029347885735348e+01, // 0x402CFDBFAAF96FE5 + 3.16662317504781540833e+01, // 0x403FAA8E29FBDC4A + 1.62527075710929267416e+01, // 0x403040B171814BB4 +} +var q0S2 = [6]float64{ + 3.03655848355219184498e+01, // 0x403E5D96F7C07AED + 2.69348118608049844624e+02, // 0x4070D591E4D14B40 + 8.44783757595320139444e+02, // 0x408A664522B3BF22 + 8.82935845112488550512e+02, // 0x408B977C9C5CC214 + 2.12666388511798828631e+02, // 0x406A95530E001365 + -5.31095493882666946917e+00, // 0xC0153E6AF8B32931 +} + +func qzero(x float64) float64 { + var p, q *[6]float64 + if x >= 8 { + p = &q0R8 + q = &q0S8 + } else if x >= 4.5454 { + p = &q0R5 + q = &q0S5 + } else if x >= 2.8571 { + p = &q0R3 + q = &q0S3 + } else if x >= 2 { + p = &q0R2 + q = &q0S2 + } + 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 (-0.125 + r/s) / x +} |