// 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 cancelation, 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 cancelation, 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
}