Source file src/pkg/math/j0.go

     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	}