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Source file src/math/j0.go

Documentation: math

     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 cancelation, 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  	x = Abs(x)
   103  	if x >= 2 {
   104  		s, c := Sincos(x)
   105  		ss := s - c
   106  		cc := s + c
   107  
   108  		// make sure x+x does not overflow
   109  		if x < MaxFloat64/2 {
   110  			z := -Cos(x + x)
   111  			if s*c < 0 {
   112  				cc = z / ss
   113  			} else {
   114  				ss = z / cc
   115  			}
   116  		}
   117  
   118  		// j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
   119  		// y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
   120  
   121  		var z float64
   122  		if x > Two129 { // |x| > ~6.8056e+38
   123  			z = (1 / SqrtPi) * cc / Sqrt(x)
   124  		} else {
   125  			u := pzero(x)
   126  			v := qzero(x)
   127  			z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x)
   128  		}
   129  		return z // |x| >= 2.0
   130  	}
   131  	if x < TwoM13 { // |x| < ~1.2207e-4
   132  		if x < TwoM27 {
   133  			return 1 // |x| < ~7.4506e-9
   134  		}
   135  		return 1 - 0.25*x*x // ~7.4506e-9 < |x| < ~1.2207e-4
   136  	}
   137  	z := x * x
   138  	r := z * (R02 + z*(R03+z*(R04+z*R05)))
   139  	s := 1 + z*(S01+z*(S02+z*(S03+z*S04)))
   140  	if x < 1 {
   141  		return 1 + z*(-0.25+(r/s)) // |x| < 1.00
   142  	}
   143  	u := 0.5 * x
   144  	return (1+u)*(1-u) + z*(r/s) // 1.0 < |x| < 2.0
   145  }
   146  
   147  // Y0 returns the order-zero Bessel function of the second kind.
   148  //
   149  // Special cases are:
   150  //	Y0(+Inf) = 0
   151  //	Y0(0) = -Inf
   152  //	Y0(x < 0) = NaN
   153  //	Y0(NaN) = NaN
   154  func Y0(x float64) float64 {
   155  	const (
   156  		TwoM27 = 1.0 / (1 << 27)             // 2**-27 0x3e40000000000000
   157  		Two129 = 1 << 129                    // 2**129 0x4800000000000000
   158  		U00    = -7.38042951086872317523e-02 // 0xBFB2E4D699CBD01F
   159  		U01    = 1.76666452509181115538e-01  // 0x3FC69D019DE9E3FC
   160  		U02    = -1.38185671945596898896e-02 // 0xBF8C4CE8B16CFA97
   161  		U03    = 3.47453432093683650238e-04  // 0x3F36C54D20B29B6B
   162  		U04    = -3.81407053724364161125e-06 // 0xBECFFEA773D25CAD
   163  		U05    = 1.95590137035022920206e-08  // 0x3E5500573B4EABD4
   164  		U06    = -3.98205194132103398453e-11 // 0xBDC5E43D693FB3C8
   165  		V01    = 1.27304834834123699328e-02  // 0x3F8A127091C9C71A
   166  		V02    = 7.60068627350353253702e-05  // 0x3F13ECBBF578C6C1
   167  		V03    = 2.59150851840457805467e-07  // 0x3E91642D7FF202FD
   168  		V04    = 4.41110311332675467403e-10  // 0x3DFE50183BD6D9EF
   169  	)
   170  	// special cases
   171  	switch {
   172  	case x < 0 || IsNaN(x):
   173  		return NaN()
   174  	case IsInf(x, 1):
   175  		return 0
   176  	case x == 0:
   177  		return Inf(-1)
   178  	}
   179  
   180  	if x >= 2 { // |x| >= 2.0
   181  
   182  		// y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
   183  		//     where x0 = x-pi/4
   184  		// Better formula:
   185  		//     cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
   186  		//             =  1/sqrt(2) * (sin(x) + cos(x))
   187  		//     sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
   188  		//             =  1/sqrt(2) * (sin(x) - cos(x))
   189  		// To avoid cancelation, use
   190  		//     sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
   191  		// to compute the worse one.
   192  
   193  		s, c := Sincos(x)
   194  		ss := s - c
   195  		cc := s + c
   196  
   197  		// j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
   198  		// y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
   199  
   200  		// make sure x+x does not overflow
   201  		if x < MaxFloat64/2 {
   202  			z := -Cos(x + x)
   203  			if s*c < 0 {
   204  				cc = z / ss
   205  			} else {
   206  				ss = z / cc
   207  			}
   208  		}
   209  		var z float64
   210  		if x > Two129 { // |x| > ~6.8056e+38
   211  			z = (1 / SqrtPi) * ss / Sqrt(x)
   212  		} else {
   213  			u := pzero(x)
   214  			v := qzero(x)
   215  			z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x)
   216  		}
   217  		return z // |x| >= 2.0
   218  	}
   219  	if x <= TwoM27 {
   220  		return U00 + (2/Pi)*Log(x) // |x| < ~7.4506e-9
   221  	}
   222  	z := x * x
   223  	u := U00 + z*(U01+z*(U02+z*(U03+z*(U04+z*(U05+z*U06)))))
   224  	v := 1 + z*(V01+z*(V02+z*(V03+z*V04)))
   225  	return u/v + (2/Pi)*J0(x)*Log(x) // ~7.4506e-9 < |x| < 2.0
   226  }
   227  
   228  // The asymptotic expansions of pzero is
   229  //      1 - 9/128 s**2 + 11025/98304 s**4 - ..., where s = 1/x.
   230  // For x >= 2, We approximate pzero by
   231  // 	pzero(x) = 1 + (R/S)
   232  // where  R = pR0 + pR1*s**2 + pR2*s**4 + ... + pR5*s**10
   233  // 	  S = 1 + pS0*s**2 + ... + pS4*s**10
   234  // and
   235  //      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
   236  
   237  // for x in [inf, 8]=1/[0,0.125]
   238  var p0R8 = [6]float64{
   239  	0.00000000000000000000e+00,  // 0x0000000000000000
   240  	-7.03124999999900357484e-02, // 0xBFB1FFFFFFFFFD32
   241  	-8.08167041275349795626e+00, // 0xC02029D0B44FA779
   242  	-2.57063105679704847262e+02, // 0xC07011027B19E863
   243  	-2.48521641009428822144e+03, // 0xC0A36A6ECD4DCAFC
   244  	-5.25304380490729545272e+03, // 0xC0B4850B36CC643D
   245  }
   246  var p0S8 = [5]float64{
   247  	1.16534364619668181717e+02, // 0x405D223307A96751
   248  	3.83374475364121826715e+03, // 0x40ADF37D50596938
   249  	4.05978572648472545552e+04, // 0x40E3D2BB6EB6B05F
   250  	1.16752972564375915681e+05, // 0x40FC810F8F9FA9BD
   251  	4.76277284146730962675e+04, // 0x40E741774F2C49DC
   252  }
   253  
   254  // for x in [8,4.5454]=1/[0.125,0.22001]
   255  var p0R5 = [6]float64{
   256  	-1.14125464691894502584e-11, // 0xBDA918B147E495CC
   257  	-7.03124940873599280078e-02, // 0xBFB1FFFFE69AFBC6
   258  	-4.15961064470587782438e+00, // 0xC010A370F90C6BBF
   259  	-6.76747652265167261021e+01, // 0xC050EB2F5A7D1783
   260  	-3.31231299649172967747e+02, // 0xC074B3B36742CC63
   261  	-3.46433388365604912451e+02, // 0xC075A6EF28A38BD7
   262  }
   263  var p0S5 = [5]float64{
   264  	6.07539382692300335975e+01, // 0x404E60810C98C5DE
   265  	1.05125230595704579173e+03, // 0x40906D025C7E2864
   266  	5.97897094333855784498e+03, // 0x40B75AF88FBE1D60
   267  	9.62544514357774460223e+03, // 0x40C2CCB8FA76FA38
   268  	2.40605815922939109441e+03, // 0x40A2CC1DC70BE864
   269  }
   270  
   271  // for x in [4.547,2.8571]=1/[0.2199,0.35001]
   272  var p0R3 = [6]float64{
   273  	-2.54704601771951915620e-09, // 0xBE25E1036FE1AA86
   274  	-7.03119616381481654654e-02, // 0xBFB1FFF6F7C0E24B
   275  	-2.40903221549529611423e+00, // 0xC00345B2AEA48074
   276  	-2.19659774734883086467e+01, // 0xC035F74A4CB94E14
   277  	-5.80791704701737572236e+01, // 0xC04D0A22420A1A45
   278  	-3.14479470594888503854e+01, // 0xC03F72ACA892D80F
   279  }
   280  var p0S3 = [5]float64{
   281  	3.58560338055209726349e+01, // 0x4041ED9284077DD3
   282  	3.61513983050303863820e+02, // 0x40769839464A7C0E
   283  	1.19360783792111533330e+03, // 0x4092A66E6D1061D6
   284  	1.12799679856907414432e+03, // 0x40919FFCB8C39B7E
   285  	1.73580930813335754692e+02, // 0x4065B296FC379081
   286  }
   287  
   288  // for x in [2.8570,2]=1/[0.3499,0.5]
   289  var p0R2 = [6]float64{
   290  	-8.87534333032526411254e-08, // 0xBE77D316E927026D
   291  	-7.03030995483624743247e-02, // 0xBFB1FF62495E1E42
   292  	-1.45073846780952986357e+00, // 0xBFF736398A24A843
   293  	-7.63569613823527770791e+00, // 0xC01E8AF3EDAFA7F3
   294  	-1.11931668860356747786e+01, // 0xC02662E6C5246303
   295  	-3.23364579351335335033e+00, // 0xC009DE81AF8FE70F
   296  }
   297  var p0S2 = [5]float64{
   298  	2.22202997532088808441e+01, // 0x40363865908B5959
   299  	1.36206794218215208048e+02, // 0x4061069E0EE8878F
   300  	2.70470278658083486789e+02, // 0x4070E78642EA079B
   301  	1.53875394208320329881e+02, // 0x40633C033AB6FAFF
   302  	1.46576176948256193810e+01, // 0x402D50B344391809
   303  }
   304  
   305  func pzero(x float64) float64 {
   306  	var p *[6]float64
   307  	var q *[5]float64
   308  	if x >= 8 {
   309  		p = &p0R8
   310  		q = &p0S8
   311  	} else if x >= 4.5454 {
   312  		p = &p0R5
   313  		q = &p0S5
   314  	} else if x >= 2.8571 {
   315  		p = &p0R3
   316  		q = &p0S3
   317  	} else if x >= 2 {
   318  		p = &p0R2
   319  		q = &p0S2
   320  	}
   321  	z := 1 / (x * x)
   322  	r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
   323  	s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))))
   324  	return 1 + r/s
   325  }
   326  
   327  // For x >= 8, the asymptotic expansions of qzero is
   328  //      -1/8 s + 75/1024 s**3 - ..., where s = 1/x.
   329  // We approximate pzero by
   330  //      qzero(x) = s*(-1.25 + (R/S))
   331  // where R = qR0 + qR1*s**2 + qR2*s**4 + ... + qR5*s**10
   332  //       S = 1 + qS0*s**2 + ... + qS5*s**12
   333  // and
   334  //      | qzero(x)/s +1.25-R/S | <= 2**(-61.22)
   335  
   336  // for x in [inf, 8]=1/[0,0.125]
   337  var q0R8 = [6]float64{
   338  	0.00000000000000000000e+00, // 0x0000000000000000
   339  	7.32421874999935051953e-02, // 0x3FB2BFFFFFFFFE2C
   340  	1.17682064682252693899e+01, // 0x402789525BB334D6
   341  	5.57673380256401856059e+02, // 0x40816D6315301825
   342  	8.85919720756468632317e+03, // 0x40C14D993E18F46D
   343  	3.70146267776887834771e+04, // 0x40E212D40E901566
   344  }
   345  var q0S8 = [6]float64{
   346  	1.63776026895689824414e+02,  // 0x406478D5365B39BC
   347  	8.09834494656449805916e+03,  // 0x40BFA2584E6B0563
   348  	1.42538291419120476348e+05,  // 0x4101665254D38C3F
   349  	8.03309257119514397345e+05,  // 0x412883DA83A52B43
   350  	8.40501579819060512818e+05,  // 0x4129A66B28DE0B3D
   351  	-3.43899293537866615225e+05, // 0xC114FD6D2C9530C5
   352  }
   353  
   354  // for x in [8,4.5454]=1/[0.125,0.22001]
   355  var q0R5 = [6]float64{
   356  	1.84085963594515531381e-11, // 0x3DB43D8F29CC8CD9
   357  	7.32421766612684765896e-02, // 0x3FB2BFFFD172B04C
   358  	5.83563508962056953777e+00, // 0x401757B0B9953DD3
   359  	1.35111577286449829671e+02, // 0x4060E3920A8788E9
   360  	1.02724376596164097464e+03, // 0x40900CF99DC8C481
   361  	1.98997785864605384631e+03, // 0x409F17E953C6E3A6
   362  }
   363  var q0S5 = [6]float64{
   364  	8.27766102236537761883e+01,  // 0x4054B1B3FB5E1543
   365  	2.07781416421392987104e+03,  // 0x40A03BA0DA21C0CE
   366  	1.88472887785718085070e+04,  // 0x40D267D27B591E6D
   367  	5.67511122894947329769e+04,  // 0x40EBB5E397E02372
   368  	3.59767538425114471465e+04,  // 0x40E191181F7A54A0
   369  	-5.35434275601944773371e+03, // 0xC0B4EA57BEDBC609
   370  }
   371  
   372  // for x in [4.547,2.8571]=1/[0.2199,0.35001]
   373  var q0R3 = [6]float64{
   374  	4.37741014089738620906e-09, // 0x3E32CD036ADECB82
   375  	7.32411180042911447163e-02, // 0x3FB2BFEE0E8D0842
   376  	3.34423137516170720929e+00, // 0x400AC0FC61149CF5
   377  	4.26218440745412650017e+01, // 0x40454F98962DAEDD
   378  	1.70808091340565596283e+02, // 0x406559DBE25EFD1F
   379  	1.66733948696651168575e+02, // 0x4064D77C81FA21E0
   380  }
   381  var q0S3 = [6]float64{
   382  	4.87588729724587182091e+01,  // 0x40486122BFE343A6
   383  	7.09689221056606015736e+02,  // 0x40862D8386544EB3
   384  	3.70414822620111362994e+03,  // 0x40ACF04BE44DFC63
   385  	6.46042516752568917582e+03,  // 0x40B93C6CD7C76A28
   386  	2.51633368920368957333e+03,  // 0x40A3A8AAD94FB1C0
   387  	-1.49247451836156386662e+02, // 0xC062A7EB201CF40F
   388  }
   389  
   390  // for x in [2.8570,2]=1/[0.3499,0.5]
   391  var q0R2 = [6]float64{
   392  	1.50444444886983272379e-07, // 0x3E84313B54F76BDB
   393  	7.32234265963079278272e-02, // 0x3FB2BEC53E883E34
   394  	1.99819174093815998816e+00, // 0x3FFFF897E727779C
   395  	1.44956029347885735348e+01, // 0x402CFDBFAAF96FE5
   396  	3.16662317504781540833e+01, // 0x403FAA8E29FBDC4A
   397  	1.62527075710929267416e+01, // 0x403040B171814BB4
   398  }
   399  var q0S2 = [6]float64{
   400  	3.03655848355219184498e+01,  // 0x403E5D96F7C07AED
   401  	2.69348118608049844624e+02,  // 0x4070D591E4D14B40
   402  	8.44783757595320139444e+02,  // 0x408A664522B3BF22
   403  	8.82935845112488550512e+02,  // 0x408B977C9C5CC214
   404  	2.12666388511798828631e+02,  // 0x406A95530E001365
   405  	-5.31095493882666946917e+00, // 0xC0153E6AF8B32931
   406  }
   407  
   408  func qzero(x float64) float64 {
   409  	var p, q *[6]float64
   410  	if x >= 8 {
   411  		p = &q0R8
   412  		q = &q0S8
   413  	} else if x >= 4.5454 {
   414  		p = &q0R5
   415  		q = &q0S5
   416  	} else if x >= 2.8571 {
   417  		p = &q0R3
   418  		q = &q0S3
   419  	} else if x >= 2 {
   420  		p = &q0R2
   421  		q = &q0S2
   422  	}
   423  	z := 1 / (x * x)
   424  	r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
   425  	s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))))
   426  	return (-0.125 + r/s) / x
   427  }
   428  

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