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 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  	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 cancellation, 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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