j1.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 one.
		 9  */
		10  
		11  // The original C code and the long comment below are
		12  // from FreeBSD's /usr/src/lib/msun/src/e_j1.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_j1(x), __ieee754_y1(x)
		26  // Bessel function of the first and second kinds of order one.
		27  // Method -- j1(x):
		28  //			1. For tiny x, we use j1(x) = x/2 - x**3/16 + x**5/384 - ...
		29  //			2. Reduce x to |x| since j1(x)=-j1(-x),	and
		30  //				 for x in (0,2)
		31  //							j1(x) = x/2 + x*z*R0/S0,	where z = x*x;
		32  //				 (precision:	|j1/x - 1/2 - R0/S0 |<2**-61.51 )
		33  //				 for x in (2,inf)
		34  //							j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
		35  //							y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
		36  //				 where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
		37  //				 as follow:
		38  //							cos(x1) =	cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
		39  //											=	1/sqrt(2) * (sin(x) - cos(x))
		40  //							sin(x1) =	sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
		41  //											= -1/sqrt(2) * (sin(x) + cos(x))
		42  //				 (To avoid cancellation, use
		43  //							sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
		44  //				 to compute the worse one.)
		45  //
		46  //			3 Special cases
		47  //							j1(nan)= nan
		48  //							j1(0) = 0
		49  //							j1(inf) = 0
		50  //
		51  // Method -- y1(x):
		52  //			1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
		53  //			2. For x<2.
		54  //				 Since
		55  //							y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x**3-...)
		56  //				 therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
		57  //				 We use the following function to approximate y1,
		58  //							y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x**2
		59  //				 where for x in [0,2] (abs err less than 2**-65.89)
		60  //							U(z) = U0[0] + U0[1]*z + ... + U0[4]*z**4
		61  //							V(z) = 1	+ v0[0]*z + ... + v0[4]*z**5
		62  //				 Note: For tiny x, 1/x dominate y1 and hence
		63  //							y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
		64  //			3. For x>=2.
		65  //							 y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
		66  //				 where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
		67  //				 by method mentioned above.
		68  
		69  // J1 returns the order-one Bessel function of the first kind.
		70  //
		71  // Special cases are:
		72  //	J1(±Inf) = 0
		73  //	J1(NaN) = NaN
		74  func J1(x float64) float64 {
		75  	const (
		76  		TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000
		77  		Two129 = 1 << 129				// 2**129 0x4800000000000000
		78  		// R0/S0 on [0, 2]
		79  		R00 = -6.25000000000000000000e-02 // 0xBFB0000000000000
		80  		R01 = 1.40705666955189706048e-03	// 0x3F570D9F98472C61
		81  		R02 = -1.59955631084035597520e-05 // 0xBEF0C5C6BA169668
		82  		R03 = 4.96727999609584448412e-08	// 0x3E6AAAFA46CA0BD9
		83  		S01 = 1.91537599538363460805e-02	// 0x3F939D0B12637E53
		84  		S02 = 1.85946785588630915560e-04	// 0x3F285F56B9CDF664
		85  		S03 = 1.17718464042623683263e-06	// 0x3EB3BFF8333F8498
		86  		S04 = 5.04636257076217042715e-09	// 0x3E35AC88C97DFF2C
		87  		S05 = 1.23542274426137913908e-11	// 0x3DAB2ACFCFB97ED8
		88  	)
		89  	// special cases
		90  	switch {
		91  	case IsNaN(x):
		92  		return x
		93  	case IsInf(x, 0) || x == 0:
		94  		return 0
		95  	}
		96  
		97  	sign := false
		98  	if x < 0 {
		99  		x = -x
	 100  		sign = true
	 101  	}
	 102  	if x >= 2 {
	 103  		s, c := Sincos(x)
	 104  		ss := -s - c
	 105  		cc := s - c
	 106  
	 107  		// make sure x+x does not overflow
	 108  		if x < MaxFloat64/2 {
	 109  			z := Cos(x + x)
	 110  			if s*c > 0 {
	 111  				cc = z / ss
	 112  			} else {
	 113  				ss = z / cc
	 114  			}
	 115  		}
	 116  
	 117  		// j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
	 118  		// y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
	 119  
	 120  		var z float64
	 121  		if x > Two129 {
	 122  			z = (1 / SqrtPi) * cc / Sqrt(x)
	 123  		} else {
	 124  			u := pone(x)
	 125  			v := qone(x)
	 126  			z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x)
	 127  		}
	 128  		if sign {
	 129  			return -z
	 130  		}
	 131  		return z
	 132  	}
	 133  	if x < TwoM27 { // |x|<2**-27
	 134  		return 0.5 * x // inexact if x!=0 necessary
	 135  	}
	 136  	z := x * x
	 137  	r := z * (R00 + z*(R01+z*(R02+z*R03)))
	 138  	s := 1.0 + z*(S01+z*(S02+z*(S03+z*(S04+z*S05))))
	 139  	r *= x
	 140  	z = 0.5*x + r/s
	 141  	if sign {
	 142  		return -z
	 143  	}
	 144  	return z
	 145  }
	 146  
	 147  // Y1 returns the order-one Bessel function of the second kind.
	 148  //
	 149  // Special cases are:
	 150  //	Y1(+Inf) = 0
	 151  //	Y1(0) = -Inf
	 152  //	Y1(x < 0) = NaN
	 153  //	Y1(NaN) = NaN
	 154  func Y1(x float64) float64 {
	 155  	const (
	 156  		TwoM54 = 1.0 / (1 << 54)						 // 2**-54 0x3c90000000000000
	 157  		Two129 = 1 << 129										// 2**129 0x4800000000000000
	 158  		U00		= -1.96057090646238940668e-01 // 0xBFC91866143CBC8A
	 159  		U01		= 5.04438716639811282616e-02	// 0x3FA9D3C776292CD1
	 160  		U02		= -1.91256895875763547298e-03 // 0xBF5F55E54844F50F
	 161  		U03		= 2.35252600561610495928e-05	// 0x3EF8AB038FA6B88E
	 162  		U04		= -9.19099158039878874504e-08 // 0xBE78AC00569105B8
	 163  		V00		= 1.99167318236649903973e-02	// 0x3F94650D3F4DA9F0
	 164  		V01		= 2.02552581025135171496e-04	// 0x3F2A8C896C257764
	 165  		V02		= 1.35608801097516229404e-06	// 0x3EB6C05A894E8CA6
	 166  		V03		= 6.22741452364621501295e-09	// 0x3E3ABF1D5BA69A86
	 167  		V04		= 1.66559246207992079114e-11	// 0x3DB25039DACA772A
	 168  	)
	 169  	// special cases
	 170  	switch {
	 171  	case x < 0 || IsNaN(x):
	 172  		return NaN()
	 173  	case IsInf(x, 1):
	 174  		return 0
	 175  	case x == 0:
	 176  		return Inf(-1)
	 177  	}
	 178  
	 179  	if x >= 2 {
	 180  		s, c := Sincos(x)
	 181  		ss := -s - c
	 182  		cc := s - c
	 183  
	 184  		// make sure x+x does not overflow
	 185  		if x < MaxFloat64/2 {
	 186  			z := Cos(x + x)
	 187  			if s*c > 0 {
	 188  				cc = z / ss
	 189  			} else {
	 190  				ss = z / cc
	 191  			}
	 192  		}
	 193  		// y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
	 194  		// where x0 = x-3pi/4
	 195  		//		 Better formula:
	 196  		//				 cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
	 197  		//								 =	1/sqrt(2) * (sin(x) - cos(x))
	 198  		//				 sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
	 199  		//								 = -1/sqrt(2) * (cos(x) + sin(x))
	 200  		// To avoid cancellation, use
	 201  		//		 sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
	 202  		// to compute the worse one.
	 203  
	 204  		var z float64
	 205  		if x > Two129 {
	 206  			z = (1 / SqrtPi) * ss / Sqrt(x)
	 207  		} else {
	 208  			u := pone(x)
	 209  			v := qone(x)
	 210  			z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x)
	 211  		}
	 212  		return z
	 213  	}
	 214  	if x <= TwoM54 { // x < 2**-54
	 215  		return -(2 / Pi) / x
	 216  	}
	 217  	z := x * x
	 218  	u := U00 + z*(U01+z*(U02+z*(U03+z*U04)))
	 219  	v := 1 + z*(V00+z*(V01+z*(V02+z*(V03+z*V04))))
	 220  	return x*(u/v) + (2/Pi)*(J1(x)*Log(x)-1/x)
	 221  }
	 222  
	 223  // For x >= 8, the asymptotic expansions of pone is
	 224  //			1 + 15/128 s**2 - 4725/2**15 s**4 - ..., where s = 1/x.
	 225  // We approximate pone by
	 226  //			pone(x) = 1 + (R/S)
	 227  // where R = pr0 + pr1*s**2 + pr2*s**4 + ... + pr5*s**10
	 228  //			 S = 1 + ps0*s**2 + ... + ps4*s**10
	 229  // and
	 230  //			| pone(x)-1-R/S | <= 2**(-60.06)
	 231  
	 232  // for x in [inf, 8]=1/[0,0.125]
	 233  var p1R8 = [6]float64{
	 234  	0.00000000000000000000e+00, // 0x0000000000000000
	 235  	1.17187499999988647970e-01, // 0x3FBDFFFFFFFFFCCE
	 236  	1.32394806593073575129e+01, // 0x402A7A9D357F7FCE
	 237  	4.12051854307378562225e+02, // 0x4079C0D4652EA590
	 238  	3.87474538913960532227e+03, // 0x40AE457DA3A532CC
	 239  	7.91447954031891731574e+03, // 0x40BEEA7AC32782DD
	 240  }
	 241  var p1S8 = [5]float64{
	 242  	1.14207370375678408436e+02, // 0x405C8D458E656CAC
	 243  	3.65093083420853463394e+03, // 0x40AC85DC964D274F
	 244  	3.69562060269033463555e+04, // 0x40E20B8697C5BB7F
	 245  	9.76027935934950801311e+04, // 0x40F7D42CB28F17BB
	 246  	3.08042720627888811578e+04, // 0x40DE1511697A0B2D
	 247  }
	 248  
	 249  // for x in [8,4.5454] = 1/[0.125,0.22001]
	 250  var p1R5 = [6]float64{
	 251  	1.31990519556243522749e-11, // 0x3DAD0667DAE1CA7D
	 252  	1.17187493190614097638e-01, // 0x3FBDFFFFE2C10043
	 253  	6.80275127868432871736e+00, // 0x401B36046E6315E3
	 254  	1.08308182990189109773e+02, // 0x405B13B9452602ED
	 255  	5.17636139533199752805e+02, // 0x40802D16D052D649
	 256  	5.28715201363337541807e+02, // 0x408085B8BB7E0CB7
	 257  }
	 258  var p1S5 = [5]float64{
	 259  	5.92805987221131331921e+01, // 0x404DA3EAA8AF633D
	 260  	9.91401418733614377743e+02, // 0x408EFB361B066701
	 261  	5.35326695291487976647e+03, // 0x40B4E9445706B6FB
	 262  	7.84469031749551231769e+03, // 0x40BEA4B0B8A5BB15
	 263  	1.50404688810361062679e+03, // 0x40978030036F5E51
	 264  }
	 265  
	 266  // for x in[4.5453,2.8571] = 1/[0.2199,0.35001]
	 267  var p1R3 = [6]float64{
	 268  	3.02503916137373618024e-09, // 0x3E29FC21A7AD9EDD
	 269  	1.17186865567253592491e-01, // 0x3FBDFFF55B21D17B
	 270  	3.93297750033315640650e+00, // 0x400F76BCE85EAD8A
	 271  	3.51194035591636932736e+01, // 0x40418F489DA6D129
	 272  	9.10550110750781271918e+01, // 0x4056C3854D2C1837
	 273  	4.85590685197364919645e+01, // 0x4048478F8EA83EE5
	 274  }
	 275  var p1S3 = [5]float64{
	 276  	3.47913095001251519989e+01, // 0x40416549A134069C
	 277  	3.36762458747825746741e+02, // 0x40750C3307F1A75F
	 278  	1.04687139975775130551e+03, // 0x40905B7C5037D523
	 279  	8.90811346398256432622e+02, // 0x408BD67DA32E31E9
	 280  	1.03787932439639277504e+02, // 0x4059F26D7C2EED53
	 281  }
	 282  
	 283  // for x in [2.8570,2] = 1/[0.3499,0.5]
	 284  var p1R2 = [6]float64{
	 285  	1.07710830106873743082e-07, // 0x3E7CE9D4F65544F4
	 286  	1.17176219462683348094e-01, // 0x3FBDFF42BE760D83
	 287  	2.36851496667608785174e+00, // 0x4002F2B7F98FAEC0
	 288  	1.22426109148261232917e+01, // 0x40287C377F71A964
	 289  	1.76939711271687727390e+01, // 0x4031B1A8177F8EE2
	 290  	5.07352312588818499250e+00, // 0x40144B49A574C1FE
	 291  }
	 292  var p1S2 = [5]float64{
	 293  	2.14364859363821409488e+01, // 0x40356FBD8AD5ECDC
	 294  	1.25290227168402751090e+02, // 0x405F529314F92CD5
	 295  	2.32276469057162813669e+02, // 0x406D08D8D5A2DBD9
	 296  	1.17679373287147100768e+02, // 0x405D6B7ADA1884A9
	 297  	8.36463893371618283368e+00, // 0x4020BAB1F44E5192
	 298  }
	 299  
	 300  func pone(x float64) float64 {
	 301  	var p *[6]float64
	 302  	var q *[5]float64
	 303  	if x >= 8 {
	 304  		p = &p1R8
	 305  		q = &p1S8
	 306  	} else if x >= 4.5454 {
	 307  		p = &p1R5
	 308  		q = &p1S5
	 309  	} else if x >= 2.8571 {
	 310  		p = &p1R3
	 311  		q = &p1S3
	 312  	} else if x >= 2 {
	 313  		p = &p1R2
	 314  		q = &p1S2
	 315  	}
	 316  	z := 1 / (x * x)
	 317  	r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
	 318  	s := 1.0 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))))
	 319  	return 1 + r/s
	 320  }
	 321  
	 322  // For x >= 8, the asymptotic expansions of qone is
	 323  //			3/8 s - 105/1024 s**3 - ..., where s = 1/x.
	 324  // We approximate qone by
	 325  //			qone(x) = s*(0.375 + (R/S))
	 326  // where R = qr1*s**2 + qr2*s**4 + ... + qr5*s**10
	 327  //			 S = 1 + qs1*s**2 + ... + qs6*s**12
	 328  // and
	 329  //			| qone(x)/s -0.375-R/S | <= 2**(-61.13)
	 330  
	 331  // for x in [inf, 8] = 1/[0,0.125]
	 332  var q1R8 = [6]float64{
	 333  	0.00000000000000000000e+00,	// 0x0000000000000000
	 334  	-1.02539062499992714161e-01, // 0xBFBA3FFFFFFFFDF3
	 335  	-1.62717534544589987888e+01, // 0xC0304591A26779F7
	 336  	-7.59601722513950107896e+02, // 0xC087BCD053E4B576
	 337  	-1.18498066702429587167e+04, // 0xC0C724E740F87415
	 338  	-4.84385124285750353010e+04, // 0xC0E7A6D065D09C6A
	 339  }
	 340  var q1S8 = [6]float64{
	 341  	1.61395369700722909556e+02,	// 0x40642CA6DE5BCDE5
	 342  	7.82538599923348465381e+03,	// 0x40BE9162D0D88419
	 343  	1.33875336287249578163e+05,	// 0x4100579AB0B75E98
	 344  	7.19657723683240939863e+05,	// 0x4125F65372869C19
	 345  	6.66601232617776375264e+05,	// 0x412457D27719AD5C
	 346  	-2.94490264303834643215e+05, // 0xC111F9690EA5AA18
	 347  }
	 348  
	 349  // for x in [8,4.5454] = 1/[0.125,0.22001]
	 350  var q1R5 = [6]float64{
	 351  	-2.08979931141764104297e-11, // 0xBDB6FA431AA1A098
	 352  	-1.02539050241375426231e-01, // 0xBFBA3FFFCB597FEF
	 353  	-8.05644828123936029840e+00, // 0xC0201CE6CA03AD4B
	 354  	-1.83669607474888380239e+02, // 0xC066F56D6CA7B9B0
	 355  	-1.37319376065508163265e+03, // 0xC09574C66931734F
	 356  	-2.61244440453215656817e+03, // 0xC0A468E388FDA79D
	 357  }
	 358  var q1S5 = [6]float64{
	 359  	8.12765501384335777857e+01,	// 0x405451B2FF5A11B2
	 360  	1.99179873460485964642e+03,	// 0x409F1F31E77BF839
	 361  	1.74684851924908907677e+04,	// 0x40D10F1F0D64CE29
	 362  	4.98514270910352279316e+04,	// 0x40E8576DAABAD197
	 363  	2.79480751638918118260e+04,	// 0x40DB4B04CF7C364B
	 364  	-4.71918354795128470869e+03, // 0xC0B26F2EFCFFA004
	 365  }
	 366  
	 367  // for x in [4.5454,2.8571] = 1/[0.2199,0.35001] ???
	 368  var q1R3 = [6]float64{
	 369  	-5.07831226461766561369e-09, // 0xBE35CFA9D38FC84F
	 370  	-1.02537829820837089745e-01, // 0xBFBA3FEB51AEED54
	 371  	-4.61011581139473403113e+00, // 0xC01270C23302D9FF
	 372  	-5.78472216562783643212e+01, // 0xC04CEC71C25D16DA
	 373  	-2.28244540737631695038e+02, // 0xC06C87D34718D55F
	 374  	-2.19210128478909325622e+02, // 0xC06B66B95F5C1BF6
	 375  }
	 376  var q1S3 = [6]float64{
	 377  	4.76651550323729509273e+01,	// 0x4047D523CCD367E4
	 378  	6.73865112676699709482e+02,	// 0x40850EEBC031EE3E
	 379  	3.38015286679526343505e+03,	// 0x40AA684E448E7C9A
	 380  	5.54772909720722782367e+03,	// 0x40B5ABBAA61D54A6
	 381  	1.90311919338810798763e+03,	// 0x409DBC7A0DD4DF4B
	 382  	-1.35201191444307340817e+02, // 0xC060E670290A311F
	 383  }
	 384  
	 385  // for x in [2.8570,2] = 1/[0.3499,0.5]
	 386  var q1R2 = [6]float64{
	 387  	-1.78381727510958865572e-07, // 0xBE87F12644C626D2
	 388  	-1.02517042607985553460e-01, // 0xBFBA3E8E9148B010
	 389  	-2.75220568278187460720e+00, // 0xC006048469BB4EDA
	 390  	-1.96636162643703720221e+01, // 0xC033A9E2C168907F
	 391  	-4.23253133372830490089e+01, // 0xC04529A3DE104AAA
	 392  	-2.13719211703704061733e+01, // 0xC0355F3639CF6E52
	 393  }
	 394  var q1S2 = [6]float64{
	 395  	2.95333629060523854548e+01,	// 0x403D888A78AE64FF
	 396  	2.52981549982190529136e+02,	// 0x406F9F68DB821CBA
	 397  	7.57502834868645436472e+02,	// 0x4087AC05CE49A0F7
	 398  	7.39393205320467245656e+02,	// 0x40871B2548D4C029
	 399  	1.55949003336666123687e+02,	// 0x40637E5E3C3ED8D4
	 400  	-4.95949898822628210127e+00, // 0xC013D686E71BE86B
	 401  }
	 402  
	 403  func qone(x float64) float64 {
	 404  	var p, q *[6]float64
	 405  	if x >= 8 {
	 406  		p = &q1R8
	 407  		q = &q1S8
	 408  	} else if x >= 4.5454 {
	 409  		p = &q1R5
	 410  		q = &q1S5
	 411  	} else if x >= 2.8571 {
	 412  		p = &q1R3
	 413  		q = &q1S3
	 414  	} else if x >= 2 {
	 415  		p = &q1R2
	 416  		q = &q1S2
	 417  	}
	 418  	z := 1 / (x * x)
	 419  	r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
	 420  	s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))))
	 421  	return (0.375 + r/s) / x
	 422  }
	 423
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