j1.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 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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