Source file src/pkg/math/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	}