The Go Programming Language

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		// TODO(rsc): Remove manual inlining of IsNaN, IsInf
    90		// when compiler does it for us
    91		// special cases
    92		switch {
    93		case x != x: // IsNaN(x)
    94			return x
    95		case x < -MaxFloat64 || x > MaxFloat64 || x == 0: // IsInf(x, 0) || x == 0:
    96			return 0
    97		}
    98	
    99		sign := false
   100		if x < 0 {
   101			x = -x
   102			sign = true
   103		}
   104		if x >= 2 {
   105			s, c := Sincos(x)
   106			ss := -s - c
   107			cc := s - c
   108	
   109			// make sure x+x does not overflow
   110			if x < MaxFloat64/2 {
   111				z := Cos(x + x)
   112				if s*c > 0 {
   113					cc = z / ss
   114				} else {
   115					ss = z / cc
   116				}
   117			}
   118	
   119			// j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
   120			// y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
   121	
   122			var z float64
   123			if x > Two129 {
   124				z = (1 / SqrtPi) * cc / Sqrt(x)
   125			} else {
   126				u := pone(x)
   127				v := qone(x)
   128				z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x)
   129			}
   130			if sign {
   131				return -z
   132			}
   133			return z
   134		}
   135		if x < TwoM27 { // |x|<2**-27
   136			return 0.5 * x // inexact if x!=0 necessary
   137		}
   138		z := x * x
   139		r := z * (R00 + z*(R01+z*(R02+z*R03)))
   140		s := 1.0 + z*(S01+z*(S02+z*(S03+z*(S04+z*S05))))
   141		r *= x
   142		z = 0.5*x + r/s
   143		if sign {
   144			return -z
   145		}
   146		return z
   147	}
   148	
   149	// Y1 returns the order-one Bessel function of the second kind.
   150	//
   151	// Special cases are:
   152	//	Y1(+Inf) = 0
   153	//	Y1(0) = -Inf
   154	//	Y1(x < 0) = NaN
   155	//	Y1(NaN) = NaN
   156	func Y1(x float64) float64 {
   157		const (
   158			TwoM54 = 1.0 / (1 << 54)             // 2**-54 0x3c90000000000000
   159			Two129 = 1 << 129                    // 2**129 0x4800000000000000
   160			U00    = -1.96057090646238940668e-01 // 0xBFC91866143CBC8A
   161			U01    = 5.04438716639811282616e-02  // 0x3FA9D3C776292CD1
   162			U02    = -1.91256895875763547298e-03 // 0xBF5F55E54844F50F
   163			U03    = 2.35252600561610495928e-05  // 0x3EF8AB038FA6B88E
   164			U04    = -9.19099158039878874504e-08 // 0xBE78AC00569105B8
   165			V00    = 1.99167318236649903973e-02  // 0x3F94650D3F4DA9F0
   166			V01    = 2.02552581025135171496e-04  // 0x3F2A8C896C257764
   167			V02    = 1.35608801097516229404e-06  // 0x3EB6C05A894E8CA6
   168			V03    = 6.22741452364621501295e-09  // 0x3E3ABF1D5BA69A86
   169			V04    = 1.66559246207992079114e-11  // 0x3DB25039DACA772A
   170		)
   171		// TODO(rsc): Remove manual inlining of IsNaN, IsInf
   172		// when compiler does it for us
   173		// special cases
   174		switch {
   175		case x < 0 || x != x: // x < 0 || IsNaN(x):
   176			return NaN()
   177		case x > MaxFloat64: // IsInf(x, 1):
   178			return 0
   179		case x == 0:
   180			return Inf(-1)
   181		}
   182	
   183		if x >= 2 {
   184			s, c := Sincos(x)
   185			ss := -s - c
   186			cc := s - c
   187	
   188			// make sure x+x does not overflow
   189			if x < MaxFloat64/2 {
   190				z := Cos(x + x)
   191				if s*c > 0 {
   192					cc = z / ss
   193				} else {
   194					ss = z / cc
   195				}
   196			}
   197			// y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
   198			// where x0 = x-3pi/4
   199			//     Better formula:
   200			//         cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
   201			//                 =  1/sqrt(2) * (sin(x) - cos(x))
   202			//         sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
   203			//                 = -1/sqrt(2) * (cos(x) + sin(x))
   204			// To avoid cancellation, use
   205			//     sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
   206			// to compute the worse one.
   207	
   208			var z float64
   209			if x > Two129 {
   210				z = (1 / SqrtPi) * ss / Sqrt(x)
   211			} else {
   212				u := pone(x)
   213				v := qone(x)
   214				z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x)
   215			}
   216			return z
   217		}
   218		if x <= TwoM54 { // x < 2**-54
   219			return -(2 / Pi) / x
   220		}
   221		z := x * x
   222		u := U00 + z*(U01+z*(U02+z*(U03+z*U04)))
   223		v := 1 + z*(V00+z*(V01+z*(V02+z*(V03+z*V04))))
   224		return x*(u/v) + (2/Pi)*(J1(x)*Log(x)-1/x)
   225	}
   226	
   227	// For x >= 8, the asymptotic expansions of pone is
   228	//      1 + 15/128 s**2 - 4725/2**15 s**4 - ..., where s = 1/x.
   229	// We approximate pone by
   230	//      pone(x) = 1 + (R/S)
   231	// where R = pr0 + pr1*s**2 + pr2*s**4 + ... + pr5*s**10
   232	//       S = 1 + ps0*s**2 + ... + ps4*s**10
   233	// and
   234	//      | pone(x)-1-R/S | <= 2**(-60.06)
   235	
   236	// for x in [inf, 8]=1/[0,0.125]
   237	var p1R8 = [6]float64{
   238		0.00000000000000000000e+00, // 0x0000000000000000
   239		1.17187499999988647970e-01, // 0x3FBDFFFFFFFFFCCE
   240		1.32394806593073575129e+01, // 0x402A7A9D357F7FCE
   241		4.12051854307378562225e+02, // 0x4079C0D4652EA590
   242		3.87474538913960532227e+03, // 0x40AE457DA3A532CC
   243		7.91447954031891731574e+03, // 0x40BEEA7AC32782DD
   244	}
   245	var p1S8 = [5]float64{
   246		1.14207370375678408436e+02, // 0x405C8D458E656CAC
   247		3.65093083420853463394e+03, // 0x40AC85DC964D274F
   248		3.69562060269033463555e+04, // 0x40E20B8697C5BB7F
   249		9.76027935934950801311e+04, // 0x40F7D42CB28F17BB
   250		3.08042720627888811578e+04, // 0x40DE1511697A0B2D
   251	}
   252	
   253	// for x in [8,4.5454] = 1/[0.125,0.22001]
   254	var p1R5 = [6]float64{
   255		1.31990519556243522749e-11, // 0x3DAD0667DAE1CA7D
   256		1.17187493190614097638e-01, // 0x3FBDFFFFE2C10043
   257		6.80275127868432871736e+00, // 0x401B36046E6315E3
   258		1.08308182990189109773e+02, // 0x405B13B9452602ED
   259		5.17636139533199752805e+02, // 0x40802D16D052D649
   260		5.28715201363337541807e+02, // 0x408085B8BB7E0CB7
   261	}
   262	var p1S5 = [5]float64{
   263		5.92805987221131331921e+01, // 0x404DA3EAA8AF633D
   264		9.91401418733614377743e+02, // 0x408EFB361B066701
   265		5.35326695291487976647e+03, // 0x40B4E9445706B6FB
   266		7.84469031749551231769e+03, // 0x40BEA4B0B8A5BB15
   267		1.50404688810361062679e+03, // 0x40978030036F5E51
   268	}
   269	
   270	// for x in[4.5453,2.8571] = 1/[0.2199,0.35001]
   271	var p1R3 = [6]float64{
   272		3.02503916137373618024e-09, // 0x3E29FC21A7AD9EDD
   273		1.17186865567253592491e-01, // 0x3FBDFFF55B21D17B
   274		3.93297750033315640650e+00, // 0x400F76BCE85EAD8A
   275		3.51194035591636932736e+01, // 0x40418F489DA6D129
   276		9.10550110750781271918e+01, // 0x4056C3854D2C1837
   277		4.85590685197364919645e+01, // 0x4048478F8EA83EE5
   278	}
   279	var p1S3 = [5]float64{
   280		3.47913095001251519989e+01, // 0x40416549A134069C
   281		3.36762458747825746741e+02, // 0x40750C3307F1A75F
   282		1.04687139975775130551e+03, // 0x40905B7C5037D523
   283		8.90811346398256432622e+02, // 0x408BD67DA32E31E9
   284		1.03787932439639277504e+02, // 0x4059F26D7C2EED53
   285	}
   286	
   287	// for x in [2.8570,2] = 1/[0.3499,0.5]
   288	var p1R2 = [6]float64{
   289		1.07710830106873743082e-07, // 0x3E7CE9D4F65544F4
   290		1.17176219462683348094e-01, // 0x3FBDFF42BE760D83
   291		2.36851496667608785174e+00, // 0x4002F2B7F98FAEC0
   292		1.22426109148261232917e+01, // 0x40287C377F71A964
   293		1.76939711271687727390e+01, // 0x4031B1A8177F8EE2
   294		5.07352312588818499250e+00, // 0x40144B49A574C1FE
   295	}
   296	var p1S2 = [5]float64{
   297		2.14364859363821409488e+01, // 0x40356FBD8AD5ECDC
   298		1.25290227168402751090e+02, // 0x405F529314F92CD5
   299		2.32276469057162813669e+02, // 0x406D08D8D5A2DBD9
   300		1.17679373287147100768e+02, // 0x405D6B7ADA1884A9
   301		8.36463893371618283368e+00, // 0x4020BAB1F44E5192
   302	}
   303	
   304	func pone(x float64) float64 {
   305		var p [6]float64
   306		var q [5]float64
   307		if x >= 8 {
   308			p = p1R8
   309			q = p1S8
   310		} else if x >= 4.5454 {
   311			p = p1R5
   312			q = p1S5
   313		} else if x >= 2.8571 {
   314			p = p1R3
   315			q = p1S3
   316		} else if x >= 2 {
   317			p = p1R2
   318			q = p1S2
   319		}
   320		z := 1 / (x * x)
   321		r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
   322		s := 1.0 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))))
   323		return 1 + r/s
   324	}
   325	
   326	// For x >= 8, the asymptotic expansions of qone is
   327	//      3/8 s - 105/1024 s**3 - ..., where s = 1/x.
   328	// We approximate qone by
   329	//      qone(x) = s*(0.375 + (R/S))
   330	// where R = qr1*s**2 + qr2*s**4 + ... + qr5*s**10
   331	//       S = 1 + qs1*s**2 + ... + qs6*s**12
   332	// and
   333	//      | qone(x)/s -0.375-R/S | <= 2**(-61.13)
   334	
   335	// for x in [inf, 8] = 1/[0,0.125]
   336	var q1R8 = [6]float64{
   337		0.00000000000000000000e+00,  // 0x0000000000000000
   338		-1.02539062499992714161e-01, // 0xBFBA3FFFFFFFFDF3
   339		-1.62717534544589987888e+01, // 0xC0304591A26779F7
   340		-7.59601722513950107896e+02, // 0xC087BCD053E4B576
   341		-1.18498066702429587167e+04, // 0xC0C724E740F87415
   342		-4.84385124285750353010e+04, // 0xC0E7A6D065D09C6A
   343	}
   344	var q1S8 = [6]float64{
   345		1.61395369700722909556e+02,  // 0x40642CA6DE5BCDE5
   346		7.82538599923348465381e+03,  // 0x40BE9162D0D88419
   347		1.33875336287249578163e+05,  // 0x4100579AB0B75E98
   348		7.19657723683240939863e+05,  // 0x4125F65372869C19
   349		6.66601232617776375264e+05,  // 0x412457D27719AD5C
   350		-2.94490264303834643215e+05, // 0xC111F9690EA5AA18
   351	}
   352	
   353	// for x in [8,4.5454] = 1/[0.125,0.22001]
   354	var q1R5 = [6]float64{
   355		-2.08979931141764104297e-11, // 0xBDB6FA431AA1A098
   356		-1.02539050241375426231e-01, // 0xBFBA3FFFCB597FEF
   357		-8.05644828123936029840e+00, // 0xC0201CE6CA03AD4B
   358		-1.83669607474888380239e+02, // 0xC066F56D6CA7B9B0
   359		-1.37319376065508163265e+03, // 0xC09574C66931734F
   360		-2.61244440453215656817e+03, // 0xC0A468E388FDA79D
   361	}
   362	var q1S5 = [6]float64{
   363		8.12765501384335777857e+01,  // 0x405451B2FF5A11B2
   364		1.99179873460485964642e+03,  // 0x409F1F31E77BF839
   365		1.74684851924908907677e+04,  // 0x40D10F1F0D64CE29
   366		4.98514270910352279316e+04,  // 0x40E8576DAABAD197
   367		2.79480751638918118260e+04,  // 0x40DB4B04CF7C364B
   368		-4.71918354795128470869e+03, // 0xC0B26F2EFCFFA004
   369	}
   370	
   371	// for x in [4.5454,2.8571] = 1/[0.2199,0.35001] ???
   372	var q1R3 = [6]float64{
   373		-5.07831226461766561369e-09, // 0xBE35CFA9D38FC84F
   374		-1.02537829820837089745e-01, // 0xBFBA3FEB51AEED54
   375		-4.61011581139473403113e+00, // 0xC01270C23302D9FF
   376		-5.78472216562783643212e+01, // 0xC04CEC71C25D16DA
   377		-2.28244540737631695038e+02, // 0xC06C87D34718D55F
   378		-2.19210128478909325622e+02, // 0xC06B66B95F5C1BF6
   379	}
   380	var q1S3 = [6]float64{
   381		4.76651550323729509273e+01,  // 0x4047D523CCD367E4
   382		6.73865112676699709482e+02,  // 0x40850EEBC031EE3E
   383		3.38015286679526343505e+03,  // 0x40AA684E448E7C9A
   384		5.54772909720722782367e+03,  // 0x40B5ABBAA61D54A6
   385		1.90311919338810798763e+03,  // 0x409DBC7A0DD4DF4B
   386		-1.35201191444307340817e+02, // 0xC060E670290A311F
   387	}
   388	
   389	// for x in [2.8570,2] = 1/[0.3499,0.5]
   390	var q1R2 = [6]float64{
   391		-1.78381727510958865572e-07, // 0xBE87F12644C626D2
   392		-1.02517042607985553460e-01, // 0xBFBA3E8E9148B010
   393		-2.75220568278187460720e+00, // 0xC006048469BB4EDA
   394		-1.96636162643703720221e+01, // 0xC033A9E2C168907F
   395		-4.23253133372830490089e+01, // 0xC04529A3DE104AAA
   396		-2.13719211703704061733e+01, // 0xC0355F3639CF6E52
   397	}
   398	var q1S2 = [6]float64{
   399		2.95333629060523854548e+01,  // 0x403D888A78AE64FF
   400		2.52981549982190529136e+02,  // 0x406F9F68DB821CBA
   401		7.57502834868645436472e+02,  // 0x4087AC05CE49A0F7
   402		7.39393205320467245656e+02,  // 0x40871B2548D4C029
   403		1.55949003336666123687e+02,  // 0x40637E5E3C3ED8D4
   404		-4.95949898822628210127e+00, // 0xC013D686E71BE86B
   405	}
   406	
   407	func qone(x float64) float64 {
   408		var p, q [6]float64
   409		if x >= 8 {
   410			p = q1R8
   411			q = q1S8
   412		} else if x >= 4.5454 {
   413			p = q1R5
   414			q = q1S5
   415		} else if x >= 2.8571 {
   416			p = q1R3
   417			q = q1S3
   418		} else if x >= 2 {
   419			p = q1R2
   420			q = q1S2
   421		}
   422		z := 1 / (x * x)
   423		r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
   424		s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))))
   425		return (0.375 + r/s) / x
   426	}

release.r60.3. Except as noted, this content is licensed under a Creative Commons Attribution 3.0 License.