diff --git a/README.md b/README.md index 9da6213a..e2298b11 100644 --- a/README.md +++ b/README.md @@ -1,8 +1,8 @@ # SpecialFunctions.jl -Special mathematical functions in Julia, including Bessel, Hankel, Airy, error, Dawson, eta, zeta, -digamma, inverse digamma, trigamma, and polygamma functions. -These functions were formerly part of Base. +Special mathematical functions in Julia, including Bessel, Hankel, Airy, error, Dawson, sine and cosine integrals, +eta, zeta, digamma, inverse digamma, trigamma, and polygamma functions. +Most of these functions were formerly part of Base. [![Travis](https://travis-ci.org/JuliaMath/SpecialFunctions.jl.svg?branch=master)](https://travis-ci.org/JuliaMath/SpecialFunctions.jl) [![AppVeyor](https://ci.appveyor.com/api/projects/status/ccfgkm2cjcggu158/branch/master?svg=true)](https://ci.appveyor.com/project/ararslan/specialfunctions-jl/branch/master) diff --git a/docs/src/index.md b/docs/src/index.md index 753e8b30..11a25a49 100644 --- a/docs/src/index.md +++ b/docs/src/index.md @@ -13,6 +13,8 @@ libraries. | [`erfi(x)`](@ref SpecialFunctions.erfi) | imaginary error function defined as `-im * erf(x * im)`, where `im` is the imaginary unit | | [`erfcx(x)`](@ref SpecialFunctions.erfcx) | scaled complementary error function, i.e. accurate `exp(x^2) * erfc(x)` for large `x` | | [`dawson(x)`](@ref SpecialFunctions.dawson) | scaled imaginary error function, a.k.a. Dawson function, i.e. accurate `exp(-x^2) * erfi(x) * sqrt(pi) / 2` for large `x` | +| [`sinint(x)`](@ref SpecialFunctions.sinint) | [sine integral](https://en.wikipedia.org/wiki/Trigonometric_integral) at `x` | +| [`cosint(x)`](@ref SpecialFunctions.cosint) | [cosine integral](https://en.wikipedia.org/wiki/Trigonometric_integral) at `x` | | [`digamma(x)`](@ref SpecialFunctions.digamma) | [digamma function](https://en.wikipedia.org/wiki/Digamma_function) (i.e. the derivative of `lgamma` at `x`) | | [`eta(x)`](@ref SpecialFunctions.eta) | [Dirichlet eta function](https://en.wikipedia.org/wiki/Dirichlet_eta_function) at `x` | | [`zeta(x)`](@ref SpecialFunctions.zeta) | [Riemann zeta function](https://en.wikipedia.org/wiki/Riemann_zeta_function) at `x` | @@ -51,7 +53,7 @@ from the Julia REPL. ## Note -Prior to Julia 0.6, these functions were available in Julia's Base module. +Prior to Julia 0.6, most of these functions were available in Julia's Base module. Because of this, the symbols from this package are not exported on Julia 0.5 to avoid name conflicts. In this case, the symbols will need to be explicitly imported or called diff --git a/docs/src/special.md b/docs/src/special.md index 32722b44..f8bb48b1 100644 --- a/docs/src/special.md +++ b/docs/src/special.md @@ -12,6 +12,8 @@ SpecialFunctions.erfi SpecialFunctions.dawson SpecialFunctions.erfinv SpecialFunctions.erfcinv +SpecialFunctions.sinint +SpecialFunctions.cosint SpecialFunctions.digamma SpecialFunctions.invdigamma SpecialFunctions.trigamma diff --git a/src/SpecialFunctions.jl b/src/SpecialFunctions.jl index 7dc0c529..829e1f18 100644 --- a/src/SpecialFunctions.jl +++ b/src/SpecialFunctions.jl @@ -59,6 +59,9 @@ if VERSION >= v"0.6.0-dev.2767" end end +export sinint, + cosint + if isdefined(Base.Math, :openspecfun) const openspecfun = Base.Math.openspecfun else @@ -67,6 +70,7 @@ end include("bessel.jl") include("erf.jl") +include("sincosint.jl") include("gamma.jl") include("deprecated.jl") diff --git a/src/sincosint.jl b/src/sincosint.jl new file mode 100644 index 00000000..b839052f --- /dev/null +++ b/src/sincosint.jl @@ -0,0 +1,258 @@ +using Base.Math.@horner + +# Compute the sine integral: ∫_0^x sin(t)/t dt, +# and the cosine integral: γ + log x + ∫_0^x (cos(t)-1)/t dt, +# using the rational approximants tabulated in: +# A.J. MacLeod, "Rational approximations, software and test methods for +# sine and cosine integrals," Numer. Algor. 12, pp. 259--272 (1996). +# http://dx.doi.org/10.1007/BF02142806 +# https://link.springer.com/article/10.1007/BF02142806 +# +# Note: the second zero of Ci(x) has a typo that is fixed: +# +# r1 = 3.38418 04228 51186 42639 78511 46402 in the article, but is in fact: +# +# r1 = 3.38418 04225 51186 42639 78511 46402. +# + +function sinint(x::Float64) + t = x*x + if t ≤ 36.0 + return x * @horner(t, 1.00000000000000000000E0, + -0.44663998931312457298E-1, + 0.11209146443112369449E-2, + -0.13276124407928422367E-4, + 0.85118014179823463879E-7, + -0.29989314303147656479E-9, + 0.55401971660186204711E-12, + -0.42406353433133212926E-15) / + @horner(t, 1.00000000000000000000E0, + 0.10891556624243098264E-1, + 0.59334456769186835896E-4, + 0.21231112954641805908E-6, + 0.54747121846510390750E-9, + 0.10378561511331814674E-11, + 0.13754880327250272679E-14, + 0.10223981202236205703E-17) + elseif t ≤ 144.0 + invt = inv(t) + return copysign(π/2, x) - cos(x) * + @horner(invt, 0.99999999962173909991E0, + 0.36451060338631902917E3, + 0.44218548041288440874E5, + 0.22467569405961151887E7, + 0.49315316723035561922E8, + 0.43186795279670283193E9, + 0.11847992519956804350E10, + 0.45573267593795103181E9) / + (x * @horner(invt, 1.00000000000000000000E0, + 0.36651060273229347594E3, + 0.44927569814970692777E5, + 0.23285354882204041700E7, + 0.53117852017228262911E8, + 0.50335310667241870372E9, + 0.16575285015623175410E10, + 0.11746532837038341076E10)) - + sin(x)*invt * @horner(invt, 0.99999999920484901956E0, + 0.51385504875307321394E3, + 0.92293483452013810811E5, + 0.74071341863359841727E7, + 0.28142356162841356551E9, + 0.49280890357734623984E10, + 0.35524762685554302472E11, + 0.79194271662085049376E11, + 0.17942522624413898907E11) / + @horner(invt, 1.00000000000000000000E0, + 0.51985504708814870209E3, + 0.95292615508125947321E5, + 0.79215459679762667578E7, + 0.31977567790733781460E9, + 0.62273134702439012114E10, + 0.54570971054996441467E11, + 0.18241750166645704670E12, + 0.15407148148861454434E12) + elseif t < Inf + invt = inv(t) + return copysign(π/2, x) - cos(x) / x * (1.0 - + @horner(invt, 0.19999999999999978257E1, + 0.22206119380434958727E4, + 0.84749007623988236808E6, + 0.13959267954823943232E9, + 0.10197205463267975592E11, + 0.30229865264524075951E12, + 0.27504053804288471142E13, + 0.21818989704686874983E13) / + @horner(invt, 1.00000000000000000000E0, + 0.11223059690217167788E4, + 0.43685270974851313242E6, + 0.74654702140658116258E8, + 0.58580034751805687471E10, + 0.20157980379272098841E12, + 0.26229141857684496445E13, + 0.87852907334918467516E13)*invt) - + sin(x) * invt * (1.0 - @horner(invt, 0.59999999999999993089E1, + 0.96527746044997139158E4, + 0.56077626996568834185E7, + 0.15022667718927317198E10, + 0.19644271064733088465E12, + 0.12191368281163225043E14, + 0.31924389898645609533E15, + 0.25876053010027485934E16, + 0.12754978896268878403E16) / + @horner(invt, 1.00000000000000000000E0, + 0.16287957674166143196E4, + 0.96636303195787870963E6, + 0.26839734750950667021E9, + 0.37388510548029219241E11, + 0.26028585666152144496E13, + 0.85134283716950697226E14, + 0.11304079361627952930E16, + 0.42519841479489798424E16)*invt) + elseif isnan(x) + return NaN + else + return copysign(π/2, x) + end +end + +function cosint(x::Float64) + t, r0, r1 = x*x, 0.616505485620716233797110404100, 3.384180422551186426397851146402 + r01, r02 = 0.6162109375, 0.29454812071623379711E-3 + r11, r12 = 3.3837890625, 0.39136005118642639785E-3 + if x < 0.0 + return throw(DomainError()) + elseif x ≤ 3.0 + return log(x/r0) + ((x - r01) - r02) * (x + r0) * + @horner(t, -0.24607411378767540707E0, + 0.72113492241301534559E-2, + -0.11867127836204767056E-3, + 0.90542655466969866243E-6, + -0.34322242412444409037E-8, + 0.51950683460656886834E-11) / + @horner(t, 1.00000000000000000000E0, + 0.12670095552700637845E-1, + 0.78168450570724148921E-4, + 0.29959200177005821677E-6, + 0.73191677761328838216E-9, + 0.94351174530907529061E-12) + elseif x ≤ 6.0 + return log(x/r1) + ((x - r11) - r12) * (x + r1) * + @horner(t, -0.15684781827145408780E0, + 0.66253165609605468916E-2, + -0.12822297297864512864E-3, + 0.12360964097729408891E-5, + -0.66450975112876224532E-8, + 0.20326936466803159446E-10, + -0.33590883135343844613E-13, + 0.23686934961435015119E-16) / + @horner(t, 1.00000000000000000000E0, + 0.96166044388828741188E-2, + 0.45257514591257035006E-4, + 0.13544922659627723233E-6, + 0.27715365686570002081E-9, + 0.37718676301688932926E-12, + 0.27706844497155995398E-15) + elseif x ≤ 12.0 + invt = inv(t) + return sin(x) * @horner(invt, 0.99999999962173909991E0, + 0.36451060338631902917E3, + 0.44218548041288440874E5, + 0.22467569405961151887E7, + 0.49315316723035561922E8, + 0.43186795279670283193E9, + 0.11847992519956804350E10, + 0.45573267593795103181E9) / + (x * @horner(invt, 1.00000000000000000000E0, + 0.36651060273229347594E3, + 0.44927569814970692777E5, + 0.23285354882204041700E7, + 0.53117852017228262911E8, + 0.50335310667241870372E9, + 0.16575285015623175410E10, + 0.11746532837038341076E10)) - + cos(x) * invt * @horner(invt, 0.99999999920484901956E0, + 0.51385504875307321394E3, + 0.92293483452013810811E5, + 0.74071341863359841727E7, + 0.28142356162841356551E9, + 0.49280890357734623984E10, + 0.35524762685554302472E11, + 0.79194271662085049376E11, + 0.17942522624413898907E11) / + @horner(invt, 1.00000000000000000000E0, + 0.51985504708814870209E3, + 0.95292615508125947321E5, + 0.79215459679762667578E7, + 0.31977567790733781460E9, + 0.62273134702439012114E10, + 0.54570971054996441467E11, + 0.18241750166645704670E12, + 0.15407148148861454434E12) + elseif x < Inf + invt = inv(t) + return sin(x)/x * (1.0 - @horner(invt, 0.19999999999999978257E1, + 0.22206119380434958727E4, + 0.84749007623988236808E6, + 0.13959267954823943232E9, + 0.10197205463267975592E11, + 0.30229865264524075951E12, + 0.27504053804288471142E13, + 0.21818989704686874983E13) / + @horner(invt, 1.00000000000000000000E0, + 0.11223059690217167788E4, + 0.43685270974851313242E6, + 0.74654702140658116258E8, + 0.58580034751805687471E10, + 0.20157980379272098841E12, + 0.26229141857684496445E13, + 0.87852907334918467516E13)*invt) - + cos(x)*invt * (1.0 - @horner(invt, 0.59999999999999993089E1, + 0.96527746044997139158E4, + 0.56077626996568834185E7, + 0.15022667718927317198E10, + 0.19644271064733088465E12, + 0.12191368281163225043E14, + 0.31924389898645609533E15, + 0.25876053010027485934E16, + 0.12754978896268878403E16) / + @horner(invt, 1.00000000000000000000E0, + 0.16287957674166143196E4, + 0.96636303195787870963E6, + 0.26839734750950667021E9, + 0.37388510548029219241E11, + 0.26028585666152144496E13, + 0.85134283716950697226E14, + 0.11304079361627952930E16, + 0.42519841479489798424E16)*invt) + elseif isnan(x) + return NaN + else + return 0.0 + end +end + +for f in (:sinint, :cosint) + @eval begin + ($f)(x::Float32) = Float32(($f)(Float64(x))) + ($f)(x::Float16) = Float16(($f)(Float64(x))) + ($f)(x::Real) = ($f)(float(x)) + ($f)(x::AbstractFloat) = error("not implemented for ", typeof(x)) + end +end + + +""" + sinint(x) + +Compute the sine integral function of `x`, defined by ``\\operatorname{Si}(x) := \\int_0^x\\frac{\\sin t}{t} dt`` +for real `x`. +""" +sinint + +""" + cosint(x) + +Compute the cosine integral function of `x`, defined by ``\\operatorname{Ci}(x) := \\gamma + \\log x + \\int_0^x \\frac{\\cos t - 1}{t} dt`` +for real `x > 0`, where ``\\gamma`` is the Euler-Mascheroni constant. +""" +cosint diff --git a/test/runtests.jl b/test/runtests.jl index e043141f..747cf724 100644 --- a/test/runtests.jl +++ b/test/runtests.jl @@ -5,6 +5,12 @@ using Base.Test const SF = SpecialFunctions +# useful test functions for relative error, which differ from isapprox +# in that relerrc separately looks at the real and imaginary parts +relerr(z, x) = z == x ? 0.0 : abs(z - x) / abs(x) +relerrc(z, x) = max(relerr(real(z),real(x)), relerr(imag(z),imag(x))) +≅(a,b) = relerrc(a,b) ≤ 1e-13 + @testset "error functions" begin @test SF.erf(Float16(1)) ≈ 0.84270079294971486934 @test SF.erf(1) ≈ 0.84270079294971486934 @@ -48,6 +54,40 @@ const SF = SpecialFunctions @test SF.erfcinv(one(Int)) == SF.erfcinv(1.0) end +@testset "sine and cosine integrals" begin + # Computed via wolframalpha.com: SinIntegral[SetPrecision[Table[x,{x, 1,20,1}],20]] and CosIntegral[SetPrecision[Table[x,{x, 1,20,1}],20]] + sinintvals = [0.9460830703671830149, 1.605412976802694849, 1.848652527999468256, 1.75820313894905306, 1.54993124494467414, 1.4246875512805065, 1.4545966142480936, 1.5741868217069421, 1.665040075829602, 1.658347594218874, 1.578306806945727416, 1.504971241526373371, 1.499361722862824564, 1.556211050077665054, 1.618194443708368739, 1.631302268270032886, 1.590136415870701122, 1.536608096861185462, 1.518630031769363932, 1.548241701043439840] + cosintvals = [0.3374039229009681347, 0.4229808287748649957, 0.119629786008000328, -0.14098169788693041, -0.19002974965664388, -0.06805724389324713, 0.07669527848218452, 0.122433882532010, 0.0553475313331336, -0.045456433004455, -0.08956313549547997948, -0.04978000688411367560, 0.02676412556403455504, 0.06939635592758454727, 0.04627867767436043960, -0.01420019012019002240, -0.05524268226081385053, -0.04347510299950100478, 0.00515037100842612857, 0.04441982084535331654] + for x in 1:20 + @test SF.sinint(x) ≅ sinintvals[x] + @test SF.sinint(-x) ≅ -sinintvals[x] + @test SF.cosint(x) ≅ cosintvals[x] + end + + @test SF.sinint(1.f0) == Float32(SF.sinint(1.0)) + @test SF.cosint(1.f0) == Float32(SF.cosint(1.0)) + + @test SF.sinint(Float16(1.0)) == Float16(SF.sinint(1.0)) + @test SF.cosint(Float16(1.0)) == Float16(SF.cosint(1.0)) + + @test SF.sinint(1//2) == SF.sinint(0.5) + @test SF.cosint(1//2) == SF.cosint(0.5) + + @test SF.sinint(1e300) ≅ π/2 + @test SF.cosint(1e300) ≅ -8.17881912115908554103E-301 + @test SF.sinint(1e-300) ≅ 1.0E-300 + @test SF.cosint(1e-300) ≅ -690.1983122333121 + + @test SF.sinint(Inf) == π/2 + @test SF.cosint(Inf) == 0.0 + @test isnan(SF.sinint(NaN)) + @test isnan(SF.cosint(NaN)) + + @test_throws ErrorException SF.sinint(big(1.0)) + @test_throws ErrorException SF.cosint(big(1.0)) + @test_throws DomainError SF.cosint(-1.0) +end + @testset "airy" begin @test_throws SF.AmosException SF.airyai(200im) @test_throws SF.AmosException SF.airybi(200) @@ -259,12 +299,6 @@ end end end -# useful test functions for relative error, which differ from isapprox -# in that relerrc separately looks at the real and imaginary parts -relerr(z, x) = z == x ? 0.0 : abs(z - x) / abs(x) -relerrc(z, x) = max(relerr(real(z),real(x)), relerr(imag(z),imag(x))) -≅(a,b) = relerrc(a,b) ≤ 1e-13 - @testset "gamma and friends" begin @testset "digamma" begin @testset "$elty" for elty in (Float32, Float64)