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Adding jinc function (#266)
* Add jinc (also called sombrero, besinc) function * Add tests for jinc function * Add more jinc tests to verify that polynomial works as well * Add redefinition of fastabs for jinc and clean jinc body * Some code comments regarded cleaning of jinc function * Add fastabs to SpecialFunctions * Add relative error comparison to tests * Add proper relative error testing * Adapt jinc function to get type stability * Add test cases for jinc function * Use fastabs in expint and gamma for complex number checking * Adapt \cong for complex numbers * Add tests for fastabs * Replace relerrc with complex relerr in tests for gamma and expint * Fix test cases for jinc with correct big expression * Revert \cong threshold back to 1e-6 for Float32 * Relax accuracy threshold for Float32 slightly * Use \isapprox for all complex number tests of jinc * Remove complex argument handling of \cong in runtest * Remove explicit zero handling in jinc because evalpoly already does * Add complex handling for relerr in runtests * Add jinc function to documentation
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‎docs/src/functions_list.md

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@@ -50,6 +50,7 @@ SpecialFunctions.besseli
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SpecialFunctions.besselix
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SpecialFunctions.besselk
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SpecialFunctions.besselkx
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SpecialFunctions.jinc
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SpecialFunctions.ellipk
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SpecialFunctions.ellipe
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SpecialFunctions.eta

‎docs/src/functions_overview.md

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@@ -78,6 +78,7 @@ Here the *Special Functions* are listed according to the structure of [NIST Digi
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| [`besselix(nu,z)`](@ref SpecialFunctions.besselix) | scaled modified Bessel function of the first kind of order `nu` at `z` |
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| [`besselk(nu,z)`](@ref SpecialFunctions.besselk) | modified [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the second kind of order `nu` at `z` |
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| [`besselkx(nu,z)`](@ref SpecialFunctions.besselkx) | scaled modified Bessel function of the second kind of order `nu` at `z` |
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| [`jinc(x)`](@ref SpecialFunctions.jinc) | scaled [Bessel function of the first kind divided by `x`](https://en.wikipedia.org/wiki/Sombrero_function). A.k.a. sombrero or besinc |
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## [Elliptic Integrals](https://dlmf.nist.gov/19)

‎src/SpecialFunctions.jl

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@@ -27,6 +27,7 @@ export
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bessely0,
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bessely1,
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besselyx,
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jinc,
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dawson,
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ellipk,
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ellipe,
@@ -83,4 +84,13 @@ for f in (:beta, :lbeta)
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end
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polygamma(m::Integer, x::Missing) = missing
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# In future just use `fastabs` from Base.Math
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# https://github.com/JuliaLang/julia/blob/93fb785831dcfcc442f82fab8746f0244c5274ae/base/special/trig.jl#L1057
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if isdefined(Base.Math, :fastabs)
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import Base.Math: fastabs
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else
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fastabs(x::Number) = abs(x)
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fastabs(x::Complex) = abs(real(x)) + abs(imag(x))
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end
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end # module

‎src/bessel.jl

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@@ -758,3 +758,33 @@ External links: [DLMF](https://dlmf.nist.gov/10.2.6), [Wikipedia](https://en.wik
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See also: [`hankelh2(nu,x)`](@ref SpecialFunctions.hankelh2)
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"""
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hankelh2x(nu, z) = besselhx(nu, 2, z)
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"""
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jinc(x)
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Bessel function of the first kind divided by `x`.
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Following convention: ``\\operatorname{jinc}{x} = \\frac{2 \\cdot J_1{\\pi x}}{\\pi x}``.
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Sometimes known as sombrero or besinc function.
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External links: [Wikipedia](https://en.wikipedia.org/wiki/Sombrero_function)
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"""
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jinc(x::Number) = _jinc(float(x))
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_jinc(x::Number) = iszero(x) ? one(x) : _jinc_core(x)
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_jinc(x::Float16) = Float16(_jinc(Float32(x)))
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_jinc(x::ComplexF16) = ComplexF16(_jinc(ComplexF32(x)))
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# below these thresholds we evaluate a fourth order Taylor polynomial
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_jinc_threshold(::Type{Float64}) = 0.002
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_jinc_threshold(::Type{Float32}) = 0.05f0
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# for small arguments, a Taylor series is faster
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@inline function _jinc(x::Union{T,Complex{T}}) where {T<:Union{Float32,Float64}}
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if fastabs(x) < _jinc_threshold(T)
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return @evalpoly(x^2, T(1), -T(π)^2/8, T(π)^4/192)
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else
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return _jinc_core(x)
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end
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end
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# the actual definition of jinc
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_jinc_core(x::Number) = 2 * besselj1(π*x) / (π*x)

‎src/expint.jl

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@@ -178,7 +178,7 @@ function En_cf_nogamma(ν::Number, z::Number, n::Int=1000)
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conv && i > 4 && break
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# rescale
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if max(abs(real(A)), abs(imag(A))) > scale
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if fastabs(A) > scale
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A /= scale
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Aprev /= scale
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B /= scale

‎src/gamma.jl

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@@ -474,7 +474,7 @@ Dirichlet eta function ``\\eta(s) = \\sum^\\infty_{n=1}(-1)^{n-1}/n^{s}``.
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"""
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function eta(z::ComplexOrReal{Float64})
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δz = 1 - z
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if abs(real(δz)) + abs(imag(δz)) < 7e-3 # Taylor expand around z==1
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if fastabs(δz) < 7e-3 # Taylor expand around z==1
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return 0.6931471805599453094172321214581765 *
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@evalpoly(δz,
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1.0,

‎test/bessel.jl

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@test f(0,1) ≈ f(0,Complex{Float16}(1))
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end
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end
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@testset "jinc function" begin
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x1 = big"1.8"
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res1 = big"-0.116401402395234261196060221416100814277048989838340270527643894539781608507904800871249915340068849141209639297793338757544621191431028495364540423087719279432811811"
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x2 = big"0.00001"
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res2 = big"0.9999999998766299449914564074224855053704639002961349883799411022244488435839201448375732357069707105107902868318079100001712780899305533775708875568993566765287247356"
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x3 = big"0.02"
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res3 = big"0.9995066009475120760121935544369072906921634576001177613216005027724298234311657036029970452969291362640681823594159438475599412434802474217121139032145806264666152465837468052082455102325311416327533"
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root1 = big"1.21966989126650445492653884746525517787935933077511212945638126557694328028076014425087191879391333148570410142424433731699"
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for T in [Float16, Float32, Float64]
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@test jinc(T(x1)) ≅ T(res1)
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@test jinc(T(0)) ≅ T(1)
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x = root1
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# check the first root for negative and positive
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@test jinc(T(x)) + T(1) ≅ T(1)
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@test jinc(T(-x)) + T(1) ≅ T(1)
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# check for symmetry
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for x = -5.0:1.0:3.0
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@test jinc(T(10^x)) ≅ jinc(T(- 10^x))
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end
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x = T(x2)
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res = T(res2)
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@test jinc(x) ≅ res
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x = T(x3)
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res = T(res3)
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@test jinc(x) ≅ res
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# type stability for 0
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@test typeof(jinc(T(0))) === T
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end
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# test big float
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begin
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T = BigFloat
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@test T(jinc(T(x1))) ≈ T(res1)
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#= @test jinc(T(0)) ≈ T(1) =#
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x = root1
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# check the first root for negative and positive
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@test jinc(T(x)) + T(1) ≈ T(1)
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@test jinc(T(-x)) + T(1) ≈ T(1)
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# check for symmetry
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for x = -5.0:1.0:3.0
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@test jinc(T(10^x)) ≈ jinc(T(- 10^x))
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end
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x = T(x2)
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res = T(res2)
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@test T(jinc(x)) ≈ res
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x = T(x3)
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res = T(res3)
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@test T(jinc(x)) ≈ res
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# type stability for 0
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@test typeof(jinc(T(0))) === T
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end
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x1 = big"1.0" + big"0.0001"*1im
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res1 = big"0.1811917493658519438199345187518953631147425189362440669941643986059307908067224704314102961998438798566229144099942611910610729246007355473812330821120666591291397178705295331963573655500130149361351" - big"0.0000970867870757251899047609572999879575662488432474075980361828257989869563037501527523384396635449088435640500216646530541742421182815911923875286404692082399771721882711636718746219759416544606077"*1im
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x2 = big"0.00001" + big"10"*1im
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res2 = big"1.971105539196620210542904558564976891448851717386016064466205835417372529649732623778146798009632930245313240967710133352055225553347549825667896762467087506209685330050397030027682762563950964962993e11" - big"5.89917589196246805438832769814196828580119495400655210987269285701113079909408122244782544425726443894463237062342568390611353700647229983489787030738912900107228049806443988000825445873073676204e6"*1im
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x3 = big"0.00002"*1im
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res3 = big"1.0000000004934802201356421734767362282021102443681623763651414603818274528461323973841772525693725599437763705213335500965076723534424132751557185487068200219257886657409530057472519875335446749379510"
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# check also some complex numbers
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# besselj1 is not type stable and returns Complex{Float64}
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# the stricter ≅ doesn't hold here
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# besselj1 seems to produce inaccurate results for some Float32 cases
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for T in [Complex{Float16}, Complex{Float32}, Complex{Float64}]
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x = T(x1)
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res = T(res1)
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@test T(jinc(x)) ≈ res
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x = T(x2)
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res = T(res2)
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@test T(jinc(x)) ≈ res
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x = T(x3)
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res = T(res3)
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@test T(jinc(x)) + T(1im) ≅ res + T(1im)
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end
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end
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end

‎test/expint.jl

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@@ -32,7 +32,7 @@ using Base.MathConstants
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, 18.922784338495287 - 0.029203673105875738im, 18.422784340698875 - 0.029203673041504553im, 17.92278434433198 - 0.029203672935374417im, 17.422784350321955 - 0.029203672760395406im, 16.922784360197753 - 0.029203672471903787im, 16.42278437648019 - 0.029203671996261525im, 15.922784403325396 - 0.02920367121206002im, 15.422784447585654 - 0.029203669919130378im, 14.922784520558482 - 0.029203667787449886im, 14.42278464087033 - 0.029203664272903267im, 13.922784839231014 - 0.029203658478396427im, 13.422785166272453 - 0.02920364892487225im, 12.922785705472512 - 0.029203633173780545im, 12.422786594462801 - 0.029203607204639166im, 11.922788060159139 - 0.029203564388813802im, 11.422790476681525 - 0.029203493797588825im, 10.922794460847014 - 0.029203377412707156im, 10.422801029608085 - 0.029203185527489635im, 9.92281185961712 - 0.029202869165002185im, 9.422829715155434 - 0.029202347578921022im, 8.922859153613475 - 0.02920148764919001im, 8.422907688480091 - 0.02920006992002703im, 7.922987706377455 - 0.02919773263004735im, 7.4231196266023955 - 0.029193879498721455im, 6.923337107299312 - 0.02918752786907706im, 6.423695620754588 - 0.029177058818573274im, 5.924286569298801 - 0.029159806468544838im, 5.425260497751445 - 0.02913138441743693im, 4.926865199419432 - 0.029084584836757257im, 4.429508095337361 - 0.029007589432432816im, 3.9338578706686858 - 0.02888109015248326im, 3.4410087804823353 - 0.028673731153572076im, 2.9527427760716067 - 0.028335098763862032im, 2.4719383098140226 - 0.02778549583478644im, 2.0031828640491396 - 0.026902526581439178im, 1.5536245933117467 - 0.025507584670758247im, 1.1339838503079331 - 0.023363827088483065im, 0.7593072088540906 - 0.020215469144043718im, 0.4483165720604514 - 0.015923527983522815im, 0.21922704942196994 - 0.010743431043978405im, 0.08026140877749337 - 0.0056148853587022im, 0.01865598084904698 - 0.0019258134104280294im, 0.0020975029211462598 - 0.00032968586251067337im, 7.260678405534787e-5 - 1.7926254192086143e-5im, 3.6394029588254425e-7 - 1.4661826414313507e-7im, 7.41111524749911e-11 - 5.228655363875185e-11im, 6.721230570877014e-17 - 1.0361363222368277e-16im, -1.864169823362996e-27 - 3.5720968579640937e-26im, -8.139775526992696e-42 - 4.2816798952068195e-42im, -8.5705278203875e-68 + 2.3494757832291492e-67im, 1.5313896916407397e-109 - 1.893701952982838e-109im, 1.3253076410048056e-178 + 1.2528740964916723e-178im, 2.2308653435430693e-292 - 1.5308786810615063e-292im, 0.0 - 0.0im, -0.0 - 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, -0.0 + 0.0im, 0.0 - 0.0im, -0.0 - 0.0im, 19.42278433697267 - 0.42920367234736084im, 18.922784338188503 - 0.42920367179092506im, 18.422784340193076 - 0.42920367087351746im, 17.922784343498055 - 0.42920366936096793im, 17.422784348947047 - 0.42920366686719563im, 16.92278435793092 - 0.42920366275565996im, 16.422784372742814 - 0.42920365597688387im, 15.922784397163502 - 0.4292036448005718im, 15.42278443742641 - 0.4292036263739489im, 14.922784503808721 - 0.4292035959935854im, 14.422784613254645 - 0.42920354590483817im, 13.922784793700458 - 0.4292034633224673im, 13.422785091205279 - 0.42920332716718834im, 12.92278558170773 - 0.4292031026851722im, 12.422786390409346 - 0.42920273257713776im, 11.922787723732338 - 0.4292021223728026im, 11.42278992200878 - 0.4292011163177121im, 10.922793546349741 - 0.4291994576181139im, 10.422799521866436 - 0.4291967228979313im, 9.922809373797097 - 0.4291922141422782im, 9.422825616800965 - 0.4291847805579199im, 8.922852396759225 - 0.4291725249129162im, 8.422896548827053 - 0.42915231948695526im, 7.92296934159787 - 0.4291190083193409im, 7.4230893520139265 - 0.4290640927834337im, 6.923287203307911 - 0.4289735667620085im, 6.423613371154225 - 0.4288243536951455im, 5.924151039178316 - 0.42857844921551824im, 5.425037254279619 - 0.42817331000498887im, 4.926497697671408 - 0.4275061325334096im, 4.428903718874863 - 0.42640827127518255im, 3.93286557186195 - 0.4246039756763536im, 3.4393839757047893 - 0.42164482116812363im, 2.95009415537207 - 0.41680819592206714im, 2.467652457856141 - 0.4089474035324931im, 1.9963316101738395 - 0.39629004411950314im, 1.5428903310988162 - 0.37622082443650384im, 1.117715990264714 - 0.345200709719881im, 0.7359758618950316 - 0.29924118518393067im, 0.41778887902644557 - 0.23578449257048073im, 0.1849728717554206 - 0.15794164682101264im, 0.050624891660021015 - 0.07977458336698368im, 0.002271877217438789 - 0.02412005211237192im, -0.001975490551566147 - 0.0025306436416951427im, -0.0001392658852773198 + 4.621052039983524e-5im, 8.145281857187584e-7 + 8.59924915803196e-7im, -4.4236745140561634e-10 - 3.40054561329938e-10im, -1.5072272094878889e-16 - 2.45592289751446e-15im, -2.0153450436042867e-24 + 4.5231338334005084e-24im, 3.0603115497994336e-38 - 5.757093007467674e-39im, 1.3186180582171105e-61 + 9.894484316990687e-62im, -1.4612609528592482e-100 - 9.442607791153564e-100im, 2.828756024728357e-163 + 1.1630396096583835e-162im, 2.335195331306485e-266 - 2.241880282600362e-266im, -0.0 + 0.0im, 0.0 + 0.0im, -0.0 + 0.0im, -0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 19.4227843364907 - 0.8292036716852221im, 18.92278433739387 - 0.8292036706992425im, 18.42278433888295 - 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3333
for (x, y) in zip(inputs, sympy_outputs)
3434
if y == 5.7333170913660523e-42 + 4.019319913618439e-39im
35-
@test relerrc(expint(x), y) ≤ 5e-12
35+
@test relerr(expint(x), y) ≤ 5e-12
3636
else
3737
@test expint(x) ≅ y
3838
end
@@ -46,7 +46,7 @@ using Base.MathConstants
4646

4747
@testset "En" begin
4848
# ≅ but worse
49-
⩭(x, y) = relerrc(x, y) ≤ 5e-12
49+
⩭(x, y) = relerr(x, y) ≤ 5e-12
5050

5151
@test expint(0, 1) ⩭ 1/e
5252
@test expint(0, 2) ⩭ 1/(2e^2)

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