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Laguerre Derivatives #113

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Laguerre Derivatives #113

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5472eb4
Implement Hilbert transform of LaguerreWeight (and thus all of Laguerre)
Nov 24, 2022
35195f6
add conversion operators between (weighted) Laguerre polys
Nov 24, 2022
e608c5f
improve sig figs. in tests
Nov 24, 2022
b36ded7
fix tests
Nov 25, 2022
42631c6
added derivatives for weighted Laguerre
Nov 25, 2022
a8af5be
Merge branch 'JuliaApproximation:main' into jp/hilbert-laguerre
ioannisPApapadopoulos Dec 12, 2022
e30638d
Merge branch 'main' into pr/113
dlfivefifty Apr 19, 2023
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3 changes: 2 additions & 1 deletion src/classical/laguerre.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -9,7 +9,8 @@ end

LaguerreWeight{T}() where T = LaguerreWeight{T}(zero(T))
LaguerreWeight() = LaguerreWeight{Float64}()
axes(::LaguerreWeight{T}) where T = (Inclusion(ℝ),)
# axes(::LaguerreWeight{T}) where T = (Inclusion(ℝ),)
axes(::LaguerreWeight{T}) where T = (Inclusion(HalfLine{T}()),)
function getindex(w::LaguerreWeight, x::Number)
x ∈ axes(w,1) || throw(BoundsError())
x^w.α * exp(-x)
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20 changes: 18 additions & 2 deletions src/stieltjes.jl
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Expand Up @@ -94,9 +94,25 @@ end
ChebyshevT{T}() * _BandedMatrix(Fill(convert(T,π),1,∞), ℵ₀, 1, -1)
end

@simplify function *(H::Hilbert{<:Any,<:HalfLine,<:HalfLine}, w::LaguerreWeight)
T = promote_type(eltype(H), eltype(w))
α = convert(T, w.α)
z = axes(w,1)

α == 0 && return real(exp.(log.(expinti.(z) .+ zero(T)*im) .-z))

β = -1 < α < 0 ? α : α - ceil(α)
# S = (2* π * im) * (-z)^β*exp(-z)*gamma(-β,-z)/(gamma(-β)*(exp(im*π*β)-exp(-im*π*β)))
S = (2 * convert(T,π) * im) .* exp.(β*log.(-z .+ zero(T)*im) .- z + loggamma.(-β, -z .+ zero(T)*im)) ./ (gamma(-β)*(exp(im*convert(T,π)*β)-exp(-im*convert(T,π)*β)))
if α > 0
for j in 1:Int(ceil(α))
S = -gamma(β+j) .+ z .* S
end
end
real(S)
end


@simplify function *(H::Hilbert{<:Any,<:ChebyshevInterval,<:ChebyshevInterval}, wP::Weighted{<:Any,<:OrthogonalPolynomial})
@simplify function *(H::Hilbert{<:Any,<:Any,<:Any}, wP::Weighted{<:Any,<:OrthogonalPolynomial})
P = wP.P
w = orthogonalityweight(P)
A = recurrencecoefficients(P)[1]
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9 changes: 9 additions & 0 deletions test/test_stieltjes.jl
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Expand Up @@ -141,6 +141,15 @@ end
@test H * ChebyshevTWeight() ≡ QuasiZeros{Float64}((x,))
@test H * ChebyshevUWeight() == π*x
@test (H * LegendreWeight())[0.1] ≈ log((0.1+1)/(1-0.1))

x = axes(Laguerre(), 1)
@test (inv.(x .- x') * LaguerreWeight())[0.1] ≈ exp(-0.1)*expinti(0.1)
@test isapprox((inv.(x .- x') * LaguerreWeight(-0.5))[0.1], 3.317769, rtol=1e-4) # Mathematica
@test isapprox((inv.(x .- x') * LaguerreWeight(-0.5))[2.1], 1.0829, rtol=1e-4) # Mathematica
@test isapprox((inv.(x .- x') * LaguerreWeight(0.5))[0.1], -1.44067, rtol=1e-4) # Mathematica
@test isapprox((inv.(x .- x') * LaguerreWeight(0.5))[2.1], 0.5017, rtol=1e-3) # Mathematica
@test isapprox((inv.(x .- x') * LaguerreWeight(3.5))[0.1], -3.46658, rtol=1e-4) # Mathematica
@test isapprox((inv.(x .- x') * LaguerreWeight(3.5))[2.1], -5.37663, rtol=1e-4) # Mathematica
end

@test (Ultraspherical(1) \ (H*wT))[1:10,1:10] == diagm(1 => fill(-π,9))
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