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Support Jacobi(-1,0)\Ultraspherical(-1/2) and Jacobi(0,-1) #245

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Merged
merged 4 commits into from
Sep 7, 2025
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Support Jacobi(-1,0)\Ultraspherical(-1/2) and Jacobi(0,-1) #245

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a21a0b8
Support Jacobi(-1,0)\Ultraspherical(-1/2)
DanielVandH Sep 3, 2025
2f91512
Fix slash
DanielVandH Sep 3, 2025
00fb966
Add Jacobi(0,-1)
DanielVandH Sep 3, 2025
cbc7424
Clarify
DanielVandH Sep 3, 2025
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2 changes: 1 addition & 1 deletion Project.toml
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Original file line number Diff line number Diff line change
@@ -1,7 +1,7 @@
name = "ClassicalOrthogonalPolynomials"
uuid = "b30e2e7b-c4ee-47da-9d5f-2c5c27239acd"
authors = ["Sheehan Olver <[email protected]>"]
version = "0.15.3"
version = "0.15.4"


[deps]
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15 changes: 13 additions & 2 deletions src/classical/ultraspherical.jl
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Expand Up @@ -199,8 +199,19 @@ function \(A::Ultraspherical, B::Jacobi)
Diagonal(Ã[1,:]./A[1,:]) * (Ã\B)
end
function \(A::Jacobi, B::Ultraspherical)
B̃ = Jacobi(B)
(A\B̃)*Diagonal(B[1,:]./B̃[1,:])
if B == Ultraspherical(-1/2) && (A == Jacobi(-1, 0) || A == Jacobi(0, -1))
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Add a comment why this special case is needed.

Also, where in (A\B̃)*Diagonal(B[1,:]./B̃[1,:]) are the NaNs coming in?

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@DanielVandH DanielVandH Sep 3, 2025 •
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The NaNs come from B̃[1,:], since

julia> Jacobi(-1, -1)[1, :]
ℵ₀-element view(::Jacobi{Float64, Int64}, 1.0, :) with eltype Float64 with indices OneToInf():
   1.0
   0.0
 NaN
 NaN
 NaN
 NaN
 NaN
 NaN
 NaN
 NaN
   ⋮

(since we haven't yet given a definition to Jacobi(-1, -1) in the code that works, although it'd be all zero anyway -except for the first two entries- I think?)

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P_k^(-1,-1)(x) are well-defined:

image

I think using the series or Hypergeometric formula. They are just degenerate since P_1^(-1,-1)(x) = 0, so we cannot use the 3-term recurrence to build them.

So really we should be fixing evaluation for degenerate Jacobi polynomials.

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@DanielVandH DanielVandH Sep 4, 2025 •
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Won't it still lead to a NaN anyway since $P_1^{(-1,-1)}(x) = 0$? We are dividing by it

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Touché

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update the comment to mention even if it is defined the above won't work because of division by zero.

# In this case, Jacobi(-1, -1) is (currently) undefined, so the conversion via B̃ = Jacobi(B) leads to NaNs
# from evaluating in B̃[1, :]
T = promote_type(eltype(A), eltype(B))
n = -2one(T) ./ (2 .* (2:∞) .- one(T))
sgn = A == Jacobi(-1, 0) ? one(T) : -one(T)
dv = Vcat(one(T), -2one(T), n)
ev = Vcat(-sgn, sgn .* n)
LazyBandedMatrices.Bidiagonal(dv, ev, :U)
else
B̃ = Jacobi(B)
(A\B̃)*Diagonal(B[1,:]./B̃[1,:])
end
end

function \(wA::Weighted{<:Any,<:Ultraspherical}, wB::Weighted{<:Any,<:Jacobi})
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11 changes: 11 additions & 0 deletions test/test_ultraspherical.jl
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Expand Up @@ -207,4 +207,15 @@ using ClassicalOrthogonalPolynomials: grammatrix
@testset "ChebyshevInterval constructior" begin
@test ultraspherical(2,ChebyshevInterval()) ≡ Ultraspherical(2)
end
end

@testset "Jacobi(-1, 0) \\ Ultraspherical(-1/2) and Jacobi(0, -1) \\ Ultraspherical(-1/2)" begin
for A in (Jacobi(-1,0),Jacobi(0,-1))
B = Ultraspherical(-1/2)
R = A \ B
AR = A * R
lhs = AR[0.3918, 1:100]
rhs = B[0.3918, 1:100]
@test lhs ≈ rhs
end
end
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