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add missing methods for division of Hessenberg matrices #1322

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May 4, 2025
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add missing methods for division of Hessenberg matrices #1322

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4578357
add missing methods for division of Hessenberg matrices
stevengj Apr 29, 2025
4844670
Update test/hessenberg.jl
stevengj Apr 30, 2025
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23 changes: 23 additions & 0 deletions src/hessenberg.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -177,6 +177,29 @@ function \(U::UnitUpperTriangular, H::UpperHessenberg)
UpperHessenberg(HH)
end

function (\)(H::Union{UpperHessenberg,AdjOrTrans{<:Any,<:UpperHessenberg}}, B::AbstractVecOrMat)
TFB = typeof(oneunit(eltype(H)) \ oneunit(eltype(B)))
return ldiv!(H, copy_similar(B, TFB))
end

function (/)(B::AbstractMatrix, H::Union{UpperHessenberg,AdjOrTrans{<:Any,<:UpperHessenberg}})
TFB = typeof(oneunit(eltype(B)) / oneunit(eltype(H)))
return rdiv!(copy_similar(B, TFB), H)
end

ldiv!(H::AdjOrTrans{<:Any,<:UpperHessenberg}, B::AbstractVecOrMat) =
(rdiv!(wrapperop(H)(B), parent(H)); B)
rdiv!(B::AbstractVecOrMat, H::AdjOrTrans{<:Any,<:UpperHessenberg}) =
(ldiv!(parent(H), wrapperop(H)(B)); B)

# fix method ambiguities for right division, from adjtrans.jl:
/(u::AdjointAbsVec, A::UpperHessenberg) = adjoint(adjoint(A) \ u.parent)
/(u::TransposeAbsVec, A::UpperHessenberg) = transpose(transpose(A) \ u.parent)
/(u::AdjointAbsVec, A::Adjoint{<:Any,<:UpperHessenberg}) = adjoint(adjoint(A) \ u.parent)
/(u::TransposeAbsVec, A::Transpose{<:Any,<:UpperHessenberg}) = transpose(transpose(A) \ u.parent)
/(u::AdjointAbsVec, A::Transpose{<:Any,<:UpperHessenberg}) = adjoint(conj(A.parent) \ u.parent) # technically should be adjoint(copy(adjoint(copy(A))) \ u.parent)
/(u::TransposeAbsVec, A::Adjoint{<:Any,<:UpperHessenberg}) = transpose(conj(A.parent) \ u.parent)

# Solving (H+µI)x = b: we can do this in O(m²) time and O(m) memory
# (in-place in x) by the RQ algorithm from:
#
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7 changes: 7 additions & 0 deletions test/hessenberg.jl
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Expand Up @@ -66,6 +66,13 @@ let n = 10
H = UpperHessenberg(Areal)
@test Array(Hc + H) == Array(Hc) + Array(H)
@test Array(Hc - H) == Array(Hc) - Array(H)
@testset "ldiv and rdiv" begin
for b in (b_, B_), H in (H, Hc, H', Hc', transpose(Hc))
@test H * (H \ b) ≈ b
@test (b' / H) * H ≈ (Matrix(b') / H) * H ≈ b'
@test (transpose(b) / H) * H ≈ (Matrix(transpose(b)) / H) * H ≈ transpose(b)
end
end
@testset "Preserve UpperHessenberg shape (issue #39388)" begin
H = UpperHessenberg(Areal)
A = rand(n,n)
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