Do we have any eigensolvers laying around that are specialized for hermitian tridiagonal arrays?
We have eigen(::SymTridiagonal)
but SymTridiagonal
only seems to support being symmetric but not Hermitian which is a really weird choice because symmetric, non-hermitian matrices are quite rare in linear algebra
Okay, so using https://web.archive.org/web/20201028202318/http://icl.cs.utk.edu/lapack-forum/viewtopic.php?f=2&t=3592#p9189 it seems the solution is to use a similarity transform.
real_form(A::Hermitian{T, <:Tridiagonal}) where {T <: Real} = (;T = A, S = Diagonal(ones(T, size(A, 1))), )
function real_form(A::Hermitian{T, <:Tridiagonal}) where {T}
(; dl, d) = parent(A)
N = length(d)
if N == 1
return (; T = SymTridiagonal(real.(d), real.(dl)), S = Diagonal([one(T)]))
end
S = Vector{ComplexF64}(undef, N)
E = dl
Er = abs.(E)
S[1] = 1
S[2] = Er[1] == 0 ? one(T) : S[1] * E[1]/Er[1]
for i ∈ 2:N-1
S[i+1] = Er[i] == 0 ? one(T) : S[i] * E[i]/Er[i]
end
(; T = SymTridiagonal(real.(d), Er), S = Diagonal(S))
end
This will produce a real SymTridiagonal
matrix T
which has the same eigenvalues as the hermitian matrix A
, as well as a diagonal matrix S
such that S * T * S' ≈ A
, so that if U
are the eigenvectors of T
, then
S * U * Diagonal(eigvals(T)) * U' * S' ≈ A
Mason Protter has marked this topic as resolved.
Last updated: Nov 06 2024 at 04:40 UTC