Diagonality and bandedness checks with InfiniteArrays.jl?

[TSGut]

The methods in InfiniteArrays.jl and InfiniteLinearAlgebra.jl don't currently seem to agree on what is a diagonal and what is a banded matrix. To give some concrete examples:

A minimal working example for Diagonal is

julia> LinearAlgebra.isdiag(InfiniteArrays.Diagonal(0:∞))
true

but the following never finishes computing, i.e. does not even return false:

julia> LinearAlgebra.isdiag(InfiniteLinearAlgebra.BandedMatrix(0=>(0:∞)))

Furthermore, we have a similar situation for banded matrix types:

julia> InfiniteLinearAlgebra.BandedMatrix(0 => (1:∞)) isa InfiniteLinearAlgebra.BandedMatrix
true

julia> InfiniteArrays.Diagonal(1:∞) isa InfiniteLinearAlgebra.BandedMatrix
false

What one wants these to evaluate to might be up to the design philosophy between the packages to some degree but the current behavior does seem odd.

[dlfivefifty]

julia> LinearAlgebra.isdiag(InfiniteLinearAlgebra.BandedMatrix(0=>(0:∞)))

This should be added to BandedMatrices.jl, but note we need to check off-diagonal bands too:

julia> isdiag(zeros(5,5))
true

something like

function isdiag(A::AbstractBandedMatrix)
   bandwidths(A) == (0,0) && return true
   Base.invoke(isdiag, Tuple{AbstractMatrix}, A)
end

julia> InfiniteArrays.Diagonal(1:∞) isa InfiniteLinearAlgebra.BandedMatrix
false 

This isn't changeble since Diagonal is a StdLib type.

[dlfivefifty]

Actually, note isdiag calls isbanded which calls istriu and istril so its really the last two that need overloading. So we can modify the code from StdLib, say in ArrayLayouts.jl. Something like:

function istriu(A::LayoutMatrix, k::Integer = 0)
    require_one_based_indexing(A)
    m, n = size(A)
    for j in 1:min(n, m + k - 1)
        for i in (max(1, j - k + 1):m) ∩ colsupport(A,j)
            iszero(A[i, j]) || return false
        end
    end
    return true
end

and similar for istril.

[dlfivefifty]

@TSGut do you want to have a go making a PR for this?

[TSGut]

Sure. You said you think it would fit into BandedMatrices.jl?

EDIT: Actually just saw you also mentioned ArrayLayouts.jl. I suppose I'll try to get it working and passing tests for now. I'll submit the PR to the package you think it would most fit into, so just let me know.

[dlfivefifty]

Hmm actually you could just put it in BandedMatrices.jl since we can do it a bit smarter by adding:

function istriu(A::AbstractBandedMatrix, k::Integer)
    l,_ = bandwidths(A)
    -l ≥ k && return true
    ...
end