start fixing ambiguities

[ArnoStrouwen]

Fixes many of the method ambiguities in this package.
Things I'm still unsure about:

First, for defaultalg there is dispatch for AbstractSciMLOperator as well as dispatch for OperatorAssumptions{Nothing}.
AbstractSciMLOperator dispatches based on A.A.
OperatorAssumptions{Nothing} dispatches based on issquare(A).
If both occur together, I don't know which order to apply them.
Currently, it first gets rid of the Nothing, but this means issquare has to be defined for Operators.

Second, there are remaining ambiguities regarding:

function init_cacheval(alg::GenericFactorization,
    A::Union{Hermitian{T, <:SparseMatrixCSC},
        Symmetric{T, <:SparseMatrixCSC}}, b, u, Pl, Pr,
    maxiters::Int, abstol, reltol, verbose::Bool,
    assumptions::OperatorAssumptions) where {T}
    newA = copy(convert(AbstractMatrix, A))
    do_factorization(alg, newA, b, u)
end

e.g. with:

function init_cacheval(alg::GenericFactorization{typeof(svd!)}, A::AbstractMatrix, b, u, Pl, Pr,
    maxiters::Int,
    abstol, reltol, verbose::Bool, assumptions::OperatorAssumptions)
    ArrayInterface.svd_instance(A)
end

I'm not sure how to resolve these, I'm assuming we want as little do_factorization as possible?

Third, for some special matrix types, such as diagonal, there is dispatch to always use the same method, no matter the algorithm. But there are also fallback methods for most algorithms to work for general A. This design choice leads to tons of additional function definitions to resolve ambiguities between the two.

[codecov]

Codecov Report

Attention: 47 lines in your changes are missing coverage. Please review.

Comparison is base (f87583e) 66.50% compared to head (3099e72) 65.44%.

Files	Patch %	Lines
src/factorization.jl	12.24%	43 Missing ⚠️
src/default.jl	63.63%	4 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #437      +/-   ##
==========================================
- Coverage   66.50%   65.44%   -1.06%     
==========================================
  Files          27       27              
  Lines        2039     2072      +33     
==========================================
  Hits         1356     1356              
- Misses        683      716      +33     

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[ChrisRackauckas]

what about the operator dispatches?

[ArnoStrouwen]

This should still catch them:

LinearSolve.jl/src/factorization.jl

Lines 1458 to 1466 in d80fade

	for alg in InteractiveUtils.subtypes(AbstractFactorization)
	@eval function init_cacheval(alg::$alg, A::MatrixOperator, b, u, Pl, Pr,
	maxiters::Int, abstol, reltol, verbose::Bool,
	assumptions::OperatorAssumptions)
	init_cacheval(alg, A.A, b, u, Pl, Pr,
	maxiters::Int, abstol, reltol, verbose::Bool,
	assumptions::OperatorAssumptions)
	end
	end

[ChrisRackauckas]

That's only for matrix operator. What about other abstract scimloperators?

[ArnoStrouwen]

There is none for that,
but note that this is currently not working anyway:

using LinearSolve
using SciMLOperators

N = 4
f = (u, p, t) -> u .* u

M = MatrixOperator(rand(N, N))
D = DiagonalOperator(rand(N))
F = FunctionOperator(f, zeros(N), zeros(N))

b = rand(4)

solve(LinearProblem(rand(N, N), b),FastQRFactorization()) # works
solve(LinearProblem(M, b),FastQRFactorization()) # fails
solve(LinearProblem(D, b),FastQRFactorization()) # fails
solve(LinearProblem(F, b),FastQRFactorization()) # fails

julia> solve(LinearProblem(M, b),FastQRFactorization())
ERROR: MethodError: init_cacheval(::FastQRFactorization{LinearAlgebra.NoPivot}, ::MatrixOperator{Float64, Matrix{Float64}, SciMLOperators.FilterKwargs{typeof(SciMLOperators.DEFAULT_UPDATE_FUNC), Tuple{}}, SciMLOperators.FilterKwargs{typeof(SciMLOperators.DEFAULT_UPDATE_FUNC), Tuple{}}}, ::Vector{Float64}, ::Vector{Float64}, ::IdentityOperator, ::IdentityOperator, ::Int64, ::Float64, ::Float64, ::Bool, ::OperatorAssumptions{Bool}) is ambiguous.

Candidates:
init_cacheval(alg::FastQRFactorization, A::MatrixOperator, b, u, Pl, Pr, maxiters::Int64, abstol, reltol, verbose::Bool, assumptions::OperatorAssumptions)
@ LinearSolve C:\Users\arno.julia\packages\LinearSolve\8EgR1\src\factorization.jl:1180
init_cacheval(alg::FastQRFactorization{LinearAlgebra.NoPivot}, A, b, u, Pl, Pr, maxiters::Int64, abstol, reltol, verbose::Bool, assumptions::OperatorAssumptions)
@ LinearSolve C:\Users\arno.julia\packages\LinearSolve\8EgR1\src\factorization.jl:1066

Possible fix, define
init_cacheval(::FastQRFactorization{LinearAlgebra.NoPivot}, ::MatrixOperator, ::Any, ::Any, ::Any, ::Any, ::Int64, ::Any, ::Any, ::Bool, ::OperatorAssumptions)

[ChrisRackauckas]

Oh, that was working before?

[ArnoStrouwen]

For it to ever have worked for FunctionOperators it needed to have this defined

WorkspaceAndFactors(ws, ArrayInterface.qr_instance(A))

Anyway, the solution for DiagonalOperators seems straightforward to me. Just add the same block of code as for MatrixOperators. But how do you want to handle FunctionOperators?

[ChrisRackauckas]

Resolve the conflicts?

[ChrisRackauckas]

@avik-pal can you review this one?

[avik-pal]

When can issquare be nothing? If we can't resolve the dimensions can't we assume rectangular

[avik-pal]

Oh I see it is to disambiguate between no assumption provided from the user. This PR seems fine in that case