Pure-Julia Sparse Cholesky

[samuelsonric]

This PR adds the Cholesky factorization algorithm implemented in CliqueTrees.jl. It is written in pure-Julia and accepts arbitrary numeric types. Somewhat slower than CHOLMOD. See this discourse thread.

Example

using BenchmarkTools, CliqueTrees, LinearSolve, MatrixMarket, SuiteSparseMatrixCollection
ssmc = ssmc_db(); name = "nasasrb";
A = mmread(joinpath(fetch_ssmc(ssmc[ssmc.name .== name, :]; format="MM")[1], "$(name).mtx"))
b = rand(size(A, 2))
problem = LinearProblem(A, b)

println("CholeskyFactorization:")
@btime solve(problem, CholeskyFactorization())

println("CliqueTreesFactorization:")
@btime solve(problem, CliqueTreesFactorization());

output:

CholeskyFactorization:
  227.126 ms (137 allocations: 327.23 MiB)
CliqueTreesFactorization:
  255.163 ms (103 allocations: 251.56 MiB)

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.
• Any new documentation only uses public API

Additional context

Add any other context about the problem here.

[ChrisRackauckas]

Is this not supposed to be a sparse matrix?

[samuelsonric]

Sparse, now. Converted internally in any case: I was trying to avoid using SparseArrays from some reason.

[ChrisRackauckas]

Suggested change

sol = solve(prob) # in case cliquetrees is used as default

It's not.

[samuelsonric]

I had copied that from the Sparsepak extension. Gone, now.