Pivoted sparse Cholesky for QuadtoSOCBridge.

[samuelsonric]

This PR adds the pivoted sparse Cholesky factorization algorithm in CliqueTrees.jl as a fallback in QuadtoSOCBridge. This algorithm can handle singular matrices, and it is MIT licensed.

Example

julia> using CliqueTrees.Multifrontal, LinearAlgebra, LDLFactorizations

julia> A = [1 1 0; 1 1 0; 0 0 1]
3×3 Matrix{Int64}:
 1  1  0
 1  1  0
 0  0  1

The CliqueTrees factorization correctly reconstructs the matrix.

julia> F = cholesky!(ChordalCholesky(A), RowMaximum());

julia> F.L * F.D * F.L'
3×3 Matrix{Float64}:
 1.0  1.0  0.0
 1.0  1.0  0.0
 0.0  0.0  1.0

The LDLFactorizations factorization does not.

julia> G = ldl(A);

julia> UnitLowerTriangular(G.L) * G.D * UnitLowerTriangular(G.L)'
3×3 SparseArrays.SparseMatrixCSC{Int64, Int64} with 5 stored entries:
 1  1           ⋅
 1  1           ⋅
 ⋅  ⋅  5873643920

Related Pull Requests

• [Bridges] Fix computing Q=U'U in QuadtoSOCBridge when Q has 0 eigenvalue #2931
• Add LDLFactorizations.jl as a package extension #2933