GitHub

SelectedInversion

Quickly compute selected entries of the inverse of a sparse matrix.

Introduction

Sparse matrices are one of the pillars of scientific computing. Sparse factorization methods allow us to solve linear systems involving the inverse efficiently. But in some applications, we might need a bunch of entries of the inverse. Selected inversion algorithms efficiently compute those entries of the inverse that correspond to non-zero entries in the factorization.

SelectedInversion.jl directly interfaces with CHOLMOD-based Cholesky factorizations, which are the default for sparse symmetric positive-definite matrices in Julia.

Installation

SelectedInversion.jl is not yet a registered Julia package. Until it is, you can install it from this GitHub repository. To do so:

  1. Download Julia (>= version 1.10).

  2. Launch the Julia REPL and type ] add https://github.com/timweiland/SelectedInversion.jl.

SelInv API

Make sure to also check the Tutorial.

SelectedInversion.selinv — Function
selinv(A::SparseMatrixCSC; depermute=false)
    -> @NamedTuple{Z::AbstractMatrix, p::Vector{Int64}}

Compute the selected inverse Z of A. The sparsity pattern of Z corresponds to that of the Cholesky factor of A, and the nonzero entries of Z match the corresponding entries in A⁻¹.

Arguments

  • A::SparseMatrixCSC: The sparse symmetric positive definite matrix for which the selected inverse will be computed.

Keyword arguments

  • depermute::Bool: Whether to depermute the selected inverse or not.

Returns

A named tuple Zp. Zp.Z is the selected inverse, and Zp.p is the permutation vector of the corresponding sparse Cholesky factorization.

source
selinv(F::SparseArrays.CHOLMOD.Factor; depermute=false)
    -> @NamedTuple{Z::AbstractMatrix, p::Vector{Int64}}

Compute the selected inverse Z of some matrix A based on its sparse Cholesky factorization F.

Arguments

  • F::SparseArrays.CHOLMOD.Factor: Sparse Cholesky factorization of some matrix A. F will be used internally for the computations underlying the selected inversion of A.

Keyword arguments

  • depermute::Bool: Whether to depermute the selected inverse or not.

Returns

A named tuple Zp. Zp.Z is the selected inverse, and Zp.p is the permutation vector of the corresponding sparse Cholesky factorization.

source