SparseInverseSubset.jl

This is a lightweight, experimental package implementing Takahashi's algorithm for calculating a subset of the inverse of a sparse symmetrical matrix $A$ based on its Cholesky factorization $A = L L^T$. The partial inverse matrix $Z$ has the same sparsity pattern as $L + L'$ and is equal to the full inverse $A^{-1}$ on these elements.

The package exports a single function, sparseinv, which takes a SparseMatrixCSC and returns the sparse inverse matrix and the permutation vector from the Cholesky decomposition:

using SparseInverseSubset
using SparseArrays
using LinearAlgebra

A = sparse(
    [1, 5, 2, 3, 4, 2, 3, 4, 2, 3, 4, 5, 1, 4, 5],
    [1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5],
    [1.371743277584265, 0.15319636423142857, 1.2974555319793244, 0.334648233917893,
        0.5630908120922691, 0.334648233917893, 2.1530446960842218, 0.36210585506955384,
        0.5630908120922691, 0.36210585506955384, 1.5043721186197785, 0.5858861097478401,
        0.15319636423142857, 0.5858861097478401, 1.543614692728164]
)
Z, P = sparseinv(A)

# check against full inverse
sparsity_pattern = Z .!= 0
Zdense = inv(Matrix(A)[P, P])
all(Z .≈ (Zdense .* sparsity_pattern)) # true

If the keyword argument depermute is set to true, the sparse inverse matrix is de-permuted before returning, so its rows and columns match those of inv(Matrix(A)):

Z, P = sparseinv(A; depermute=true)
sparsity_pattern = Z .!= 0
all(Z .≈ (inv(Matrix(A)) .* sparsity_pattern)) # true

This implementation is based on the formulas in Erisman and Tinney (1975). It has not been optimized or tested very thoroughly, but is already significantly faster than calculating a dense inverse once the matrix is bigger than ~100 x 100.

References

Erisman, A.M. and Tinney, W.F. (1975). "On computing certain elements of the inverse of a sparse matrix," Communications of the ACM 18(3) 177-179 (https://dl.acm.org/doi/10.1145/360680.360704).

Takahashi, K., Fagan, J., and Chin, M-S. (1973). "Formation of a sparse bus impedance matrix and its application to short circuit study. 8th PICA Conference Proceedings, 4-6 June 1973, Minneapolis, MN.