A New Method for Computing the Diagonal of a Matrix Inverse

Abstract

The computation of some entries of a matrix inverse arises in several important applications in practice. This paper presents a probing method for determining the diagonal of the inverse of a sparse matrix in the common situation when its inverse exhibits a decay property. A few simple properties of the inverse suggest a way to determine effective probing vectors based on standard graph theory results. An iterative method is then applied to solve the resulting sequence of linear systems, from which the diagonal of the matrix inverse is extracted. Results of numerical experiments are provided to demonstrate the effectiveness of the proposed method.

Keywords:

• Matrix Diagonal Extraction
• Probing
• Sparse Approximate Inverses
• Graph Theory
• Krylov-subspace Methods
• Green’s Functions
• Covariance Matrices