A finite dimensional analog of the Krein formula

TL;DR

This paper introduces a simple formula for the resolvent of small rank perturbations of large matrices, with applications to solving difference boundary value problems and analyzing numerical algorithms.

Contribution

It provides a new finite-dimensional analog of the Krein formula, facilitating analytical and numerical solutions for boundary value problems involving large matrices.

Findings

Efficient algorithms for difference boundary value problems

Applications to difference Laplacian problems

Numerical efficiency estimates of the proposed methods

Abstract

I offer a simple and useful formula for the resolvent of a small rank perturbation of large matrices. I discuss applications of this formula, in particular, to analytical and numerical solving of difference boundary value problems. I present examples connected with such problems for the difference Laplacian and estimate numerical efficiency of the corresponding algorithms.