On the Inversion of Polynomials of Discrete Laplace Matrices

TL;DR

This paper presents a method for efficiently inverting matrix polynomials of multidimensional Laplace matrices using eigenvector and eigenvalue expansions, extending previous one-dimensional results.

Contribution

It introduces a novel eigen-based procedure for inverting polynomial functions of multidimensional Laplace matrices, generalizing prior one-dimensional approaches.

Findings

Method successfully computes inverses for multidimensional Laplace matrix polynomials.

The approach aligns with existing one-dimensional inverse expressions.

Several examples demonstrate the method's effectiveness.

Abstract

The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on eigenvector and eigenvalue expansions. The method is consistent with previous expressions of the inverse discretized Laplacian in one spatial dimension \citep{Vermolen_2022}. Several examples are given.