Inverse properties of a class of seven-diagonal (near) Toeplitz matrices

TL;DR

This paper derives an explicit inverse formula for a specific class of seven-diagonal Toeplitz matrices, aiding in the numerical solution of nonlinear fourth-order differential equations and analyzing their convergence properties.

Contribution

It introduces a non-recurrence explicit inverse formula for these matrices and establishes their positivity and norm bounds, enhancing numerical analysis methods.

Findings

Explicit inverse formula derived using Sherman-Morrison

Proved positivity of the inverse matrix

Constructed upper bounds for inverse matrix norms

Abstract

This paper discusses the explicit inverse of a class of seven-diagonal (near) Toeplitz matrices, which arises in the numerical solutions of nonlinear fourth-order differential equation with a finite difference method. A non-recurrence explicit inverse formula is derived using the Sherman-Morrison formula. Related to the fixed-point iteration used to solve the differential equation, we show the positivity of the inverse matrix and construct an upper bound for the norms of the inverse matrix, which can be used to predict the convergence of the method.