Inverse properties of a class of pentadiagonal matrices related to higher order difference operators

TL;DR

This paper investigates the inverse properties of specific pentadiagonal matrices linked to higher order difference operators, providing explicit formulas, analyzing convergence of fixed-point iterations, and demonstrating computational efficiency with numerical results.

Contribution

It introduces explicit formulas for the inverse of pentadiagonal matrices related to fourth-order difference operators and analyzes their properties in the context of nonlinear beam equations.

Findings

Explicit inverse formulas for the matrices.

Convergence analysis of fixed-point iterations.

Numerical validation of computational efficiency.

Abstract

This paper analyzes the convergence of fixed-point iterations of the form u = f(u) and the properties of the inverse of the related pentadiagonal matrices, associated with the fourth-order nonlinear beam equation. This nonlinear problem is discretized using the finite difference method with the clamped-free and clamped-clamped boundary conditions in the one dimension. Explicit formulas for the inverse of the matrices and norms of the inverse are derived. In iterative process, the direct computation of inverse matrix allows to achieve an efficiency. Numerical results were provided.