Explicit inverse of symmetric, tridiagonal near Toeplitz matrices with strictly diagonally dominant Toeplitz part

TL;DR

This paper derives explicit inverses and bounds for symmetric, near-Toeplitz tridiagonal matrices with diagonally dominant Toeplitz parts, improving numerical methods for solving related linear systems.

Contribution

It provides the exact inverse formulas and upper bounds for the inverse norms of these matrices, including cases with non-dominant corners, enhancing numerical analysis techniques.

Findings

Exact inverse formulas for specific matrix cases.

Upper bounds closely match actual inverse norms.

Improved convergence rates for fixed-point iterations.

Abstract

This study investigates tridiagonal near-Toeplitz matrices in which the Toeplitz part is strictly diagonally dominant. The focus is on determining the exact inverse of these matrices and establishing upper bounds for the infinite norms of the inverse matrices. For cases with $b > 2$ and $b < -2$, we derive the compact form of the entries of the exact inverse. These results remain valid even when the matrices' corners are not diagonally dominant, specifically when $|\widetilde{b}| < 1$. Furthermore, we calculate the traces and row sums of the inverse matrices. Afterwards, we present upper bound theorems for the infinite norms of the inverse matrices. To demonstrate the effectiveness of the bounds and their application, we provide numerical results for solving Fisher's problem. Our findings reveal that the converging rates of fixed-point iterations closely align with the expected rates, and there is minimal disparity between the upper bounds and infinite norm of the inverse matrix. Specifically, this observation holds true when $b > 2$ with $\widetilde{b} \leq 1$ and $b < -2$ with $\widetilde{b} \geq -1$. For other cases, there is potential for further improvement in the obtained upper bounds. This study contributes to the field of numerical analysis of fixed-point iterations by improving the convergence rate of iterations and reducing the computing time of the inverse matrices.