Explicit inverse of symmetric, tridiagonal near Toeplitz matrices Part II: with weakly diagonally dominant Toeplitz

TL;DR

This paper derives explicit inverse formulas for symmetric near-Toeplitz matrices with weak diagonal dominance, extends results to non-dominant corners, and validates the theoretical findings through numerical experiments on the Fisher problem.

Contribution

It provides new explicit inverse formulas for a class of symmetric near-Toeplitz matrices and analyzes their properties, including bounds and numerical applications.

Findings

Explicit inverse formulas for weakly diagonally dominant near-Toeplitz matrices.

Upper bounds for the inverse matrix norms and their accuracy.

Numerical validation showing convergence rates match theoretical expectations.

Abstract

In this paper, we provide explicit formulas for the exact inverses of the symmetric tridiagonal near-Toeplitz matrices characterized by weak diagonal dominance in the Toeplitz part. Furthermore, these findings extend to scenarios where the corners of the near Toeplitz matrices lack diagonal dominance ($|\widetilde{b}| < 1$). Additionally, we compute the row sums and traces of the inverse matrices, thereby deriving upper bounds for their infinite norms. To demonstrate the practical applicability of our theoretical results, we present numerical examples addressing numerical solution of the Fisher problem using the fixed point method. Our findings reveal that the convergence rates of fixed-point iterations closely align with the expected rates, and there is minimal disparity between the upper bounds and the infinite norm of the inverse matrix. Specifically, this observation holds true for $|b| = 2$ with $|\widetilde{b}| \geq 1$. In other cases, there exists potential to enhance the obtained upper bounds.