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

TL;DR

This work addresses the explicit inversion and norm-bounding of symmetric tridiagonal near-Toeplitz matrices with a strictly diagonally dominant Toeplitz part, a scenario that arises in discretizations of differential equations and fixed-point iterations. The authors decompose the target matrix $ ilde{T}_n$ into a Toeplitz part $T_n$ and a rank-2 perturbation, apply Sherman–Morrison to obtain an explicit inverse, and derive closed-form expressions for its entries, its trace, and row sums, leveraging auxiliary sequences $oldsymbol{eta}_k$ based on the roots of $p(r)=-r^2+br-1$. They provide sharp upper bounds for $ orm{ ilde{T}_n^{-1}}_ty$ across regimes $|b|>2$ with $ ilde{b}$ in relation to $b$, along with positivity results and symmetry arguments; results are extended to $b<-2$ via a sign-symmetry transformation. Numerical experiments on Fisher’s equation corroborate the theoretical bounds and show that the observed fixed-point convergence rates align closely with the predicted rates, indicating practical benefits in convergence analysis and reduced inverse-computation time.

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.

Figures (4)

• Figure 1: Comparison of $\|\widetilde{T}^{-1}_{n}\|_{\infty}$ vs. Upper bounds when $b > 2$ and $\widetilde{b} \geq 1$.
• Figure 2: Comparison of $\|\widetilde{T}^{-1}_{n}\|_{\infty}$ vs. Upper bounds when $b>2$ and $\widetilde{b} < 1$.
• Figure 3: Comparison of $\|\widetilde{T}^{-1}_{n}\|_{\infty}$ vs. Upper bounds when $b < -2$ and $\widetilde{b} \leq -1$.
• Figure 4: Comparison of $\|\widetilde{T}^{-1}_{n}\|_{\infty}$ vs. Upper bounds when $b < -2$ and $\widetilde{b} > -1$.

Theorems & Definitions (30)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 3.1: Non-singularity of $\widetilde{T}_n$
• proof
• Lemma 3.2
• Lemma 3.3
• proof