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

Bakytzhan Kurmanbek, Yogi Erlangga, Yerlan Amanbek

TL;DR

The paper derives explicit inverses for symmetric tridiagonal near-Toeplitz matrices with weak diagonal dominance in the Toeplitz part, including corner cases where $| ilde{b}|\

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.

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

TL;DR

The paper derives explicit inverses for symmetric tridiagonal near-Toeplitz matrices with weak diagonal dominance in the Toeplitz part, including corner cases where $| ilde{b}|\

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 (). 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 with . In other cases, there exists potential to enhance the obtained upper bounds.
Paper Structure (21 sections, 15 theorems, 125 equations, 3 figures, 2 tables)

This paper contains 21 sections, 15 theorems, 125 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. The case $b = 2$
  4. Bounds for rowsums of the inverse
  5. Bounds for norms of the inverse
  6. The subcase $\widetilde{b} > 1$
  7. The subcase $\widetilde{b} \le 0$
  8. The subcase $0 < \widetilde{b} < 1$
  9. Subcase 2.1:
  10. Subcase 2.2:
  11. Subcase 2.3:
  12. The case $b = -2$
  13. Numerical Experiments
  14. Conclusion
  15. Part of Proof of Theorem \ref{['theor_upper_bound_1']}
  16. ...and 6 more sections

Key Result

Lemma 2.1

Let $\widetilde{T}_{n}$ be invertible with $b = \pm 2$. Then the elements of the inverse matrix $\widetilde{T}_n^{-1}$ are, for $i \geq j$, where $\beta = \widetilde{b} - b$, and $\widetilde{t}^{-1}_{i,j} = \widetilde{t}^{-1}_{j,i}$ for $i < j$.

Figures (3)

  • Figure 1: Comparison of $\|\widetilde{T}^{-1}_{n}\|_{\infty}$ vs. Lower bounds when $b = 2$.
  • Figure 2: Comparison of $\|\widetilde{T}^{-1}_{n}\|_{\infty}$ vs. Upper Bounds when $b = 2$.
  • Figure 3: Comparison of $\|\widetilde{T}_{n}^{-1}\|_{\infty}$ vs. Upper bounds when $b = -2$.

Theorems & Definitions (28)

  • Lemma 2.1
  • proof
  • Lemma 2.2: Singularity
  • proof
  • Lemma 2.3: General Trace
  • proof
  • Theorem 3.1: Row Sums
  • proof
  • Theorem 3.2: Bounds for Row Sums
  • proof
  • ...and 18 more