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

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.

More by Bakytzhan Kurmanbek

• Deep Learning Approaches for Autoregressive Fracture Network Path Prediction: A Comprehensive Study on the Teapot Dome DatasetIEEE Access · 2026
• Explicit inverse of symmetric, tridiagonal near Toeplitz matrices with strictly diagonally dominant Toeplitz partSpecial Matrices · 2025
• Prediction of natural fracture network patterns using feature engineering and machine learning approachesComputational energy science. · 2024 · Citations: 2
• Explicit inverse of symmetric, tridiagonal near Toeplitz matrices with strictly diagonally dominant Toeplitz partarXiv · 2024