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.

翻译：本研究探讨了Toeplitz部分严格对角占优的三对角近Toeplitz矩阵。重点在于确定这些矩阵的精确逆，并建立逆矩阵无穷范数的上界。针对$b > 2$和$b < -2$的情形，我们推导出精确逆矩阵元素的紧凑形式。这些结果在矩阵角元素非对角占优时依然成立，特别是当$|\widetilde{b}| < 1$时。此外，我们计算了逆矩阵的迹和行和。随后，我们提出了逆矩阵无穷范数的上界定理。为说明所获上界的有效性及其应用，我们给出了求解Fisher问题的数值结果。研究结果表明：不动点迭代的收敛速率与预期速率高度吻合，且上界与逆矩阵无穷范数之间的差异极小。具体而言，这一现象在$b > 2$且$\widetilde{b} \leq 1$，以及$b < -2$且$\widetilde{b} \geq -1$的条件下成立。对于其他情形，所得上界仍有进一步优化的空间。本研究通过提升迭代收敛速率和减少逆矩阵计算时间，为不动点迭代的数值分析领域作出了贡献。