具有严格对角占优 Toeplitz 部分的对称三对角近 Toeplitz 矩阵的显式逆
Explicit inverse of symmetric, tridiagonal near Toeplitz matrices with strictly diagonally dominant Toeplitz part
AI总结：
本文研究了具有严格对角占优 Toeplitz 部分的三对角近 Toeplitz 矩阵，推导了其精确逆的紧凑形式及逆矩阵无穷范数的上界，并通过数值实验验证了其在不动点迭代中提升收敛速率与减少计算时间的有效性。
摘要：
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，并为逆矩阵的无穷范数建立上界。对于 $b > 2$ 和 $b < -2$ 的情况，我们推导了精确逆元素的紧凑形式。即使矩阵的角不满足对角占优，特别是当 $|\widetilde{b}| < 1$ 时，这些结果依然成立。此外，我们计算了逆矩阵的迹和行和。随后，我们给出了逆矩阵无穷范数的上界定理。为了展示这些界限的有效性及其应用，我们提供了解决 Fisher 问题的数值结果。我们的发现表明，不动点迭代的收敛速率与预期速率密切吻合，且上界与逆矩阵的无穷范数之间差异极小。具体而言，当 $b > 2$ 且 $\widetilde{b} \leq 1$ 以及 $b < -2$ 且 $\widetilde{b} \geq -1$ 时，这一观察成立。对于其他情况，所得上界仍有进一步改进的潜力。本研究通过提高迭代收敛速率并减少逆矩阵的计算时间，为不动点迭代的数值分析领域做出了贡献。
英文摘要：
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.
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