Matrix inverse computation is one of the fundamental mathematical problems of linear algebra and has been widely used in many fields of science and engineering. In this paper, we consider the inverse computation of an opposite-bordered tridiagonal matrix which has attracted much attention in recent years. By exploiting the low-rank structure of the matrix, first we show that an explicit formula for the inverse of the opposite-bordered tridiagonal matrix can be obtained based on the combination of a specific matrix splitting and the generalized Sherman–Morrison–Woodbury formula. Accordingly, a numerical algorithm is outlined. Second, we present a breakdown-free symbolic algorithm of $$O(n^2)$$ O ( n 2 ) for computing the inverse of an n -by- n opposite-bordered tridiagonal matrix, which is based on the use of GTINV algorithm and the generalized symbolic Thomas algorithm. Finally, we have provided the results of some numerical experiments for the sake of illustration.

中文翻译：

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基于广义 Sherman-Morrison-Woodbury 公式的对边三对角矩阵的逆算法

矩阵逆计算是线性代数的基本数学问题之一，已广泛应用于科学和工程的许多领域。在本文中，我们考虑了近年来备受关注的对边三对角矩阵的逆计算。通过利用矩阵的低秩结构，首先我们证明了基于特定矩阵分裂和广义 Sherman-Morrison-Woodbury 公式的组合可以获得对边三对角矩阵的逆的显式公式。因此，概述了数值算法。其次，我们提出了 $$O(n^2)$$O ( n 2 ) 的无故障符号算法，用于计算 n × n 对边三对角矩阵的逆矩阵，它基于使用 GTINV 算法和广义符号 Thomas 算法。最后，为了说明起见，我们提供了一些数值实验的结果。