Generalized Sherman–Morrison–Woodbury formula based algorithm for the inverses of opposite-bordered tridiagonal matrices

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(n2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n^2)$$\end{document} 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.