An incomplete block-diagonalization approach for evaluating the determinants of bordered k-tridiagonal matrices

作     者：Jia, Ji-Teng Wang, Jie Yuan, Ting-Feng Zhang, Kai-Kai Zhong, Bao-Ming 

作者机构：Xidian Univ Sch Math & Stat Xian 710071 Peoples R China 

出 版 物：《JOURNAL OF MATHEMATICAL CHEMISTRY》 (数学化学杂志)

年 卷 期：2022年第60卷第8期

页      面：1658-1673页

核心收录：

学科分类：07[理学] 0703[理学-化学] 0701[理学-数学] 

基　　金：Natural Science Foundation of China (NSFC) Fundamental Research Funds for the Central Universities [JB210720] 

主　　题：Bordered tridiagonal matrices k-Tridiagonal matrices Determinants Block-diagonalizations Breakdown-free algorithm 

摘      要：In this paper, we present an efficient numerical algorithm for evaluating the determinants of general bordered k-tridiagonal matrices in linear time. The algorithm is based on a novel incomplete block-diagonalization (IBD) approach which preserves the low-rank structure and sparsity of the original matrix, and a reliable algorithm for the determinants of general bordered tridiagonal matrices. An advantage of the proposed algorithm is that there is no division operation involved and hence the algorithm will never suffer from breakdown. Moreover, the algorithm theoretically produces exact values of the determinants when all entries of the bordered k-tridiagonal matrix are given in integer. Some numerical results with simulations in Matlab implementation are provided in order to illustrate the accuracy and effectiveness of the proposed algorithm, and its competitiveness with Gaussian elimination algorithm and MATLAB built-in function.