A novel numerical algorithm for solving linear systems with periodic pentadiagonal Toeplitz coefficient matrices

In the present paper, we mainly consider the direct solution of periodic pentadiagonal Toeplitz linear systems. By exploiting the low-rank and Toeplitz structure of the coefficient matrix, we derive a new matrix decomposition of periodic pentadiagonal Toeplitz matrices. Based on this matrix decomposition form and combined with the Sherman-Morrison-Woodbury formula, we propose an efficient algorithm for numerically solving periodic pentadiagonal Toeplitz linear systems. Furthermore, we present a fast and reliable algorithm for evaluating the determinants of periodic pentadiagonal Toeplitz matrices by a certain type of matrix reordering and partitioning, and linear transformation. Numerical examples are given to demonstrate the performance and effectiveness of our algorithms. All of the experiments are performed on a computer with the aid of programs written in Matlab.