Numerical algorithms for the fast and reliable solution of periodic tridiagonal Toeplitz linear systems

In the present paper, we consider the solution of periodic tridiagonal Toeplitz linear systems. By exploiting the special low-rank and Toeplitz structure, we give a new structure-preserving decomposition form of the coefficient matrix. Based on this matrix decomposition form and combined with elementary column operations, we give different algorithms for the fast and reliable solutions of three types of periodic tridiagonal Toeplitz linear systems. Compared with the \(\varvec{LU}\) $\mathit{LU}$ decomposition method, our proposed algorithms take less computational costs and need less memory storage and data transmission. Numerical experiments are given to demonstrate the accuracy and efficiency of our algorithms. All of the experiments are performed on a computer with the aid of programs written in MATLAB.