Fast numerical algorithms for solving opposite-bordered tridiagonal Toeplitz linear systems and their applications

This paper presents two fast numerical algorithms for solving opposite-bordered tridiagonal Toeplitz linear systems. Both algorithms are designed to solve a system of $n$ equations in linear time. The first algorithm uses a block $2\times 2$-LU factorization combined with a fast approach for solving upper quasi-triangular Toeplitz systems. The second algorithm applies a splitting technique to the opposite-bordered tridiagonal Toeplitz matrix, along with a fast algorithm for solving tridiagonal Toeplitz systems. The effectiveness of the proposed algorithms is demonstrated through numerical experiments.