Let A be a tridiagonal Toeplitz matrix denoted by \(A = {\text {Tritoep}} (\beta , \alpha , \gamma )\). The matrix A is said to be: strictly diagonally dominant if \(|\alpha |>|\beta |+|\gamma |\), weakly diagonally dominant if \(|\alpha |\ge |\beta |+|\gamma |\), subdiagonally dominant if \(|\beta |\ge |\alpha |+|\gamma |\), and superdiagonally dominant if \(|\gamma |\ge |\alpha |+|\beta |\). In this paper, we consider the solution of a tridiagonal Toeplitz system \(A\mathbf {x}= \mathbf {b}\), where A is subdiagonally dominant, superdiagonally dominant, or weakly diagonally dominant, respectively. We first consider the case of A being subdiagonally dominant. We transform A into a block \(2\times 2\) matrix by an elementary transformation and then solve such a linear system using the block LU factorization. Compared with the LU factorization method with pivoting, our algorithm takes less flops, and needs less memory storage and data transmission. In particular, our algorithm outperforms the LU factorization method with pivoting in terms of computing efficiency. Then, we deal with superdiagonally dominant and weakly diagonally dominant cases, respectively. Numerical experiments are finally given to illustrate the effectiveness of our algorithms.

设A为\（A = {\ text {Tritoep}}（\ beta，\ alpha，\ gamma）\）表示的三对角Toeplitz矩阵。矩阵A据说是：如果\（| \ alpha |> | \ beta | + | \γ| \）严格对角占优，如果\（| \ alpha | \ ge | \ beta | + | \ γ| \），如果\（| \ beta | \ ge | \ alpha | + | \ gamma | \）是对角线占优的，如果\（| \γ| \ ge | \ alpha | + | \ beta | \ ）。在本文中，我们考虑三对角Toeplitz系统\（A \ mathbf {x} = \ mathbf {b} \）的解，其中A分别是对角下优势，超对角优势或弱对角优势。我们首先考虑A对角线占优势的​​情况。我们通过基本变换将A转换成一个块\（2 × 2 \）矩阵，然后使用块LU分解求解这样的线性系统。与采用透视的LU分解方法相比，我们的算法所需的触发器更少，所需的存储器存储和数据传输也更少。特别是，我们的算法在计算效率方面具有枢纽优势，胜过LU分解方法。然后，我们分别处理超对角占优和弱对角占优的情况。最后通过数值实验说明了算法的有效性。