Fast solvers for tridiagonal Toeplitz linear systems
Zhongyun Liu, Shan Li, Yin Yi, Yulin Zhang
Abstract
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.