The upper and lower bounds for generalized minimal residual method on a tridiagonal Toeplitz linear system
Reza Doostaki, Hossein Sadeghi Goughery · International Journal of Computer Mathematics · 2015
The generalized minimal residual (GMRES) method is widely used to solve a linear system Ax=b. This paper establishes upper and lower bounds for GMRES residuals for solving an N×N tridiagonal Toeplitz linear system. For normal matrix A, this problem has been studied previously by Li [Convergence of CG and GMRES on a tridiagonal Toeplitz linear system, BIT 47(3) (2007), 577–599.]. Also, Li and Zhang [The rate of convergence of GMRES on a tridiagonal Toeplitz linear system, Numer. Math. 112 (2009), pp. 267–293.] for non-symmetric matrix A, presented upper bound for GMRES residuals. In fact, our main goal in this paper is to find the upper and lower bounds for GMRES residuals on normal tridiagonal Toeplitz linear systems, and lower bounds for residuals of GMRES on solving non-normal tridiagonal Toeplitz linear systems.