The Upper Bound for GMRES on Normal Tridiagonal Toeplitz Linear System

The Generalized Minimal Residual method (GMRES) is often used to solve a large and sparse system Ax = b. This paper establishes error bound for residuals of GMRES on solving an N × N normal tridiagonal Toeplitz linear system. This problem has been studied previously by Li [R.-C. Li, Convergence of CG and GMRES on a tridiagonal Toeplitz linear system, BIT 47 (3) (2007) 577-599.], for two special right-hand sides b = e1 , eN . Also, Li and Zhang [R.-C. Li, W. Zhang, The rate of convergence of GMRES on a tridiagonal Toeplitz linear system, Numer. Math. 112 (2009) 267-293.] for non-symmetric matrix A, presented upper bound for GMRES residuals. But in this paper we establish the upper bound on normal tridiagonal Toeplitz linear systems for special right-hand sides b = b(l)el, for 1  l  N .