The upper and lower bounds for generalized minimal residual method on a tridiagonal Toeplitz linear system

Abstract

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 $N\times N$ 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.

Keywords:

• GMRES
• tridiagonal Toeplitz matrix
• Krylov subspace
• Chebyshev polynomial
• linearsystem

2000 AMS Subject Classification::

• 65F10

Disclosure statement

No potential conflict of interest was reported by the author(s).