Convergence of GMRES for Tridiagonal Toeplitz Matrices

Abstract. We analyze the residuals of GMRES [9], when the method is applied to tridiagonal Toeplitz matrices. We first derive formulas for the residuals as well as their norms when GMRES is applied to scaled Jordan blocks. This problem has been studied previously by Ipsen [5], Eiermann and Ernst [2], but we formulate and prove our results in a different way. We then extend the (lower) bidiagonal Jordan blocks to tridiagonal Toeplitz matrices and study extensions of our bidiagonal analysis to the tridiagonal case. Intuitively, when a scaled Jordan block is extended to a tridiagonal Toeplitz matrix by a superdiagonal of small modulus (compared to the modulus of the subdiagonal), the GMRES residual norms for both matrices and the same initial residual should be close to each other. We confirm and quantify this intuitive statement. We also demonstrate principal difficulties of any GMRES convergence analysis which is based on eigenvector expansion of the initial residual when the eigenvector matrix is ill-conditioned. Such analyses are complicated by a cancellation of possibly huge components due to close eigenvectors, which can prevent achieving well-justified conclusions.