Abstract
The Generalized Minimal Residual method (GMRES) is often used to solve a nonsymmetric linear system Ax = b. But its convergence analysis is a rather difficult task in general. A commonly used approach is to diagonalize A = XΛ X −1 and then separate the study of GMRES convergence behavior into optimizing the condition number of X and a polynomial minimization problem over A’s spectrum. This artificial separation could greatly overestimate GMRES residuals and likely yields error bounds that are too far from the actual ones. On the other hand, considering the effects of both A’s spectrum and the conditioning of X at the same time poses a difficult challenge, perhaps impossible to deal with in general but only possible for certain particular linear systems. This paper will do so for a (nonsymmetric) tridiagonal Toeplitz system. Sharp error bounds on and sometimes exact expressions for residuals are obtained. These expressions and/or bounds are in terms of the three parameters that define A and Chebyshev polynomials of the first kind.
Similar content being viewed by others
References
Beckermann B., Goreinov S.A., Tyrtyshnikov E.E.: Some remarks on the Elman estimate for GMRES. SIAM J. Matrix Anal. Appl. 27, 772–778 (2006)
Beckermann B., Kuijlaars A.B.J.: Superlinear CG convergence for special right-hand sides. Electron. Trans. Numer. Anal. 14, 1–19 (2002)
Demmel J.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)
Driscoll T.A., Toh K.-C., Trefethen L.N.: From potential theory to matrix iterations in six steps. SIAM Rev. 40, 547–578 (1998)
Eiermann M.: Fields of values and iterative methods. Linear Algebra Appl. 180, 167–197 (1993)
Eiermann M., Ernst O.G.: Geometric aspects in the theory of Krylov subspace methods. Acta Numer. 10, 251–312 (2001)
Eisenstat S.C., Elman H.C., Schultz M.H.: Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20, 345–357 (1983)
Elman, H.C.: Iterative Methods for Large, Sparse Nonsymmetric Systems of Linear Equations. Ph.D. thesis, Department of Computer Science, Yale University (1982)
Ernst O.G.: Residual-minimizing Krylov subspace methods for stabilized discretizations of convection-diffusion equations. SIAM J. Matrix Anal. Appl. 21, 1079–1101 (2000)
Greenbaum A.: Iterative Methods for Solving Linear Systems. SIAM, Philadelphia (1997)
Greenbaum A., Pták V., Strakoš Z.: Any nonincreasing convergence curve is possible for GMRES. SIAM J. Matrix Anal. Appl. 17, 465–469 (1996)
Ipsen I.C.F.: Expressions and bounds for the GMRES residual. BIT 40, 524–535 (2000)
Li, R.-C.: Sharpness in rates of convergence for CG and symmetric Lanczos methods, Technical Report 2005-01 (Department of Mathematics, University of Kentucky, 2005). http://www.ms.uky.edu/~math/MAreport/
Li R.-C.: Convergence of CG and GMRES on a tridiagonal Toeplitz linear system. BIT 47, 577–599 (2007)
Li R.-C.: Hard cases for conjugate gradient method. Int. J. Info. Sys. Sci. 4, 15–29 (2008)
Li, R.-C.: On Meinardus’ examples for the conjugate gradient method. Math. Comp. 77, 335–352 (2008) (Electronically published on September 17, 2007)
Liesen J., Rozlozník M., Strakoš Z.: Least squares residuals and minimal residual methods. SIAM J. Sci. Comput. 23, 1503–1525 (2002)
Liesen J., Strakoš Z.: Convergence of GMRES for tridiagonal Toeplitz matrices. SIAM J. Matrix Anal. Appl. 26, 233–251 (2004)
Liesen J., Strakoš Z.: GMRES convergence analysis for a convection-diffusion model problem. SIAM J. Sci. Comput. 26, 1989–2009 (2005)
Saad Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)
Saad Y., Schultz M.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)
Smith G.D.: Numerical Solution of Partial Differential Equations, 2nd edn. Clarendon Press, Oxford (1978)
Trefethen, L.N., Pseudospectra of matrices. In: Griffiths, D.F., Watson, G.A. (eds.) Numerical Analysis 1991: Proceedings of the 14th Dundee Conference, June, 1991. Research Notes in Mathematics Series. Longman Press, London (1992)
Trefethen L.N., Bau D. III: Numerical Linear Algebra. SIAM, Philadelphia (1997)
Zavorin I., O’Leary D.P., Elman H.: Complete stagnation of GMRES. Linear Algebra Appl. 367, 165–183 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, RC., Zhang, W. The rate of convergence of GMRES on a tridiagonal Toeplitz linear system. Numer. Math. 112, 267–293 (2009). https://doi.org/10.1007/s00211-008-0206-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-008-0206-2