Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T17:07:31.766Z Has data issue: false hasContentIssue false
  • English
  • Français

A Convergence Analysis of the MINRES Method for Some Hermitian Indefinite Systems

Part of: Numerical linear algebra Numerical analysis: Ordinary differential equations Partial differential equations, boundary value problems

Published online by Cambridge University Press:  31 January 2018

Ze-Jia Xie ,
Xiao-Qing Jin  and
Zhi Zhao
Ze-Jia Xie*
Affiliation:
Department of Mathematics and Data Science, Dongguan University of Technology, Dongguan, 523808, China
Xiao-Qing Jin*
Affiliation:
Department of Mathematics, University of Macau, Macao
Zhi Zhao*
Affiliation:
Department of Mathematics, School of Sciences, Hangzhou Dianzi University, Hangzhou, 310018, China
*
*Corresponding author. Email addresses:xiezejia2012@gmail.com (Z.-J. Xie), xqjin@umac.mo (X.-Q. Jin), zhaozhi231@163.com (Z. Zhao)
*Corresponding author. Email addresses:xiezejia2012@gmail.com (Z.-J. Xie), xqjin@umac.mo (X.-Q. Jin), zhaozhi231@163.com (Z. Zhao)
*Corresponding author. Email addresses:xiezejia2012@gmail.com (Z.-J. Xie), xqjin@umac.mo (X.-Q. Jin), zhaozhi231@163.com (Z. Zhao)
Get access

Abstract

Some convergence bounds of the minimal residual (MINRES) method are studied when the method is applied for solving Hermitian indefinite linear systems. The matrices of these linear systems are supposed to have some properties so that their spectra are all clustered around ±1. New convergence bounds depending on the spectrum of the coefficient matrix are presented. Some numerical experiments are shown to demonstrate our theoretical results.

Keywords

MINRESConvergence boundHermitian indefiniteToeplitz system

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65F15: Eigenvalues, eigenvectors 65L05: Initial value problems 65N22: Solution of discretized equations
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 4 , November 2017 , pp. 827 - 836
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Axelsson, O., A class of iterative methods for finite element equations, Comput. Methods Appl. Mech. Engrg. 9, 123–127 (1976).Google Scholar
[2] Axelsson, O., Lindskog, G., On the rate of convergence of the preconditioned conjugate gradient method, Numer. Math. 48, 499–523 (1986).Google Scholar
[3] Axelsson, O., Solution of linear systems of equations: Iterative methods, In Barker, V., editor, Sparse Matrix Techniques, v. 572 of Lecture Notes in Mathematics, Springer, 1997, pp. 1–51.Google Scholar
[4] Bai, Z., Jin, X., Yao, T., Superoptimal preconditioners for functions of matrices, Numerical Mathematics: Theory, Methods and Applications 8, 515–529 (2015).Google Scholar
[5] Beckermann, B., Kuijlaars, A., Superlinear convergence of conjugate gradients, SIAM J. Numer. Anal. 39, 300–329 (2001).Google Scholar
[6] Beckermann, B., Discrete orthogonal polynomials and superlinear convergence of Krylov sub-space methods in numerical linear algebra, In Orthogonal Polynomials and Special Functions, Springer, 2006.Google Scholar
[7] Campbell, S., Ipsen, I., Kelley, C., Meyer, C., GMRES and the minimal polynomial, BIT 36, 664–675 (1996).Google Scholar
[8] Chan, T., An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Stat. Comput. 9, 766–771 (1988).Google Scholar
[9] Chan, R., Ng, M., Conjugate gradient methods for Toeplitz systems, SIAM Rev. 38, 427–482 (1996).Google Scholar
[10] Chan, R., Potts, D., Steidl, G., Preconditioners for nondefinite Hermitian Toeplitz systems, SIAM J. Matrix Anal. Appl. 22, 647–665 (2000).Google Scholar
[11] Chan, R., Jin, X., An Introduction to Iterative Toeplitz Solvers, SIAM, Philadelphia, 2007.Google Scholar
[12] Elman, H., Silvester, D., Wathen, A., Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, Oxford University Press, Oxford, 2005.Google Scholar
[13] Fischer, B., Polynomial Based Iteration Methods for Symmetric Linear Systems, Willey and Teubner, Chichester, West Essex, England, and Stuttgart, 1996.Google Scholar
[14] Gergelits, T., Strakoš, Z., Composite convergence bounds based on Chebyshev polynomials and finite precision conjugate gradient computations, Numer. Algor. 65, 759–782 (2014).Google Scholar
[15] Greenbaum, A., Iterative Methods for Solving Linear Systems, SIAM, Philadelphia, 1997.Google Scholar
[16] Herzog, R., Sachs, E., Superlinear convergence of Krylov subspace methods for self-adjoint problems in Hilbert space, SIAM J. Numer. Anal. 53, 1304–1324 (2015).Google Scholar
[17] Jennings, A., Influence of the eigenvalue spectrum on the convergence rate of the conjugate gradient method, J. Inst. Math. Appl. 20, 61–72 (1977).Google Scholar
[18] Jin, X., Preconditioning Techniques for Toeplitz Systems, Higher Education Press, Beijing, 2010.Google Scholar
[19] Jin, X., Wei, Y., Zhao, Z., Numerical Linear Algebra and Its Applications, 2nd edition, Science Press, Beijing, 2015.Google Scholar
[20] Jin, X., Vong, S., An Introduction to Applied Matrix Analysis, Higher Education Press, Beijing, and World Scientific Publishing, Singapore, 2016.Google Scholar
[21] Lei, S., Sun, H., A circulant preconditioner for fractional diffusion equations, J. Comput. Phys. 242, 715–725 (2013).Google Scholar
[22] Lin, F., Yang, S., Jin, X., Preconditioned iterative methods for fractional diffusion equation, J. Comput. Phys. 256, 109–117 (2014).Google Scholar
[23] Moret, I., A note on the superlinear convergence of GMRES, SIAM J. Numer. Anal. 34, 513–516 (1997).Google Scholar
[24] Ng, M., Iterative Methods for Toeplitz Systems, Oxford University Press, Oxford, UK, 2004.Google Scholar
[25] Olshanskii, M., Tyrtyshnikov, E., Iterative Methods for Linear Systems: Theory and Applications, SIAM, Philadelphia, 2014.Google Scholar
[26] Paige, C., Saunders, M., Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal. 12, 617–629 (1975).Google Scholar
[27] Pestana, J., Wathen, A., A preconditioned MINRES method for nonsymmetric Toeplitz matrices, SIAM J. Matrix Anal. Appl. 36, 273–288 (2015).Google Scholar
[28] Simoncini, V., Szyld, D., On the occurrence of superlinear convergence of exact and inexact Krylov subspace methods, SIAM Rev. 47, 247–272 (2005).Google Scholar
[29] Simoncini, V., Szyld, D., On the superlinear convergence of MINRES, In Numerical Mathematics and Advanced Applications 2011, Springer, 2013.Google Scholar
[30] Sleijpen, G., van der Sluis, A., Further results on the convergence behavior of conjugate-gradients and Ritz values, Linear Algebra Appl. 246, 233–278 (1996).Google Scholar
[31] Strang, G., A proposal for Toeplitz matrix calculations, Stud. Appl. Math. 74, 171–176 (1986).Google Scholar
[32] Tyrtyshnikov, E., Optimal and superoptimal circulant preconditioners, SIAMJ.Matrix Anal. Appl. 13, 459–473 (1992).Google Scholar
[33] van der Sluis, A., van der Vorst, H., The rate of convergence of conjugate gradients, Numer. Math. 48, 543–560 (1986).Google Scholar
[34] van der Vorst, H., Vuik, C., The superlinear convergence behaviour of GMRES, J. Comput. Appl. Math. 48, 327–341 (1993).Google Scholar
[35] Wathen, A., Preconditioning, Acta Numer. 24, 329–376 (2015).Google Scholar
[36] Winther, R., Some superlinear convergence results for the conjugate gradient method, SIAM J. Numer. Anal. 17, 14–17 (1980).Google Scholar