Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

IMPACT FACTOR 2018: 1.218
5-year IMPACT FACTOR: 1.411

CiteScore 2018: 1.42

SCImago Journal Rank (SJR) 2018: 0.947
Source Normalized Impact per Paper (SNIP) 2018: 0.939

Mathematical Citation Quotient (MCQ) 2018: 1.22

See all formats and pricing
More options …
Volume 18, Issue 2


GMRES Convergence Bounds for Eigenvalue Problems

Melina A. Freitag / Patrick Kürschner
  • Corresponding author
  • Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstraße 1, 39106 Magdeburg, Germany
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Jennifer Pestana
Published Online: 2017-06-07 | DOI: https://doi.org/10.1515/cmam-2017-0017


The convergence of GMRES for solving linear systems can be influenced heavily by the structure of the right-hand side. Within the solution of eigenvalue problems via inverse iteration or subspace iteration, the right-hand side is generally related to an approximate invariant subspace of the linear system. We give detailed and new bounds on (block) GMRES that take the special behavior of the right-hand side into account and explain the initial sharp decrease of the GMRES residual. The bounds motivate the use of specific preconditioners for these eigenvalue problems, e.g., tuned and polynomial preconditioners, as we describe. The numerical results show that the new (block) GMRES bounds are much sharper than conventional bounds and that preconditioned subspace iteration with either a tuned or polynomial preconditioner should be used in practice.

Keywords: GMRES; Convergence Analysis; Inexact Inverse Iteration; Inexact Subspace Iteration; Krylov Subspace Methods; Block Krylov Methods; Preconditioning

MSC 2010: 15A18; 65F08; 65F10; 65F15; 65N25


  • [1]

    M. I. Ahmad, D. B. Szyld and M. B. van Gijzen, Preconditioned multishift BiCG for 2-optimal model reduction, SIAM J. Matrix Anal. Appl. 38 (2017), no. 2, 401–424. Google Scholar

  • [2]

    M. Arioli, V. Pták and Z. Strakoš, Krylov sequences of maximal length and convergence of GMRES, BIT 38 (1998), 636–643. CrossrefGoogle Scholar

  • [3]

    S. F. Ashby, T. A. Manteuffel and J. S. Otto, A comparison of adaptive Chebyshev and least squares polynomial preconditioning for Hermitian positive definite linear systems, SIAM J. Sci. Statist. Comput. 13 (1992), no. 1, 1–29. CrossrefGoogle Scholar

  • [4]

    R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. A. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed., SIAM, Philadelphia, 1994. Google Scholar

  • [5]

    M. Baumann and M. van Gijzen, Nested Krylov methods for shifted linear systems, SIAM J. Sci. Comput. 37 (2015), 90–112. CrossrefWeb of ScienceGoogle Scholar

  • [6]

    J. Berns-Müller, I. G. Graham and A. Spence, Inexact inverse iteration for symmetric matrices, Linear Algebra Appl. 416 (2006), no. 2, 389–413. CrossrefGoogle Scholar

  • [7]

    L. Du, T. Sogabe and S.-L. Zhang, IDR(s) for solving shifted nonsymmetric linear systems, J. Comput. Appl. Math. 274 (2015), no. 0, 35–43. Web of ScienceCrossrefGoogle Scholar

  • [8]

    J. Duintjer Tebbens and G. Meurant, Prescribing the behavior of early terminating GMRES and Arnoldi iterations, Numer. Algorithms 65 (2014), 69–90. CrossrefWeb of ScienceGoogle Scholar

  • [9]

    M. A. Freitag, Inner-outer iterative methods for eigenvalue problems – Convergence and preconditioning, Ph.D. thesis, University of Bath, 2007. Google Scholar

  • [10]

    M. A. Freitag and A. Spence, Convergence theory for inexact inverse iteration applied to the generalised nonsymmetric eigenproblem, Electron. Trans. Numer. Anal. 28 (2007), 40–64. Google Scholar

  • [11]

    M. A. Freitag and A. Spence, A tuned preconditioner for inexact inverse iteration applied to Hermitian eigenvalue problems, IMA J. Numer. Anal. 28 (2008), no. 3, 522–551. Web of ScienceGoogle Scholar

  • [12]

    M. A. Freitag, A. Spence and E. Vainikko, Rayleigh quotient iteration and simplified Jacobi–Davidson with preconditioned iterative solves for generalised eigenvalue problems, Technical Report, University of Bath, 2008. Google Scholar

  • [13]

    R. Freund, On conjugate gradient type methods and polynomial preconditioners for a class of complex non-hermitian matrices, Numer. Math. 57 (1990), no. 1, 285–312. CrossrefGoogle Scholar

  • [14]

    G. H. Golub and Q. Ye, Inexact inverse iteration for generalized eigenvalue problems, BIT 40 (2000), no. 4, 671–684. CrossrefGoogle Scholar

  • [15]

    I. C. Ipsen, Computing an eigenvector with inverse iteration, SIAM Rev. 39 (1997), no. 2, 254–291. CrossrefGoogle Scholar

  • [16]

    Q. Liu, R. B. Morgan and W. Wilcox, Polynomial Preconditioned GMRES and GMRES-DR, SIAM J. Sci. Comput. 37 (2015), no. 5, S407–S428. Google Scholar

  • [17]

    A. Martinez, Tuned preconditioners for the eigensolution of large SPD matrices arising in engineering problems, Numer. Linear Algebra Appl. 23 (2016), no. 3, 427–443. CrossrefWeb of ScienceGoogle Scholar

  • [18]

    G. Meurant and J. Duintjer Tebbens, The role eigenvalues play in forming GMRES residual norms with non-normal matrices, Numer. Algorithms 68 (2015), 143–165. CrossrefWeb of ScienceGoogle Scholar

  • [19]

    C. C. Paige and M. A. Saunders, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal. 12 (1975), no. 4, 617–629. CrossrefGoogle Scholar

  • [20]

    M. L. Parks, K. M. Soodhalter and D. B. Szyld, A block Recycled GMRES method with investigations into aspects of solver performance, preprint (2016), https://arxiv.org/abs/1604.01713.

  • [21]

    G. Peters and J. H. Wilkinson, Inverse iteration, ill-conditioned equations and Newton’s method, SIAM Rev. 21 (1979), no. 3, 339–360. CrossrefGoogle Scholar

  • [22]

    M. Robbé, M. Sadkane and A. Spence, Inexact inverse subspace iteration with preconditioning applied to non-hermitian eigenvalue problems, SIAM J. Matrix Anal. Appl. 31 (2009), no. 1, 92–113. Web of ScienceCrossrefGoogle Scholar

  • [23]

    Y. Saad, A Flexible Inner-Outer Preconditioned GMRES Algorithm, SIAM J. Sci. Comput. 14 (1993), no. 2, 461–469. CrossrefGoogle Scholar

  • [24]

    Y. Saad and M. Schultz, GMRES a generalised minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986), 856–869. CrossrefGoogle Scholar

  • [25]

    V. Simoncini and L. Eldén, Inexact Rayleigh quotient-type methods for eigenvalue computations, BIT 42 (2002), no. 1, 159–182. CrossrefGoogle Scholar

  • [26]

    K. Soodhalter, A block MINRES algorithm based on the banded Lanczos method, Numer. Algorithms 69 (2015), 473–494. CrossrefGoogle Scholar

  • [27]

    D. Szyld and F. Xue, Efficient preconditioned inner solves for inexact Rayleigh quotient iteration and their connections to the single-vector Jacobi–Davidson method, SIAM J. Matrix Anal. A 32 (2011), no. 3, 993–1018. Web of ScienceCrossrefGoogle Scholar

  • [28]

    D. Titley-Peloquin, J. Pestana and A. J. Wathen, GMRES convergence bounds that depend on the right-hand-side vector, IMA J. Numer. Anal. 34 (2014), 462–479. CrossrefWeb of ScienceGoogle Scholar

  • [29]

    M. B. van Gijzen, A polynomial preconditioner for the GMRES algorithm, J. Comput. Appl. Math. 59 (1995), no. 1, 91–107. CrossrefGoogle Scholar

  • [30]

    M. B. van Gijzen, G. L. G. Sleijpen and J.-P. M. Zemke, Flexible and multi-shift induced dimension reduction algorithms for solving large sparse linear systems, Numer. Linear Algebra Appl. 22 (2015), no. 1, 1–25. CrossrefWeb of ScienceGoogle Scholar

  • [31]

    F. Xue and H. C. Elman, Fast inexact subspace iteration for generalized eigenvalue problems with spectral transformation, Linear Algebra Appl. 435 (2011), no. 3, 601–622. Web of ScienceCrossrefGoogle Scholar

About the article

Received: 2016-09-19

Revised: 2017-03-03

Accepted: 2017-05-24

Published Online: 2017-06-07

Published in Print: 2018-04-01

Citation Information: Computational Methods in Applied Mathematics, Volume 18, Issue 2, Pages 203–222, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2017-0017.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Rayan Nasser and Miloud Sadkane
Computational Methods in Applied Mathematics, 2019, Volume 0, Number 0

Comments (0)

Please log in or register to comment.
Log in