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Inexact GMRES for singular linear systems

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Abstract

Inexact Krylov subspace methods have been shown to be practical alternatives for the solution of certain linear systems of equations. In this paper, the solution of singular systems with inexact matrix-vector products is explored. Criteria are developed to prescribe how inexact the matrix-vector products can be, so that the computed residual remains close to the true residual, thus making the inexact method of practical applicability. Cases are identified for which the methods work well, and this is the case in particular for systems representing certain Markov chains. Numerical experiments illustrate the effectiveness of the inexact approach.

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References

  1. V. A. Barker, Numerical solutions of sparse singular systems of equations arising from ergodic Markov chains, Commun. Stat. Stochastic Models, 5 (1989), pp. 335–381.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Benzi and B. Uçar, Block triangular preconditioners for M-matrices and Markov chains, Electron. Trans. Numer. Anal., 26 (2007), pp. 209–227.

    MathSciNet  MATH  Google Scholar 

  3. Å. Björck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996.

  4. A. Bouras and V. Frayssé, Inexact matrix-vector products in Krylov methods for solving linear systems: a relaxation strategy, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 660–678.

    Article  MATH  MathSciNet  Google Scholar 

  5. P. Brown and H. Walker, GMRES on (nearly) singular systems, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 37–51.

    Article  MATH  MathSciNet  Google Scholar 

  6. D. Calvetti, B. W. Lewis, and L. Reichel, GMRES-type methods for inconsistent systems, Linear Algebra Appl., 316 (2000), pp. 157–169.

    Article  MATH  MathSciNet  Google Scholar 

  7. Z. H. Cao and M. Wang, A note on Krylov subspace methods for singular systems, Linear Algebra Appl., 350 (2002), pp. 285–288.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Van den Eshof and G. Sleijpen, Inexact Krylov subspace methods for linear systems, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 125–153.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. W. Freund and M. Hochbruck, On the use of two QMR algorithms for solving singular systems and applications in Markov chain modeling, Numer. Linear Algebra Appl., 1 (1994), pp. 403–420.

    Article  MATH  MathSciNet  Google Scholar 

  10. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, NY, 1999.

    Google Scholar 

  11. G. Karypis and V. Kumar, Metis: A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices version 4.0, 1998.

  12. C. C. Paige and Z. Strakǒs, Scaled total least squares fundamentals, Numer. Math., 91 (2002), pp. 117–146.

    Article  MATH  MathSciNet  Google Scholar 

  13. B. Philippe, Y. Saad, and W. J. Stewart, Numerical methods in Markov chain modeling, Oper. Res., 40 (1992), pp. 1156–1179.

    Article  MATH  Google Scholar 

  14. L. Reichel and Q. Ye, Breakdown-free GMRES for singular systems, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 1001–1021.

    Article  MATH  MathSciNet  Google Scholar 

  15. Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edn., The PWS Publishing Company, Boston, USA 1996, SIAM, Philadelphia, PA, 2003.

  16. Y. Saad and M. H. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, J. Sci. Stat. Comput., 7 (1986), pp. 856–869.

    Article  MATH  MathSciNet  Google Scholar 

  17. V. Simoncini and D. B. Szyld, Theory of inexact Krylov subspace methods and applications to scientific computing, SIAM J. Sci. Comput., 25 (2003), pp. 454–477.

  18. V. Simoncini and D. B. Szyld, Recent computational developments in Krylov subspace methods for linear systems, Numer. Linear Algebra Appl., 14 (2007), pp. 1–59.

    Article  MathSciNet  MATH  Google Scholar 

  19. L. Smoch, Some results about GMRES in the singular case, Numer. Algorithms, 22 (1999), pp. 193–212.

    Article  MATH  MathSciNet  Google Scholar 

  20. W. J. Stewart, Introduction to the Numerical Solution of Markov Chains, Princeton University Press, Princeton, NJ, 1994.

    MATH  Google Scholar 

  21. W. J. Stewart, Marca models: A collection of Markov chain models, Available at http://www4.ncsu.edu/∼billy/MARCA_Models/MARCA_Models.html (accessed August 2006).

  22. Y. Wei and H. Wu, Convergence properties of Krylov subspace methods for singular linear systems with arbitrary index, J. Comput. Appl. Math., 114 (2000), pp. 305–318.

    Article  MATH  MathSciNet  Google Scholar 

  23. J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, UK, 1965.

    MATH  Google Scholar 

  24. N. Zhang and Y. Wei, A note on solving EP inconsistent linear systems, Appl. Math. Comput., 169 (2005), pp. 8–15.

    Article  MATH  MathSciNet  Google Scholar 

  25. N. Zhang and Y. Wei, Solving EP singular linear systems, Int. J. Comput. Math., 81 (2005), pp. 1395–1405.

    Article  MathSciNet  Google Scholar 

  26. J. Zhou and Y. Wei, DFOM algorithm and error analysis for projection methods for solving singular linear system, Appl. Math. Comput., 157 (2004), pp. 313–329.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Temple University (038-16), 1805 N. Broad Street, Philadelphia, Pennsylvania, 19122-6094, USA

    Xiuhong Du & Daniel B. Szyld

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  1. Xiuhong Du
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  2. Daniel B. Szyld
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Correspondence to Daniel B. Szyld.

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AMS subject classification (2000)

65F10

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Du, X., Szyld, D. Inexact GMRES for singular linear systems . Bit Numer Math 48, 511–531 (2008). https://doi.org/10.1007/s10543-008-0171-2

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  • DOI: https://doi.org/10.1007/s10543-008-0171-2

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