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Expressions and Bounds for the GMRES Residual

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Abstract

Expressions and bounds are derived for the residual norm in GMRES. It is shown that the minimal residual norm is large as long as the Krylov basis is well-conditioned. For scaled Jordan blocks the minimal residual norm is expressed in terms of eigenvalues and departure from normality. For normal matrices the minimal residual norm is expressed in terms of products of relative eigenvalue differences.

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REFERENCES

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

    Google Scholar 

  2. P. Brown, A theoretical comparison of the Arnoldi and GMRES algorithms, SIAM J. Sci. Stat. Comput., 12(1991), pp. 58–78.

    Google Scholar 

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

    Google Scholar 

  4. S. Campbell, I. Ipsen, C. Kelley, and C. Meyer, GMRES and the minimal polynomial, BIT, 36(1996), pp. 664–675.

    Google Scholar 

  5. Z. Cao, A note on the convergence behaviour of GMRES, Appl. Numer. Math., 25 (1997), pp. 13–20.

    Google Scholar 

  6. S. Chandrasekaran and I. Ipsen, Perturbation theory for the solution of systems of linear equations, Research Report 866, Department of Computer Science, Yale University, New Haven, CT, 1991.

  7. S. Chandrasekaran and I. Ipsen, On the sensitivity of solution components in linear systems of equations, SIAM J. Matrix Anal. Appl., 16(1995), pp. 93–112.

    Google Scholar 

  8. T. Driscoll, K. Toh, and L. Trefethen, From potential theory to matrix iterations in six steps, SIAM Rev., 40(1998), pp. 547–578.

    Google Scholar 

  9. V. Faber, W. Joubert, E. Knill, and T. Manteuffel, Minimal residual method stronger than polynomial preconditioning, SIAM J. Matrix Anal. Appl, 17(1996), pp. 707–729.

    Google Scholar 

  10. G. Golub and C. van Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore, MD, 1996.

    Google Scholar 

  11. W. Govaerts and J. Pryce,A singular value inequality for block matrices, Linear Algebra Appl., 125(1989), pp. 141–148.

    Google Scholar 

  12. A. Greenbaum, Comparison of splittings used with the conjugate gradient algorithm, Numer. Math., 33(1979), pp. 181–194.

    Google Scholar 

  13. A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM, Philadelphia, PA, 1997.

  14. A. Greenbaum and L. Gurvits, Max-min properties of matrix factor norms, SIAM J. Sci. Comput., 15(1994), pp. 348–358.

    Google Scholar 

  15. A. Greenbaum, V. Pták, and Z. Strakoš, Any nonincreasing convergence curve is possible for GMRES, SIAM J. Matrix Anal. Appl., 17(1996), pp. 465–469.

    Google Scholar 

  16. A. Greenbaum and L. Trefethen, GMRES/CR and Arnoldi/Lanczos as matrix approximation problems, SIAM J. Sci. Comput., 15(1994), pp. 359–368.

    Google Scholar 

  17. M. Gu and S. Eisenstat, Efficient algorithms for computing a strong rank-revealing QR factorization, SIAM J. Sci. Comput., 17(1996), pp. 848–69.

    Google Scholar 

  18. P. Henrici, Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices, Numer. Math., 4(1962), pp. 24–40.

    Google Scholar 

  19. N. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, 1996.

  20. M. Hochbruck and C. Lubich, Error analysis of Krylov methods in a nutshell, SIAM J. Sci. Comput., 19(1998), pp. 695–701.

    Google Scholar 

  21. I. Ipsen, A different approach to bounding the minimal residual norm in Krylov methods, Tech. Rep. CRSC-TR98-19, Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, 1998. http://www4.ncsu.edu/~ipsen/linsys.html.

  22. I. Ipsen and C. Meyer, The idea behind Krylov methods, Amer. Math. Monthly, 105 (1998), pp. 889–899.

    Google Scholar 

  23. I. Moret, A note on the superlinear convergence of GMRES, SIAM J. Numer. Anal., 34(1997), pp. 513–516.

    Google Scholar 

  24. N. Nachtigal, S. Reddy, and L. Trefethen, How fast are nonsymmetric matrix iterations?, SIAM J. Matrix Anal. Appl., 13(1992), pp. 778–795.

    Google Scholar 

  25. N. Nachtigal, L. Reichel, and L. Trefethen, A hybrid GMRES algorithm for nonsymmetric linear systems, SIAM J. Matrix Anal. Appl., 13(1992), pp. 796–825.

    Google Scholar 

  26. C. Paige, B. Parlett, and H. Van der Vorst, Approximate solutions and eigenvalue bounds from Krylov subspaces, Numer. Linear Algebra, 2(1995), pp. 115–133.

    Google Scholar 

  27. C. Paige and M. Saunders, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12(1975), pp. 617–629.

    Google Scholar 

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

    Google Scholar 

  29. G. Stewart, Collinearity and least squares regression, Statist. Sci., 2(1987), pp. 68–100.

    Google Scholar 

  30. K. Toh, GMRES vs. ideal GMRES, SIAM J. Matrix Anal. Appl., 18(1997), pp. 30–36.

    Google Scholar 

  31. H. Van der Vorst and C. Vuik, The superlinear convergence behaviour of GMRES, J. Comput. Appl. Math., 48(1993), pp. 327–341.

    Google Scholar 

  32. H.Walker and L. Zhou, A simpler GMRES, Numer. Linear Algebra Appl., 1(1994), pp. 571–581.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695-8205, USA.

    I. C. F. Ipsen

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  1. I. C. F. Ipsen
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Ipsen, I.C.F. Expressions and Bounds for the GMRES Residual. BIT Numerical Mathematics 40, 524–535 (2000). https://doi.org/10.1023/A:1022371814205

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