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The role eigenvalues play in forming GMRES residual norms with non-normal matrices

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Abstract

In this paper we give explicit expressions for the norms of the residual vectors generated by the GMRES algorithm applied to a non-normal matrix. They involve the right-hand side of the linear system, the eigenvalues, the eigenvectors and, in the non-diagonalizable case, the principal vectors. They give a complete description of how eigenvalues contribute in forming residual norms and offer insight in what quantities can prevent GMRES from being governed by eigenvalues.

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References

  1. Arioli, M., Pták, V., Strakoš, Z.: Krylov sequences of maximal length and convergence of GMRES. BIT 38(4), 636–643 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baglama, J., Calvetti, D., Golub, G.H., Reichel, L.: Adaptively preconditioned GMRES algorithms. SIAM J. Sci. Comput. 20(1), 243–269 (1998)

    Article  MathSciNet  Google Scholar 

  3. Bellalij, M., Jbilou, K., Sadok, H.: New convergence results on the global GMRES method for diagonalizable matrices. J. Comput. Appl. Math. 219, 350–358 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Campbell, S.L., Ipsen, I.C.F., Kelley, C.T., Meyer, C.D.: GMRES and the minimal polynomial. BIT 36(4), 664–675 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Carpentieri, B., Duff, I.S., Giraud, L.: A class of spectral two-level preconditioners. SIAM J. Sci. Comput. 25(2), 749–765 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Carpentieri, B., Giraud, L., Gratton, S.: Additive and multiplicative two-level spectral preconditioning for general linear systems. SIAM J. Sci. Comput. 29(4), 1593–1612 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chapman, A., Saad, Y.: Deflated and augmented Krylov subspace techniques. Numer. Linear Algebra Appl. 4(1), 43–66 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Duintjer Tebbens, J., Meurant, G.: Any Ritz value behavior is possible for Arnoldi and for GMRES. SIAM J. Matrix Anal. Appl. 33(3), 958–978 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Duintjer Tebbens, J., Meurant, G.: Prescribing the behavior of early terminating GMRES and Arnoldi iterations. Num. Alg. 65(1), 69–90 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  10. Duintjer Tebbens, J., Meurant, G., Sadok, H., Strakoš, Z.: On investigating GMRES convergence using unitary matrices. Lin. Alg. Appl. 450, 83–107 (2014)

    Article  MATH  Google Scholar 

  11. Eiermann, M.: Fields of values and iterative methods. Lin. Alg. Appl. 180, 167–197 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  12. Erhel, J., Burrage, K., Pohl, B.: Restarted GMRES preconditioned by deflation. J. Comput. Appl. Math. 69(2), 303–318 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gautschi, W.: On inverses of Vandermonde and confluent Vandermonde matrices. III. Numer. Math. 29, 445–450 (1977/78)

    Article  MathSciNet  Google Scholar 

  14. Giraud, L., Gratton, S., Martin, E.: Incremental spectral preconditioners for sequences of linear systems. Appl. Numer. Math. 57(11–12), 1164–1180 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Giraud, L., Gratton, S., Pinel, X., Vasseur, X.: Flexible GMRES with deflated restarting. SIAM J. Sci. Comput. 32(4), 1858–1878 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. Greenbaum, A.: Generalizations of the field of values useful in the study of polynomial functions of a matrix. Lin. Alg. Appl. 347, 233–249 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Greenbaum, A., Pták, V., Strakoš, Z.: Any nonincreasing convergence curve is possible for GMRES. SIAM J. Matrix Anal. Appl. 17(3), 465–469 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Greenbaum, A., Strakoš, Z.: Matrices that generate the same Krylov residual spaces. In: Recent Advances in Iterative Methods, volume 60 of IMA Vol. Math. Appl., pp. 95–118. Springer, New York (1994)

    Google Scholar 

  19. Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Research Nat. Bur. Standards 49, 409–436 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  20. Huhtanen, M., Nevanlinna, O.: Minimal decompositions and iterative methods. Numer. Math. 86(2), 257–281 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ipsen, I.C.F.: Expressions and bounds for the GMRES residual. BIT 40(3), 524–535 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kharchenko, S.A., Yu. Yeremin, A.: Eigenvalue translation based preconditioners for the GMRES (k) method. Numer. Linear Algebra Appl. 2(1), 51–77 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kuijlaars, A.B.J.: Convergence analysis of Krylov subspace iterations with methods from potential theory. SIAM Rev. 48(1), 3–40 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Le Calvez, C, Molina, B.: Implicitly restarted and deflated GMRES. Numer. Algorithms 21(1–4), 261–285 (1999). Numerical methods for partial differential equations (Marrakech, 1998)

    Article  MATH  MathSciNet  Google Scholar 

  25. Liesen, J., Rozložník, M., Strakoš, Z.: Least squares residuals and minimal residual methods. SIAM J. Sci. Comput. 23(5), 1503–1525 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  26. Liesen, J., Strakoš, Z.: Convergence of GMRES for tridiagonal Toeplitz matrices. SIAM J. Matrix Anal. Appl. 26(1), 233–251 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. Liesen, J., Tichý, P.: Convergence analysis of Krylov subspace methods. GAMM Mitt. Ges. Angew Math. Mech. 27(2), 153–173 (2004)

    MATH  MathSciNet  Google Scholar 

  28. Liesen, J., Tichý, P.: The worst-case GMRES for normal matrices. BIT 44(1), 79–98 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Loghin, D., Ruiz, D., Touhami, A.: Adaptive preconditioners for nonlinear systems of equations. J. Comput. Appl. Math. 189(1–2), 362–374 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  30. Morgan, R.B.: A restarted GMRES method augmented with eigenvectors. SIAM J. Matrix Anal. Appl. 16(4), 1154–1171 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  31. Morgan, R.B.: Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations. SIAM J. Matrix Anal. Appl. 21(4), 1112–1135 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  32. Morgan, R.B.: GMRES with deflated restarting. SIAM J. Sci. Comput. 24(1), 20–37 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  33. Nachtigal, N.M., Reddy, S.C., Trefethen, L.N.: How fast are nonsymmetric matrix iterations? SIAM J. Matrix Anal. Appl. 13(3), 778–795 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  34. Paige, C.C., Saunders, M.A.: LSQR: An algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. 8(1), 43–71 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  35. Parks, M.L., de Sturler, E., Mackey, G., Johnson, D.D., Maiti, S.: Recycling Krylov subspaces for sequences of linear systems. SIAM J. Sci. Comput. 28(5), 1651–1674 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  36. Pestana, J., Wathen, A.: On choice of preconditioner for minimum residual methods for non-hermitian matrices. J. Comput. Appl. Math. 249, 57–68 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  37. Saad, Y.: Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics, 2nd edn. Philadelphia (2003)

  38. Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  39. Sadok, H.: Analysis of the convergence of the minimal and the orthogonal residual methods. Numer. Algorithms 40(2), 201–216 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  40. Stewart, G.W.: Collinearity and least squares regression. Stat. Sci. 2(1), 68–100 (1987)

    Article  Google Scholar 

  41. Stynes, M.: Steady-state convection-diffusion problems. Acta Numer. 14, 445–508 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  42. Tichý, P., Liesen, J., Faber, V.: On worst-case GMRES, ideal GMRES, and the polynomial numerical hull of a Jordan block. Electron. Trans. Numer. Anal. 26, 453–473 (2007)

    MATH  MathSciNet  Google Scholar 

  43. Titley-Peloquin, D., Pestana, J., Wathen, A.: GMRES convergence bounds that depend on the right-hand side vector. IMA J. Numer. Anal. 34(2), 462–479 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  44. Trefethen, L.N., Embree, M.: Spectra and Pseudospectra. Princeton University Press, Princeton (2005)

    MATH  Google Scholar 

  45. Zítko, J.: Generalization of convergence conditions for a restarted GMRES. Numer. Linear Algebra Appl. 7, 117–131 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. 30 rue du sergent Bauchat, 75012, Paris, France

    Gérard Meurant

  2. Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou věží 2, 18 207 Praha 8, Libeň, Prague, Czech Republic

    Jurjen Duintjer Tebbens

  3. Faculty of Pharmacy in Hradec Králové, Charles University in Prague, Heyrovského 1203, 500 05, Hradec Králové, Czech Republic

    Jurjen Duintjer Tebbens

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  1. Gérard Meurant
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Correspondence to Jurjen Duintjer Tebbens.

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Meurant, G., Duintjer Tebbens, J. The role eigenvalues play in forming GMRES residual norms with non-normal matrices. Numer Algor 68, 143–165 (2015). https://doi.org/10.1007/s11075-014-9891-3

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