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Local convergence analysis of several inexact Newton-type algorithms for general nonlinear eigenvalue problems

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Abstract

We study the local convergence of several inexact numerical algorithms closely related to Newton’s method for the solution of a simple eigenpair of the general nonlinear eigenvalue problem \(T(\lambda )v=0\). We investigate inverse iteration, Rayleigh quotient iteration, residual inverse iteration, and the single-vector Jacobi–Davidson method, analyzing the impact of the tolerances chosen for the approximate solution of the linear systems arising in these algorithms on the order of the local convergence rates. We show that the inexact algorithms can achieve the same order of convergence as the exact methods if appropriate sequences of tolerances are applied to the inner solves. We discuss the connections and emphasize the differences between the standard inexact Newton’s method and these inexact algorithms. When the local symmetry of \(T(\lambda )\) is present, the use of a nonlinear Rayleigh functional is shown to be fundamental in achieving higher order of convergence rates. The convergence results are illustrated by numerical experiments.

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Fig. 1

Notes

  1. For a \(k\)th-order matrix polynomial \(T(\mu )\), \({\text{ rev}}(T(\mu ))=\mu ^kT(1/\mu )\) is the reversal of \(T(\mu )\) (see [6]).

References

  1. Antoniou, E.N., Vologiannidis, S.: A new family of companion forms of polynomial matrices. Electron. J. Linear Algebra 11, 78–87 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Bai, Z., Su, Y.: SOAR: a second-order Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 26, 640–659 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berns-Müller, J., Spence, A.: Inexact inverse iteration with variable shift for nonsymmetric generalized eigenvalue problems. SIAM J. Matrix Anal. Appl. 28, 1069–1082 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berns-Müller, J., Graham, I.G., Spence, A.: Inexact inverse iteration for symmetric matrices. Linear Algebra Appl. 416, 389–413 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berns-Müller, J.: Inexact inverse iteration using Galerkin Krylov solvers. PhD thesis, Department of Mathematics, University of Bath, UK (2003)

  6. Betcke, T., Higham, N.J., Mehrmann, V., Schröder, C., Tisseur F.: NLEVP: a collection of nonlinear eigenvalue problems. MIMS EPrint 2010.98, School of Mathematics, University of Manchester (2010)

  7. Betcke, T., Voss, H.: A Jacobi–Davidson type projection method for nonlinear eigenvalue problems. Future Gener. Comput. Syst. 20, 363–372 (2004)

    Article  Google Scholar 

  8. Day, D., Walsh, T.: Quadratic eigenvalue problems. Sandia Report, SAND2007-2072, Sandia National Lab (2007)

  9. Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19, 400–408 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dennis, J.E., Jr., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Classics in Applied Mathematics, vol. 16. SIAM, Philadelphia (1996)

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

    MathSciNet  MATH  Google Scholar 

  12. Freitag, M.A., Spence, A.: Convergence rates for inexact inverse iteration with application to preconditioned iterative solves. BIT Numer. Math. 47, 27–44 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Freitag, M.A., Spence, A.: A tuned preconditioner for inexact inverse iteration applied to Hermitian eigenvalue problems. IMA J. Numer. Anal. 28, 522–551 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Freitag, M.A. Inner–outer iterative methods for eigenvalue problems—convergence and preconditioning. PhD thesis, Department of Mathematics, University of Bath, UK (2007)

  15. Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials. Academic Press, New York (1982)

    MATH  Google Scholar 

  16. Guo, C.-H., Higham, N., Tisseur, F.: Detecting and solving hyperbolic quadratic eigenvalue problems. SIAM J. Matrix Anal. Appl. 30, 1593–1613 (2009)

    Google Scholar 

  17. Guo, C.-H., Lancaster, P.: Algorithms for hyperbolic quadratic eigenvalue problems. Math. Comput. 74, 1777–1791 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hechme, G.: Convergence analysis of the Jacobi–Davidson method applied to a generalized eigenproblem. Comptes rendus Mathématique Académie des Sciences Paris 345, 293–296 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Higham, N., Mackey, S., Tisseur, F., Garvey, S.: Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems. Int. J. Numer. Methods Eng. 73, 344–360 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hochstenbach, M.E., Notay, Y.: Controlling inner iterations in the Jacobi–Davidson method. SIAM J. Matrix Anal. Appl. 31, 460–477 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hwang, T.-M., Lin, W.-W., Mehrmann, V.: Numerical solution of quadratic eigenvalue problems with structure-preserving methods. SIAM J. Sci. Comput. 24, 1283–1302 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jarlebring, E., Michiels, W.: Analyzing the convergence factor of residual inverse iteration. BIT Numer. Math 51, 937–957 (2011)

    Google Scholar 

  23. Jia, Z., Zeng, W.: A convergence analysis of the inexact Rayleigh quotient iteration and simplified Jacobi–Davidson method for the large Hermitian matrix eigenproblem. Sci. China Ser. A Math. 51, 2205–2216 (2009)

    Article  Google Scholar 

  24. Keller, H.B.: Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Rabinowitz, P.H. (ed.) Applications of Bifurcation Theory, pp.359–384. Academic Press, New York (1977)

  25. Kozlov, V., Maz’ya, V.: Differential Equations with Operator Coefficients: with Applications to Boundary Value Problems for Partial Differential Equations. Springer Monographs in Mathematics. Springer, Berlin (1999)

    Google Scholar 

  26. Mackey, S.D., Mackey, N., Mehl, C., Mehrmann, V.: Vector spaces of linearizations for matrix polynomials. SIAM J. Matrix Anal. Appl. 28, 971–1004 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Meerbergen, K.: The quadratic Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 30, 1463–1482 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mehrmann, V.: Private communication (2011)

  29. Mehrmann, V., Schröder, C.: Nonlinear eigenvalue and frequency response problems in industrial practice. J. Math. Ind. 1, 7 (2011)

    Google Scholar 

  30. Mehrmann, V., Voss, H.: Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods. Mitteilungen der Gesellschaft für Angewandte Mathematik und Mechanik 27, 121–152 (2004)

    MathSciNet  MATH  Google Scholar 

  31. Neumaier, A.: Residual inverse iteration for the nonlinear eigenvalue problem. SIAM J. Numer. Anal. 22, 914–923 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ruhe, A.: A Rational Krylov algorithm for nonlinear matrix eigenvalue problems. Rossiĭskaya Akademiya Nauk. Sankt-Peterburgskoe Otdelenie. Matematicheskiĭ Institut im. V. A. Steklova. Zapiski Nauchnykh Seminarov (POMI), 268, 176–180 (2000); translation in J. Math. Sci. (New York) 114, 1854–1856 (2003)

  33. Schreiber, K.: Nonlinear eigenvalue problems: Newton-type methods and nonlinear Rayleigh functionals. PhD thesis, Department of Mathematics, TU Berlin (2008)

  34. Schwetlick, H., Schreiber, K.: Nonlinear Rayleigh functionals. Linear Algebra Appl. 436, 3991–4016 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sonneveld, P., van Gijzen, M.B.: IDR(s): a family of simple and fast algorithms for solving large nonsymmetric linear systems. SIAM J. Sci. Comput. 31, 1035–1062 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Su, Y., Bai, Z.: Solving rational eigenvalue problems via linearization. SIAM J. Matrix Anal. Appl. 32, 201–216 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Szyld, D.B.: Criteria for combining inverse and Rayleigh quotient iteration. SIAM J. Numer. Anal. 25, 1369–1375 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  38. Szyld, D.B., Xue, F.: Local convergence analysis of several inexact Newton-type algorithms for general nonlinear eigenvalue problems. Research Report 11-08-09, Department of Mathematics, Temple University (2011)

  39. Tisseur, F., Meerbergen, K.: The quadratic eigenvalue problem. SIAM Rev. 43, 235–286 (2001)

    Google Scholar 

  40. van den Eshof, J.: The convergence of Jacobi–Davidson iterations for Hermitian eigenproblems. Numer. Linear Algebra Appl. 9, 163–179 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  41. van den Eshof, J.: Nested iteration methods for nonlinear matrix problems. PhD thesis, Department of Mathematics, Utrecht University, The Netherlands (2003)

  42. Xue, F.: Numerical solution of eigenvalue problems with spectral transformations. PhD thesis, Applied Mathematics, Statistics, and Scientific Computing, Department of Mathematics, University of Maryland, College Park (2009)

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Acknowledgments

We thank Volker Mehrmann for useful comments and pointers to the literature, and the two referees whose comments helped us to improve our original manuscript.

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Correspondence to Daniel B. Szyld.

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This work was supported by the US Department of Energy under Grant DEFG0205ER25672, and the US National Science Foundation under Grant DMS-1115520.

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Szyld, D.B., Xue, F. Local convergence analysis of several inexact Newton-type algorithms for general nonlinear eigenvalue problems. Numer. Math. 123, 333–362 (2013). https://doi.org/10.1007/s00211-012-0489-1

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