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nonlinear eigenvalue problem; rational Krylov method; Arnoldi method; projection method
In recent papers Ruhe suggested a rational Krylov method for nonlinear eigenproblems knitting together a secant method for linearizing the nonlinear problem and the Krylov method for the linearized problem. In this note we point out that the method can be understood as an iterative projection method. Similarly to the Arnoldi method the search space is expanded by the direction from residual inverse iteration. Numerical methods demonstrate that the rational Krylov method can be accelerated considerably by replacing an inner iteration by an explicit solver of projected problems.
[1] W. E.  Arnoldi: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q. Appl. Math. 9 (1951), 17–29. MR 0042792 | Zbl 0042.12801
[2] Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. A. van der  Vorst (eds.), SIAM, Philadelphia, 2000. MR 1792141 | Zbl 0965.65058
[3] T. Betcke, H. Voss: A Jacobi-Davidson-type projection method for nonlinear eigenvalue problems. Future Generation Computer Systems 20 (2004), 363–372. DOI 10.1016/j.future.2003.07.003
[4] C. Conca, J. Planchard, and M. Vanninathan: Existence and location of eigenvalues for fluid-solid structures. Comput. Methods Appl. Mech. Eng. 77 (1989), 253–291. DOI 10.1016/0045-7825(89)90078-9 | MR 1031134
[5] P.  Hager: Eigenfrequency Analysis. FE-Adaptivity and a Nonlinear Eigenvalue Problem. PhD. thesis, Chalmers University of Technology, Göteborg, 2001.
[6] P. Hager, N. E. Wiberg: The rational Krylov algorithm for nonlinear eigenvalue problems. In: Computational Mechanics for the Twenty-First Century, B. H. V. Topping (ed.), Saxe-Coburg Publications, Edinburgh, 2000, pp. 379–402.
[7] E. Jarlebring: Krylov Methods for Nonlinear Eigenvalue Problems. Master thesis, Royal Institute of Technology. Dept. Numer. Anal. Comput. Sci., Stockholm, 2003.
[8] V. N. Kublanovskaya: On an application of Newton’s method to the determination of eigenvalues of $\lambda $-matrices. Dokl. Akad. Nauk. SSSR 188 (1969), 1240–1241. MR 0250470 | Zbl 0242.65042
[9] V. N. Kublanovskaya: On an approach to the solution of the generalized latent value problem for $\lambda $-matrices. SIAM. J. Numer. Anal. 7 (1970), 532–537. DOI 10.1137/0707043 | MR 0281333 | Zbl 0225.65048
[10] C. Lanczos: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J.  Res. Nat. Bur. Standards 45 (1950), 255–282. DOI 10.6028/jres.045.026 | MR 0042791
[11] A.  Neumaier: Residual inverse iteration for the nonlinear eigenvalue problem. SIAM J. Numer. Anal. 22 (1985), 914–923. DOI 10.1137/0722055 | MR 0799120 | Zbl 0594.65026
[12] J. Planchard: Eigenfrequencies of a tube bundle placed in a confined fluid. Comput. Methods Appl. Mech. Eng. 30 (1982), 75–93. DOI 10.1016/0045-7825(82)90055-X | MR 0659568 | Zbl 0483.70016
[13] A. Ruhe: Algorithms for the nonlinear eigenvalue problem. SIAM J. Numer. Anal. 10 (1973), 674–689. DOI 10.1137/0710059 | MR 0329231 | Zbl 0261.65032
[14] A. Ruhe: Computing nonlinear eigenvalues with spectral transformation Arnoldi. Z.  Angew. Math. Mech. 76 (1996), 17–20. Zbl 0886.65055
[15] A. Ruhe: Rational Krylov: A practical algorithm for large sparse nonsymmetric matrix pencils. SIAM J.  Sci. Comput. 19 (1998), 1535–1551. DOI 10.1137/S1064827595285597 | MR 1618804 | Zbl 0914.65036
[16] A. Ruhe: The rational Krylov algorithm for nonlinear matrix eigenvalue problems. Zap. Nauchn. Semin. POMI 268 (2000), 176–180. MR 1795855 | Zbl 1029.65035
[17] G. L. Sleijpen, G. L. Booten, D. R. Fokkema, and H. A. van der Vorst: Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT 36 (1996), 595–633. DOI 10.1007/BF01731936 | MR 1410100
[18] G. L. Sleijpen, H. A. van der Vorst: A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17 (1996), 401–425. DOI 10.1137/S0895479894270427 | MR 1384515
[19] H. Voss: An Arnoldi method for nonlinear eigenvalue problems. BIT 44 (2004), 387-401. DOI 10.1023/B:BITN.0000039424.56697.8b | MR 2093512 | Zbl 1066.65059
[20] H. Voss: An Arnoldi method for nonlinear symmetric eigenvalue problems. In: Online Proceedings of the SIAM Conference on Applied Linear Algebra, Williamsburg, 2003,
[21] H. Voss: Initializing iterative projection methods for rational symmetric eigenproblems. In: Online Proceedings of the Dagstuhl Seminar Theoretical and Computational Aspects of Matrix Algorithms, Schloss Dagstuhl 2003,, 2003.
[22] H. Voss: A Jacobi-Davidson method for nonlinear eigenproblems. In: Computational Science—ICCS  2004, 4th  International Conference, Kraków, Poland, June 6–9, 2004, Proceedings, Part  II, Vol. 3037 of Lecture Notes in Computer Science, M. Buback, G. D.  van Albada, P. M. A.  Sloot, and J. J.  Dongarra (eds.), Springer-Verlag, Berlin, 2004, pp. 34–41. MR 2213179 | Zbl 1080.65535
[23] H. Voss, B. Werner: A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems. Math. Methods Appl. Sci. 4 (1982), 415–424. DOI 10.1002/mma.1670040126 | MR 0669135
[24] W. H.  Yang: A method for eigenvalues of sparse $\lambda $-matrices. Int. J.  Numer. Methods Eng. 19 (1983), 943–948. DOI 10.1002/nme.1620190613 | MR 0708826 | Zbl 0517.65018
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