Abstract
One of the most efficient methods for solving the polynomial eigenvalue problem (PEP) is the Sakurai-Sugiura method with Rayleigh-Ritz projection (SS-RR), which finds the eigenvalues contained in a certain domain using the contour integral. The SS-RR method converts the original PEP to a small projected PEP using the Rayleigh-Ritz projection. However, the SS-RR method suffers from backward instability when the norms of the coefficient matrices of the projected PEP vary widely. To improve the backward stability of the SS-RR method, we combine it with a balancing technique for solving a small projected PEP. We then analyze the backward stability of the SS-RR method. Several numerical examples demonstrate that the SS-RR method with the balancing technique reduces the backward error of eigenpairs of PEP.
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References
J. Asakura, T. Sakurai, H. Tadano, T. Ikegami, K. Kimura: A numerical method for non-linear eigenvalue problems using contour integrals. JSIAM Lett. 1 (2009), 52–55.
J. Asakura, T. Sakurai, H. Tadano, T. Ikegami, K. Kimura: A numerical method for polynomial eigenvalue problems using contour integral. Japan J. Ind. Appl. Math. 27 (2010), 73–90.
T. Betcke, N. J. Higham, V. Mehrmann, C. Schröder, F. Tisseur: NLEVP: A collection of nonlinear eigenvalue problems. ACM Trans. Math. Softw. 39 (2013), Paper No. 7, 28 pages.
H. Chen, Y. Maeda, A. Imakura, T. Sakurai, F. Tisseur: Improving the numerical stability of the Sakurai-Sugiura method for quadratic eigenvalue problems. JSIAM Lett. 9 (2017), 17–20.
N. J. Higham, R. Li, F. Tisseur: Backward error of polynomial eigenproblems solved by linearization. SIAM J. Matrix Anal. Appl. 29 (2007), 1218–1241.
N. J. Higham, D. S. Mackey, F. Tisseur, S. D. Garvey: Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems. Int. J. Numer. Methods Eng. 73 (2008), 344–360.
T. Ikegami, T. Sakurai: Contour integral eigensolver for non-Hermitian systems: a Rayleigh-Ritz-type approach. Taiwanese J. Math. 14 (2010), 825–837.
T. Ikegami, T. Sakurai, U. Nagashima: A filter diagonalization for generalized eigenvalue problems based on the Sakurai-Sugiura projection method. J. Comput. Appl. Math. 233 (2010), 1927–1936.
E. E. Osborne: On preconditioning of matrices. J. Assoc. Comput. Math. 7 (1960), 338–345.
B. Parlett, C. Reinsch: Balancing a matrix for calculation of eigenvalues and eigenvectors. Numer. Math. 13 (1969), 293–304.
T. Sakurai, H. Sugiura: A projection method for generalized eigenvalue problems using numerical integration. J. Comput. Appl. Math. 159 (2003), 119–128.
F. Tisseur: Backward error and condition of polynomial eigenvalue problems. Linear Algebra Appl. 309 (2000), 339–361.
F. Tisseur, K. Meerbergen: The quadratic eigenvalue problem. SIAM Rev. 43 (2001), 235–286.
S. Yokota, T. Sakurai: A projection method for nonlinear eigenvalue problems using contour integrals. JSIAM Lett. 5 (2013), 41–44.
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This research was supported partly by JST/ACT-I (No. JPMJPR16U6), JST/CREST, KAKENHI (No. 17K12690) and University of Tsukuba Basic Research Support Program Type A.
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Chen, H., Imakura, A. & Sakurai, T. Improving backward stability of Sakurai-Sugiura method with balancing technique in polynomial eigenvalue problem. Appl Math 62, 357–375 (2017). https://doi.org/10.21136/AM.2017.0016-17
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DOI: https://doi.org/10.21136/AM.2017.0016-17