Abstract
We propose a numerical method using contour integral to solve polynomial eigenvalue problems (PEPs). The method finds eigenvalues contained in a certain domain which is defined by a surrounding integral path. By evaluating the contour integral numerically along the path, the method reduces the original PEP into a small generalized eigenvalue problem, which has the identical eigenvalues in the domain. When the contour integral is approximated numerically, eigenvalues on the periphery of the path are also obtained. Error analysis shows that, even though condition numbers of those exterior eigenvalues can be huge, the interior eigenvalues are calculated less erroneously. Four numerical examples are presented, which confirm the theoretical predictions.
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References
Bai Z., Demmel J., Dongarra J., Ruhe A., Van Der Vorst H.: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia (2000)
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)
Beckermann B., Golub G.H., Labahn G.: On the numerical condition of a generalized Hankel eigenvalue problem. Numer. Math. 106, 41–68 (2007)
Betcke, T., Higham, N.J., Mehrmann, V., Schröder, C., Tisseur, F.: NLEVP: A Collection of Nonlinear Eigenvalue Problem. MIMS EPrint 2008.40 (2008)
Frayssé V., Toumazou V.: A note on the normwise perturbation theory for the regular generalized eigenvalueproblem Ax = λ Bx. Numer. Linear Algebra Appl. 5, 1–10 (1998)
Gantmacher, F.R.: The Theory of Matrices, vols. 1–2. Chelsea Publishing Company, New York (1977)
Gel’fand, I.M.: Lecture on Linear Algebra. Dover Publications, INC, Mineola (1961)
Gohberg I., Kaashoek M.A., Lancaster P.: General theory of regular matrix polynomials and band toeplitz operators. Integr. Equ. Oper. Theory 11, 776–882 (1988)
Gohberg I., Lancaster P., Rodman F.: Matrix polynomials. Academic Press, New York (1982)
Heeg, R.S.: Stability and transition of attachment-line flow, Ph.D.thesis, Universiteit Twente, Enschede, the Netherlands (1998)
Higham D.J., Higham N.J.: Structured backward error and condition of generalized eigenvalue problems. SIAM J. Matrix Anal. Appl. 20, 493–512 (1998)
Higham N.J., Tisseur F.: Bounds for eigenvalues of matrix polynomials. Linear Algebra Appl. 358, 5–22 (2003)
Hwang T., Lin W., Liu J., Wang W.: Jacobi–Davidson methods for cubic eigenvalue problems. Numer. Linear Algebra Appl. 12, 605–624 (2005)
Ikegami, T., Sakurai, T., Nagashima, U.: A filter diagonalization for generalized eigenvalue problems based on the Sakurai-Sugiura projection method. J. Comput. Appl. Math. (2010, in press)
Mackey D.S., Mackey N., Mehl C., Mehrmann V.: Vector spaces of linearizations for matrix polynomials. SIAM J. Matrix Anal. Appl. 28, 971–1004 (2006)
Mehrmann V., Watkins D.: Polynomial eigenvalue problems with Hamiltonian structure. Electron. Trans. Numer. Anal. 13, 106–118 (2002)
Pereira E.: On solvents of matrix polynomials. Appl. Numer. Math. 47, 197–208 (2003)
Sakurai T., Sugiura H.: A projection method for generalized eigenvalue problems. J. Comput. Appl. Math. 159, 119–128 (2003)
Sleijpen G.L.G., Booten A.G.L., Fokkema D.R., Van Der Vorst H.A.: Jacobi–Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT 36, 595–633 (1996)
Tisseur F., Meerbergen K.: The quadratic eigenvalue problem. SIAM Rev. 43, 235–286 (2001)
Tyrtyshnikov, E.E.: How bad are Hankel matrices? Numer. Math. 67, 261–269 (1994)
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Asakura, J., Sakurai, T., Tadano, H. et al. A numerical method for polynomial eigenvalue problems using contour integral. Japan J. Indust. Appl. Math. 27, 73–90 (2010). https://doi.org/10.1007/s13160-010-0005-x
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DOI: https://doi.org/10.1007/s13160-010-0005-x