An integral method for solving nonlinear eigenvalue problems

Linear Algebra and Its Applications

Volumn 436, Issue 10, 2012, Pages 3839-3863

  Beyn, Wolf Jürgen ^a  

^a BIELEFELD UNIVERSITY   (Germany)

Author keywords

Contour integrals; Nonlinear eigenvalue problems; Numerical methods

Indexed keywords

CLOSED CONTOURS; COLUMN VECTOR; COMPLEX INTEGRALS; COMPLEX PLANES; CONTOUR INTEGRALS; EIGENVALUE PROBLEM; EIGENVALUES; EIGENVALUES AND EIGENVECTORS; INITIAL APPROXIMATION; INTEGRAL METHOD; MINIMAL DISTANCE; NONLINEAR EIGENVALUE PROBLEM; QUADRATURE ERROR; RESOLVENT OPERATORS;

NUMERICAL METHODS; SWITCHING SYSTEMS;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 84858764451     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2011.03.030     Document Type: Article

References (36)

1. J. Asakura, T. Sakurai, H. Tadano, T. Ikegami, K. Kimura, A numerical method for polynomial eigenvalue problems using contour integral, Technical Report CS-TR-08-15, Department of Computer Science, University of Tsukuba, 2008.
2. J. Asakura, T. Sakurai, H. Tadano, T. Ikegami, and K. Kimura A numerical method for nonlinear eigenvalue problems using contour integrals JSIAM Lett. 1 2009 52 55
3. Z. Bai, and Y. Su A second-order Arnoldi method for the solution of the quadratic eigenvalue problem SIAM J. Matrix Anal. Appl. 26 3 2005 640 659 (Pubitemid 41049837)
4. T. Betcke, N.J. Higham, V. Mehrmann, C. Schröder, F. Tisseur, NLEVP: a collection of nonlinear eigenvalue problems, Technical Report 2008.40, MIMS, University of Manchester, April 2008.
5. T. Betcke, D. Kressner, Perturbation, computation and refinement of invariant matrix pairs, Technical Report 2009-21, SAM, ETH Zürich, 2009.
6. T. Betcke, and H. Voss A Jacobi-Davidson type projection method for nonlinear eigenvalue problems Future Gener. Comput. Syst. 20 2004 363 372
7. D. Chu, and G. Golub On a generalized eigenvalue problem for nonsquare pencils SIAM J. Matrix Anal. Appl. 28 2006 770 787 (Pubitemid 47109361)
8. P.J. Davis, On the numerical integration of periodic analytic functions, in: R.E. Langer (Ed.), Proceedings of a Symposium On Numerical Approximation, Madison, April 21-23, 1958, Publication No. 1 of the Mathematics Research Center, US Army, the University of Wisconsin, The University of Wisconsin Press, Madison, 1959, pp. 45-59.
9. P.J. Davis, and P. Rabinowitz Methods of Numerical Integration 2007 Dover Publications Inc. Mineola, NY Corrected reprint of the second (1984) edition
10. I. Gohberg, P. Lancaster, and L. Rodman Matrix Polynomials Computer Science and Applied Mathematics 1982 Academic Press Inc. New York
11. I. Gohberg, and L. Rodman Analytic matrix functions with prescribed local data J. Anal. Math. 40 1982 1981 90 128
12. I.C. Gohberg, and E.I. Sigal An operator generalization of the logarithmic residue theorem and Rouché's theorem Mat. Sb. 84 126 1971 607 629
13. G.H. Golub, and C.F. Van Loan Matrix Computations third ed. 1996 Johns Hopkins University Press Baltimore, MD
14. R.E. Greene, and S.G. Krantz Function Theory of One Complex Variable third ed. Graduate Studies in Mathematics vol. 40 2006 American Mathematical Society Providence, RI
15. Aα, log (A), and related matrix functions by contour integrals SIAM J. Numer. Anal. 46 2008 2505 2523
16. N.J. Higham Functions of Matrices 2008 SIAM
17. M.V. Keldysh On the characteristic values and characteristic functions of certain classes of non-self-adjoint equations Dokl. Akad. Nauk SSSR (N.S.) 77 1951 11 14
18. M.V. Keldysh The completeness of eigenfunctions of certain classes of nonselfadjoint linear operators Uspehi Mat. Nauk 26 4160 1971 15 41
19. S. Kirkup, and S. Amini Soluton of the Helmholtz eigenvalue problem via the boundary element method Internat. J. Numer. Methods Engrg. 36 1993 321 330 (Pubitemid 23618441)
20. D. Kressner A block Newton method for nonlinear eigenvalue problems Numer. Math. 114 2009 355 372
21. D.S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann Vector spaces of linearizations for matrix polynomials SIAM J. Matrix Anal. Appl. 28 4 2006 971 1004 (electronic) (Pubitemid 47610046)
22. A.S. Markus, and E.I. Sigal The multiplicity of the characteristic number of an analytic operator function Mat. Issled. 5 3(17) 1970 129 147
23. K. Meerbergen The quadratic Arnoldi method for the solution of the quadratic eigenvalue problem SIAM J. Matrix Anal. Appl. 30 4 2008 1463 1482
24. V. Mehrmann, and H. Voss Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods GAMM Mitt. 27 2004
25. R. Mennicken, and M. Möller Root functions, eigenvectors, associated vectors and the inverse of a holomorphic operator function Arch. Math. (Basel) 42 5 1984 455 463
26. R. Mennicken, and M. Möller Non-Self-Adjoint Boundary Eigenvalue Problems North-Holland Mathematics Studies vol. 192 2003 North-Holland Publishing Co. Amsterdam
27. W. Michiels, and S.-I. Niculescu Stability and Stabilization of Time-delay Systems Advances in Design and Control vol. 12 2007 Society for Industrial and Applied Mathematics (SIAM)
28. E. Polizzi Density-matrix-based algorithms for solving eigenvalue problems Phys. Rev. B 2009 115112
29. T. Sakurai, and H. Sugiura A projection method for generalized eigenvalue problems J. Comput. Appl. Math. 159 2003 119 128
30. S. Solovev Preconditioned iterative methods for a class of nonlinear eigenvalue problems Linear Algebra Appl. 415 2006 210 229
31. G. Stewart, and J.G. Sun Matrix Perturbation Theory 1990 Academic Press Inc. Boston, MA
32. V.P. Trofimov The root subspaces of operators that depend analytically on a parameter Mat. Issled. 3 vyp. 3 (9) 1968 117 125
33. H. Voss A maxmin principle for nonlinear eigenvalue problems with application to a rational spectral problem in fluid-solid vibration Appl. Math 48 2003 607 622
34. H. Voss An Arnoldi method for nonlinear eigenvalue problems BIT 44 2004 387 401 (Pubitemid 39187343)
35. H. Voss A Jacobi-Davidson method for nonlinear and nonsymmetric eigenvalue problems Comput. Struct. 85 2007 1284 1292 (Pubitemid 47048810)
36. H. Voss, and B. Werner A minmax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems Math. Methods Appl. Sci. 4 1982 415 424