Numerical solution of a quadratic eigenvalue problem

Linear Algebra and Its Applications

Volumn 385, Issue 1-3, 2004, Pages 391-406

  Guo, Chun Hua ^a  

^a UNIVERSITY OF REGINA   (Canada)

Author keywords

Cyclic reduction; Quadratic eigenvalue problem; Quadratic matrix equation; Solvent

Indexed keywords

GYROSCOPES; HAMILTONIANS; LINEAR EQUATIONS; MATRIX ALGEBRA; POLYNOMIALS; PROBLEM SOLVING;

CYCLIC REDUCTION; QUADRATIC EIGENVALUE PROBLEM (QEP); QUADRUPLETS; QYUADRATIC MATRIX SOLUTIONS;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 2542480153     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2003.12.010     Document Type: Article

References (25)

1. -1B* Linear Algebra Appl. 134:1990;53-62.
2. Benner P., Byers R., Mehrmann V., Xu H. Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils. SIAM J. Matrix Anal. Appl. 24:2002;165-190.
3. Bini D.A., Gemignani L., Meini B. Computations with infinite Toeplitz matrices and polynomials. Linear Algebra Appl. 343-344:2002;21-61.
4. Davis G.J. Numerical solution of a quadratic matrix equation. SIAM J. Sci. Statist. Comput. 2:1981;164-175.
5. -1A=Q Linear Algebra Appl. 186:1993;255-275.
6. Gohberg I., Lancaster P., Rodman L. Matrix Polynomials. 1982;Academic Press, New York.
7. Gohberg I., Lancaster P., Rodman L. Invariant Subspaces of Matrices with Applications. 1986;John Wiley, New York.
8. Guo C.-H. Newton's method for discrete algebraic Riccati equations when the closed-loop matrix has eigenvalues on the unit circle. SIAM J. Matrix Anal. Appl. 20:1998;279-294.
9. Guo C.-H. Convergence rate of an iterative method for a nonlinear matrix equation. SIAM J. Matrix Anal. Appl. 23:2001;295-302.
10. Guo C.-H., Lancaster P. Iterative solution of two matrix equations. Math. Comput. 68:1999;1589-1603.
11. C.-H. Guo, P. Lancaster, Algorithms for hyperbolic quadratic eigenvalue problems, submitted for publication.
12. Higham N.J., Kim H.-M. Numerical analysis of a quadratic matrix equation. IMA J. Numer. Anal. 20:2000;499-519.
13. Higham N.J., Kim H.-M. Solving a quadratic matrix equation by Newton's method with exact line searches. SIAM J. Matrix Anal. Appl. 23:2001;303-316.
14. Higham N.J., Tisseur F., Van Dooren P.M. Detecting a definite Hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem, and associated nearness problems. Linear Algebra Appl. 351-352:2002;455-474.
15. Lancaster P. Lambda-Matrices and Vibrating Systems. 1966;Pergamon Press, Oxford.
16. Lancaster P. Strongly stable gyroscopic systems. Electron. J. Linear Algebra. 5:1999;53-66.
17. Langer H. Factorization of operator pencils. Acta Sci. Math. (Szeged). 38:1976;83-96.
18. Mehrmann V., Watkins D. Structure-preserving methods for computing eigenpairs of large sparse skew-Hamiltonian/Hamiltonian pencils. SIAM J. Sci. Comput. 22:2001;1905-1925.
19. -1A=Q Math. Comput. 71:2002;1189-1204.
20. Miranda M., Tilli P. Asymptotic spectra of Hermitian block Toeplitz matrices and preconditioning results. SIAM J. Matrix Anal. Appl. 21:2000;867-881.
21. Moro J., Burke J.V., Overton M.L. On the Lidskii-Vishik-Lyusternik perturbation theory for eigenvalues of matrices with arbitrary Jordan structure. SIAM J. Matrix Anal. Appl. 18:1997;793-817.
22. Serra S. Asymptotic results on the spectra of block Toeplitz preconditioned matrices. SIAM J. Matrix Anal. Appl. 20:1998;31-44.
23. -1A=P . II Linear Algebra Appl. 362:2003;211-228.
24. Tisseur F., Meerbergen K. The quadratic eigenvalue problem. SIAM Rev. 43:2001;235-286.
25. -1A=I Linear Algebra Appl. 247:1996;337-345.