The quadratic arnoldi method for the solution of the quadratic eigenvalue problem

SIAM Journal on Matrix Analysis and Applications

Volumn 30, Issue 4, 2008, Pages 1463-1482

  Meerbergen, Karl ^a  

^a UNIVERSITY OF LEUVEN   (Belgium)

Author keywords

Arnoldi method; Quadratic eigenvalue problem; Schur decomposition; SOAR method

Indexed keywords

ARNOLDI; ARNOLDI ALGORITHM; ARNOLDI METHOD; KRYLOV METHOD; KRYLOV VECTORS; MEMORY REQUIREMENTS; OPEN PROBLEMS; QUADRATIC EIGENVALUE PROBLEM; QUADRATIC EIGENVALUE PROBLEMS; SCHUR DECOMPOSITION; SCHUR FORM; SCHUR VECTORS; SECOND ORDERS; SOAR METHOD;

LINEARIZATION; SWITCHING SYSTEMS;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 70449371821     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/07069273X     Document Type: Article

References (26)

1. W. Arnoldi, The principle of minimized iterations in the solution of the matrix eigenvalue problem, Quart. Appl. Math., 9 (1951), pp. 17-29.
2. Z. Bai, D. Day, and Q. Ye, ABLE: An adaptive block Lanczos method for non-Hermitian eigenvalue problems, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 1060-1082.
3. Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, Software Environ. Tools 11, SIAM, Philadelphia, 2000.
4. Z. Bai and Y. Su, SOAR: A second-order Arnoldi method for the solution of the quadratic eigenvalue problem, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 640-659.
5. Free Field Technologies, MSC.ACTRAN 2006, User's Manual, Free Field Technologies, Louvain-La-Neuve, Belgium, 2006.
6. G. Golub and C. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
7. R. Grimes, J. Lewis, and H. Simon, A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 228-272.
8. N. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, 1996.
9. C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Stand., 45 (1950), pp. 255-282.
10. C. Lanczos, Solution of systems of linear equations by minimized iterations, J. Res. Nat. Bur. Stand., 49 (1952), pp. 33-53.
11. R. Lehoucq and D. Sorensen, Deflation techniques within an implicitly restarted Arnoldi iteration, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 789-821.
12. R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users' Guide: Solution of Large- Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, Software Environ. Tools 6, SIAM, Philadelphia, 1998.
13. D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, Structured polynomial eigenvalue problems: Good vibrations from good linearizations, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 1029-1051
14. D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, Vector spaces of linearizations for matrix polynomials, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 971-1004.
15. K. Meerbergen, Locking and restarting quadratic eigenvalue solvers, SIAM J. Sci. Comput., 22 (2001), pp. 1814-1839.
16. K. Meerbergen, The rational Lanczos method for Hermitian eigenvalue problems, Numer. Linear Algebra Appl., 8 (2001), pp. 33-52.
17. K. Meerbergen and A. Spence, Implicitly restarted Arnoldi and purification for the shiftinvert transformation, Math. Comp., 66 (1997), pp. 667-689. (Pubitemid 127383117)
18. V. Mehrmann and D. Watkins, Structure-preserving methods for computing eigenpairs of large sparse skew-Hamiltonian/Hamiltonian pencils, SIAM J. Sci. Comput., 22 (2001), pp. 1905-1925.
19. R. Morgan, On restarting the Arnoldi method for large nonsymmetric eigenvalue problems, Math. Comp., 65 (1996), pp. 1213-1230.
20. B. N. Parlett and H. C. Chen, Use of indefinite pencils for computing damped natural modes, Linear Algebra Appl., 140 (1990), pp. 53-88.
21. Y. Saad, Numerical methods for large eigenvalue problems, Algorithms and Architectures for Advanced Scientific Computing, Manchester University Press, Manchester, UK, 1992.
22. D. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357-385.
23. G. W. Stewart, A Krylov-Schur algorithm for large eigenproblems, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 601-614.
24. F. Tisseur, Backward error and condition of polynomial eigenvalue problems, Linear Algebra Appl., 309 (2000), pp. 339-361.
25. F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), pp. 235-286.
26. K. Wu and H. Simon, Thick-restart Lanczos method for large symmetric eigenvalue problems, SIAM J. Matrix Anal. Appl., 22 (2000), pp. 602-616.