Convergence of the dominant pole algorithm and Rayleigh quotient iteration

SIAM Journal on Matrix Analysis and Applications

Volumn 30, Issue 1, 2008, Pages 346-363

  Rommes, Joost ^a,b   Sleijpen, Gerard L G ^b  

^a NXP SEMICONDUCTORS   (Netherlands)

^b UTRECHT UNIVERSITY   (Netherlands)

Author keywords

Dominant pole algorithm; Dominant poles; Eigenvalues; Eigenvectors; Modal model reduction; Rate of convergence; Subspace accelerated Newton method; Transfer function; Two sided Rayleigh quotient iteration

Indexed keywords

DOMINANT POLE ALGORITHM; DOMINANT POLES; EIGENVALUES; EIGENVECTORS; MODAL MODEL REDUCTION; RATE OF CONVERGENCE; SUBSPACE ACCELERATED NEWTON METHOD; TWO-SIDED RAYLEIGH QUOTIENT ITERATION;

ALGORITHMS; CONVERGENCE OF NUMERICAL METHODS; DYNAMICAL SYSTEMS; EIGENVALUES AND EIGENFUNCTIONS; NEWTON-RAPHSON METHOD; POLES;

TRANSFER FUNCTIONS;

EID: 43849108552     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/060671401     Document Type: Article

References (29)

1. P.-A. ABSIL, R. SEPULCHRE, P. VAN DOOREN, AND R. MAHONY, Cubically convergent iterations for invariant subspace computation, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 70-96.
2. L. A. AGUIRRE, Quantitative measure of modal dominance for continuous systems, in Proceedings of the 32nd IEEE Conference on Decision and Control, IEEE Control Systems Society, Piscataway, NJ, 1993, pp. 2405-2410.
3. A. C. ANTOULAS, Approximation of Large-Scale Dynamical Systems, Adv. Des. Control 6, SIAM, Philadelphia, 2005.
4. S. BATTERSON AND J. SMILLIE, The dynamics of Rayleigh quotient iteration, SIAM J. Numer. Anal., 26 (1989), pp. 624-636.
5. C. BEATTIE AND D. W. FOX, Localization criteria and containment for Rayleigh quotient iteration, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 80-93.
6. L. H. BEZERRA, Written discussion to [15], IEEE Trans. Power Syst., 11 (1996), p. 168.
7. L. H. BEZERRA, An eigenvalue method for calculating dominant poles of a transfer function, Appl. Math. Lett., 21 (2008), pp. 244-247.
8. Y. CHAHLAOUI AND P. VAN DOOREN, A Collection of Benchmark Examples for Model Reduction of Linear Time Invariant Dynamical Systems, SLICOT Working Note 2002-2, National Institute for Research and Development in Informatics, Bucharest, Romania, 2002.
9. V. FRAYSSÉ AND V. TOUMAZOU, A note on the normwise perturbation theory for the regular generalized eigenproblem Ax = λBx, Numer. Linear Algebra Appl., 5 (1998), pp. 1-10.
10. M. GREEN AND D. J. N. LIMEBEER, Linear Robust Control, Prentice-Hall, Englewood Cliffs, NJ, 1995.
11. A. M. A. HAMDAN AND A. H. NAYFEH, Measures of modal controllability and observability for first- and second-order linear systems, J. Guidance Control Dynam., 12 (1989), pp. 421-428.
12. D. J. HIGHAM AND N. J. HIGHAM, Structured backward error and condition of generalized eigenvalue problems, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 493-512.
13. M. E. HOCHSTENBACH AND G. L. G. SLEUPEN, Two-sided and alternating Jacobi-Davidson, Linear Algebra Appl., 358 (2003), pp. 145-172.
14. T. KAILATH, Linear Systems, Prent ice-Hall, Englewood Cliffs, NJ, 1980.
15. N. MARTINS, L. T. G. LIMA, AND H. J. C. P. PINTO, Computing dominant poles of power system transfer functions, IEEE Trans. Power Syst., 11 (1996), pp. 162-170.
16. N. MARTINS AND P. E. M. QUINTÃO, Computing dominant poles of power system multivariable transfer functions, IEEE Trans. Power Syst., 18 (2003), pp. 152-159.
17. A. M. OSTROWSKI, On the convergence of the Rayleigh quotient iteration for the computation of the characteristic roots and vectors. I, Arch. Ration. Mech. Anal., 1 (1958), pp. 233-241.
18. A. M. OSTROWSKI, On the convergence of the Rayleigh quotient iteration for the computation of the characteristic roots and vectors. III, Arch. Ration. Mech. Anal., 3 (1959), pp. 325-340.
19. R. D. PANTAZIS AND D. B. SZYLD, Regions of convergence of the rayleigh quotient iteration method, Numer. Linear Algebra Appl., 2 (1995), pp. 251-269.
20. B. N. PARLETT, The Rayleigh quotient iteration and some generalizations for nonnormal matrices, Math. Comp., 28 (1974), pp. 679-693.
21. J. ROMMES, Methods for Eigenvalue Problems with Applications in Model Order Reduction, Ph.D. thesis, Utrecht University, Utrecht, The Netherlands, 2007.
22. J. ROMMES AND N. MARTINS, Efficient computation of multivariable transfer function dominant poles using subspace acceleration, IEEE Trans. Power Syst., 21 (2006), pp. 1471-1483.
23. J. ROMMES AND N. MARTINS, Efficient computation of transfer function dominant poles using subspace acceleration, IEEE Trans. Power Syst., 21 (2006), pp. 1218-1226.
24. G. L. G. SLEUPEN AND H. A. VAN DER VORST, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 401-425.
25. J. R. SMITH, J. F. HAUER, D. J. TRUDNOWSKI, F. FATEHI, AND C. S. WOODS, Transfer function identification in power system application, IEEE Trans. Power Syst., 8 (1993), pp. 1282-1290.
26. A. STATHOPOULOS, A case for a biorthogonal Jacobi-Davidson method: Restarting and correction equation, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 238-259.
27. D. B. SZYLD, Criteria for combining inverse and Rayleigh quotient iteration, SIAM J. Numer. Anal., 25 (1988), pp. 1369-1375.
28. THE MATHWORKS, INC., MATLAB 7, The MathWorks, Natick, MA, 2005.
29. J A. VARGA, Enhanced modal approach for model reduction, Math. Model Systems, 1 (1995), pp. 91-105.