Iterative computation of derivatives of repeated eigenvalues and the corresponding eigenvectors

Numerical Linear Algebra with Applications

Volumn 7, Issue 4, 2000, Pages 151-167

  Andrew, Alan L ^a   Tan, Roger C E ^b  

^a LA TROBE UNIVERSITY   (Australia)

^b NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Close eigenvalues; Eigenvalue and eigenvector sensitivities; Multiple eigenvalues; Simultaneous iteration

Indexed keywords

EID: 0034396233     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/1099-1506(200005)7:4<151::AID-NLA191>3.0.CO;2-M     Document Type: Article

References (39)

1. Haug EJ, Choi KK, Komkov V. Design Sensitivity Analysis and Structural Systems. Academic Press: New York, 1986.
2. Mottershead JE, Friswell MI. Model updating in structural dynamics: a survey. Journal of Sound and Vibration 1993; 167:347-375.
3. Messina A, Williams EJ, Contursi T. Structural damage detection by a sensitivity and statistical-based method. Journal of Sound and Vibration 1998; 216:791-808.
4. Andrew AL, Tan RCE. Computation of derivatives of repeated eigenvalues and the corresponding eigenvectors of symmetric matrix pencils. SIAM Journal on Matrix Analysis and Applications 1998; 20:78-100.
5. Haftka RT, Adelman HM. Recent developments in structural sensitivity analysis. Structural Optimization 1989; 1:137-151.
6. Zhang O, Zerva A. Accelerated iterative procedure for calculating eigenvector derivatives. American Institute of Aeronautics and Astronautics Journal 1997; 35:340-348.
7. Mills-Curran WC. Calculation of eigenvector derivatives for structures with repeated eigenvalues. American Institute of Aeronautics and Astronautics Journal 1988; 26:867-871.
8. Lewis AS, Overton ML. Eigenvalue optimization. Acta Numerica 1996; 5:149-190.
9. Seyranian AP, Lund E, Olhoff N. Multiple eigenvalues in structural optimization problems. Structural Optimization 1994; 8:207-227.
10. Tan RCE, Andrew AL. Computing derivatives of eigenvalues and eigenvectors by simultaneous iteration. IMA Journal of Numerical Analysis 1989; 9:111-122.
11. Andrew AL, Tan RCE. Computation of derivatives of repeated eigenvalues and corresponding eigenvectors by simultaneous iteration. American Institute of Aeronautics and Astronautics Journal 1996; 34:2214-2216.
12. Sun J-G. Multiple eigenvalue sensitivity analysis. Linear Algebra and its Applications 1990; 137/138:183-211.
13. Chu K-WE. On multiple eigenvalues of matrices depending on several parameters. SIAM Journal on Numerical Analysis 1990; 27:1368-1385.
14. Lee I-W, Jung G-H, Lee J-W. Numerical method for sensitivity analysis of eigensystems with non-repeated and repeated eigenvalues. Journal of Sound and Vibration 1996; 195:17-32.
15. Alvin KF. Efficient computation of eigenvector sensitivities for structural dynamics. American Institute of Aeronautics and Astronautics Journal 1997; 35:1760-1766.
16. Béliveau J-G, Cogan S, Lallement G, Ayer F. Iterative least-squares calculation for modal eigenvector sensitivity. American Institute of Aeronautics and Astronautics Journal 1996; 34:385-391.
17. Ting T. Accelerated subspace iteration for eigenvector derivatives. American Institute of Aeronautics and Astronautics Journal 1992; 30:2114-2118.
18. Allgower EL, Georg K. Numerical path following. In Handbook of Numerical Analysis vol. 5, Ciarlet PG, Lions JL (eds). North-Holland, 1997; 3-207.
19. Li T-Y, Rhee NH. Homotopy algorithm for symmetric eigenvalue problems. Numerische Mathematik 1989; 55:265-280.
20. Li T-Y, Zeng Z. Homotopy-determinant algorithm for solving non-symmetric eigenvalue problems. Mathematics of Computation 1992; 59:483-502.
21. Chu MT. Inverse eigenvalue problems. SIAM Review 1998; 40:1-39.
22. Li T-Y, Zhang H, Sun Z-H. Parallel homotopy algorithm for the symmetric tridiagonal eigenvalue problem. SIAM Journal on Scientific and Statistical Computing 1991; 12:469-487.
23. Rellich F. Störungstheorie der Spektralzerlegung - I: Analytische Störung der isolierten Punkteigenwerte einesbeschränkten Operators. Mathematische Annalen 1937; 113:600-619.
24. Andrew AL, Chu K-WE, Lancaster P. Derivatives of eigenvalues and eigenvectors of matrix functions. SIAM Journal on Matrix Analysis and Applications 1993; 14:903-926.
25. Arnold VI. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer-Verlag: New York, 1983.
26. Lancaster P. Lambda-matrices and Vibrating Systems. Pergamon: Oxford, 1966.
27. Baumgärtel H. Analytic Perturbation Theory for Matrices and Operators. Birkhäuser Verlag: Basel, 1985.
28. Vishik MI, Lyusternik LA. The solution of some perturbation problems for matrices and selfadjoint or non-selfadjoint differential equations - I: Russian Mathematical Surveys 1960; 15:1-73.
29. Lancaster P, Tismenetsky M. The Theory of Matrices (2nd edn). Academic Press: New York, 1985.
30. Lidskii VB. Perturbation theory of non-conjugate operators. USSR Computational Mathematics and Mathematical Physics 1966; 6(1):73-85.
31. Moro J, Burke JV, Overton ML. On the Lidskii-Vishik-Lyusternik perturbation theory for eigenvalues of matrices with arbitrary Jordan structure. SIAM Journal on Matrix Analysis and Applications 1997; 18:793-817.
32. Andrew AL. Convergence of an iterative method for derivatives of eigensystems. Journal of Computational Physics 1978; 26:107-112.
33. Andrew AL, Tan RCE. Computation of mixed partial derivatives of eigenvalues and eigenvectors by simultaneous iteration. Communications in Numerical Methods in Engineering 1999; 15:641-649.
34. Tan RCE, Andrew AL. Henrici's transformation and its application to the computation of derivatives of eigensystems. Journal of Computational and Applied Mathematics 1995; 59:1-8.
35. Brezinski C, Redivo Zaglia M. Extrapolation Methods: Theory and Practice. North-Holland: Amsterdam, 1991.
36. Tan RCE. Computing derivatives of eigensystems by the vector ε-algorithm. IMA Journal of Numerical Analysis 1987; 7:485-494.
37. Tan RCE. Computing derivatives of eigensystems by the topological ε-algorithm. Applied Numerical Mathematics 1987; 3:539-550.
38. Andrew AL. Iterative computation of derivatives of eigenvalues and eigenvectors. Journal of the Institute of Mathematics and its Applications 1979; 24:209-218.
39. Andrew AL. Eigenvectors of certain matrices. Linear Algebra and its Applications 1973; 7:151-162.