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Iterative Computation of Higher Derivatives of Repeated Eigenvalues and the Corresponding Eigenvectors

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Linear Operators and Matrices

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 130))

Abstract

This paper is concerned with iterative methods for computing partial derivatives of eigenvalues and eigenvectors of matrix-valued functions of several real variables. First, an analysis is given of a previously announced method which computes mixed partial derivatives of simple eigenvalues and the corresponding eigenvectors and also second order mixed partial derivatives of repeated eigenvalues. Next a new method for computing third order partial derivatives of repeated eigenvalues and second order derivatives of the corresponding eigenvectors is presented and its key properties are established. Efficiency and numerical stability are considered as well as theoretical convergence.

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References

  1. Alvin, K. F., Efficient computation of eigenvector sensitivities for structural dynamics, AIAA J. 35 (1997), 1760–1766.

    Article  MATH  Google Scholar 

  2. Andrew, A. L., Iterative computation of derivatives of eigenvalues and eigenvectors, J. Inst. Math. Appl. 24 (1979), 209–218.

    Article  MathSciNet  MATH  Google Scholar 

  3. Andrew, A. L., Chu, K.-W. E., Lancaster, P., Derivatives of eigenvalues and eigenvectors of matrix functions, SIAM J. Matrix Anal. Appl. 14 (1993), 903–926.

    Article  MathSciNet  MATH  Google Scholar 

  4. Andrew, A. L., Chu, K.-W. E., Lancaster, P., On the numerical solution of nonlinear eigenvalue problems, Computing 55 (1995), 91–111.

    Article  MathSciNet  MATH  Google Scholar 

  5. Andrew, A. L., Tan, R. C.-E., Computation of derivatives of repeated eigenvalues and the corresponding eigenvectors of symmetric matrix pencils, SIAM J. Matrix Anal. Appl. 20 (1998), 78–100.

    Article  MathSciNet  MATH  Google Scholar 

  6. Andrew, A. L., Tan, R. C.-E., Computation of mixed partial derivatives of eigenvalues and eigenvectors by simultaneous iteration, Comm. Numer. Methods Engrg. 15 (1999), 641–649.

    Article  MathSciNet  MATH  Google Scholar 

  7. Andrew, A. L., Tan, R. C.-E., Iterative computation of derivatives of repeated eigenvalues and the corresponding eigenvectors, Numer. Linear Algebra Appl. 7 (2000), 151–167.

    Article  MathSciNet  MATH  Google Scholar 

  8. Baumgärtel, H., Analytic perturbation theory for matrices and operators, Birkhauser, Basel 1985.

    MATH  Google Scholar 

  9. Brandon, J. A., Second order design sensitivities to asses the applicability of sensitivity analysis, AIAA J 29 (1991), 135–139.

    Article  Google Scholar 

  10. Brezinski, C., Redivo Zaglia, M., Extrapolation methods: theory and practice, North-Holland, Amsterdam 1991.

    MATH  Google Scholar 

  11. Dai, H., Lancaster, P., Numerical methods for finding multiple eigenvalues of matrices depending on parameters, Numer. Math. 76 (1997), 189–208.

    Article  MathSciNet  MATH  Google Scholar 

  12. Friswell, M. I., Calculation of second and higher order derivatives, J. Guidance Control Dyn. 18 (1995), 919–921.

    Article  MATH  Google Scholar 

  13. Gohberg, I. C., Lancaster, P., Rodman, L., Matrices and indefinite scalar products, Birkhäuser, Basel 1983.

    MATH  Google Scholar 

  14. Haug, E. J., Choi, K. K., Komkov, V., Design sensitivity analysis of structural sys-tems,Academic Press, New York 1986.

    Google Scholar 

  15. Hryniv, R., Lancaster, P., On the perturbation of analytic matrix functions, Integral Equations Oper. Theory 34 (1999), 325–338.

    Article  MathSciNet  MATH  Google Scholar 

  16. Lancaster, P., Tismenetsky, M., The theory of matrices,2nd ed., Academic Press, New York 1985.

    Google Scholar 

  17. Mills-Curran, W. C., Calculation of eigenvector derivatives for structures with repeated eigenvalues, AIAA J. 26 (1988), 867–871.

    Article  MATH  Google Scholar 

  18. Mottershead, J. E., Friswell, M. I., Model updating in structural dynamics: a survey, J. Sound Vibration 167 (1993), 347–375.

    Article  MATH  Google Scholar 

  19. Sun, J.-G., Multiple eigenvalue sensitivity analysis, Linear Algebra Appl. 137/138 (1990), 183–211.

    Article  MathSciNet  Google Scholar 

  20. Tan, R. C.-E., Andrew, A. L., Computing derivatives of eigenvalues and eigenvectors by simultaneous iteration, IMA J. Numer. Anal. 9 (1989), 111–122.

    Article  MathSciNet  MATH  Google Scholar 

  21. Tan, R. C.-E., Andrew, A. L., Some numerical tests of algorithms for computing derivatives of eigenvalues and eigenvectors, Math. Res. Paper 98–23, La Trobe University, Melbourne 1999.

    Google Scholar 

  22. Vishik, M. I., Lyusternik, The solution of some perturbation problems for matrices and selfadjoint or non-selfadjoint differential equations I, Russian Math. Surveys 15 (1960), 1–73.

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, O., Zerva, A., Accelerated iterative procedure for calculating eigenvector derivatives, AIAA J. 35 (1997), 340–348.

    Article  MATH  Google Scholar 

  24. Zhang, Y.-Q., Wang, W.-L., Eigenvector derivatives of generalized nondefective eigenproblems with repeated eigenvalues, Trans. ASME J. Engrg. Gas Turbines Power 117 (1995), 207–212.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematics Department, La Trobe University, Bundoora, Victoria, 3083, Australia

    Alan L. Andrew & Roger C. E. Tan

  2. Mathematics Department, National University of Singapore, 119260, Singapore

    Alan L. Andrew & Roger C. E. Tan

Authors
  1. Alan L. Andrew
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  2. Roger C. E. Tan
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Editor information

Editors and Affiliations

  1. School of Mathematical Sciences Raymond and Beverly Sackler, Faculty of Exact Sciences Tel Aviv University, Ramat Aviv, 69978, Israel

    I. Gohberg

  2. Mathematik, Technische Universität Wien, Wiedner Hauptstrasse 8–10/1411, Wien, 1040, Austria

    H. Langer

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© 2002 Springer Basel AG

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Andrew, A.L., Tan, R.C.E. (2002). Iterative Computation of Higher Derivatives of Repeated Eigenvalues and the Corresponding Eigenvectors. In: Gohberg, I., Langer, H. (eds) Linear Operators and Matrices. Operator Theory: Advances and Applications, vol 130. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8181-4_6

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  • DOI: https://doi.org/10.1007/978-3-0348-8181-4_6

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9467-8

  • Online ISBN: 978-3-0348-8181-4

  • eBook Packages: Springer Book Archive

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