ANZIAM J. 46(E) pp.C1327--C1335, 2006.

Some remarks on the inverse eigenvalue problem for real symmetric Toeplitz matrices

N. Li

(Received 8 October 2004, revised 17 November 2005)

Abstract

A theorem about the bounds of solutions of the Toeplitz Inverse Eigenvalue Problem is introduced and proved. It can be applied to make a better starting generator for iterative numerical methods. This application is tested through a short Mathematica program. Also an optimisation method for solving the Toeplitz Inverse Eigenvalue Problem with a global convergence property is presented. A global convergence theorem is proved.

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Authors

N. Li
Mathematics Discipline, Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, Melbourne, Australia. mailto:nli@swin.edu.au

Published January 6, 2006. ISSN 1446-8735

References

  1. Chu, M. T. and Golub, G. H., Structured inverse eigenvalue problems, Acta Numerica, 2001.
  2. Chu, M. T., On a Newton method for the inverse Toeplitz eigenvalue problem. http://www4.ncsu.edu/ mtchu/Research/Papers/itep.ps
  3. Garcia, C. B. and Li, T. Y., On the numbers of solutions to polynomial systems of equations, SIAM J. Numer. Anal., 17(4), 1980, 540--546.
  4. Gill, P., Murray, W. and Wright, M., Practical Optimization, page 100, Academic Press, 1981.
  5. Friedland, S., Nocedal, J. and Overton, M. L., The formulaton and analysis of numerical methods for inverse eigenvalue problems, SIAM J. Numer. Anal., 24(3), 1987, 634--667.
  6. Friedland, S., Inverse eigenvalue problems for symmetric Toeplitz matrices, SIAM J. Matrix Anal. Appl., 13(4), 1992, 1142--1153.
  7. Landau, H. J., The inverse eigenvalue problem for real symmetric Toeplitz matrices, J. Amer. Math. Soc., 7(3), 1994, 749--767.
  8. Laurie, D. P., A numerical approach to the inverse Toeplitz eigenproblem, SIAM J. Sci. Stat. Comput., 9(2), 1988, 401--405.
  9. Laurie, D. P., Initial values for the inverse Toeplitz eigenvalue problem, SIAM J. Sci. Stat. Comput., 22, 2001, 2239--2255.
  10. Li, N., A matrix inverse eigenvalue problem and its application, Linear Algebra And Its Applications, 266, 1997, 143--152.
  11. Li, N., and Chu, K-W. E., Designing the Hopfield neural network via pole assignment, Internat. J. Systems Sci., 25, 1994, 669--681.
  12. Li, N., An Inverse eigenvalue problem and feedback control, In May, R. L., Fitz-Gerald, G. F. and Grundy, I. H., editors, Proceedings of the 4th Biennial Engineering Mathematics and Applications Conference, Melbourne, Australia, pages 183--186. RMIT, 2000.
  13. Osborne, M. R., Nonlinear least squares---the Levenberg--Marquardt algorithm revisited. J. Austral. Math. Soc. Ser. B, 1976, 343--357.
  14. Powell, M. J. D., A hybrid method for nonlinear equations. In P. Rabinowitz, editor, Numerical methods for nonlinear equations, pages 87--114, Gordon and Breach, London, 1970.
  15. Trench, W. F., Numerical solution of the inverse eigenvalue problem for real symmetric Toeplitz matrices, SIAM J. Sci. Comput., 18(6), 1997, 1722--1736.