About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 721346, 6 pages
http://dx.doi.org/10.1155/2014/721346
Research Article

A Generalized Inexact Newton Method for Inverse Eigenvalue Problems

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received 5 December 2013; Accepted 7 January 2014; Published 19 February 2014

Academic Editor: Chong Li

Copyright © 2014 Weiping Shen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. P. J. Brussard and P. W. Glaudemans, Shell Model Applications in Nuclear Spectroscopy, Elsevier, New York, NY, USA, 1977.
  2. C. I. Byrnes, “Pole placement by output feedback,” in Three Decades of Mathematics Systems Theory, vol. 135 of Lecture Notes in Control and Information Sciences, pp. 31–78, Springer, New York, NY, USA, 1989.
  3. M. T. Chu and G. H. Golub, “Structured inverse eigenvalue problems,” Acta Numerica, vol. 11, pp. 1–71, 2002.
  4. S. Elhay and Y. M. Ram, “An affine inverse eigenvalue problem,” Inverse Problems, vol. 18, no. 2, pp. 455–466, 2002. View at Publisher · View at Google Scholar · View at Scopus
  5. G. M. L. Gladwell, “Inverse problems in vibration,” Applied Mechanics Reviews, vol. 39, pp. 1013–1018, 1986.
  6. G. M. L. Gladwell, “Inverse problems in vibration II,” Applied Mechanics Reviews, vol. 49, pp. 25–34, 1996.
  7. O. Hald, On discrete and numerical Sturm-Liouville problems [Ph.D. thesis], Department Mathematics, New York University, New York, NY, USA, 1970.
  8. K. T. Joseph, “Inverse eigenvalue problem in structural design,” AIAA Journal, vol. 30, no. 12, pp. 2890–2896, 1992. View at Scopus
  9. N. Li, “A matrix inverse eigenvalue problem and its application,” Linear Algebra and Its Applications, vol. 266, no. 1–3, pp. 143–152, 1997. View at Scopus
  10. M. Müller, “An inverse eigenvalue problem: computing B-stable Runge-Kutta methods having real poles,” BIT, vol. 32, no. 4, pp. 676–688, 1992. View at Publisher · View at Google Scholar · View at Scopus
  11. R. L. Parker and K. A. Whaler, “Numerical methods for establishing solutions to the inverse problem of electromagnetic induction,” Journal of Geophysical Research, vol. 86, no. 10, pp. 9574–9584, 1981. View at Scopus
  12. J. Peinado and A. M. Vidal, “A new parallel approach to the Toeplitz inverse Eigenproblem using Newton-like methods,” in Vector and Parallel Processing—VECPAR 2000, Lecture Notes in Computer Science, pp. 355–368, Springer, Berlin, Germany, 2001.
  13. M. S. Ravi, J. Rosenthal, and X. A. Wang, “On decentralized dynamic pole placement and feedback stabilization,” IEEE Transactions on Automatic Control, vol. 40, no. 9, pp. 1603–1614, 1995. View at Publisher · View at Google Scholar · View at Scopus
  14. W. F. Trench, “Numerical solution of the inverse eigenvalue problem for real symmetric toeplitz matrices,” SIAM Journal on Scientific Computing, vol. 18, no. 6, pp. 1722–1736, 1997. View at Scopus
  15. S. Friedland, J. Nocedal, and M. L. Overton, “The formulation and analysis of numerical methods for inverse eigenvalue problems,” SIAM Journal on Numerical Analysis, vol. 24, no. 3, pp. 634–667, 1987. View at Scopus
  16. S. F. Xu, An Introduction to Inverse Algebric Eigenvalue Problems, Peking University Press, 1998.
  17. Z.-J. Bai, R. H. Chan, and B. Morini, “An inexact Cayley transform method for inverse eigenvalue problems,” Inverse Problems, vol. 20, no. 5, pp. 1675–1689, 2004. View at Publisher · View at Google Scholar · View at Scopus
  18. R. H. Chan, H. L. Chung, and S.-F. Xu, “The inexact Newton-like method for inverse eigenvalue problem,” BIT Numerical Mathematics, vol. 43, no. 1, pp. 7–20, 2003. View at Publisher · View at Google Scholar · View at Scopus
  19. R. H. Chan, S.-F. Xu, and H.-M. Zhou, “On the convergence rate of a quasi-Newton method for inverse eigen value problems,” SIAM Journal on Numerical Analysis, vol. 36, no. 2, pp. 436–441, 1999. View at Scopus
  20. W. P. Shen, C. Li, and X. Q. Jin, “A Ulm-like method for inverse eigenvalue problems,” Applied Numerical Mathematics, vol. 61, no. 3, pp. 356–367, 2011. View at Publisher · View at Google Scholar · View at Scopus
  21. W. N. Kublanovskaja, “On an approach to the eigenvalue problem,” Zapiski. Nauk. Sem. Lenin. Otdel. Math. Inst. (Akad. Nauk SSSR V. A. Steklova), pp. 138–149, 1970.
  22. D. Sun and J. Sun, “Strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems,” SIAM Journal on Numerical Analysis, vol. 40, no. 6, pp. 2352–2367, 2002. View at Publisher · View at Google Scholar · View at Scopus
  23. F. Clark, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, NY, USA, 1983.
  24. R. T. Rockafeliarc, Convex Analysis, Princeton University Press, Princeton, NJ, USA, 1970.
  25. R. Miffin, “Semismooth and semiconvex functions in constrained optimization,” SIAM Journal on Control and Optimization, vol. 15, pp. 957–972, 1977.
  26. L. Qi and J. Sun, “A nonsmooth version of Newton's method,” Mathematical Programming, vol. 58, no. 1–3, pp. 353–367, 1993. View at Publisher · View at Google Scholar · View at Scopus
  27. F. A. Potra, L. Qi, and D. Sun, “Secant methods for semismooth equations,” Numerische Mathematik, vol. 80, no. 2, pp. 305–324, 1998. View at Scopus
  28. G. H. Gloub and C. F. Van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, Md, USA.
  29. R. W. Freund and N. M. Nachtigal, “QMR: a quasi-minimal residual method for non-Hermitian linear systems,” Numerische Mathematik, vol. 60, no. 1, pp. 315–339, 1991. View at Publisher · View at Google Scholar · View at Scopus