Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지

Minimization Method for Solving a Quadratic Matrix Equation

  • Kim, Hyun-Min (Department of Mathematics, Pusan National University)
  • Received : 2006.01.23
  • Published : 2007.06.23
Download PDF
⟨ Previous Next ⟩

Abstract

We show how the minimization can be used to solve the quadratic matrix equation and then compare two different types of conjugate gradient method which are Polak and Ribi$\acute{e}$re version and Fletcher and Reeves version. Finally, some results of the global and local convergence are shown.

Keywords

Acknowledgement

Supported by : Pusan National University

References

  1. Mehiddin Al-Baali, Descent property and global convergence of the Fletcher-Reeves method with inexact line search, IMA J. Numer. Anal., 5(1985), 121-124. https://doi.org/10.1093/imanum/5.1.121
  2. King-wah Eric Chu, The solution of the matrix equation AXB - CXD = E and (Y A - DZ; Y C - BZ) = (E, F), Linear Algebra and Appl., 93(1987), 93-105. https://doi.org/10.1016/S0024-3795(87)90314-4
  3. H. P. Crowder and P. Wolfe. Linear convergence of the conjugate gradient method, IBM Journal of Research and Development, 16(1972), 431-433. https://doi.org/10.1147/rd.164.0431
  4. George J. Davis. Numerical solution of a quadratic matrix equation, SIAM J. Sci. Stat. Comput., 2(2)(1981), 164-175, 1981. https://doi.org/10.1137/0902014
  5. J. E. Dennis, Jr. and Robert B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1983, ISBN 0-13-627216-9.
  6. R. Fletcher and C. M. Reeves, Function minimization by conjugate gradients, Computer J., 7(1964), 149-154. https://doi.org/10.1093/comjnl/7.2.149
  7. Roger Fletcher, Practical Methods of Optimization, Wiley, Chichester, UK, 2nd edition, 1987, ISBN 0-471-91547-5.
  8. R. A. Frazer, W. J. Duncan, and A. R. Collar, Elementary Matrices and Some Applications to Dynamics and Di erential Equations, Cambridge University Press, 10th edition, 1938, 1963 printing.
  9. Philip E. Gill, Walter Murray, and Margaret H. Wright, Practical Optimization, Academic Press, London, 1981, ISBN 0-12-283952-8 (paperback).
  10. Nicholas J. Higham and Hyun-Min Kim, Numerical analysis of a quadratic matrix equation, IMA J. Numer. Anal., 20(4)(2000), 499-519. https://doi.org/10.1093/imanum/20.4.499
  11. Nicholas J. Higham and Hyun-Min Kim, Solving a quadratic matrix equation by New- ton's method with exact line searches, SIAM J. Matrix Anal. Appl., 23(2)(2001), 303-316. https://doi.org/10.1137/S0895479899350976
  12. Hyun-Min Kim, Numerical methods for solving a quadratic matrix equation, Ph.D. Thesis, University of Manchester, 2000.
  13. Peter Lancaster, Lambda-Matrices and Vibrating Systems, Pergamon Press, Oxford, 1966, xiii+196 pp.
  14. Jorge Nocedal, Conjugate gradient methods and nonlinear optimization In Loyce Adams and J. L. Nazareth, editors, Linear and Nonlinear Conjugate Gradient-Related Methods, pages 9-23, Philadelphia, PA, USA, 1996. Society for Industrial and Applied Mathematics.
  15. Jorge Nocedal and Stephen J.Wright, Numerical Optimization, Springer-Verlag, New York, 1999, xx+636 pp, ISBN 0-387-98793-2.
  16. E. Polak and G. Ribiere, Note sur la convergence des methodes de directions conjugees, Revue Fr. Inf. Rech. Oper., 16(R1)(1969), 35-43.
  17. M. J. D. Powell, A Fortran subroutine for solving systems of nonlinear algebraic equations, In Philip Rabinowitz, editor, Numerical Methods for Nonlinear Algebraic Equations, pages 115-161, London, 1970, Gordon and Breach Science.
  18. M. J. D. Powell. Nonconvex minimization calculations and the conjugate gradient method, In D. F. Griths, editor, Numerical Analysis, Dundee 1983, pages 122-141, Lecture Notes in Mathematics, 1066, Springer, Berlin-New York, 1984.
  19. G. Zoutendijk, Nonlinear programming, computational methods, In J. Abadie, editor, Integer and Nonlinear Programming, pages 37-86, North-Holland, Amsterdam, 1970.