DOI QR코드

DOI QR Code

NEWTON'S METHOD FOR SYMMETRIC AND BISYMMETRIC SOLVENTS OF THE NONLINEAR MATRIX EQUATIONS

  • Han, Yin-Huan (School of Mathematics and Physics, Qingdao University of Science and Technology) ;
  • Kim, Hyun-Min (Department of Mathematics, Pusan National University)
  • Received : 2010.08.18
  • Published : 2013.07.01

Abstract

One of the interesting nonlinear matrix equations is the quadratic matrix equation defined by $$Q(X)=AX^2+BX+C=0$$, where X is a $n{\times}n$ unknown real matrix, and A, B and C are $n{\times}n$ given matrices with real elements. Another one is the matrix polynomial $$P(X)=A_0X^m+A_1X^{m-1}+{\cdots}+A_m=0,\;X,\;A_i{\in}\mathbb{R}^{n{\times}n}$$. Newton's method is used to find the symmetric and bisymmetric solvents of the nonlinear matrix equations Q(X) and P(X). The method does not depend on the singularity of the Fr$\acute{e}$chet derivative. Finally, we give some numerical examples.

Keywords

quadratic matrix equation;matrix polynomial;solvent;Newton's method;iterative algorithm;symmetric;bisymmetric

References

  1. J. S. Arora, Introduction to Optimum Design, McGraw-Hill Book Co., New York, 19-th edition, 1989.
  2. G. J. Davis, Numerical solution of a quadratic matrix equation, SIAM J. Sci. Statist. Comput. 2 (1981), no. 2, 164-175. https://doi.org/10.1137/0902014
  3. G. J. Davis, Algorithm 598: An algorithm to compute solvents of the matrix equation $AX^{2}$ + BX + C = 0, ACM Trans. Math. Software 9 (1983), no. 2, 246-254. https://doi.org/10.1145/357456.357463
  4. C.-H. Guo and A. J. Laub, On the iterative solution of a class of nonsymmetric algebraic equations, SIAM J. Matrix Anal. Appl. 22 (2000), no. 2, 376-391. https://doi.org/10.1137/S089547989834980X
  5. N. J. Higham and H.-M. Kim, Numerical analysis of a quadratic matrix equation, IMA J. Numer. Anal. 20 (2000), no. 4, 499-519. https://doi.org/10.1093/imanum/20.4.499
  6. N. J. Higham and H.-M. Kim, Solving a quadratic matrix equation by Newton's method with exact line searches, SIAM J. Matrix Anal. Appl. 23 (2001), no. 2, 303-316. https://doi.org/10.1137/S0895479899350976
  7. H.-M. Kim, Convergence of Newton's method for solving a class of quadratic matrix equations, Honam Math. J. 30 (2008), no. 2, 399-409. https://doi.org/10.5831/HMJ.2008.30.2.399
  8. W. Kratz and E. Stickel, Numerical solution of matrix polynomial equations by Newton's method, IMA J. Numer. Anal. 7 (1987), no. 3, 355-369. https://doi.org/10.1093/imanum/7.3.355
  9. G. M. Phillips and P. J. Taylor, Theory and Applications of Numerical Analysis, Academic Press. 2nd edition, London, 1996.
  10. D. X. Xie, L. Zhang, and X. Y. Hu, The solvability conditions for the inverse problem of bisymmetric nonnegative definite matrices, J. Comput. Math. 6 (2000), no. 6, 597-608.
  11. X. Y. Peng, X. Y. Hu, and L. Zhang, The bisymmetric solutions of the matrix equation $A_1X_1B_1+A_1X_1B_1+{\cdot}{\cdot}{\cdot}+A_lX_lB_l=C$ and its optimal approximation, Linear Algebra Appl. 426 (2007), no. 2-3, 583-595. https://doi.org/10.1016/j.laa.2007.05.034
  12. L. Zhao, X. Hu and L. Zhang, Least square solutions to AX = B for bisymmetric matrices under a central principal submatrix constrain and the optimal approximation, Linear Algebra Appl. 428 (2008), no. 4, 871-880. https://doi.org/10.1016/j.laa.2007.08.019

Cited by

  1. Diagonal update method for a quadratic matrix equation vol.283, 2016, https://doi.org/10.1016/j.amc.2016.02.016