- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 685753, 7 pages
On the Iterative Method for the System of Nonlinear Matrix Equations
Faculty of Science and Arts, Qassim University, Buraydah 51431, P.O. Box 1162, Saudi Arabia
Received 15 November 2012; Accepted 20 February 2013
Academic Editor: Mohammad T. Darvishi
Copyright © 2013 Asmaa M. Al-Dubiban. 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.
- W. N. Anderson Jr., T. D. Morley, and G. E. Trapp, “Positive solutions to ,” Linear Algebra and Its Applications, vol. 134, pp. 53–62, 1990.
- P. Lancaster and L. Rodman, Algebraic Riccati Equations, Oxford Science, Oxford, UK, 1995.
- B. Meini, “Matrix equations and structures: efficient solution of special discrete algebraic Riccati equations,” in Proceedings of the WLSSCOO, 2000.
- A. M. Aldubiban, Iterative algorithms for computing the positive definite solutions for nonlinear matrix equations [Ph.D. thesis], Riyadh University for Girls, Riyadh, Saudi Arabia, 2008.
- A. M. Aldubiban, “Iterative algorithm for solving a system of nonlinear matrix equations,” Journal of Applied Mathematics, vol. 2012, Article ID 461407, 15 pages, 2012.
- O. L. V. Costa and R. P. Marques, “Maximal and stabilizing Hermitian solutions for discrete-time coupled algebraic Riccati equations,” Mathematics of Control, Signals, and Systems, vol. 12, no. 2, pp. 167–195, 1999.
- O. L. V. Costa and J. C. C. Aya, “Temporal difference methods for the maximal solution of discrete-time coupled algebraic Riccati equations,” Journal of Optimization Theory and Applications, vol. 109, no. 2, pp. 289–309, 2001.
- A. Czornik and A. Swierniak, “Lower bounds on the solution of coupled algebraic Riccati equation,” Automatica, vol. 37, no. 4, pp. 619–624, 2001.
- A. Czornik and A. Świerniak, “Upper bounds on the solution of coupled algebraic Riccati equation,” Journal of Inequalities and Applications, vol. 6, no. 4, pp. 373–385, 2001.
- R. Davies, P. Shi, and R. Wiltshire, “Upper solution bounds of the continuous and discrete coupled algebraic Riccati equations,” Automatica, vol. 44, no. 4, pp. 1088–1096, 2008.
- F. Ding and T. Chen, “Iterative least-squares solutions of coupled Sylvester matrix equations,” Systems & Control Letters, vol. 54, no. 2, pp. 95–107, 2005.
- F. Ding and T. Chen, “Gradient based iterative algorithms for solving a class of matrix equations,” IEEE Transactions on Automatic Control, vol. 50, no. 8, pp. 1216–1221, 2005.
- F. Ding and T. Chen, “On iterative solutions of general coupled matrix equations,” SIAM Journal on Control and Optimization, vol. 44, no. 6, pp. 2269–2284, 2006.
- F. Ding, P. X. Liu, and J. Ding, “Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 41–50, 2008.
- B. Hashemi and M. Dehghan, “The interval Lyapunov matrix equation: analytical results and an efficient numerical technique for outer estimation of the united solution set,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 622–633, 2012.
- J. Ding, Y. Liu, and F. Ding, “Iterative solutions to matrix equations of the form ,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3500–3507, 2010.
- A. Liu and G. Chen, “On the Hermitian positive definite solutions of nonlinear matrix equation ,” Mathematical Problems in Engineering, vol. 2011, Article ID 163585, 18 pages, 2011.
- L. Xie, J. Ding, and F. Ding, “Gradient based iterative solutions for general linear matrix equations,” Computers & Mathematics with Applications, vol. 58, no. 7, pp. 1441–1448, 2009.
- L. Xie, Y. Liu, and H. Yang, “Gradient based and least squares based iterative algorithms for matrix equations ,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 2191–2199, 2010.
- F. Yin and G.-X. Huang, “An iterative algorithm for the least squares generalized reflexive solutions of the matrix equations ,” Abstract and Applied Analysis, vol. 2012, Article ID 857284, 18 pages, 2012.
- W. Zhao, H. Li, X. Liu, and F. Xu, “Necessary and sufficient conditions for the existence of a Hermitian positive definite solution of a type of nonlinear matrix equations,” Mathematical Problems in Engineering, vol. 2009, Article ID 672695, 13 pages, 2009.
- H. Mukaidani, “Newton's method for solving cross-coupled sign-indefinite algebraic Riccati equations for weakly coupled large-scale systems,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 103–115, 2007.
- H. Mukaidani, S. Yamamoto, and T. Yamamoto, “A Numerical algorithm for finding solution of cross-coupled algebraic Riccati equations,” IEICE Transactions, vol. 91, pp. 682–685, 2008.
- H. Mukaidani, “Numerical computation of cross-coupled algebraic Riccati equations related to -constrained LQG control problem,” Applied Mathematics and Computation, vol. 199, no. 2, pp. 663–676, 2008.
- I. G. Ivanov, “A method to solve the discrete-time coupled algebraic Riccati equations,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 34–41, 2008.
- I. G. Ivanov, “Stein iterations for the coupled discrete-time Riccati equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 12, pp. 6244–6253, 2009.
- R. Bhatia, Matrix Analysis, vol. 169 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1997.
- M. Fujii and Y. Seo, “Reverse inequalities of Araki, Cordes and Löwner-Heinz inequalities,” Nihonkai Mathematical Journal, vol. 16, no. 2, pp. 145–154, 2005.
- P. Lancaster, Theory of Matrices, Academic Press, New York, NY, USA, 1969.
- C.-H. Guo and N. J. Higham, “A Schur-Newton method for the matrix pth root and its inverse,” SIAM Journal on Matrix Analysis and Applications, vol. 28, no. 3, pp. 788–804, 2006.