Korean J. Math. Vol. 22 No. 1 (2014) pp.123-131
DOI: https://doi.org/10.11568/kjm.2014.22.1.123

Perturbation anaysis for the matrix equation $X=I-A^{*}X^{-1}A+B^{*}X^{-1}B$

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Published: Mar 27, 2014
Mathematics Subject Classification:
15A24, 65F35

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Hosoo Lee
Louisiana State University

Abstract

The purpose of this paper is to study the perturbation analysis of the matrix equation $X=I-A^{*}X^{-1}A+B^{*}X^{-1}B.$ Based on the matrix differentiation, we give a precise perturbation bound for the positive definite solution. A numerical example is presented to illustrate the shrpness of the perturbation bound.


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References

[1] P. Benner and H. Fabbender, On the solution of the rational matrix equation $Xu2009=u2009Qu2009+u2009LX^{-1} L^{*}$, EURASIP J. Adv. Signal Pro. 1 (2007), 1–10. Google Scholar

[2] M. Berzig, X. Duan and B. Samet, Positive definite solution of the matrix equation $X=Q-A^{*} X^{-1} A+B^{*}X^{-1} B$ via Bhaskar-Lakshmikantham fixed point theorem, Math. Sci. (Springer) 6 (2012), Art. 27. Google Scholar

[3] T.G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006), 13791393. Google Scholar

[4] X. Duan and Q. Wang, Perturbation analysis for the matrix equation $X-sum_{i=1}^{m}A_{i}^{*}XA_{i}+sum_{j=1}^{n}B_{j}^{*}XB_{j}=I$, J. Appl. Math. 2012 (2012), Art. ID 784620. Google Scholar

[5] J.C. Engwerda, A.C.M. Ran, and A.L. Rijkeboer, Necessary and sufficient con- ditions for the existence of a positive definite solution of the matrix equation Google Scholar

[6] $Xu2009+u2009A^{*} X^{-1} Au2009=u2009Q.$., Linear Algebra Appl. 186 (1993), 255–275. Google Scholar

[7] B.R. Fang, J.D. Zhou, and Y.M. Li, Matrix Theory, Tsinghua University Press, Beijing, China, 2006. Google Scholar

[8] A. Ferrante and B.C. Levy, Hermitian solutions of the equations $Xu2009=u2009Qu2009+u2009NX^{-1} N^{*}$, Linear Algebra Appl. 247 (1996), 359–373. Google Scholar

[9] V.I. Hasanov, Positive definite solutions of the matrix equations $Xu2009pm A^{*} X^{-q} Au2009=u2009Q$, Linear Algebra Appl. 404 (2005), 166–182. Google Scholar

[10] T.P. Minka, Old and new matrix algebra useful for statistics, December 2000. Notes. Google Scholar

[11] X. Zhan, Computing the extreme positive definite solutions of a matrix equation, SIAM J. Sci. Comput. 17 (1996), 632–645. Google Scholar