JOURNAL BROWSE
Search
Advanced SearchSearch Tips
ON THE NONLINEAR MATRIX EQUATION (0<q≤1)
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
ON THE NONLINEAR MATRIX EQUATION (0<q≤1)
Yin, Xiaoyan; Wen, Ruiping; Fang, Liang;
  PDF(new window)
 Abstract
In this paper, the nonlinear matrix equation < is investigated. Some necessary conditions and sufficient conditions for the existence of positive definite solutions for the matrix equation are derived. Two iterative methods for the maximal positive definite solution are proposed. A perturbation estimate and an explicit expression for the condition number of the maximal positive definite solution are obtained. The theoretical results are illustrated by numerical examples.
 Keywords
nonlinear matrix equation;positive definite solution;perturbation estimate;condition number;
 Language
English
 Cited by
1.
Convergence analysis of some iterative methods for a nonlinear matrix equation, Computers & Mathematics with Applications, 2016, 72, 4, 1164  crossref(new windwow)
2.
Positive definite solutions and perturbation analysis of a class of nonlinear matrix equations, Journal of Applied Mathematics and Computing, 2017, 53, 1-2, 245  crossref(new windwow)
 References
1.
R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics, Springer, Verlag, 1997.

2.
B. L. Buzbee, G. H. Golub, and C. W. Nilson, On direct methods for solving Poisson's equations, SIAM J. Numer. Anal. 7 (1970), no. 4, 627-656. crossref(new window)

3.
X. F. Duan and A. P. Liao, On Hermitian positive definite solution of the matrix equation $X-{\Sigma}^m_{i=1}A_i^*X^rA_i=Q$, J. Comput. Appl. Math. 229 (2009), no. 1, 27-36. crossref(new window)

4.
X. F. Duan, C. M. Li, and A. P. Liao, Solutions and perturbation analysis for the nonlinear matrix equation $X+{\Sigma}^m_{i=1}A_i^*X^{-1}A_i=I$, Appl. Math. Comput. 218 (2011), no. 8, 4458-4466. crossref(new window)

5.
X. F. Duan, A. P. Liao, and B. Tang, On the nonlinear matrix equation $X-{\Sigma}^m_{i=1}A_i^*X^{{\delta}_i}A_i=Q$, Linear Algebra Appl. 429 (2008), no. 1, 110-121. crossref(new window)

6.
J. C. Engwerda, On the existence of a positive definite solution of the matrix equation $X+A^TX^{-1}A=I$, Linear Algebra Appl. 194 (1993), 91-108. crossref(new window)

7.
T. Furuta, Operator inequalities associated with Holder-Mccarthy and Kantorovich inequalities, J. Inequal. Appl. 6 (1998), no. 2, 137-148.

8.
C. H. Guo, Y. C. Kuo, andW. W. Lin, Numerical solution of nonlinear matrix equations arising from Green's function calculations in nano research, J. Comput. Appl. Math. 236 (2012), no. 17, 4166-4180. crossref(new window)

9.
V. I. Hasanov, Positive definite solutions of the matrix equations $X{\pm}A^*X^{-q}A=Q$, Linear Algebra Appl. 404 (2005), 166-182. crossref(new window)

10.
V. I. Hasanov and S. M. El-Sayed, On the positive definite solutions of nonlinear matrix equation $X+A^*X^{-\delta}A=Q$, Linear Algebra Appl. 412 (2006), no. 2, 154-160. crossref(new window)

11.
Y. He and J. Long, On the Hermitian positive definite solution of the nonlinear matrix equation $X+{\Sigma}^m_{i=1}A_i^*iX^{-1}A_i=I$, Appl. Math. Comput. 216 (2010), no. 12, 3480-3485. crossref(new window)

12.
I. G. Ivanov, On positive definite solutions of the family of matrix equations $X+A^*X^{-n}A=Q$, J. Comput. Appl. Math. 193 (2006), no. 1, 277-301. crossref(new window)

13.
P. Lancaster and L. Rodman, Algebraic Riccati Equations, Oxford Science Publishers, Oxford, 1995.

14.
J. Li and Y. H. Zhang, Perturbation analysis of the matrix equation $X-A^*X^{-p}A=Q$, Linear Algebra Appl. 431 (2009), no. 9, 936-945.

15.
Y. Lim, Solving the nonlinear matrix equation X = $Q+{{\Sigma}_{i=1}^m}M_iX^{{\delta}_i}M_i^*$ via a contraction principal, Linear Algebra Appl. 430 (2009), no. 4, 1380-1383. crossref(new window)

16.
W. W. Lin and S. F. Xu, Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations, SIAM J. Matrix Anal. Appl. 28 (2006), no. 1, 26-39. crossref(new window)

17.
X. G. Liu and H. Gao, On the positive definite solutions of the matrix equation $X^s{\pm}A^*X^{-t}A=I_n$, Linear Algebra Appl. 368 (2003), 83-79. crossref(new window)

18.
B. Meini, Efficient computation of the extreme solutions of $X+A^*X^{-1}A=Q$ and $X-A^*X^{-1}A=Q$, Math. Comput. 71 (2002), no. 239, 1189-1204.

19.
Z. Y. Peng and S. M. El-Sayed, On positive definite solution of a nonlinear matrix equation, Numer. Linear Algebra Appl. 14 (2007), no. 2, 99-113. crossref(new window)

20.
Z. Y. Peng, S. M. El-Sayed, and X. L. Zhang, Iterative methods for the extremal positive definite solution of the matrix equation $X+A^*X^{-{\alpha}}A=Q$, J. Comput. Appl. Math. 2000 (2007), no. 2, 520-527.

21.
C. M. Ran and C. B. R. Martine, On the nonlinear matrix equation $X+A^*F(X)A=Q$ : solutions and perturbation theory, Linear Algebra Appl. 346 (2002), 15-26. crossref(new window)

22.
J. R. Rice, A theory of condition, SIAM J. Numer. Anal. 3 (1966), no. 2, 287-310. crossref(new window)

23.
A. M. Sarhan, N. M. El-Shazy, and E. M. Shehata, On the existence of extremal positive definite solutions of the nonlinear matrix equation $X^r+{\Sigma}_{i=1}^mA_i^*X^{{\delta}_i}Ai=I$, Math. Comput. Model. 51 (2010), no. 9-10, 1107-1117. crossref(new window)

24.
S. M. El-Sayed and M. G. Petkov, Iterative methods for nonlinear matrix equation $X+A^*X^{-\alpha}A=I$, Linear Algebra Appl. 403 (2005), no. 1, 45-52. crossref(new window)

25.
J. G. Sun and S. F. Xu, Perturbation analysis of the maximal solution of the matrix equation $X+A^*X^{-1}A=P.{\Pi}$, Linear Algebra Appl. 362 (2003), 211-228. crossref(new window)

26.
J.Wang, Y. H. Zhang, and B. R. Zhu, On Hermitian positive definite solutions of matrix equation $X+A^*X^{-q}A=I(q>0)$, Math. Num. Sin. 26 (2004), no. 1, 61-72.

27.
X. Y. Yin, S. Y. Liu, and L.Fang, Solutions and perturbation estimates for the matrix equation $X^s+A^*X^{-t}A=Q$, Linear Algebra Appl. 431 (2009), no. 9, 1409-1421. crossref(new window)

28.
X. Zhan, Computing the extremal positive definite solutions of a matrix equation, SIAM J. Scient. Comput. 17 (1996), no. 5, 1167-1174. crossref(new window)