DOI QR코드

DOI QR Code

ON THE NONLINEAR MATRIX EQUATION $X+\sum_{i=1}^{m}A_i^*X^{-q}A_i=Q$(0<q≤1)

  • Yin, Xiaoyan ;
  • Wen, Ruiping ;
  • Fang, Liang
  • Received : 2013.03.27
  • Published : 2014.05.31

Abstract

In this paper, the nonlinear matrix equation $$X+\sum_{i=1}^{m}A_i^*X^{-q}A_i=Q(0<q{\leq}1)$$ 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

References

  1. 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. https://doi.org/10.1016/j.laa.2005.01.010
  2. 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.
  3. 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. https://doi.org/10.1016/j.mcm.2009.12.021
  4. 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. https://doi.org/10.1016/S0024-3795(01)00508-0
  5. 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. https://doi.org/10.1016/S0024-3795(02)00490-1
  6. 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.
  7. J. R. Rice, A theory of condition, SIAM J. Numer. Anal. 3 (1966), no. 2, 287-310. https://doi.org/10.1137/0703023
  8. 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. https://doi.org/10.1016/j.laa.2009.05.010
  9. X. Zhan, Computing the extremal positive definite solutions of a matrix equation, SIAM J. Scient. Comput. 17 (1996), no. 5, 1167-1174. https://doi.org/10.1137/S1064827594277041
  10. 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. https://doi.org/10.1016/j.amc.2010.04.041
  11. 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. https://doi.org/10.1016/j.cam.2012.05.012
  12. V. I. Hasanov, Positive definite solutions of the matrix equations $X{\pm}A^*X^{-q}A=Q$, Linear Algebra Appl. 404 (2005), 166-182. https://doi.org/10.1016/j.laa.2005.02.024
  13. 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. https://doi.org/10.1016/j.laa.2005.06.026
  14. 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. https://doi.org/10.1016/j.cam.2005.06.007
  15. P. Lancaster and L. Rodman, Algebraic Riccati Equations, Oxford Science Publishers, Oxford, 1995.
  16. 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.
  17. 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. https://doi.org/10.1016/j.laa.2008.10.034
  18. 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. https://doi.org/10.1137/040617650
  19. T. Furuta, Operator inequalities associated with Holder-Mccarthy and Kantorovich inequalities, J. Inequal. Appl. 6 (1998), no. 2, 137-148.
  20. 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. https://doi.org/10.1016/S0024-3795(02)00661-4
  21. 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.
  22. 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. https://doi.org/10.1002/nla.510
  23. 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. https://doi.org/10.1016/j.amc.2011.10.026
  24. R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics, Springer, Verlag, 1997.
  25. 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. https://doi.org/10.1137/0707049
  26. 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. https://doi.org/10.1016/j.cam.2008.10.018
  27. 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. https://doi.org/10.1016/j.laa.2008.02.014
  28. 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. https://doi.org/10.1016/0024-3795(93)90115-5

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