Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 4 (2009), 179 -- 190

POSITIVE DEFINITE SOLUTION OF TWO KINDS OF NONLINEAR MATRIX EQUATIONS

Xuefeng Duan, Zhenyun Peng and Fujian Duan

Abstract. Based on the elegant properties of the Thompson metric, we prove that the following two kinds of nonlinear matrix equations X=Σi=1m  Ai* XδiAi and X=Σi=1m (Ai* XAi)δi,   (0<|δi|<1) MORE WE EFFECTIVE POSITIVE A THAT HAVE AS ITERATIVE δ="max{|δi|,  i=1,2, ..., m} decreases. Perturbation bounds for the unique positive definite solution are derived in the end.

2000 Mathematics Subject Classification: 15A24; 65H05.
Keywords: Nonlinear matrix equation; Positive definite solution; Iterative method; Perturbation bound; Thompson metric.

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Acknowledgment. The work was supported by National Natural Science Foundation of China (10861005), and Provincial Natural Science Foundation of Guangxi (0991238).

References

  1. W. N. Anderson, Jr., T. D. Morley, G. E. Trapp, Positive solutions to X=A-BX-1B* , Linear Algebra Appl. 134 (1990), 53-62. MR1060009(91c:47031). Zbl 0702.15009.

  2. M. S. Chen, S. F. Xu, Perturbation analysis of the Hermitian positive definite solution of the matrix equation X-A* X-2A=I, Linear Algebra Appl. 394 (2005), 39-51. MR2100575(2005g:15026). Zbl 1063.15010.

  3. X. F. Duan, A. P. Liao, B. Tang, On the nonlinear matrix equation X-\sum\limitsi=1mAi* Xδ iAi=Q, Linear Algebra Appl. 429 (2008), 110-121. MR2419144(2009c:15019). Zbl 1148.15012.

  4. X. F. Duan, A. P. Liao, On the nonlinear matrix equation X+A* X-qA=Q(q≥ 1), Math. Comput. Mod. 49 (2009), 936-945. MR2495010. Zbl 1165.15302.

  5. X. F. Duan, A. P. Liao, On Hermitian positive definite solution of the matrix equation X-\sum\limitsi=1mA*iXrAi=Q , J. Comput. Appl. Math. 229 (2009), 27-36. MR2522496. Zbl 1170.15005.

  6. S. P. Du, J. C. Hou, Positive definite solutions of operator equations Xm+ A*X-nA=I, Linear and Multilinear Algebra 51 (2003),163-173. MR1976862(2004b:47019). Zbl 1046.47019.

  7. S.M. El-Sayed, Andre C.M. Ran, On an iterative method for solving a class of nonlinear matrix equations, SIAM J. Matrix Anal. Appl. 23 (2001), 632-645. MR1896810(2002m:15023). Zbl 1002.65061.

  8. C. H. Guo, P. Lancaster, Iterative solution of two matrix equations, Math. Comput. 68 (1999), 1589-1603. MR16511757(99m:65061). Zbl 0940.65036.

  9. V. I. Hasanov, Positive definite solutions of the matrix equations X± ATX-qA=Q, Linear Algebra Appl. 404 (2005), 166-182. MR2149658(2006c:15026). Zbl 1078.15012.

  10. M. Huang, C. Huang, T. Tsai, Applications of Hilbert's projective metric to a class of positive nonlinear operators, Linear Algebra Appl. 413 (2006), 202-211. MR2202103(2007i:47062). Zbl 1092.47053 .

  11. I. G. Ivanov, V. I. Hasanov, F. Uhlig, Improved methods and starting values to solve the matrix equations X± A* X-1A=I iteratively, Math. comput. 74 (2005), 263-278. MR2085410 (2005h:65083). Zbl 1058.65051.

  12. L. V. Kantorovich, G. P. Akilov, Functional analysis, Pergamon Press, Elmsford, NY, 1982.

  13. Y. Lim, Solving the nonlinear matrix equation X=Q+\sum\limitsi=1mAi* Xδ iAi via a contraction principle, Linear Algebra Appl. 430 (2009), 1380-1383. MR2489400(2009j:15078). Zbl 1162.15008 .

  14. R. D. Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Memoirs of Amer. Math. Soc. 391, 1988. MR0961211(89m:47046). Zbl 0666.47028.

  15. J. H. Long, X. Y. Hu, L. Zhang, On the Hermitian positive definite solution of the nonlinear matrix equation X+A* X-1A+B* X-1B=I, Bull. Braz. Math. Soc. 39 (3), 371-386 (2008). MR2473853. Zbl 1175.65052.

  16. Andre C. M. Ran, M. C. B. Reurings, A. L. Rodman, A perturbation analysis for nonlinear selfadjoint operators, SIAM J. Matrix Anal. Appl. 28 (2006), 89-104. MR2218944(2007c:47082). Zbl 1105.47053.

  17. X. Q. Shi, F. S. Liu, H. Umoh, F. Gibson, Two kinds of Nonlinear matrix equations and their corresponding matrix sequences, Linear and Multilinear Algebra 52 (2004), 1-15. MR2030786(2005a:1506). Zbl 1057.15016 .

  18. A. C. Thompson, On certain contraction mappings in a partially ordered vector space, Pro. Amer. Math. Soc. 14 (1963), 438-443. MR0149237(266727). Zbl 0147.34903.

  19. X. X. Zhan, J. J. Xie, On the matrix equation X+ATX-1A=I, Linear Algebra Appl. 247 (1996), pp.337-345. MR1412759(97k:15036). Zbl 0863.15005.




Xuefeng Duan
College of Mathematics and Computational Science,
Guilin University of Electronic Technology,
Guilin 541004, P.R. China.
and
 Department of Mathematics,
Shanghai University,
Shanghai 200444, P.R. China.
e-mail: duanxuefenghd@yahoo.com.cn; duanxuefeng@guet.edu.cn
http://www2.gliet.edu.cn/dept7/last/TeacherDetail.Asp?TeacherID=369


Zhenyun Peng
College of Mathematics and Computational Science,
Guilin University of Electronic Technology,
Guilin 541004, P.R. China.

Fujian Duan
College of Mathematics and Computational Science,
Guilin University of Electronic Technology,
Guilin 541004, P.R. China.


http://www.utgjiu.ro/math/sma