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) always have a unique positive definite solution. Iterative methods are proposed to compute the unique positive definite solution. We show that the iterative methods are more effective as 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.

Full text


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

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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