Positive definite solution of two kinds of nonlinear matrix equations
Journal Title: Surveys in Mathematics and its Applications - Year 2009, Vol 4, Issue 0
Abstract
Based on the elegant properties of the Thompson metric, we prove that the following two kinds of nonlinear matrix equations X=<I>Σ<SUB>i=1</SUB><SUP>m</SUP></I> A<SUB>i</SUB><SUP>*</SUP> X<SUP>δ<SUB>i</SUB></SUP>A<SUB>i</SUB> and X=<I>Σ<SUB>i=1</SUB><SUP>m</SUP></I> (A<SUB>i</SUB><SUP>*</SUP> XA<SUB>i</SUB>)<SUP>δ<SUB>i</SUB></SUP>, (0<|δ<SUB>i</SUB>|<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{|δ<SUB>i</SUB>|, i=1,2, ..., m} decreases. Perturbation bounds for the unique positive definite solution are derived in the end.
Authors and Affiliations
Xuefeng Duan, Zhenyun Peng
Keywords
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