On nonlinear matrix equations X ± ∑ i = 1 m A i ∗ X − n i A i = I $Xpmsum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i}=I$

摘要

We study the nonlinear matrix equations X + ∑ i = 1 m A i ∗ X − n i A i = I $X+sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i}=I$ and X − ∑ i = 1 m A i ∗ X − n i A i = I $X-sum_{i=1}^{m}A_{i}^{*}X^{-n_{i}}A_{i}=I$ , where n i $n_{i}$ are positive integers for i = 1 , 2 , … , m $i=1,2,ldots,m$ . The iterative algorithms for obtaining positive definite solutions for these equations are proposed. The necessary and sufficient conditions for the existence of positive definite solutions of these equations are derived. Moreover, the rate of convergence of the sequences generated from the algorithms is studied. The efficiency of proposed algorithms is illustrated by numerical examples.