Solving system of nonlinear matrix equations over Hermitian positive definite matrices

Abstract

In this article, we first introduce the necessary and sufficient conditions for the existence of the Hermitian positive definite solution group of a system of nonlinear matrix equations X1p1+A1∗X1−s11A1+A2∗X2−s12A2+⋯+Am−1∗Xm−1−s1m−1Am−1+Am∗Xm−s1mAm=I,X2p2+A1∗X2−s21A1+A2∗X3−s22A2+⋯+Am−1∗Xm−s2m−1Am−1+Am∗X1−s2mAm=I,⋮Xmpm+A1∗Xm−sm1A1+A2∗X1−sm2A2+⋯+Am−1∗Xm−2−smm−1Am−1+Am∗Xm−1−smmAm=I. Then we propose an inversion-free iteration method to find the Hermitian positive definite solution group of the system of nonlinear matrix equations. Finally, the accuracy and effectiveness of the proposed method in compare to some existing algorithms are demonstrated by various numerical examples.

Topics & Concepts