Solving system of nonlinear matrix equations over Hermitian positive definite matrices. Issue 4 (4th March 2023) - Erfanifar, Raziyeh; Sayevand, Khosro; Hajarian, Masoud
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 1 X 1 p 1 + A 1 ∗ X 1 − s 11 A 1 + A 2 ∗ X 2 − s 12 A 2 + ⋯ + A m − 1 ∗ X m − 1 − s 1 m − 1 A m − 1 + A m ∗ X m − s 1 m A m = I, X 2 p 2 + A 1 ∗ X 2 − s 21 A 1 + A 2 ∗ X 3 − s 22 A 2 + ⋯ + A m − 1 ∗ X m − s 2 m − 1 A m − 1 + A m ∗ X 1 − s 2 m A m = I, ⋮ X m p m + A 1 ∗ X m − s m 1 A 1 + A 2 ∗ X 1 − s m 2 A 2 + ⋯ + A m − 1 ∗ X m − 2 − s m m − 1 A m − 1 + A m ∗ X m − 1 − s m m A m = 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.