Solving system of nonlinear matrix equations over Hermitian positive definite matrices. Issue 4 (4th March 2023)

• Record Type: Journal Article
• Authors: Erfanifar, Raziyeh; Sayevand, Khosro; Hajarian, Masoud
• Is Part Of: Linear & multilinear algebra. Volume 71:Issue 4(2023)
• Page Start: 597
• Page End: 630
• Publication Date: 2023-03-04
• Subjects: System of nonlinear matrix equations -- iterative method -- Hermitian positive definite solution -- inversion-free iteration method; 34A08 -- 49S05; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2022.2038058
• Languages: English
• ISSNs: 0308-1087

Abstract

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.