Fixed Point Theorems and Sensitivity Analysis in Solving Nonlinear Matrix Equations: A Study Involving Partially Ordered Sets

Abstract

Nonlinear matrix equations of the form   , where  represents an unknown matrix, and     and  are given square matrices, are encountered in various fields. Understanding the characteristics and behaviour of these equations is essential for developing efficient computational methods and obtaining reliable solutions. This paper explores the use of partially ordered sets within fixed point schemes to solve the targeted equations. Furthermore, it derives perturbation estimates of the solutions and evaluates them computationally. Finally, the numerical simulations are provided to validate our theoretical claims and affirm the effectiveness of the proposed fixed-point scheme.

Keywords:    Fixed point, partially ordered sets, Nonlinear matrix equation, Positive definite solution, Perturbation