A class of efficient inversion-free iterative methods based on fixed point theory for solving nonlinear matrix equations with applications in control theory

In this paper, we propose an innovative class of iterative methods designed to solve the nonlinear matrix equation (NME): <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>X</mml:mi> <mml:mo>+</mml:mo> <mml:munderover accent="true" accentunder="false"> <mml:mrow> <mml:mo>∑</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:munderover> <mml:msubsup> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>l</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> </mml:msubsup> <mml:msup> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msub> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>l</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>,</mml:mo> </mml:math> without the need for matrix inversion. These methods are based on the fixed point iteration approach, providing a computationally efficient framework for obtaining the solution. Then, building upon this process, the iterative methods are extended to address the tensor version of the nonlinear equation, incorporating the Einstein product to manage the inherent multidimensional structure of tensor data. This extension is particularly relevant for applications in areas such as signal processing, machine learning, image reconstruction, and quantum mechanics, where tensor formulations are prevalent. A thorough convergence analysis is conducted to guarantee the reliability of the proposed methods under suitable conditions. The results illustrate that the proposed methods are not only accurate and efficient but also scalable to high-dimensional problems. By eliminating the need for direct matrix inversion, the methods significantly reduce computational costs, making them suitable for large-scale systems. This work highlights the potential of the proposed iterative approaches as a versatile tool for solving complex nonlinear matrix and tensor equations in various scientific and engineering applications.