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2018•Linear and Multilinear AlgebraRequires access

Newton's iterative method to solve a nonlinear matrix equation

Jingjing Peng, Anping Liao, Zhenyun Peng, Zhencheng Chen

Open publisher page 8 citations

Abstract

In this paper, Newton's iterative method to solve the nonlinear matrix equation X+A∗X−nA=Q is studied. For the given initial matrix Q, the main results that the matrix sequence generated by the iterative method is contained in a fixed open ball, and that the matrix sequence generated by the iterative method converges to the only solution of the nonlinear matrix equation X+A∗X−nA=Q in a fixed closed ball are proved. In addition, the error estimate of the approximate solution in the closed ball and a numerical example to illustrate the convergence results are given.

About this research paper

What this paper is about

In this paper, Newton's iterative method to solve the nonlinear matrix equation X+A∗X−nA=Q is studied. For the given initial matrix Q, the main results that the matrix sequence generated by the iterative method is contained in a fixed open ball, and that the matrix sequence generated by the iterative method converges to the only solution of the nonlinear matrix equation X+A∗X−nA=Q in a fixed closed ball are proved. In addition, the error estimate of the approximate solution in the closed ball and a numerical example to illustrate the convergence results are given.

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OpenAlex reports 8 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

In this paper, Newton's iterative method to solve the nonlinear matrix equation X+A∗X−nA=Q is studied. For the given initial matrix Q, the main results that the matrix sequence generated by the iterative method is contained in a fixed open ball, and that the matrix sequence generated by the iterative method converges to the only solution of the nonlinear matrix equation X+A∗X−nA=Q in a fixed closed ball are proved. In addition, the error estimate of the approximate solution in the closed ball and a numerical example to illustrate the convergence results are given.

Key concepts: Mathematics, Newton's method, Iterative method, Applied mathematics, Nonlinear system, Matrix (chemical analysis), Newton's method in optimization, Mathematical analysis

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