ScholarLens Back to search
1978•SIAM Journal on Numerical AnalysisRequires access

On Newton-Iterative Methods for the Solution of Systems of Nonlinear Equations

Andrew H. Sherman

Open publisher page 141 citations

Abstract

In this paper we consider the local rates of convergence of Newton-iterative methods for the solution of systems of nonlinear equations. We show that under certain conditions on the inner, linear iterative method, Newton-iterative methods can be made to converge quadratically in a certain sense by computing a sufficient number of inner iterates at each step. As examples of this phenomenon, we consider the Newton-SOR and Newton-Richardson methods, particularly as applied to semilinear partial differential equations. Numerical results are included to illustrate the theory.

About this research paper

What this paper is about

In this paper we consider the local rates of convergence of Newton-iterative methods for the solution of systems of nonlinear equations. We show that under certain conditions on the inner, linear iterative method, Newton-iterative methods can be made to converge quadratically in a certain sense by computing a sufficient number of inner iterates at each step. As examples of this phenomenon, we consider the Newton-SOR and Newton-Richardson methods, particularly as applied to semilinear partial differential equations. Numerical results are included to illustrate the theory.

Why it matters

OpenAlex reports 141 citations for this work. Citation counts describe recorded attention and do not establish research quality.

Key contribution

A contribution statement is not available in the OpenAlex record.

Method / approach

Method details are not available in the OpenAlex metadata.

Main findings

Findings are not separately available in the OpenAlex metadata.

Limitations

Limitations are not available in the OpenAlex metadata.

Applications

Application details are not available in the OpenAlex metadata.

Available abstract

In this paper we consider the local rates of convergence of Newton-iterative methods for the solution of systems of nonlinear equations. We show that under certain conditions on the inner, linear iterative method, Newton-iterative methods can be made to converge quadratically in a certain sense by computing a sufficient number of inner iterates at each step. As examples of this phenomenon, we consider the Newton-SOR and Newton-Richardson methods, particularly as applied to semilinear partial differential equations. Numerical results are included to illustrate the theory.

Key concepts: Mathematics, Local convergence, Iterated function, Newton's method, Iterative method, Newton's method in optimization, Nonlinear system, Quadratic growth

Related papers

Related paper 1Related paper 2Related paper 3
Back to paper searchBrowse research topicsOriginal source
On Newton-Iterative Methods for the Solution of Systems of Nonlinear Equations — Research Paper | ScholarLens