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We present a local convergence analysis of inexact Newton method for solving singular systems of equations. Under the hypothesis that the derivative of the function associated with the singular systems satisfies a majorant condition, we obtain that the method is well defined and converges. Our analysis provides a clear relationship between the majorant function and the function associated with the singular systems. It also allows us to obtain an estimate of convergence ball for inexact Newton method and some important special cases.

Newton’s method and its variations, including the inexact Newton methods, are the most efficient methods known for solving the following nonlinear equation:

It is clear that the residual control (

To study the local convergence of inexact Newton method and inexact Newton-like method (called inexact methods for short below), Morini presented in [

Recent attentions are focused on the study of finding zeros of singular nonlinear systems by Gauss-Newton’s method, which is defined as follows. For a given initial point

In the last years, some authors have studied the convergence behaviour of inexact versions of Gauss-Newton’s method for singular nonlinear systems. For example, Chen [

In the present paper, we are interested in the local convergence analysis; that is, based on the information in a neighborhood of a solution of (

The rest of this paper is organized as follows. In Section

For

Let

The following lemma gives a perturbation bound for Moore-Penrose inverse, which is stated in [

Let

Also, we need the following useful lemma about elementary convex analysis.

Let

For a positive real

For fixed

Set

The constant

Since

The constant

On one hand, by Lemma

Let

The sequence

Since

Based on the researches in [

Let

In the case when

Let

In this section, we state our main results of local convergence for inexact Newton method (

Our first result concerns the local convergence properties of inexact Newton’s method for general singular systems with constant rank derivatives.

Let

If

If

Suppose that

In the case when

Suppose that

When

Suppose that

In the case when

Let

In this section, we prove our main results of local convergence for inexact Newton method (

Suppose that

Since

We will prove by induction that

Suppose that

Since

Let

Suppose that

Since

Using Lemma

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was supported by Quzhou City Science and Technology Bureau Project of Zhejiang Province of China (Grant no. 20111046).