#### Abstract

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.

#### 1. Introduction

Newton’s method and its variations, including the inexact Newton methods, are the most efficient methods known for solving the following nonlinear equation: where is a nonlinear operator with its Fréchet derivative denoted by ; is open and convex. In the case , the inexact Newton method was introduced in [1] denoting any method which, given an initial point , generates the sequence as follows: where the residual control satisfies and is a sequence of forcing terms such that . Let be a solution of (1) such that is invertible. As shown in [1], if , then, there exists such that, for any initial guess , the sequence is well defined and converges to a solution in the norm , where is any norm in . Moreover, the rate of convergence of to is characterized by the rate of convergence of to 0. It is worth noting that, in [1], no Lipschitz condition is assumed on the derivative to prove that is well defined and linearly converging. However, no estimate of the convergence radius is provided. As pointed out by [2] the result of [1] is difficult to apply due to dependence of the norm , which is not computable.

It is clear that the residual control (3) is not affine invariant (see [3] for more details about the affine invariant). To this end, Ypma used in [4] the affine invariant condition of residual control in the form to study the local convergence of inexact Newton method (2). And the radius of convergent result is also obtained.

To study the local convergence of inexact Newton method and inexact Newton-like method (called inexact methods for short below), Morini presented in [2] the following variation for the residual controls: where is a sequence of invertible operators from to and is the forcing term. If and for each , (5) reduces to (3) and (4), respectively. Both proposed inexact methods are linearly convergent under Lipschitz condition. It is worth noting that the residual controls (5) are used in iterative methods if preconditioning is applied and lead to a relaxation on the forcing terms. And we also note that the results obtained in [2] cannot make us clearly see how big the radius of the convergence ball is. To this end, Chen and Li [5] obtained the local convergence properties of inexact methods for (1) under weak Lipschitz condition, which was first introduced by Wang in [6] to study the local convergence behavior of Newton’s method. The results in [5] easily provide an estimate of convergence ball for the inexact methods. Furthermore, Ferreira and Gonçalves presented in [7] a new local convergence analysis for inexact Newton-like method under the so-called majorant condition.

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 , define where denotes the Moore-Penrose inverse of the linear operator (or matrix) . For example, Shub and Smale extended in [8] the Smale point estimate theory (including -theory and -theory) to Gauss-Newton's methods for underdetermined analytic systems with surjective derivatives. For overdetermined systems, Dedieu and Shub studied in [9] the local linear convergence properties of Gauss-Newton’s method for analytic systems with injective derivatives and provided estimates of the radius of convergence balls for Gauss-Newton’s method. Dedieu and Kim in [10] generalized the results of both the underdetermined case and the overdetermined case to such case where is of constant rank (not necessarily full rank), which has been improved by some authors in [11–16].

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 [17] employed the ideas of [5] to study the local convergence properties of several inexact Gauss-Newton-type methods where a scaled relative residual control is performed at each iteration under weak Lipschitz conditions. Ferreira et al. presented in their recent paper [18] a local convergence analysis of an inexact version of Gauss-Newton's method for solving nonlinear least squares problems. Moreover, the radii of the convergence balls under the corresponding conditions were estimated in these two papers.

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 (1), we determine the convergence ball of the method. Following the ideas of [7, 14, 18–20], we will present a new local convergence analysis for inexact Newton’s method for solving singular systems with constant rank derivatives under majorant condition. The convergence analysis presented provides a clear relationship between the majorizing function, which relaxes the Lipschitz continuity of the derivative, and the function associated with the nonlinear operator equation. Besides, the results presented here have the conditions and the proof of convergence in quite a simple manner. Moreover, two unrelated previous results pertaining to inexact Newton method are unified, namely, the result for analytical functions and the classical one for functions with Lipschitz derivative.

The rest of this paper is organized as follows. In Section 2, we introduce some preliminary notions and properties of the majorizing function. The main results about the local convergence are stated in Section 3. And finally in Section 4, we prove the local convergence results given in Section 3.

#### 2. Notations and Auxiliary Results

For and a positive number , throughout the whole paper, we use to stand for the open ball with radius and center , and let denote its closure.

Let be a linear operator (or an matrix). Recall that an operator (or matrix) is the Moore-Penrose inverse of if it satisfies the following four equations: where denotes the adjoint of . Let and denote the kernel and image of , respectively. For a subspace of , we use to denote the projection onto . Then, it is clear that In particular, in the case when is full row rank (or equivalently, when is surjective), ; when is full column rank (or equivalently, when is injective), .

The following lemma gives a perturbation bound for Moore-Penrose inverse, which is stated in [21].

Lemma 1 (see [21], Corollary 7.1.1 and Corollary 7.1.2 ). *Let and be matrices and let . Suppose that , , and . Then, and
*

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

Lemma 2 (see [22], Proposition 1.3). *Let . If is continuously differentiable and convex, then*(i)* for all and ,*(ii)* for all , , and .*

For a positive real , let be a continuously differentiable function of three of its arguments and satisfy the following properties:.(i) and ,(ii) is convex and strictly increasing with respect to the argument .

For fixed , we write for short below. Then the above two properties can be restated as follows:(iii)* * and ,(iv)* * is convex and strictly increasing.

Set The next two lemmas show that the constants and defined in (11) and (12), respectively, are positive.

Lemma 3. *The constant defined in (11) is positive and for all .*

*Proof. *Since , there exists such that for all . Then, we get . It remains to show that for all . Because is strictly increasing, is strictly convex. It follows from Lemma 2 (i) that
Since and for all , we obtain the desired inequality by (14).

Lemma 4. *The constant defined in (12) is positive. As a consequence, for all .*

*Proof. *On one hand, by Lemma 3, it is clear that for . On the other hand, we can obtain from Lemma 2 (i) that
Then, we conclude that there exists a such that
Therefore, is positive. This completes the proof.

Let where and are given in (12) and (13), respectively. For any starting point , let denote the sequence generated by

Lemma 5. *The sequence given by (18) is well defined, strictly decreasing, and contained in and converges to .*

*Proof. *Since , using Lemma 4, one has that is well defined, strictly decreasing, and contained in . Thus, there exists such that . That is to say, we have
If , it follows from Lemma 4 that
This is a contradiction. So as . This completes the proof.

Based on the researches in [7, 14, 18, 20] for the convergence of various Newton-type methods, we present the following modified majorant condition.

*Definition 6. *Let be such that . Then, is said to satisfy the majorant condition on if
for any and .

In the case when is not surjective (see [9, 10]), the information on may be lost. To this end, we need to modify the above notion to suit the case when is not surjective.

*Definition 7. *Let be such that . Then, is said to satisfy the modified majorant condition on if
for any and .

#### 3. Main Results

In this section, we state our main results of local convergence for inexact Newton method (2). Recall that (1) is a surjective-underdetermined (resp., injective-overdetermined) system if the number of equations is less (resp., greater) than the number of unknowns and is of full rank for each . Note that, for surjective-underdetermined systems, the fixed points of the Newton operator are the zeros of , while for injective-overdetermined systems, the fixed points of are the least square solutions of (1), which, in general, are not necessarily the zeros of .

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

Theorem 8. *Let be continuously Fréchet differentiable nonlinear operator, where is open and convex. Suppose that , and that satisfies the modified majorant condition (22) on , where is given in (17). In addition, we assume that for any and that
**
where the constant satisfies . Let be sequence generated by inexact Newton’s method with any initial point and the conditions for the residual and the forcing term :
**
where denotes the condition number of . Then, converges to a zero of in . Moreover, we have the following estimate:
**
where the sequence is defined by (18).*

*Remark 9. *If is given by
then the results obtained in Theorem 8 reduce to the one given in [19]. Moreover, if taking (in this case and ) in Theorem 8, we obtain the local convergence of Newton’s method for solving the singular systems, which has been studied by Dedieu and Kim in [10] for analytic singular systems with constant rank derivatives and Li et al. in [13] for some special singular systems with constant rank derivatives.

If is full column rank for every , then we have . Thus, That is, . We immediately have the following corollary.

Corollary 10. *Suppose that and that
**
for any . Suppose that , and that satisfies the modifed majorant condition (22). Let be sequence generated by inexact Newton’s method with any initial point and the condition (24) for the residual and the forcing term . Then, converges to a zero of in . Moreover, we have the following estimate:
**
where the sequence is defined by (18) for .*

In the case when is full row rank, the modified majorant condition (22) can be replaced by the majorant condition (21).

Theorem 11. *Suppose that and is full row rank and that satisfies the majorant condition (21) on , where is given in (17). In addition, we assume that for any and that condition (23) holds. Let be sequence generated by inexact Newton’s method with any initial point and the conditions for the residual and the forcing term :
**
Then, converges to a zero of in . Moreover, we have the following estimate:
**
where the sequence is defined by (18).*

*Remark 12. *When is given by (26), the results obtained in Theorem 11 reduce to the ones given in [19].

Theorem 13. *Suppose that is full row rank and that satisfies the majorant condition (21) on , where is given in (17). In addition, we assume that for any and that condition (23) holds. Let be sequence generated by inexact Newton method with any initial point and the conditions for the control residual and the forcing term :
**
Then, converges to a zero of in . Moreover, we have the following estimate:
**
where the sequence is defined by (18).*

*Remark 14. *In the case when is invertible in Theorem 13 and is given by (26), we obtain the local convergence results of inexact Newton method for nonsingular systems, and the convergence ball in this case satisfies
In particular, if taking , the convergence ball determined in (34) reduces to the one given in [6] by Wang and the value is the optimal radius of the convergence ball when the equality holds. Now, Theorem 13 merges into the theory of Newton’s method obtained in [6].

*Example 15. *Let be endowed with the -norm. Consider the operator defined by
Then, is analytic on and its derivative is given by
Clearly, and the Moore-Penrose inverse is
Moreover, by mathematical induction, for , we can obtain that
So, we have
Consequently,
Let
Then, for any , we have
Consider the point . Then, satisfies and
Hence, we have
That is, (23) holds with . Furthermore, it is clear that
So, we can choose the forcing term such that ; see [23] for more details. Then (24) holds with . Now, the majorizing function associated with the operator defined by (35) reduces to
Then we have the convergence radius . Therefore, Theorem 8 is applicable to concluding that, for any , the sequence generated by inexact Newton method (2) with initial point converges to a zero of .

#### 4. Proofs

In this section, we prove our main results of local convergence for inexact Newton method (2) given in Section 3.

##### 4.1. Proof of Theorem 8

Lemma 16. *Suppose that satisfies the modified majorant condition on and that , where , , and are defined in (17), (12), and (11), respectively. Then, and
*

*Proof. *Since , we have
It follows from Lemma 1 that and
This completes the proof.

*Proof of Theorem 8. *We will prove by induction that is the majorizing sequence for ; that is,
Because , (50) holds for . Now, we assume that for some . For the case , we first notice that
By using the modified majorant condition (22), Lemma 16, the inductive hypothesis (50), and Lemma 2, one has that
Thanks to (24),
Since
combining Lemma 2, Lemma 16, the modified majorant condition (22), the inductive hypothesis (50), and the condition (24), we have
Combining (23), (52), (53), and (55), we can obtain that
Note that for any ,
Thus, in view of the definition of given in (18), one has that
which implies that . Therefore, the proof by induction is complete. Since converges to 0 (by Lemma 5), it follows from (50) that converges to and the estimate (25) holds for all . This completes the proof.

##### 4.2. Proof of Theorem 11

Lemma 17. *Suppose that is full row rank, and satisfies the majorant condition (21) on . Then, for any , we have and
*

*Proof. *Since , we have
It follows from Banach lemma that exists and
Since is full row rank, we have and
which implies that is full row rank; that is, . The proof is complete.

*Proof of Theorem 11. *Let be defined by
with residual . Since
one has that coincides with the sequence generated by inexact Newton’s method (2) for . In addition, we have
and so
Because , we have
Therefore, by (21), we can obtain that
That is, satisfies the modified majorant condition (22) on . So, Theorem 8 is applicable and converges to follows. Note that and ; it follows that is a zero of .

##### 4.3. Proof of Theorem 13

Lemma 18. *Suppose that , is full row rank, and satisfies the majorant condition (21) on . Then we have
*

*Proof. *Since is full row rank, we have . It follows that
By Lemma 17, is invertible for any . Thus, in view of the equality for any matrix , one has that
Therefore, Lemma 17 is applicable to conclude that
The proof is complete.

*Proof of Theorem 13. *Using Lemma 18, majorant condition (21), and the residual condition (32), respectively, instead of Lemma 16, modified majorant condition (22), and condition (24), one can complete the proof of Theorem 13 as the same line of proof in Theorem 8.

#### Conflict of Interests

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

#### Acknowledgment

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