Abstract and Applied Analysis

Volume 2014 (2014), Article ID 601745, 8 pages

http://dx.doi.org/10.1155/2014/601745

## Third-Order Newton-Type Methods Combined with Vector Extrapolation for Solving Nonlinear Systems

^{1}Department of Foundation Courses, Hubei Vocational Technical College, Xiaogan, Hubei 432000, China^{2}School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, Hubei 432000, China

Received 22 March 2014; Accepted 31 May 2014; Published 15 June 2014

Academic Editor: Vinay Kanwar

Copyright © 2014 Wen Zhou and Jisheng Kou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present a third-order method for solving the systems of nonlinear equations. This method is a Newton-type scheme with the vector extrapolation. We establish the local and semilocal convergence of this method. Numerical results show that the composite method is more robust and efficient than a number of Newton-type methods with the other vector extrapolations.

#### 1. Introduction

Finding the solution of nonlinear equations is important in scientific and engineering computing areas. In this paper, we focus on the following nonlinear system of equations: where is differentiable. Here, and .

Some efficient methods for solving the system of (1) have been brought forward [1]. The Newton method for (1) is a second-order method. Its iterative formula is given by where is the current approximate solution and is the Jacobian matrix of at . Potra and Pták [2] propose the modified Newton method (PPM) given by In each iteration, PPM needs two evaluations of the vector function and one evaluation of the Jacobian matrix and the order is three.

Though the PPM can reduce the computational cost of Jacobian matrix, in some cases, the sequences produced by PPM converge slowly and even cannot converge because of the accumulation of the computational error. This problem limits its practical application.

In order to solve this problem, we will introduce the vector extrapolation technique to improve the convergence of PPM. Many vector extrapolation methods have been developed, such as the minimal polynomial extrapolation (MPE) method [3], the reduced rank extrapolation (RRE) method [4, 5], the modified minimal polynomial extrapolation (MMPE) method [6–8], the topological -algorithm (TEA) [6], and vector -algorithms (VEA) [9, 10]; also see [11, 12] and the references therein. These methods could be applied to the solvers of linear and nonlinear systems and accelerate their convergence.

In this paper, we construct a new extrapolation method and combine it with PPM, thus obtaining a Newton-type method. We will show by numerical results that the composite method can be of practical interest. The local and semilocal convergence are also established for the method.

#### 2. The Method

We introduce the following Newton-type method: where is Euclidean norm and .

This iteration scheme consists of a PPM iterate to get from , followed by a modified iterate to calculate from , and .

We now derive the last substep. Let be a scalar real equation; then King’s method [13] is described as In order to extend the method (5) to the case of vector functions, we define the vector inverse as The last substep is obtained by applying the above vector inverse to the scalar King method.

The following theorem will give the order of convergence of the method with given by (4).

Theorem 1. *Suppose that the function is continuously differentiable and is nonsingular, where is an open set and is the solution of . Define . Further, assume that there exists a positive number such that for any ,
**
then there exists a set such that for any , the sequence generated by (4) with converges to and the order of convergence is three.*

*Proof. *We can write (4) as where
Without loss of generality, we use the Euclidean norms as in the following. Let and . Let and .

It is obtained from (7) that
By Banach lemma, we obtain that is nonsingular and . So and are well defined.

By making use of Taylor expansion and (7), we have
So for we obtain
Similarly to (10), we get
It is obtained by (7) and (12) that
Therefore it follows that
This proves that and is a contraction mapping. Thus, for any , the sequence produced by (4) is well defined and it converges to . Finally, it is shown from (13) that the order of the method (4) is three.

#### 3. The Semilocal Convergence

In this section, we will establish the semilocal convergence of method (4). This convergence may be derived by using recurrence relations, which have been used in establishing the convergence of Newton’s method and some third-order methods [14–29]. In what follows, an attempt is made to use recurrence relations to establish the semilocal convergence for the method (4). The recurrence relations based on one constant which depend on are derived. Further, based on these recurrence relations, the error estimate is obtained for the present iterative method.

In order to establish the recurrence relations for the present iterative method, we will use the following scalar functions which are defined by where .

Let . For any positive real number , it is easy to obtain , so has at least a real zero point . Furthermore, let . It can be included and , so has at least a real zero in . Furthermore, it can be obtained that is an increasing function in . So is the unique zero of in . For the functions defined by (15), we have the following results.

Lemma 2. *Let be the unique real root of in . Then*(a)* is an increasing function in and satisfies ;*(b)* is an increasing function in and satisfies ;*(c)* is an increasing function in and satisfies ;*(d)* is an increasing function in and satisfies ;*(e)*.*

*Proof. *The results (a)–(d) can be obtained by simple derivations. We only prove the validity of (e). Noticing that
we get
which can be converted to (e).

Theorem 3. *Assume that the function is continuously differentiable where is an open set and there exists a positive number such that for any **
Let , , , and be defined by (15). Further, define , , and as
**
Let satisfy , be nonsingular, , and where and is the root of in ; then we have that*(i)* generated by the method (4) is well defined in and satisfies
*(ii)*, , and are well defined and satisfy
*

*Proof. *Without loss of generality, we use the Euclidean norms as in the following. We firstly consider the case .

Since , it is obvious that by the definition . By Taylor expansion we have
Furthermore,
It then follows that
Taking account of the relation
we have . Since
we obtain that is well defined and
This shows and the validity of (20).

By condition (18) we have
Because
by Banach lemma we obtain that is nonsingular and
This is to say that (22) holds.

Now we consider . By making use of (24), (25), and (30), we obtain

Finally, we prove (21) and (23). By making use of (34) and (35), we have
It then holds that

Now we consider the cases . By induction we can obtain the following facts.(P1)By Lemma 2, we obtain that
which leads to
It follows that
This further yields
Thus it is obtained that
Next we show that , are well defined in . By Lemma 2 and (42), we have
This means that . Furthermore, by analogous procedures to (24), (25), and (26), we obtain that
Since
we get
Hence it follows that
This shows that . Similarly to the case , we obtain that is well defined and have
By (42) we obtain
which shows .(P2)We can prove analogously to (35) that
(P3)Because
we obtain that is nonsingular and
(P4)From (50) and (52), we have
It then follows that
Thus far, we have proved all conclusions of this theorem.

The theorem given below will establish the convergence of the sequence and give the error estimate for it.

Theorem 4. *Let the conditions of Theorem 3 be satisfied. Denote and . Then the sequence generated by (4) converges to a unique solution of , and it holds that
*

*Proof. *Since , it follows from (41) that
This means that is a Cauchy sequence and thus there exists a such that . By letting in (56), we obtain (55). From (56) and (42), we can get
This shows .

From (50) and (40), we obtain that
By letting in (58), we obtain ; namely, is a solution of .

Now, we prove the uniqueness of in . Let be another zero of in . By mean value theorem, we have
where is between and . Since
it follows by Banach lemma that is invertible and hence . This ends the proof.

#### 4. Numerical Tests

In this section, we present some numerical results for the method given by (4) (NTM) and compare it with PPM on their numerical behavior. We also test the composite methods combining PPM with some known vector extrapolation methods mentioned in Section 1, which are indicated as VEA-PPM, MPE-PPM, and RRE-PPM, respectively. We use to denote the value of at the th approximate solution .

We consider the nonlinear elliptic differential equation: This equation often arises from the flow model in porous media and in this case, is the pressure, the fluid saturation, and the conductivity. The boundary conditions can be given by

In this test, we consider the one-dimensional case. The uniform cell-centered finite difference (CCFD) approximation method is used to discretize the boundary value problem. For the detailed CCFD formulations, we refer to [30] or the references therein. The values on the faces of each cell are taken as the harmonic mean of cell-central ones. Here, we take where is a positive real constant. The input boundary condition is given by , while the output boundary condition is .

The discrete scheme leads to a nonlinear equation system with variables. We test two cases with the sizes and 1000, respectively. We take in our method. All methods start from the initial approximate solutions and stop when they satisfy the given criteria. For the case , the stopping criterion is , while it is taken as for . In these tables, we show the iteration number cost by various methods.

The computational results are displayed in Tables 1 and 2. In the tables, denote and “D” indicates that the method is divergent or cannot converge in 50 steps. We use NTM to represent the proposed method.

From the numerical results, we can know that the performance of NTM is more efficient and robust than PPM.

#### 5. Conclusions

We establish the convergence of a third-order method for systems of nonlinear equations; an existence-uniqueness theorem and the error estimate for this method are also obtained. Numerical results show that this method is more robust and efficient than a number of Newton-type methods with the other vector extrapolation algorithms.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work is supported by the Scientific and Technical Research Project of Hubei Provincial Department of Education (no. D20132701).

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