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

and Applied Analysis 3 Therefore it follows that 󵄩󵄩󵄩󵄩G (x) − x ∗󵄩󵄩󵄩󵄩 ≤ ( 󵄩󵄩󵄩󵄩x − x ∗󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩z − x ∗󵄩󵄩󵄩󵄩 + (3 − ω) 󵄩󵄩󵄩󵄩z − x ∗󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩y − x ∗󵄩󵄩󵄩󵄩 + (2 − ω) 󵄩󵄩󵄩󵄩z − x ∗󵄩󵄩󵄩󵄩 2 + 2 󵄩󵄩󵄩󵄩y − x ∗󵄩󵄩󵄩󵄩 2 ) × ( 󵄩󵄩󵄩󵄩x − x ∗󵄩󵄩󵄩󵄩 − (ω + 1) 󵄩󵄩󵄩󵄩y − x ∗󵄩󵄩󵄩󵄩 − ω 󵄩󵄩󵄩󵄩z − x ∗󵄩󵄩󵄩󵄩) −1 ≤ 292700 480111 λ 2 γ 2󵄩󵄩󵄩󵄩x − x ∗󵄩󵄩󵄩󵄩 3 ≤ 2927 1920444 󵄩󵄩󵄩󵄩x − x ∗󵄩󵄩󵄩󵄩 < 󵄩󵄩󵄩󵄩x − x ∗󵄩󵄩󵄩󵄩 . (14) This proves that G(x) ∈ S and G is a contraction mapping. Thus, for any x 0 ∈ S, the sequence {x k } produced by (4) is well defined and it converges to x. 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 F 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 g 1 (t) = 1 + 1 2 t + t 2


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  : R  → R  is differentiable.Here, () = ( 1 (),  2 (), . . .,   ())  and  ∈ R  .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][7][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
Abstract and Applied Analysis

The Method
We introduce the following Newton-type method: where ‖ ⋅ ‖ is Euclidean norm and 0 <  ≤ 2. This iteration scheme consists of a PPM iterate to get   from   , followed by a modified iterate to calculate  +1 from   ,   , and   .
We now derive the last substep.Let () = 0 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 0 <  ≤ 2 given by (4).
Similarly to (10), we get It is obtained by (7) This proves that () ∈  and  is a contraction mapping.Thus, for any  0 ∈ , 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.

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][15][16][17][18][19][20][21][22][23][24][25][26][27][28][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 For any positive real number , it is easy to obtain ℎ(0)ℎ(2/) < 0, so ℎ() has at least a real zero point t ∈ (0, 2/).Furthermore, let () =  4 () − 1.It can be included (0) < 0 and ( t) → +∞, so () has at least a real zero in  * ∈ (0, t).Furthermore, it can be obtained that () is an increasing function in (0, 2/).So  * is the unique zero of () in (0, 2/).For the functions defined by (15), we have the following results.
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).
By condition (18) we have Because by Banach lemma we obtain that   ( 1 ) is nonsingular and This is to say that ( 22) holds.Now we consider ( 1 ).By making use of ( 24), (25), and Finally, we prove ( 21) and (23).By making use of ( 34) and (35), we have It then holds that Now we consider the cases  ≥ 1.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), ( Since we get Hence it follows that This shows that   ∈ .Similarly to the case  = 0, we obtain that  +1 is well defined and have ] . ( we obtain that   ( +1 ) 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.

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 ‖  ‖ 2 to denote the value of ‖()‖ 2 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, Θ 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 (Θ()) on the faces of each cell are taken as the harmonic mean of cell-central ones.Here, we take (Θ()) =  where  is a positive real constant.The input boundary condition is given by   = 1, while the output boundary condition is   = 1.
The discrete scheme leads to a nonlinear equation system with  variables.We test two cases with the sizes  = 100 and 1000, respectively.We take  = 2 in our method.All methods start from the initial approximate solutions and stop when they satisfy the given criteria.For the case  = 100, the stopping criterion is ‖‖ 2 < 1 − 12, while it is taken as ‖‖ 2 < 1 − 11 for  = 1000.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  = (1, 1, . . ., 1)  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.

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.

Theorem 3 .
Assume that the function  :  ⊂ R  → R  is continuously differentiable where  is an open set and there exists a positive number  such that for any , V ∈         () −   (V)      ≤  ‖ − V‖ .(

Theorem 1 .
Suppose that the function  :  ⊂ R  → R  is continuously differentiable and   ( * ) is nonsingular, where  is an open set and  * ∈  is the solution of () = 0. Define  = ‖  ( * ) −1 ‖.Further, assume that there exists a positive number  such that for any  ∈ ,        () −   ( * )      ≤       −  *     ;(7)then there exists a set  such that for any  0 ∈ , the sequence {  } generated by (4) with 0 <  ≤ 2 converges to  * and the order of convergence is three.

Table 1 :
Results of the case  = 100.

Table 2 :
Results of the case  = 1000.