Abstract and Applied Analysis

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

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

Research Article

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

Wen Zhou¹ and Jisheng Kou²

¹Department of Foundation Courses, Hubei Vocational Technical College, Xiaogan, Hubei 432000, China

²School 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.

Table 1: Results of the case .

PPM
VEA-PPM
MPE-PPM
RRE-PPM
NTM

0.5
D
D
8
8
5

1
D
6
7
7
5

1.5
22
6
6
6
5

2
D
5
6
6
4

3
D
5
6
6
4

Table 2: Results of the case .

PPM
VEA-PPM
MPE-PPM
RRE-PPM
NTM

0.1
D
D
10
10
7

0.5
D
D
9
9
6

1
D
30
8
8
6

1.5
D
8
7
7
5

2
D
7
7
7
5

3
D
6
7
7
5

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).

References

J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, NY, USA, 1970.

F. A. Potra and V. Pták, Nondiscrete Induction and Iterative Processes, vol. 103 of Research Notes in Mathematics, Pitman, Boston, Mass, USA, 1984.

S. Cabay and L. W. Jackson, “A polynomial extrapolation method for finding limits and antilimits of vector sequences,” SIAM Journal on Numerical Analysis, vol. 13, no. 5, pp. 734–752, 1976.

R. P. Eddy, “Extrapolation to the limit of a vector sequence,” in Information Linkage Between Applied Mathematics and Industry, P. C. C. Wang, Ed., pp. 387–396, Academic Press, New York, NY, USA, 1979.

M. Mešina, “Convergence acceleration for the iterative solution of the equations ,” Computer Methods in Applied Mechanics and Engineering, vol. 10, no. 2, pp. 165–173, 1977.

C. Brezinski, “Généralisations de la transformation de Shanks, de la table de Padé et de l'epsilon-algorithme,” Calcolo, vol. 12, no. 4, pp. 317–360, 1975.

B. P. Pugachev, “Acceleration of the convergence of iterative processes and a method of solving systems of non-linear equations,” USSR Computational Mathematics and Mathematical Physics, vol. 17, no. 5, pp. 199–207, 1977.

A. Sidi, W. F. Ford, and D. A. Smith, “Acceleration of convergence of vector sequences,” SIAM Journal on Numerical Analysis, vol. 23, no. 1, pp. 178–196, 1986.

P. Wynn, “On a device for computing the em(sn) transformation,” Mathematical Tables and Other Aids to Computation, vol. 10, pp. 91–96, 1956.

P. Wynn, “Acceleration techniques for iterated vector and matrix problems,” Mathematics of Computation, vol. 16, pp. 301–322, 1962.

C. Brezinski and M. Redivo Zaglia, Extrapolation Methods: Theory and practice, North-Holland, Amsterdam, The Netherlands, 1991.

K. Jbilou and H. Sadok, “Vector extrapolation methods. Applications and numerical comparison,” Journal of Computational and Applied Mathematics, vol. 122, no. 1-2, pp. 149–165, 2000.

R. F. King, “A family of fourth order methods for nonlinear equations,” SIAM Journal on Numerical Analysis, vol. 10, pp. 876–879, 1973.

L. B. Rall, Computational Solution of Nonlinear Operator Equations, Robert E. Krieger, New York, NY, USA, 1979.

V. Candela and A. Marquina, “Recurrence relations for rational cubic methods: I. The Halley method,” Computing, vol. 44, no. 2, pp. 169–184, 1990.

J. He, J. Wang, and J.-C. Yao, “Local convergence of Newton's method on Lie groups and uniqueness balls,” Abstract and Applied Analysis, vol. 2013, Article ID 367161, 9 pages, 2013.

V. Candela and A. Marquina, “Recurrence relations for rational cubic methods. II. The Chebyshev method,” Computing, vol. 45, no. 4, pp. 355–367, 1990.

J. M. Gutiérrez and M. A. Hernández, “Recurrence relations for the super-Halley method,” Computers & Mathematics with Applications, vol. 36, no. 7, pp. 1–8, 1998.

Y. Ling, X. Xu, and S. Yu, “Convergence behavior for Newton-Steffensen's method under -condition of second derivative,” Abstract and Applied Analysis, vol. 2013, Article ID 682167, 11 pages, 2013.

M. A. Hernández, “Chebyshev's approximation algorithms and applications,” Computers and Mathematics with Applications, vol. 41, no. 3-4, pp. 433–445, 2001.

J. A. Ezquerro and M. A. Hernández, “Recurrence relations for Chebyshev-type methods,” Applied Mathematics and Optimization, vol. 41, no. 2, pp. 227–236, 2000.

S. Amat, S. Busquier, and Á. A. Magreñán, “Reducing chaos and bifurcations in Newton-type methods,” Abstract and Applied Analysis, vol. 2013, Article ID 726701, 10 pages, 2013.

X. Wang, J. Kou, and C. Gu, “Semilocal convergence of a class of modified super-Halley methods in Banach spaces,” Journal of Optimization Theory and Applications, vol. 153, no. 3, pp. 779–793, 2012.

P. D. Proinov, “New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems,” Journal of Complexity, vol. 26, no. 1, pp. 3–42, 2010.

X. Xu, Y. Xiao, and T. Liu, “Semilocal convergence analysis for inexact Newton method under weak condition,” Abstract and Applied Analysis, vol. 2012, Article ID 982925, 13 pages, 2012.

S. Amat, Á. A. Magreñán, and N. Romero, “On a two-step relaxed Newton-type method,” Applied Mathematics and Computation, vol. 219, no. 24, pp. 11341–11347, 2013.

I. K. Argyros and S. Hilout, “Improved generalized differentiability conditions for Newton-like methods,” Journal of Complexity, vol. 26, no. 3, pp. 316–333, 2010.

J. M. Gutiérrez, Á. A. Magreñán, and N. Romero, “On the semilocal convergence of Newton-Kantorovich method under center-Lipschitz conditions,” Applied Mathematics and Computation, vol. 221, pp. 79–88, 2013.

R. Lin, Y. Zhao, Z. Šmarda, Y. Khan, and Q. Wu, “Newton-Kantorovich and Smale uniform type convergence theorem for a deformed Newton method in Banach spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 923898, 8 pages, 2013.

J. Kou, S. Sun, and B. Yu, “Multiscale time-splitting strategy for multiscale multiphysics processes of two-phase flow in fractured media,” Journal of Applied Mathematics, vol. 2011, Article ID 861905, 24 pages, 2011.