Journal of Applied Mathematics

Journal of Applied Mathematics / 2015 / Article
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Iterative Methods and Applications 2014

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Research Article | Open Access

Volume 2015 |Article ID 805278 | https://doi.org/10.1155/2015/805278

Moin-ud-Din Junjua, Saima Akram, Nusrat Yasmin, Fiza Zafar, "A New Jarratt-Type Fourth-Order Method for Solving System of Nonlinear Equations and Applications", Journal of Applied Mathematics, vol. 2015, Article ID 805278, 14 pages, 2015. https://doi.org/10.1155/2015/805278

A New Jarratt-Type Fourth-Order Method for Solving System of Nonlinear Equations and Applications

Academic Editor: D. R. Sahu
Received06 Jun 2014
Revised25 Sep 2014
Accepted24 Nov 2014
Published25 Mar 2015

Abstract

Solving systems of nonlinear equations plays a major role in engineering problems. We present a new family of optimal fourth-order Jarratt-type methods for solving nonlinear equations and extend these methods to solve system of nonlinear equations. Convergence analysis is given for both cases to show that the order of the new methods is four. Cost of computations, numerical tests, and basins of attraction are presented which illustrate the new methods as better alternates to previous methods. We also give an application of the proposed methods to well-known Burger's equation.

1. Introduction

Let us consider the problem of constructing an optimal iterative method to find a simple zero of a nonlinear equationas well as its extension for solving nonlinear system of equations . There is no ambiguity that the quadratically convergent Newton method (NM) is one of the best root finding methods based on two evaluations of function for approximating the solution of a nonlinear equation (1) and is given asThe natural extension of Newton’s method for system of nonlinear equations is given [1] as A large number of variants of the classical Newton method to obtain higher-order multistep schemes with better efficiencies have appeared using numerous techniques [24]. However, the research remained focussed on obtaining optimal and computationally efficient methods. The main objective was not only to boost up the order and speed of convergence but also to reduce the computational cost. This objective was achieved by defining two-step methods. A very few of them had less computational cost. Weerakoon and Fernando [5] proposed an accelerated cubically convergent two-step variant of Newton’s method:where . Another cubically convergent without memory method was suggested by Frontini and Sormani [2]:However, the above methods were not optimal. Jarratt [6] constructed an optimal method given byand its extension for the systems of nonlinear equations is given by [7]Several researchers have also focussed their attention on obtaining Jarratt-type iterative schemes for solving nonlinear equation [811]. Khattri and Abbasbandy [10] gave their contribution by using concept of parameter approach to develop an optimal fourth-order Jarratt-type iterative scheme requiring one evaluation of the function and two evaluations of first derivative. The following is a special case of their scheme:Soleymani et al. [11] replaced the parameter approach by the weight function approach to construct a family of optimal Jarratt-type two-step methods:where and . A special case of their scheme is as follows:Babajee et al. [12] extended (10) for the multivariate case as follows:where, and is the identity matrix. Chun et al. [8] used Halley’s method with an approximation of second derivative by using weight function approach in the second step of their scheme to obtain optimal variant of Jarratt-type fourth-order method. The classical Jarratt family of fourth-order methods was obtained as special case. A special case of his scheme is given asIn this contribution, we present and analyze an optimal family of fourth-order convergent iterative schemes using weight function in the second step. We, then, extend it for multivariate case. The rest of the paper is organized as follows. Section 2 is comprised of the construction and convergence analysis of the new family of methods for the single nonlinear equations. Section 3 consists of the extension of our method to the system of nonlinear equations. Section 4 includes numerical tests. Sections 5 and 6 provide the cost of computations and basins of attraction, respectively, for the sake of comparison of the new methods with the existing ones in this domain. In Section 7 an application of the proposed method to well-known Burger’s equation is presented and concluding remarks are given in Section 8.

2. New Family of Jarratt-Type Methods and Its Convergence Analysis

In this section, we propose a new optimal fourth-order Jarratt-type scheme for computing the zeros of a univariate nonlinear function. We give a two-step scheme in which the first step is similar to Jarratt’s scheme. In the second step, we use weight functions approach as follows:where , , and and represent real-valued weight functions chosen such that new scheme (14) achieves optimal fourth-order convergence as stated in Theorem 1.

Theorem 1. Let be a simple zero of sufficiently differentiable function in an open interval containing . Then, for , the new without-memory scheme (14) has optimal convergence of order four under the following conditions on weight functions:and it satisfies the error equation given bywhere , .

Proof. Let be the error at th computing step. By using Taylor’s expansion of about the root , we getwhere , . The first derivative in the first step of our scheme can be calculated asUsing (17) and (18) in the first step of (14), the Taylor expansions for and are given byIn the similar manner, for the second step of (14), we haveBy using (17), (18), and (20), we getAgain using Taylor’s expansion, we haveSubstituting , (21) and (23) in (14) and using (15), we have which gives the following error equation:Hence, it can be seen that the new scheme has optimal fourth-order convergence.

3. Extension of the New Family for Multivariate Case and Its Analysis

In this section, we extend our new family of optimal Jarratt-type schemes to solve systems of nonlinear equations. We give a special case for our new family by defining weight functions satisfying the conditions of Theorem 1 as follows:Thus, we achieve a new Jarratt-type fourth-order method:Now, let us define to be a sufficiently Fréchet differentiable in a convex set , where is an open convex neighborhood of the root , is th approximate root of the exact root , and ; we can write the th derivative of at , as a -linear function such that as described in [13]. Thus, it is defined asAlso, where are the Hessian matrices of . Therefore, Taylor’s series for function of variables can be written aswhere , is continuous and nonsingular, and is closer to . Now, we extend our scheme for solving a system of nonlinear equations as follows:where , , is the identity matrix, and . By using the above Taylor expansion we can prove the following theorem.

Theorem 2. Let be an open convex set containing the root of and let be four-time Fréchet differentiable in such that the Jacobian matrix is continuous and nonsingular in . Then the new method (31) has convergence of order four.

Proof. Let be the solution of nonlinear system , and let be an initial guess close to ; then by Taylor’s expansion of about , we havewhere , , and . We can calculateto getThe Jacobian matrix has the following Taylor expansion:to calculateUsing (33) and (36) in second step of (31), we attain with the following error term:which shows the proposed method (31) is fourth-order convergent.

4. Numerical Results

We, now, check the effectiveness of our new optimal fourth-order family of methods (27) (MSNF-1) by comparing it with Newton’s method (2) (NM), Khattri’s method (8) (KM), Soleymani’s method (10) (SM), and Chun’s method (13) (CM). Algorithms have been executed in Maple software and are tested for the examples given in Table 1. The approximate solutions for solving single equations are calculated by using a precision of decimal digits with the stopping criterion . In Tables 2, 3, 4, 5, and 6, number of iterations, “,” absolute values of functions and absolute values of the difference between approximated roots and exact roots, , for each iterative method are given. We also compare our proposed method (31) (MSNF-2) for solving systems of nonlinear equations with Newton’s method (3) (NM), Babajee’s method (11) (BM), and Khattri’s method (8) (KM). For solving systems of equations results are computed with a precision of decimal digits. The stopping criterion is . Table 7 consists of test functions for solving systems of nonlinear equations along with their exact zeros. Table 8 shows absolute values of the difference between two consecutive approximations of the root and absolute functional values satisfying the above stopping criterion for each of the methods.


Numerical example Exact root



NM KM CM SM MSNF-1

9 D D D 7
3.02  10 2.56  10 7.05  10 6.07  10 3.20  10
1.075  10 1.16  10 1.39  10 1.46  10 1.12  10
9.4  10 3.0  10 2.10  10 35854.07  10 4.63  10
3.72  10 2.64  10 1.03  10 21526.38  10 1.84  10
2.13  10 2.11  10 95.83  10 35853.13 8.67  10
8.54  10 1.03  10 2.23  10 10417.75  10 3.47  10
2.43  10 883.95  10 95.80  10 35853.41  10 1.95  10
9.74  10 344.38  10 8.81  10 290.79  10 7.81  10
3.61  10 884.35  10 95.80  10 35853.42  10 1.12  10
1.44  10 1.11  10 3.90  10 1.55  10 4.51  10
1.17  10 884.35  10 95.80  10 35853.42  10 7.28  10
4.71  10 5.21  10 4.56  10 1.25  10 2.91  10


NM KM CM SM MSNF-1

9 5 5 5 5
3.63  10 5.21  10 2.06  10 5.02  10 4.13  10
4.40  10 6.31  10 2.50  10 6.09  10 5.01  10
7.21  10 8.64  10 8.09  10 7.82  10 3.14  10
8.74  10 1.04  10 9.81  10 9.48  10 3.81  10
2.84  10 6.55  10 1.92  10 4.61  10 1.05  10
3.45  10 7.95  10 2.32  10 5.59  10 1.27  10
4.43  10 2.17  10 6.07  10 5.57  10 1.32  10
5.37  10 2.63  10 7.36  10 6.76  10 1.60  10
1.07  10 0 0 0 0
1.30  10 0 0 0 0
6.30  10 0 0 0 0
7.64  10 0 0 0 0


NM KM CM SM MSNF-1

  10 6 6 6 5
1.15  10 1.91  10 8.46  10 4.12  10 3.16  10
3.49  10 5.74  10 2.54  10 1.23  10 9.50  10
1.30  10 1.19  10 2.38  10 6.74  10 2.00  10
3.92  10 3.59  10 7.14  10 2.02  10 6.00  10
1.70  10 1.88  10 1.50  10 4.82  10 3.23  10
5.12  10 5.66  10 4.50  10 1.44  10 9.71  10
2.92  10 1.15  10 2.36  10 1.26  10 2.20  10
8.76  10 3.46  10 7.10  10 3.80  10 6.60  10
8.54  10 1.62  10 1.46  10 0 0
2.56  10 4.86  10 1.46  10 0 0
7.29  10 0 0 0 0
2.18  10 0 0 0 0


NM KM CM SM MSNF-1

  10 6 6 6 5
7.18   1.04   5.06   1.16   3.95  
7.76   1.10   5.36   1.22   4.18  
1.50   1.58   3.16   1.79   1.97  
1.50   1.67   3.34   1.90   2.08  
6.04   8.28   4.71   1.01   4.55  
6.39   8.77   4.98   1.07   4.81  
9.71   6.16   2.32   1.04   1.29  
1.02   6.52   2.45   1.11   1.36  
2.51   1.89   1.36   1.18   0