IMPROVING THE CONVERGENCE ORDER AND EFFICIENCY INDEX OF QUADRATURE-BASED ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS

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1 136 IMPROVING THE CONVERGENCE ORDER AND EFFICIENCY INDEX OF QUADRATURE-BASED ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS 1Ogbereyivwe, O. and 2 Ojo-Orobosa, V. O. Department of Mathematics and Statistics, Delta State Polytechnic, Ozoro 1Corresponding Author:vesenelly@yahoo.com, 2 orobosa@gmail.com Abstract The object of this paper is to modify the two-step and three-step quadrature based iterative methods for approximating the solution of nonlinear equations in Noor et al., [10] using the weight technique. The modification improved the order of convergence and the efficiency index of the iterative methods at the cost of one additional functional evaluation per iteration. The convergence of the proposed methods herein, were established using the Taylor s series technique. Numerical results obtained from implementation of the proposed methods compared with methods from which they have been derived, shows that methods developed herein are more efficient than the iterative methods from which they have been derived. Keywords: Iterative methods, quadrature based methods, convergence order, efficiency index.

2 Introduction Diverse problems emanating from the field of science and engineering are modelled into nonlinear equation of the form where. Because there is no unified analytic method for obtaining the solution of the nonlinear equation (1), iterative based approximation methods are utilized. One well known iterative methods for solving equation (1) is the Newton method convergence order, Truab [7]. It is given as: where is an iterative counter. The iterative method (2) requires the cost of evaluation of one function, and one first derivative of function. This made the computational efficiency of the Newton method to be. In recent years, plethora of iterative methods for approximating solutions of nonlinear of equations have been developed. These methods are developed via diverse techniques such as Taylor s series and homotopy {Chun [1]; Golbabai and Javidi [2]; Noor [3] }. Others have used decomposition technique { Jafari and Gejji [4]; and Noor et. al, [5] }. Another technique been utilized is the quadrature formulas technique as in the work of Cordero and Terregrossa [11], Khirallah and Hafiz [12], Ogbereyivwe and Emunefe [17], Liu [18], Liu and Fang [19] and some reference therein. In developing new iterative methods for the approximation of the solution of equation (1), the most important rule is to design iterative methods with improved convergence rate or computational efficiency [8]. Although the methods proposed in Frontini and Sormani [9], Noor et al., [10], Cordero and Terregrossa [11], Khirallah and Hafiz [12] and Darvishi and Barati [13] may be considered to have high order convergence than the Newton method, the computational efficiency may not be attractive in practice due to involvement of high number of evaluation of function and derivatives of function per iteration. In this paper a technique used in improving the order of convergence and efficiency of some existing quadrature based iterative methods is proposed. 2. The iterative method development Consider the quadrature based iterative method of convergence order four proposed as Algorithm 2.12 in Noor et al., [10] given as The iterative method in equation (3) requires the evaluation of six functions per iteration. It is established in [10] that the method in equation (3) is of convergence order with efficiency index [10]. Though the

3 138 convergence order of the iterative method in equation (3) is higher than that of the Newton method, its efficiency index is lower because of the involvement of six function evaluation per iteration. To improve the efficiency index of the iterative method in equation (3), a slight modification to its first, second and third step is made so as to increase its computational efficiency index at no additional cost of function evaluation. To achieve this, consider a family of twostep iterative methods which is a modified form of the first two-step of equation (3) given as: where and are to be specified. The following definitions and lemma will be required in establishing the convergence of the iterative method in equation (4). Definition 2.1 [11] Let be the solution of (1) and be error in the th iteration; the relation, (5) is known as error equation. If we can obtain the error equation for any iterative method, then the value is its order of convergence. Definition 2.2 Let be total number of new functional evaluation required by an iterative method. The efficiency of an iterative method is measured by efficiency index [11] and is defined by where is the order of the method. Lemma 2.1 [6] Let, then for any be -time differentiable in the domain Equation (7) is the Taylor s series expansion of. The convergence of equation (4) is investigated using the Taylor s series expansion technique. We consider the following theorem. Theeorem 2.1. Let be time differentiable in the domain containing the zero of equation (1). If the, then the sequence { } generated by the iterative method given in equation (4) converges to with order if and only if

4 139 } Furthermore, the sequence { } generated by equation (4) converges to with convergence order if and only if Proof. Expanding using Lemma 2.1 about the solution and set, yield [ ] and [ ] using equations (9) and (10), Insert equation (11) in first step of equation (4) yield Setting in the Taylor s expansion of and about yield * + and * ] Using equation (14) * ]

5 140 Inserting equations (12) and (15) in the second step of equation (4), ) for equation (4) to converge to with convergence order, the first and second term of equation (16) must vanish. It follows that } Solving the coupled equation (17) yield and which leads to the error equation Thus, equation (4) converges to with order if and only if (8) is satisfied. Moreover, from (18), the iterative method in equation (4) converges to with convergence order if and only if. This completes the proof of Theorem 2.1. Expressing the parameters and in terms of gives and. Therefore, the proposed two-step family of iterative methods for approximating the solution of nonlinear equation (4) reduces to: for all constants { }. Remark 2.1 The family of iterative methods in equation (19) is a modification of Algorithm 2.6 in Noor et al., [10].

6 Some particular methods Some particular forms of the iterative method in equation (19) are generated by assigning values to the parameter. For, a new quadrature based iterative method for approximation of solution of with convergence order is proposed as: Method1 Given an initial guess, approximate the solution by the iterative scheme: with asymptotic error The efficiency index of Method 1 is. Mathematica 9 Program Code for the Convergence Order of Algorithm 2.1 In [1]: In [2]: In [3]: * { }+, In [4]: In [5]: In [6]: (22) In[7]: In [8]: [ { }], In [9]: [ { } ], Out [10]:

7 142 The program Code above can be modified to obtain the error equations for subsequent proposed methods. for the constant, a new quadrature based iterative method for approximation of nonlinear equation of convergence order is proposed. Method 2 Given an initial guess, approximate the solution by the iterative scheme: with asymptotic error The efficiency index of Method 2 is. Observe that the iterative method in equation (23) attains convergence order with four functional evaluations per iteration. This is better than Algorithm 2.6 in Noor et al., [10] that requires same number of functional evaluation per iteration but with and converges with order. In order to improve the convergence order and efficiency index of the iterative method in equation (3) (Algorithm 2.12 proposed in Noor et al., [10]), the following modification is made as follows: The proof of the convergence of (25) is given below. Theorem 2.2 Let be time differentiable in a convex set containing the zero the root of (1). If the, then the sequence { } generated by the iterative method in equation (25) converges to with convergence order.

8 143 Proof. Let be error in the th iteration. Using (16) and set in the Taloy s expansion of about, * ( From equation (15) and equation (26) ) ( ) + * ( ) ( )+ The third step of equation (25) becomes ( )

9 144 for the relation in equation (25) to converge to with convergence order, the first and second term of equation (28) must vanish. It follows that, which leads to the asymptotic error equation This completes the proof. For the parameter in equation (25), a new quadrature based iterative method of convergence order is proposed: Method 3 Given an initial guess, approximate the solution by the iterative scheme: with asymptotic error equation and efficiency index. 4. Numerical Experimentation In this section, the proposed iterative methods (Method 1, Method 2 and Method 3, where is Method of convergence order ) are implemented on four problems (Examples 1-4) in literature in other to illustrate their efficiency. Numerical results obtained by the proposed methods ( and ), are compared with methods from which they have been derived (Algorithm 2.6 and Algorithm 2.12 in [10]) and Newton method (NM). All numerical computations presented in Tables , are carried out in a PYTHON environment with 25 digits floating arithmetic. Intel Celeron(R) CPU 1.6 GHz with 2 GB of RAM processor was used to execute all programs. The stopping criterion used for all programs is. The measurements used for comparison are number of iteration required by method to achieve convergence (IT), number of functional evaluations required by method to achieve convergence (NFE), norm of difference of last two consecutive iterations ( ), function of the last iteration (, and Computer execution time in seconds (CPU).

10 145 The following problems used for implementation of the methods are taken from Noori [16]. Example 1 Example 2 Example 3 Example 4 Table 3.1 Computational results for Example 1 Method IT NFE CPU-time NM e e e e e e e e e e e e Table 3.2 Computational results for Example 2 Method IT NFE CPU-time NM e e e e e e e e e e e e

11 146 Table 3.3 Computational results for Example 3 Method IT NFE CPU-time NM e e e e e e e e e e e e Table 3.4 Computational results for Example 4 Method IT NFE CPU-time NM e e e e e e e e e e e e From Tables , the computational results indicates that the proposed iterative methods converges faster and required less execution time compared with the iterative methods from which they have been derived. This agrees with the theoretical results obtained on convergence order and efficiency index of the proposed methods in Section Conclusion In this paper, a technique utilized to improve convergence order and efficiency i of a two-step and three-step quadrature based iterative methods (Algorithm 2.6 and Algorithm 2.12) in [10] for the approximation of solution of nonlinear equations is developed. The convergence analysis of the developed methods are established via Taylor s series expansion technique. The implementation results suggest that, the methods proposed outperformed methods from which they have been derived on account of convergence order and efficiency index. Competing Interests The authors do not have any competing interest in the manuscripts.

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