A new modified Halley method without second derivatives for nonlinear equation

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1 Applied Mathematics and Computation 189 (2007) A new modified Halley method without second derivatives for nonlinear equation Muhammad Aslam Noor *, Waseem Asghar Khan, Akhtar Hussain Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan Abstract In a recent paper, Noor and Noor [K. Inayat Noor, M. Aslam Noor, Predictor corrector Halley method for nonlinear equations, Appl. Math. Comput., in press, doi: /j.amc ] have suggested and analyzed a predictor corrector method Halley method for solving nonlinear equations. In this paper, we modified this method by using the finite difference scheme, which has a quintic convergence. We have compared this modified Halley method with some other iterative of fifth-orders convergence methods, which shows that this new method is a robust one. Several examples are given to illustrate the efficiency and the performance of this new method. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Halley method; Predictor-corrector method; Iterative methods; Convergence; Examples 1. Introduction In recent years, several iterative type methods have been developed by using the Taylor series, decomposition and quadrature formulae, see [1 11] and the references therein. Using the technique of updating the solution and Taylor series expansion, Noor and Noor [10] have suggested and analyzed a sixth-order predictor corrector iterative type Halley method for solving the nonlinear equations. Also Kou et al. [6,7] have also suggested a class of fifth-order iterative methods. In the implementation of these methods, one has to evaluate the second derivative of the function, which is a serious drawback of these methods. To overcome these drawbacks, we modify the predictor corrector Halley method by replacing the second derivatives of the function f by its finite difference scheme. We prove that the new modified predictor corrector method is of fifth-order convergence. We also present the comparison of the new method with the methods of Kou et al. [6,7]. In passing, we would like to point out the results presented by Kou et al. [6,7] are incorrect. We also rectify this error. Several examples are given to illustrate the efficiency and robustness of the new proposed method. * Corresponding author. addresses: noormaslam@hotmail.com (M.A. Noor), waseemasg@gmail.com (W.A. Khan), dr_akhtar@comsats.edu.pk (A. Hussain) /$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi: /j.amc

2 M.A. Noor et al. / Applied Mathematics and Computation 189 (2007) Iterative methods We consider iterative methods to find a simple root of a nonlinear equation f ðxþ ¼0: ð1þ We assume that a is simple root of the Eq. (1) and c is an initial guess sufficiently close to a. Use the Taylor s series, we have f ðcþþf 0 ðcþðx cþþf 00 ðx cþ2 ðcþ þ¼0 ð2þ 2! where c is the initial approximation. From (2), one can have x ¼ c f ðcþ f 0 ðcþ : This formulation is used to suggest the following iterative method. Algorithm 2.1. For a given x 0, compute the approximate solution x nþ1 by the iterative schemes. x nþ1 ¼ x n f ðx nþ f 0 ðx n Þ : Algorithm 2.1 is the well-known Newton method, which has a quadratic convergence [11]. Also, from (2), we can obtain 2f ðcþf 0 ðcþ x ¼ c 2f 02 ðcþ fðcþf 00 ðcþ : This formulation allows us to suggest the following an iterative method for solving the nonlinear equation (1). Algorithm 2.2. For a given x 0, compute the approximate solution x nþ1 by the iterative schemes. 2f ðx n Þf 0 ðx n Þ x nþ1 ¼ x n 2f 02 ðx n Þ fðx n Þf 00 ðx n Þ : This is known as Halley s method, which has cubic convergence [4,5]. Noor and Noor [10], have suggested the following two-step method, using Algorithm 2.1 method as predictor and Halley s method as a corrector. Algorithm 2.3. For a given x 0 compute the approximate solution x nþ1 by the iterative schemes: f 0 ðx n Þ ; 2f ðy x nþ1 ¼ y n n Þf 0 ðy n Þ 2f 02 ðy n Þ fðy n Þf 00 ðy n Þ : If f 00 ðy n Þ¼0, then Algorithm 2.3 is exactly Algorithm 2.2 In order to implement this method, one has to find the second derivative of this function, which may create some problems. To overcome this drawback, several authors have developed involving only the first derivative. In order to overcome drawback, one usually uses the finite difference approximation of the second derivative. This idea is very important and plays a significant part in developing some iterative methods free from second derivatives. To be more precise, we consider f 00 ðy n Þffi f 0 ðy n Þ f 0 ðx n Þ : ð4þ y n x n Combining (3) and (4), we suggest the following new iterative method for solving the nonlinear equation (1) and this is the main motivation of this paper. ð3þ

3 1270 M.A. Noor et al. / Applied Mathematics and Computation 189 (2007) Algorithm 2.4. For a given x 0, compute the approximate solution x nþ1 by the iterative schemes f 0 ðx n Þ ; 2f ðx n Þf ðy x nþ1 ¼ y n n Þf 0 ðy n Þ 2f ðx n Þf 02 ðy n Þ f 02 ðx n Þf ðy n Þþf 0 ðx n Þf 0 ðy n Þf ðy n Þ : Algorithm 2.4 is called the two-step modified Halley s method. This method does not require the second derivative. Important characteristic of this method is that per iteration they require two evaluations of the function and one of its first derivatives. The efficiency of this class of methods is better than that of the well-known other methods involving the second-order derivative of the function We now state some fifth-order iterative methods which have been suggested by Noor and Noor [9] and Kou et al. [6,7] using quite different techniques. Algorithm 2.5 (NR [9]). For a given x 0, compute the approximate solution x nþ1 by the iterative schemes: 2f ðx n Þf 0 ðx n Þ y n ¼ x n 2f 02 ðx n Þ fðx n Þf 00 ðx n Þ ; 2½f ðx n Þþfðy x nþ1 ¼ x n n ÞŠf 0 ðx n Þ 2f 02 ðx n Þ ½fðx n Þþfðy n ÞŠf 00 ðx n Þ ; which is a two-step Halley method of fifth-order convergent. In a recent paper Kou et al. [6,7] have suggested following the iterative methods. Algorithm 2.6 (SHM [7]). For a given x 0, compute the approximate solution x nþ1 by the iterative schemes: f 0 ðx n Þ f 00 ðx n Þf 2 ðx n Þ 2f 03 ðx n Þ 2fðx n Þf 0 ðx n Þf 00 ðx n Þ ; f ðy x nþ1 ¼ y n n Þ f 0 ðx n Þþf 00 ðx n Þðy n x n Þ : Algorithm 2.7 (ISHM [6]). For a given x 0, compute the approximate solution x nþ1 by the iterative schemes: f 0 ðx n Þ f 00 ðx n Þf 2 ðx n Þ 2f 03 ðx n Þ 2fðx n Þf 0 ðx n Þf 00 ðx n Þ ; x nþ1 ¼ y n f ðy nþ f 0 ðx n Þ f 00 ðx n Þf ðy n Þ : 2f 03 ðx n Þ 3. Convergence analysis In this section we consider the convergence criteria of Algorithm 2.4. Theorem 3.1. Let r 2 I be a simple zero of sufficiently differentiable function f : I R! R for an open interval I. If x 0 is sufficiently close to r, then Algorithm 2.4 has fifth-order convergence. Proof. If a is the root and e n be the error at nth iteration, than e n ¼ x n a and using Taylor series expansion, we have f ðx n Þ¼f 0 ðx n Þe n þ 1!2 f 00 ðx n Þe 2 n þ 1!3 f 000 ðx n Þe 3 n þ 1!4 f ðivþ ðx n Þe 4 n þ 1!5 f ðvþ ðx n Þe 5 n þ 1!6 f ðviþ ðx n Þe 6 n þ Oðe7 n Þ; f ðx n Þ¼f 0 ðaþ½e n þ c 2 e 2 n þ c 3e 3 n þ c 4e 4 n þ c 5e 5 n þ c 6e 6 n þ Oðe7 n ÞŠ; ð5þ f 0 ðx n Þ¼f 0 ðaþ½1 þ 2c 2 e n þ 3c 3 e 2 n þ 4c 4e 3 n þ 5c 5e 4 n þ 6c 6e 5 n þ Oðe6 n ÞŠ; ð6þ

4 M.A. Noor et al. / Applied Mathematics and Computation 189 (2007) where c k ¼ 1 f ðkþ ðaþ!k f 0 ðaþ ; k ¼ 2; 3;...; and e n ¼ x n a: From (6) (8), we have, y n ¼ a þ c 2 e 2 n þð2c 3 2c 2 2 Þe3 n þð3c 4 7c 2 c 3 þ 4c 3 2 Þe4 n þð 6c2 3 þ 20c 3c c 2c 4 þ 4c 5 8c 4 2 Þe5 n þð 17c 4 c 3 þ 28c 4 c c 2c 5 þ 5c 6 þ 33c 2 c c 3c 3 2 þ 16c5 2 Þe6 n þð 22c 5c 3 þ 36c 5 c 2 2 þ 6c 7 16c 2 c 6 12c 2 4 þ 92c 4c 2 c 3 72c 4 c 3 2 þ 18c c2 3 c2 2 þ 128c 3c c6 2 Þe7 n þ Oðe8 n Þ: Using Taylor s series, we have ð7þ and f ðy n Þ¼f 0 ðaþ½c 2 e 2 n þð2c 3 2c 2 2 Þe3 n þð3c 4 7c 2 c 3 þ 5c 3 2 Þe4 n þð 6c2 3 þ 24c 3c c 2c 4 þ 4c 5 12c 4 2 Þe5 n þð 17c 4 c 3 þ 34c 4 c c 2c 5 þ 5c 6 þ 37c 2 c c 3c 3 2 þ 28c5 2 Þe6 n þð 22c 5c 3 þ 44c 5 c 2 2 þ 6c 7 16c 2 c 6 12c 2 4 þ 104c 4c 2 c 3 104c 4 c 3 2 þ 18c c2 3 c2 2 þ 206c 3c c6 2 Þe7 n þ Oðe8 n ÞŠ ð8þ f 0 ðy n Þ¼f 0 ðaþ½1 þ 2c 2 2 e2 n þð4c 2c 3 4c 3 2 Þe3 n þð6c 2c 4 11c 3 c 2 2 þ 8c4 2 Þe4 n þð28c2 2 c3 2 20c 4c 2 2 þ 8c 2c 5 16c 5 2 Þe5 n þð 16c 4 c 2 c 3 68c 3 c 4 2 þ 12c3 3 þ 60c 4c c 5c 2 2 þ 10c 2c 6 þ 32c 6 2 Þe6 n þð 84c 2c 3 3 þ 112c 3c 4 c c 3 c 2 c 5 þ 160c 3 c 5 2 þ 36c 4c 2 3 þ 72c 5c 3 2 þ 12c 2c 7 32c 2 2 c 6 24c 2 c c 4c c7 2 Þe7 n þ Oðe8 n ÞŠ: ð9þ Now similarly f ðx n Þf ðy n Þf 0 ðy n Þ¼f 03 ðaþ½c 2 e 3 n þð2c 3 c 2 2 Þe4 n þð3c 4 4c 2 c 3 þ Þe5 n þð 6c 2 3 þ 23c 3c 2 2 6c 2c 4 þ 4c 5 13c 4 2 Þe6 n þ Oðe7 n ÞŠ ð10þ and and f 0 ðx n Þf ðy n Þf 0 ðy n Þ¼f 03 ðaþ½c 2 e 2 n þ 2c 3e 3 n þð3c 4 þ 3c 3 2 Þe4 n þð12c 3c 2 2 þ 4c 5 6c 4 2 Þe5 n þð12c 4 c 2 2 þ 5c 6 þ 5c 2 c c 3c 2 2 þ 74c 3c 3 2 4c5 2 Þe6 n þ Oðe7 n ÞŠ; f 02 ðx n Þf ðy n Þ¼f 03 ðaþ½c 2 e 2 n þð2c 3 2c 2 2 þ 4c 3c 2 Þe 3 n þð2c 4 c 3 c 2 þ 9c 3 2 þ 8c2 3 2c2 2 Þe4 n þð6c c 3c 2 2 2c 2c 4 þ 4c 5 20c 4 2 þ 12c 4c 3 28c 2 c 2 3 þ 20c3 2 c 3Þe 5 n þ Oðe6 n ÞŠ ð11þ ð12þ f 02 ðy n Þf ðx n Þ¼f 03 ðaþ½e n þ c 2 e 2 n þðc 3 2c 2 2 Þe3 n þðc 4 þ 4c 2 c 3 4c 3 2 Þe4 n þðc 5 þ 6c 2 c 4 5c 3 c 2 2 þ 4c4 2 Þe5 n þ Oðe6 n ÞŠ: Now 2f 02 ðy n Þf ðx n Þ f 02 ðx n Þf ðy n Þþf 0 ðx n Þf ðy n Þf 0 ðy n Þ ð13þ ¼ f 03 ðaþ½2e n þ 2c 2 e 2 n þð2c 3 þ 6c 2 2 4c 3c 2 Þe 3 n þð2c 4 þ 9c 3 c 2 þ 9c 3 2 þ 8c2 3 2c2 2 Þe4 n þð2c 5 þ 14c 2 c 4 þ 26c 3 c 2 2 þ 28c4 2 6c2 3 12c 4c 3 þ 8c 2 c c3 2 c 3Þe 5 n þð12c 4c 2 2 þ 5c 6 þ 5c 2 c 2 3 þ 74c 3c c 3c 2 2 þ 4c 5 2 Þe6 n þ Oðe7 n ÞŠ: Using (10) (14) in Algorithm 2.4, we have x nþ1 ¼ a þð 6c 2 3 þ 20c 3c c 2c 4 þ 4c 5 8c 4 2 Þe5 n þ Oðe6 n Þ; e nþ1 ¼ð 6c 2 3 þ 20c 3c c 2c 4 þ 4c 5 8c 4 2 Þe5 n þ Oðe6 n Þ; which shows that algorithm 2.4 is fifth-order convergence. h ð14þ

5 1272 M.A. Noor et al. / Applied Mathematics and Computation 189 (2007) Table 1 Comparison of various iterative schemes f ðx n Þ x 0 NM NR [9] SHM [7] ISHM [6] Algorithm 2.4 f (108) 19 21(285) 14(156) (10) 3 2 (9) 3 (8) (10) 3 2 (9) 3 (8) 2 f (8) 2 2 (9) 2 (8) (8) 2 2 (9) 2 (8) 2 f (10) 3 2 (9) 3 (8) (10) 3 2 (9) 3 (8) 2 f (10) 3 2 (9) 3 (8) (8) 2 2 (9) 2 (8) 2 f (10) 2 2 (9) 3 (8) (12) 3 3 (12) 3 (12) 3 f (12) 3 2 (9) 3 (8) (12) 3 2 (9) 3 (8) 2 f (10) 3 2 (9) 3 (8) (10) 3 2 (9) 3 (8) 2 f (8) 2 2 (9) 2 (8) (10) 3 2 (9) 3 (8) 3 Results enclosed in small bracket () are original results which were reported in [6]. It appeared that the results reported in [7] and [6] are misleading. 4. Numerical examples We present some examples to illustrate the efficiency of the new developed two-step iterative methods in this paper. We compare the Newton method (NM) [1], the method of Noor and Noor (NR) [9], the method of Super-Halley (SHM) [7], the method of improvement of Super-Halley (ISHM) [6], and Algorithm 2.4, introduced in this present paper. We used ¼ The following stopping criteria is used for computer programs: ðiþ jx nþ1 x n j < e: ðiiþ jf ðx nþ1 Þj < e. f 1 ¼ x 3 þ 4x 2 10; x ¼ 1: ; f 2 ¼ x 2 e x 3x þ 2; x ¼ 0: ; f 3 ¼ x 3 10; x ¼ 2: ; f 4 ¼ cos x x; x ¼ 0: ; f 5 ¼ sin 2 x x 2 þ 1; x ¼ 1: ; f 6 ¼ x 2 þ sinðx=5þ 1=4; x ¼ 0: ; f 7 ¼ e x 4x 2 ; x ¼ 0: ; f 8 ¼ e x þ cosðxþ; x ¼ 1: : Also displayed are the numbers of iterations to approximate the zero (IT), the approximate zero x *. From Table 1, it is clear that the new modified Halley method perform better than the other fifth-order iterative methods suggested in [6,7,9]. 5. Conclusions We have shown that it is possible to obtain several modifications of Halley s method free from second derivative by using a new approach by replacing the second derivative from Halley s method. We prove that the order of convergence of this method is fifth. Using the idea of this paper, one can obtain a wide class of higher-order iterative methods free from second derivatives.

6 M.A. Noor et al. / Applied Mathematics and Computation 189 (2007) Acknowledgement This research is supported by the Higher Education Commission, Pakistan, through research Grant No. I-28/HEC/HRD/2005/90. References [1] S. Amat, S. Busquier, J.M. Gutierrez, Geometric construction of iterative functions to solve nonlinear equations, J. Comput. Appl. Math. 157 (2003) [2] J.A. Ezquerro, M.A. Hernandez, A uniparametric Halley-type iteration with free Second derivative, Int. J. pure Appl. Math. 6 (1) (2003) [3] J.A. Ezquerro, M.A. Hernandez, On Halley-type iterations with free second Derivative, J. Comput. Appl. Math. 170 (2004) [4] E. Halley, A new exact and easy method for finding the roots of equations generally and without any previous reduction, Phil. Roy. Soc. London 18 (1964) [5] A. Melman, Geometry and convergence of Halley s method, SIAM Rev. 39 (4) (1997) [6] Jisheng Kou, Yitian Li, Improvements of Chebyshev Halley methods with fifth-order convergence, Appl. Math. Comput. (2006), doi: /j.amc [7] Jisheng Kou, Yitian Li, Xiuhua Wang, A family of fifth-order iterations composed of Newton and third-order methods, Appl. Math. Comput. (2006), doi: /j.amc [8] M. Aslam Noor, Numerical Analysis and Optimization, Lecture Notes, Mathematics Department, COMSATS Institute of information Technology, Islamabad, Pakistan, [9] M. Aslam Noor, K. Inayat Noor, Fifth-order iterative methods for solving nonlinear equations, Appl. Math. Comput. (2006), doi: /j.amc [10] K. Inayat Noor, M. Aslam Noor, Predictor corrector Halley method for nonlinear equations, Appl. Math. Comput., in press, doi: /j.amc [11] J.F. Traub, Iterative Methods for Solution of Equations, Prentice-Hall, Englewood, Cliffs, NJ, 1964.