British Journal of Mathematics & Computer Science 19(2): 1-9, 2016; Article no.bjmcs.2922 ISSN: 221-081 SCIENCEDOMAIN international www.sciencedomain.org Some New Three Step Iterative Methods for Solving Nonlinear Equation Using Steffensen s and Halley Method Najmuddin Ahmad 1* and Vimal Pratap Singh 1 1 Department of Mathematics, Integral University, Lucknow, Uttar Pradesh, India. Authors contributions This work was carried out in collaboration between both authors. Both authors read and approved the final manuscript. Article Information DOI: 10.97/BJMCS/2016/2922 Editor(s): (1) Andrej V. Plotnikov, Department of Applied and Calculus Mathematics and CAD, Odessa State Academy of Civil Engineering and Architecture, Ukraine. (2) Zuomao Yan, Department of Mathematics, Hexi University. China. Reviewers: (1) Nasr Al Din IDE, University of Aleppo, Syria. (2) Fayyaz Ahmad, Politecnical University Catalonia, Barcelona, Spain. Complete Peer review History: http://www.sciencedomain.org/review-history/16602 Received: rd September 2016 Accepted: 29 th September 2016 Original Research Article Published: 18 th October 2016 Abstract In this paper, we introduce the comparative study of new three step iterative methods for finding the zeros of the nonlinear equation f(x) = 0. The new method based on the Steffensen s method and Halley method with using predictor corrector technique. It is established that the new method (NT-1) has convergence order sixth and second new method (NT-2) has convergence order seventh. Numerical tests show that the new methods is comparable with the well known existing methods and gives better results. Keywords: Non linear equations; iterative methods; three step; convergence analysis; Halley method; Steffensen s method. 1 Introduction Numerical analysis is the area of mathematics and computer sciences that creates, analyzes and implements algorithms for solving numerically the problems of continuous mathematics. Such problems originate generally from real world applications of algebra, geometry and calculus and they involve variables which vary continuously: These problems occur throughout the natural sciences, social sciences, engineering, medicine and business. New three step iterative methods for finding the approximate solutions of the *Corresponding author: E-mail: najmuddinahmad@gmail.com;
nonlinear equation f(x) = 0 are being developed using several different techniques including Taylor series, quadrature formulas, homotopy and decomposition techniques, see [1-1] and the references therein. The most famous of these methods is the classical Newton s method (NM) [16]. = ( ) ( ) (1) The Newton s method (1) was modified by Steffensen s method who replaced the first derivative () in Newton s method by forward difference approximation [16]. () = ( ) ( ) ( ) and obtained the famous Steffensen s method () [17,8,16]. = ( ( ))( ) (2) Newton s method and Steffensen s method are of second order converges. For a given, compute approximation solution by the iterative scheme = - ( ) ( ) ( ) ( ) ( ) () This is known as Halley s method has cubic convergence (see [7,12]). We use Predictor corrector methods, we shall now discuss the application of the explicit and implicit multistep methods, for the solution of the initial value problems. We use explicit (predictor) method for predicting a value and then use the implicit (corrector) method iteratively until the convergence is obtained [20]. 2 Iterative Methods It is well known that a wide class of problems, which arise in various fields of pure and applied sciences can be formulated in terms of nonlinear equations of the type. () = 0 () Various numerical methods have been developed using the Taylor series and other techniques. In this paper, we use another series of the nonlinear function () which can be obtained by using the trapezoidal rule and the Fundamental Theorem of Calculus. To be more precise, we assume that α is a simple root of () and γ is an initial guess sufficiently close to α now using the trapezoidal rule and fundamental theorem of calculus, one can show that the function () can be approximated by the series [19]. ()= (γ) + γ ()+ (γ ) () where () is the differential of. From () and (), we have = γ - 2 (γ ) () ( γ) (γ) (γ) (6) 2
Using (6), one can suggest the following iterative method for solving the nonlinear equations (). For a given initial choice, find the approximate solution, by the iterative scheme [19]. = - 2 ( ) ( ) - ( ) ( ) ( ) n = 0, 1, 2,,. (7) we use the predictor corrector technique. Using the Steffensen s method as a predictor, Halley method and equation (6) as a corrector, we suggest and analyze the following iterative method for solving the nonlinear equation () and this is the main motivation of this note [17,18,1]. Theorem 1: For a given initial choice, find the approximate solution by the iterative schemes. From equation (2), () and (6). = ( ( ))( ) = - ( ) ( ) ( ) ( ) ( ) = - 2 ( ) ( ) - ( ) ( ) ( ) n=0, 1, 2,, Theorem 1 is called the NEW THREE STEP METHOD -1 (NT-1) and has sixth order convergence. Theorem 2: From equation () and equation () we can have = - ( ) ( ) ( ) This is fixed point formulation enable us to suggest the following iterative method for solution the nonlinear equation. Theorem : For a given initial choice, find the approximate solution by the iterative schemes. From equation (2), () and theorem (2) = ( ( ))( ) = - = - ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n=0, 1, 2,, Theorem is called the NEW THREE STEP METHOD -2 (NT-2) and has seventh order convergence.
Convergence Analysis Let us now discuss the convergence analysis of the above Theorem 1. Theorem.1: let be a simple zero of sufficiently differential function for an open interval I, if is sufficiently close to then the three step iterative method defined by theorem 1 sixth order convergence. Proof: Let α be a simple zero of. Than by expanding ( ) and ( ) about α we have ( ) = + + +.. (8) ( ) = 1+ + + +. (9) Where =! () (α) k=1, 2,,. and = - α from (8), we have = + 2 + +. (10) ( +( )) = +( + +2 ) +. (11) From (10) and (11), we have ( ( )) = ( + +2 ) + (12) From (12), we have = α + ( + +2 ) +. (1) Let us set A = -. Then the equation (1) can be re written in the form A = ( + +2 ) +.. (1) Now expanding ( ), ( ), ( ) about and using (1), we have ( ) = + + + (1) ( ) = + 2 + + +. (16) ( ) = 2 + 6 + 12 + (17) Combining (1) (17), we have = + ( ) (18)
Also expanding ( ) about and using (18), we have ( ) = 1+2( ) + ( ) + (19) By substituting (8), (9), (18), (19) in theorem (1) and after some simple calculations, we obtain = α +( )( + +2 ) +. (20) = ( )( + +2 ) + (21) This shows that Theorem 1 is sixth order convergence. Let us now discuss the convergence analysis of the above theorem. Theorem.2: Let be a simple zero of sufficiently differential function for an open interval I, if is sufficiently close to then the three step iterative method defined by theorem seventh order convergence. Proof: From equation (19), we have ( ) = 2 + 2 + 2 + 2 +.. (22) By substituting (8), (9), (22) in theorem ( ) and after some simple calculations, we obtain = α +2( )( + +2 ) + (2) = 2( )( + +2 ) + (2) This shows that Theorem is seventh order convergence. Numerical Examples For comparisons, we have used Steffensen s method [20] and three step predictor-corrector Newton-Halley method () [8] defined respectively by and = w = ( ) ( ), ( ( ))( ) = - ( ) ( ) ( ) ( ) ( ), = - 2 ( ) ( ) - ( ) ( ) ( ), n = 0, 1, 2,..
All computations are performed using MATLAB. The following examples are used for numerical testing. () = 1+sin() () = + sin() - () = -x () = cos() x +1 () = - 2 () = -1.2 () = cos() () = x - () = sin() -1 - () = - 1. - () As for the convergence criteria, it was required that the distance of two consecutive approximations δ and also displayed is the number of iterations to approximate the zero (IT), the approximate zero and the value ( ). Table 1. Numerical examples and comparison Method IT ( ) δ, =. NT-1 NT-2, = NT-1 NT-2, =. NT-1 NT-2, =. NT-1 NT-2 2 2 6 0.918690700268 0.918690700268 0.918690700268 0.918690700268 1.171629610060 1.171629610060 1.171629610060 1.171629610060 0.61906128679 0.61906128679 0.61906128679 0.61906128679 0.607101681012 0.607101681012 0.607101681012 0.607101681012 -.0892098006e-016 9.7699626167018e-01 2.220609201e-016-1.2267629019e-01 1.01108796971e-010 2.088100872e-008 7.02866089921e-006.7889796160e-006.286177701e-00.689861972e-011.9902711067e-012 6.7899979e-009 7.08918000209e-009 6.8987691e-011.1062929002e-007.9108918222e-008 9.9920072216261e-016.16708967896e-010 6.97689979711e-011.6966018662e-012 6
Method IT ( ) δ, = NT-1 NT-2 6 0.82600201726 0.82600201726 0.82600201726 0.82600201726-2.290617e-01.1761067279e-010 2.96606686e-008 2.6809016109e-010 2.88919999062e-011, = NT-1 NT-2, =. NT-1 NT-2, =. NT-1 NT-2, = NT-1 NT-2, = NT-1 NT-2 Conclusion 6 2.7066099769 2.7066099769 2.7066099769 2.7066099769 0.7908121161 0.7908121161 0.7908121161 0.7908121161 2.12926110 Divergent 2.12926110 2.12926110-1.2902180119-1.2902180119-1.2902180119-1.2902180119 0.7676266201279 0.7676266201279 0.7676266201279 0.7676266201279 -.108626890e-01 6.661817709e-016-1.776689002e-01 1.7016128901e-008 1.9022072002e-008.981898669102e-009 1.98129968896e-010 7.19827168e-01 7.061922072e-010.899972628798e-011 2.10971696e-012.27006179e-011 1.8260998021e-008 2.29886798862e-008-1.99801228e-01 0.0002279120800172 2.89996928e-009 1.2069920802e-008 1.01799016610e-009 0 2.279287869e-012 1.892097198e-012 9.89891967e-010 1.800997691e-010 In this paper, we have suggested and analyzed newly developed technique is faster than Steffensen s method () and predictor-corrector Newton-Halley method (). This method based on a Steffensen s method and Halley method and using predictor corrector technique. Our method can be considered as significant improvement of Steffensen s method and and can be considered as alternative method of solving nonlinear equations. Competing Interests Authors have declared that no competing interests exist. References [1] Ostrowaki AM. Solution of equations and systems of equations. Academic Press, New-York, London; 1966. [2] Noor MA. New family of iterative methods for nonlinear equations. Appl. Math. Computation. 2007; 190: 8. 7
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[19] Noor MA, Noor KI, Kshif A. Some new iterative methods for solving nonlinear equations. World Applied Sciences Journal. 2012;20(6):870-87. [20] Argyros LK, Khattri SK. An improved semi local convergence analysis for the Chebyshev method. Journal of Applied Mathematics and Computing. 201;2(1,2):09-28. DOI: 10.1007/s12190-01-067-2016 Ahmad and Singh; This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Peer-review history: The peer review history for this paper can be accessed here (Please copy paste the total link in your browser address bar) http://sciencedomain.org/review-history/16602 9