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Applied Mathematics, 2017, 8, 929-933http://www.scirp.org/journal/amISSN Online: 2152-7393ISSN Print: 2152-7385Simpson’s Method for Solution of NonlinearEquationHamideh EskandariDepartment of Mathematics, Payame Noor University, Tehran, IranHow to cite this paper: Eskandari, H.(2017) Simpson’s Method for Solution ofNonlinear Equation. Applied Mathematics,8, ved: June 5, 2017Accepted: July 11, 2017Published: July 14, 2017Copyright 2017 by author andScientific Research Publishing Inc.This work is licensed under the CreativeCommons Attribution InternationalLicense (CC BY en AccessAbstractThe programs offered for solving nonlinear equations, usually the old method,such as alpha, chordal movement, Newton, etc. have been used. Among thesemethods may Newton’s method of them all be better and higher integration.In this paper, we propose the integration method for finding the roots of nonlinear equation we use. In this way, Newton’s method uses integration methods to obtain. In previous work, [1] and [2] presented numerical integrationmethods such as integration, trapezoidal and rectangular integration methodthat are used. The new method proposed here, uses Simpson’s integration.With this method, the approximation error is reduced. The calculated resultsshow that this hypothesis is confirmed.KeywordsConvergence, Newton Method, Simpson Method, Nonlinear Equation,Iterative Method1. IntroductionLet f : R R be a smooth nonlinear function with a simple root x* , i.e.f ( x* ) 0 and f ′ ( x* ) 0. . We consider iterative methods for the calculationof x* that uses f and f ′ but not the higher derivatives of f and that generalizes the Newton method. Modifications for multiple roots will not be considered in the present contribution.To find the roots of an equation of nonlinear methods, there are many methods. Most famous method to find the approximate root of x* from the equation, non-linear and using the first derivative, is what called Newton’s method.([3]-[8]).We know that Newton’s method, an iterative procedure is to obtain an approximate root of the equation f ( x ) 0 , with an initial guess x0 R , forn 0,1, 2, valuesDOI: 10.4236/am.2017.87073July 14, 2017

H. Eskandarixn xn 1f ( xn )f ′ ( xn )(1)Calculates show that this formula is repeated, with the convergence of ordertwo.2. Elementarily MethodsNewton’s iteration formula in different ways and in many ways can be found[3]-[8]. But in this paper specific integration methods, we use. According to thedefinite integralf ′ ( t ) dt f ( x ) f ( xn )(2) f ( x ) f ( xn ) x f ′ ( t ) dt(3)x xncan writexnThe definite integral in this regard can be calculated by different methods. Ifthis is the definite integral of the square method [1] to obtain, can be writtenx xnf ′ ( t ) dt ( x xn ) f ′ ( xn )(4)After placement in relation to certain integration, we get the following statement.f ( x ) f ( xn ) ( x xn ) f ′ ( xn )(5)According to f ( x ) 0 is due to the new value x xn f ( xn )f ′ ( xn )to obtainthe same formula is repeated Newton [9].As well as to find solutions integrator can be used as [1] of midpoint method.x xn x x f ′ ( t ) dt ( x xn ) f ′ n 2 And with Placement x xn (6)f ( xn )that is Newton iteration, to new iteraf ′ ( xn )tion will reach a formula.xn xn 1f ( xn )(7) f ( xn ) f ′ xn 2 f ′ ( xn ) However, if we use trapezoidal method and midpoint method instead of rectangular method [1] [2], then the method can be writtenx xnf ′ ( t ) dt ( x xn )2 f ′ ( xn ) f ′ ( x ) (8)And the placement of certain integration, we get the following statement.f ( x ) f ( xn ) ( x xn )2 f ′ ( xn ) f ′ ( x ) x xn And according to f ( x ) 0 is the new value of 930(9)2 f ( xn )to f ′ ( xn ) f ′ ( x )

H. Eskandariobtain by replacing f ′ ( x ) with f ′ ( xn 1 ) , where xn xn 1f ( xn )is Newf ′ ( xn )ton repeated the following three methods to obtain explicit order.xn xn 12 f ( xn ) f ( xn ) f ′ ( xn ) f ′ xn f ′ ( xn ) (10)This relationship, modified Newton iteration formula [10] is.3. Preliminary ResultsNow back to the original Equation (3) return.To find the definite integral in the above equation, we use the method ofSimpson [3]. We can writex xnx xn2 f ′ ( x ) 4 f ′ x xn f ′ ( x ) f ′ ( t ) dt n 3 2 (11)By substituting the equation can be writtenf ( x) f ( xn ) x xn x xn f ′( x) 4 f ′ 6 2 f ′ ( xn ) (12)According to the f ( x ) 0 , we will gain new value6 f ( xn )and substitution x with xn 1 expli x xn x xn ′′′ fxffx4( n ) ( ) 2 f ( xn )cit method to obtain, where in xn is Newton method.xn 1f ′ ( xn )xn xn 16 f ( xn ) f ( xn ) xn xn f ′ ( xn ) f ′ x f ( xn ) 4 f ′ f ′ ( x ) n n f ′ ( xn ) 2 And then we’ll simplify.xn xn 16 f ( xn ) f ( xn ) f ( xn ) f ′ xn 4 f ′ xn f ′ ( xn ) ′′2 f ( xn ) f ( xn ) (13)This relationship, a new iterative method is a convergence of order higherthan two.Methods that have already been presented, rectangular and trapezoidal integration method is used. These methods have convergence times lower thanSimpson’s method. In the future we will see that this method is superior to othermethods and convergence is it better than before.Here, all computing software Maple is done and we have one of the followingstop conditions:931

H. EskandariTable 1. Example 1.nxnf ( xn 284564366841.071944817796 10 able 2. Example 2.nxnf ( xn 72883302121115472217.7296437954397 10 730.772882959149210112887 10 0.(I) xn 1 xn ε(II) f ( xn 1 ) εIn each of them ε 10 1000 and also all computations were done using Mapleusing 128 digit floating point arithmetic (Digits: 128).4. Numerical ExperimentsIn this section, we will test several functions in obtained iteration formula.Example 1:Consider the equation f ( x ) sin x x 1 0 . Starting from the pointx0 1.0 , we obtain the value of 1.9345632107520242676 , if1.9345632107520242676 is the exact answer. Different iterations of this method in Table 1.Example 2:Consider the equation f ( x ) x3 e x 0 . Starting from the point x0 1.0 ,we obtain the value of 0.77288295914921011285 , if0.77288295914921011285 is the exact answer. Different iterations of this method in Table 2.5. ConclusionIn this paper, to solve a nonlinear equation formula offered new iteration, wehave seen that this formula iteration was obtained using Simpson integration. Itwas observed that using examples provided, its accuracy is higher than the accuracy of Newton iterative method.AcknowledgementsThis article is supported by Payame Noor University.932

H. EskandariReferences[1]Frontini, M. and Sormani, E. (2003) Some Variant of Newton’s Method with ThirdOrder Convergence. Applied Mathematics and Computation, 140, 8-2[2]Homeier, H.H.H. (2005) On Newton-Type Methods with Cubic Convergence. Applied Mathematics and Computation, 176, 3]Atkinson, K.E. (1988) An Introduction to Numerical Analysis. John Wiley & Sons.[4]Cheney, E.W. and Kincaid, D. (2003) Numerical Mathematics and Computing.Thomson Learning.[5]Hildebrand, F.B. (1974) Introduction to Numerical Analysis. Tata McGraw-Hill.[6]Quarteroni, A., Sacco, R. and Saleri, F. (2000) Numerical Mathematics. Springer.[7]Stewart, G.W. (1996) After Notes on Numerical Analysis. SIAM.[8]Stoer, J. and Bulirsch, R. (1983) Introduction to Numerical Analysis. Springer-Verlag.[9]Weerakoom, S. and Fernando, T.G.I. (2000) A Variant of Newton s Method withAccelerated Third-Order Convergence. Applied Mathematics Letters, 13, 2[10] Kou, J.S., Li, Y.T. and Wang, X.H. (2006) On Modified Newton Methods with Cubic Convergence. Applied Mathematics and Computation, 176, ubmit or recommend next manuscript to SCIRP and we will provide bestservice for you:Accepting pre-submission inquiries through Email, Facebook, LinkedIn, Twitter, etc.A wide selection of journals (inclusive of 9 subjects, more than 200 journals)Providing 24-hour high-quality serviceUser-friendly online submission systemFair and swift peer-review systemEfficient typesetting and proofreading procedureDisplay of the result of downloads and visits, as well as the number of cited articlesMaximum dissemination of your research workSubmit your manuscript at: http://papersubmission.scirp.org/Or contact am@scirp.org933

Simpson’s Method for Solution of Nonlinear Equation Hamideh Eskandari Department of Mathematics, Payame Noor University, Tehran , Iran Abstract The programs offered for solving nonlinear equations, usually the old method, such as alpha, chordal movement, Newton, etc. have been used. Among these methods may Newton’s method of them all better and higher integration. be In this paper, we .