Research Article A New Second-Order Iteration Method for Solving Nonlinear Equations

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1 Abstract ad Applied Aalysis Volume 2013, Article ID , 4 pages Research Article A New Secod-Order Iteratio Method for Solvig Noliear Equatios Shi Mi Kag, 1 Arif Rafiq, 2 ad Youg Chel Kwu 3 1 Departmet of Mathematics ad RINS, Gyeogsag Natioal Uiversity, Jiju , Republic of Korea 2 School of Computer Sciece ad Mathematics, Hajvery Uiversity, Idustrial Area, Gulberg III, Lahore 54660, Pakista 3 Departmet of Mathematics, Dog-A Uiversity, Busa , Republic of Korea Correspodece should be addressed to Youg Chel Kwu; yckwu@dau.ac.kr Received 26 Jauary 2013; Accepted 13 March 2013 Academic Editor: Gue Lee Copyright 2013 Shi Mi Kag et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We establish a ew secod-order iteratio method for solvig oliear equatios. The efficiecy idex of the method is which is the same as the Newto-Raphso method. By usig some examples, the efficiecy of the method is also discussed. It is worth to ote that (i) our method is performig very well i compariso to the fixed poit method ad the method discussed i Babolia ad Biazar (2002) ad (ii) our method is so simple to apply i compariso to the method discussed i Babolia ad Biazar (2002) ad ivolves oly first-order derivative but showig secod-order covergece ad this is ot the case i Babolia ad Biazar (2002), where the method requires the computatios of higher-order derivatives of the oliear operator ivolved i the fuctioal equatio. 1. Itroductio Our problem, to recall, is solvig equatios i oe variable. We are give a fuctio f ad would like to fid at least oe solutio to the equatio f) = 0. Note that, priorly, we do ot put ay restrictios o the fuctio f; we eed to be able to evaluate the fuctio; otherwise, we caot eve check that agivesolutiox=αis true, that is, f(r) = 0. Ireality, the mere ability to be able to evaluate the fuctio does ot suffice. We eed to assume some kid of good behavior. The more we assume, the more potetial we have, o the oe had, to develop fast algorithms for fidig the root. At the same time, the more we assume, the fewer the fuctios are goig to satisfy our assumptios! This is a fudametal paradigm i umerical aalysis. We kow that oe of the fudametal algorithm for solvig oliear equatios is so-called fixed-poit iteratio method [1]. I the fixed-poit iteratio method for solvig the oliear equatio f) = 0, the equatio is usually rewritte as x=g), (1) where (i) there exists [a, b] such that g) [a, b] for all x [a, b], (ii) there exists [a, b] such that g ) L < 1 for all x [a,b]. Cosiderig the followig iteratio scheme: x +1 =g ), = 0, 1, 2,..., (2) ad startig with a suitable iitial approximatio x 0,webuild up a sequece of approximatios, say {x },forthesolutioof the oliear equatio, say α. Theschemewillcovergeto the root α,providedthat (i) the iitial approximatio x 0 is chose i the iterval [a, b], (ii) g has a cotiuous derivative o (a, b), (iii) g ) < 1 for all x [a,b], (iv) a g) bfor all x [a,b](see [1]). The order of covergece for the sequece of approximatios derived from a iteratio method is defied i the literature, as follows.

2 2 Abstract ad Applied Aalysis Defiitio 1. Let {x } coverge to α. If there exist a iteger costat p ad real positive costat C such that x lim +1 α α) p =C, (3) the p is called the order ad C thecostatofcovergece. To determie the order of covergece of the sequece {x }, let us cosider the Taylor expasio of g ) g )=g) + g ) 1! + g(k) ) k! x) k +. Usig (1)ad(2)i(4)we have x)+ g ) x) 2 + x +1 x=g ) x)+ g ) x) g(k) ) k! x) k +, ad we ca state the followig result [1]. Theorem 2 (see [2]). Suppose that g C p [a, b].ifg (k) ) = 0, for k = 1,2,...,p 1ad g (p) ) =0, the the sequece {x } is of order p. It is well kow that the fixed poit method has first order covergece. Durig the last may years, the umerical techiques for solvig oliear equatios have bee successfully applied (see, e.g., [2 4] ad the refereces therei). I [4], Babolia ad Biazar modified the stadard Adomia decompositio method for solvig the oliear equatio f) = 0 to derive a sequece of approximatios to the solutio, with early superliear covergece. However, their method requires the computatio of higher-order derivatives of the oliear operator ivolved i the fuctioal equatio. I this paper, a ew iteratio method extracted from the fixed poit method is proposed to solve oliear equatios. The proposed method has secod-order covergece ad the applied to solve some problems i order to assess its validity ad accuracy. It is worth to metio that our method ivolves oly first-order derivative but showig secod-order covergece. 2. New Iteratio Method Cosider the oliear equatio f ) =0, x R. (6) We assume that α is simple zero of f), adx 0 is a iitial guess sufficietly close to α. Equatio(6) isusually rewritte as x=g). (7) (4) (5) Followig the approach of [4], if g ) =1,wecamodify(7) by addig θ = 1tobothsides as follows: x+θx=θx+g), (1+θ) x=θx+g), (8) which implies that g θ x= θx+g) 1+θ =g θ ). (9) I order for (9)tobeefficiet,wecachooseθ such that ) = 0, we yields so that (9)takestheform θ= g ), (10) x= g ) x+g) 1 g. (11) ) This formulatio allows us to suggest the followig iteratio methods for solvig oliear equatio (6). Algorithm 3. For a give x 0,wecalculatetheapproximatio solutio x +1, by the iteratio scheme x +1 = g )x +g ), g 1 g ) ) =1. (12) 3. Covergece Aalysis Now we discuss the covergece aalysis of Algorithm3. Theorem 4. Let f:d R R for a ope iterval D ad cosider that the oliear equatio f) = 0 (or x = g)) has a simple root α D,whereg) : D R R be sufficietly smooth i the eighborhood of the root α; the the order of covergece of Algorithm 3 is at least 2. Proof. The iteratio scheme is give by x +1 = g )x +g ), g 1 g ) ) =1. (13) Let α be a simple zero of f, e =x α,wheree is the error term ivolved at the th step of Algorithm 3ad g =α. (14)

3 Abstract ad Applied Aalysis 3 By Taylor s expasio, we have α e +1 = g (α e )(α e )+g(α e ) 1 g (α e ) this shows that =( (g e g +e 2 g (α e )+g e g +e 2 g (1 (g e g +e 2 g 1 =( (g e g +e 2 g (α e ) +α e g +e 2 g (1 g +e g e 2 g 1 =(α αg +αe g e 2 (g +αg ) (1 g +e g e 2 g 1 =α ρ(α,g,g,g )e 2, (15) e +1 =ρ(α,g,g,g )e 2 +. (16) This completes the proof. Remark 5. For G ) = g ) x+g) 1 g ) (17) Table 1 x 1 = x 1 = x 2 = x 2 = 2 x 3 = x 4 = x 5 = x 6 = x 7 = x 8 = x 9 = x 10 = x 11 = Table 2 x 1 = x 1 = 2.12 x 2 = x 3 = x 4 = 2.12 adusigthesoftwaremaple,wecaeasilydeducethat G ) = g ) ( x+g)) (1 g )) 2, (18) G ) = ((1 g ))(g ) ( x+g)) (1 g )) 1. g ) (1 g ))) +2g 2 ) ( x+g))) (19) Now,itcabeeasilyseethatfor(14) weobtaig = α, G = 0, ad G = (1 g )g =0. Hece, accordig to Theorem 2, Algorithm 3 has secodorder covergece. 4. Applicatios Now we preset some examples [4] to illustrate the efficiecy of the developed methods amely, Algorithm 3.Wecompare the fixed poit method (FPM) with Algorithm 3. Example 6. Cosider the equatio x 3 +4x 2 +8x+8=0.We have g) = (1+(1/2)x 2 +(1/8)x 3 ) ad g ) = x (3/8)x 2. The exact solutio of this equatio is 2. Takex 0 = 1.9; the the compariso of the two methods is show i Table 1 correct up to four decimal places. Example 7. Cosider the equatio x+l 2) = 0.Wehave g) = 2+e x ad g ) = e x.thegraphicalsolutioofthis equatio is (4D). Take x 0 = 2.1, the the compariso of the two methods is show i Table 2 up to four decimal places.

4 4 Abstract ad Applied Aalysis Table 3 x 1 = x 1 = x 2 = x 2 = x 3 = x 3 = x 4 = x 5 = x 6 = x 7 = x 8 = x 9 = x 10 = x 11 = x 12 = x 13 = x 14 = x 15 = x 16 = x 17 = x 18 = x 19 = x 20 = x 21 = Table 4 x 1 = x 1 = x 2 = x 2 = x 3 = x 4 = x 5 = x 6 = x 7 = x 8 = x 9 = x 10 = x 11 = x 12 = x 13 = Coclusios A ew iteratio method for solvig oliear equatios is established. By usig some examples the performace of the method is also discussed. The method is performig very well i compariso to the fixed poit method ad the method discussed i [4]. The method ca be studied for fuctioal equatios ad ca be exteded to a system of oliear equatios. Ackowledgmets The authors would like to thak the editor ad referees for useful commets ad suggestios. This study was supported by research fuds from Dog-A Uiversity. Refereces [1] E. Isaacso ad H. B. Keller, Aalysis of Numerical Methods, Joh Wiley & Sos, New York, NY, USA, [2] E. Babolia ad J. Biazar, O the order of covergece of Adomia method, Applied Mathematics ad Computatio, vol. 130, o. 2-3, pp , [3] S. Abbasbady, Improvig Newto-Raphso method for oliear equatios by modified Adomia decompositio method, Applied Mathematics ad Computatio, vol.145,o.2-3,pp , [4] E. Babolia ad J. Biazar, Solutio of oliear equatios by modified Adomia decompositio method, Applied Mathematics ad Computatio,vol.132,o.1,pp ,2002. Example 8. Cosider the equatio x 3 +4x 2 +8x+8 = 0. We have g) = x 2 2x 3 +x 4 0.2x 5 ad g ) = 3.6x 6x 2 +4x 3 x 4. The graphical solutio of this equatio is (5D). Take x 0 = 0.28, the the compariso of the two methods is show i Table 3 correct up to five decimal places. Example 9. Cosider the equatio e x 3x 2 = 0.Wehave g) = e x /3 ad g ) = (1/2 3)e x/2. The graphical solutio of this equatio is 0.91 (2D). Take x 0 = 0.8; the the compariso of the two methods is show i Table 4 corrected up to five decimal places.

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