Applied ad Computatioal Mathematics 07; 6(6): 38-4 http://www.sciecepublishiggroup.com/j/acm doi: 0.648/j.acm.070606. ISSN: 38-5605 (Prit); ISSN: 38-563 (Olie) Solvig a Noliear Equatio Usig a New Two-Step Derivative Free Iterative Methods Alyauma Hajjah Departmet of Iformatics Techical, (Sekolah Tiggi Ilmu Komputer) STIKOM Pelita Idoesia, Pekabaru, Idoesia Email address: alyaumah@yahoo.com To cite this article: Alyauma Hajjah. Solvig a Noliear Equatio Usig a New Two-Step Derivative Free Iterative Methods. Applied ad Computatioal Mathematics. Vol. 6, No. 6, 07, pp. 38-4. doi: 0.648/j.acm.070606. Received: August 8, 07; Accepted: September 6, 07; Published: November 7, 07 Abstract: I this paper, suggest aew two step iterative method for solvig a oliear equatio, which is derivative free by approimatig a derivative i the iterative method by cetral differece with oe parameter θ. The aew derivative free iterative method has a covergece of order four ad computatioal cost the family requires three evaluatios of fuctios per iteratio. Numerical eperimets show that the proposed a method is comparable to the eistig method i terms of the umber of iteratios. Keywords: Noliear Equatio, Iterative Method, Derivative Free, Cetral Differece, Covergece of Order. Itroductio Fidig for a root of a oliear equatios f( ) = 0 () Where f: D R R is most importat problem i sciece ad egieerig. Aalytical methods for solvig such a equatios are almost oeistet ad therefore it is oly possible to obtai approimate solutios by relyig o umerical techiques based o iteratio procedures, so solvig umerically becomes a alterative. A umerical method used fidig the solutio of a oliear equatio () is a oe step iterative methods [, 6], a two step iterative methods [4, 5, 7, 8, ] ad a three step iterative methods [, 3, 9]. Some other paper have discuss about a two step iterative methods for solvig a oliear equatio, that is a methods proposed by Kigis call as Kig Method (KM), with covergece of order four ad three evaluatios of the fuctio per iteratio so it possesses.587 as the efficiecy ide, ca epress their formula as []: + f( ) y = () f ( ) f( ) + β f( y) f( y) f( ) + f( y ) f ( ) ( β ) (3) The other a two step iterative methods are is a combiatio of Newto method [] ad Helley method [6] with order of covergece three, i the form: y f( ) y = (4) f ( ) ( f y ) f y ( f y f y ) ( ) '( ) + = f y '( ) ( ) "( ) The other a two step iterative methods are the iterative method proposed by Weerako ad Ferado [7], ca epress their formula as: (5) f( ) y = (6) f ( ) + = y f( ) f'( ) + f'( y ) Methods proposed by Kig i equatio () ad equatio (3), combiatio of Newto ad Helley methods i equatio (4) ad equatio (5) ad methods proposed by Weerako ad Ferado i equatio (6) ad equatio (7) is methods for solvig a oliear equatio, they ca fast to approimatig of a roots, but they ot success fidig a root if f'( ) =0 or f'( y ) =0, therefore must use derivative free iterative method. (7)
39 Alyauma Hajjah: Solvig a Noliear Equatio Usig a New Two-Step Derivative Free Iterative Methods Some other paper have discuss about a two step iterative methods free derivative for solvig a oliear equatio is a method proposed by Dehgha-Hajaria ca be call as Dehgha Method (DM), with order of covergece three ad four evaluatios of the fuctio per iteratio, so it possesses.36 as the efficiecy ide, ca epress their formula as [5]: y = + f( ) f( + f( )) f( f( )) ( + ) f( ) f( ) f( y ) f( + f( )) f( f( )) ad the oe method proposed by Hajjah is call as Hajjah Method (HM), with order of covergece three ad three evaluatios of the fuctio per iteratio, so it possesses.44 as the efficiecy ide, ca epress their formula as [7]: + y = θ f ( ) f( + θ f( )) f( ) f( ) + βf( y) θf( ) f( y) = y f( ) + ( β ) f( y ) f( + θf( )) f( ) (8) (9) (0) () Some other papers discuss the a two step type iterative methods for solvig a oliear equatio ad their applicatios, these iclude i [4, 8, 0, ] I this paper will discuss a two step iterative method of proposed by Kig. This method is free from derivatives by approimatig derivatives by cetral differece with oe parameter θ. The ew method is proved to be covergece of order four. Some eamples are give to compare the ew method with some bechmarked methods.. Basic Defiitios I order to establish the order of covergece of the ew derivative free methods, state some of the defiitios: Defiitio (Order of Covergece). Let α R, =,,3,... the, the sequece { } is said to coverge to α if lim α = 0 If, i additio, there eists a costat c 0, a iteger 0, ad p 0 such that for all 0, 0 p α c α, + () The { } is said to coverge to α with q-order at least p = or 3, the covergece is said to be q-quadratic or q-cubic, respectively [3]. Whe e = α is the error i the relatio th iterate, the p p e ce O( e + + = + ), (3) Is called the error of a equatio. By substitutig e = α for all i ay iterative method ad after simplifyig, obtai the error equatio for that method. The value of p thus obtaied is called the order of this method. Defiitio (Efficiecy Ide). Let r be the umber of fuctio evaluatios of the ew method. The efficiecy of the ew method is measured by the cocept of efficiecy ide [6] ad defied as: r p, where p is the order of the method. Defiitio 3 (Computatio Order of Covergece). Suppose that, ad + are three successive iteratios closer to the root α of equatio (). The the computatioal order of covergece [7] may be approimated by: l( + α)/( α) COC = l( α)/( α) 3. Proposed Methods I this sectio will discussed twostep iterative method: + (4) f( ) y = (5) f ( ) f( ) + β f( y) f( y) f( ) + f( y ) f ( ) ( β ) (6) Use cetral differece with oe parameter θ [], to approimate f'( ) i equatio (5) ad equatio (6) by: f'( ) ( θ ) f + f( ) f( θf( )) θ f( ) (7) Substitutig equatio (7) ito equatio (5) ad equatio (6), ca be chage as: + y = θ f ( ) f( + θf( )) f( θf( )) f( ) + βf( y) θf( ) f( y) = y f( ) + ( β ) f( y ) f( + f( )) f( θf( )) (8) (9) Equatio (8) ad equatio (9) are called Solvig Noliear Equatio Usig A New Two Step Derivative Free Iterative Method. Theorem Let f: D R for a ope iterval D. Assume that f has sufficietly cotiuous derivative i the iterval D. If has a simple root at f( ), ad if 0 is
Applied ad Computatioal Mathematics 07; 6(6): 38-4 40 sufficietly close to, the the ew iterative method defied by equatio (8) ad equatio (9) satisfies the followig error equatio: Where Proof. Let ( ) 3 3 4 + = 3 3 + e c c c θ F c c β c e (0) ( j) f ( ) j F = f'( ), c =, j >, ad j! f'( ) f( ) = 0 ad f( ) 0 epasio of f( ) about followig: e =. is be a simple root of f( ) = 0, the. Let e =. With Taylor =, ca be obtai as Where ( ) 3 4 5 F e ce c3e c4e O e f( ) = + + + + ( ) () c j ( j) f ( ) =, j >. j! f'( ) The, computig θf ( ) usig equatio () the multiplied byθ, ca be obtai as followig: ( ) 3 4 5 3 θ f ( ) = θ F e + 4θ c F e + 4θ c + θ c F e + O( e ) () Computig f(( ) + θf( )), ca be get after simplifyig: 3 + θ = + θ + + θ + θ + 3 + θ 3 + θ 3 4 3 3 3 4 5 3 4 +θ c3f + θ cf + 3 θ c3f ) e + ( θ c4f + c4f + 3θ cc3f f(( ) f( )) ( F F ) e ( c F 3 F c F c ) e ( c F 4 c F c F 3 4 3 3 c4f cc3f c4f cc3f c4f + 4θ + 5θ + 5θ + 8θ + 6θ 3 3 4 5 c F e O e The, computig f(( ) θ f( )), ca be have after simplifyig: +θ ) + ( ) (3) 3 θ = θ + θ + θ + 3 θ 3 θ f(( ) f( )) ( F F ) e ( c F 3 F c F c ) e ( c F 4 c F c F 3 4 3 3 3 4 5 3 4 c3f cf 3 c3f ) e ( c4f c4f 3 cc3f +θ + θ + θ + θ + θ 3 3 4 5 c F e O e 3 4 3 3 c4f cc3f c4f cc3f c4f 4θ 5θ 5θ + 8θ + 6θ +θ ) + ( ) (4) Usig equatio (3) ad equatio (4), compute f(( ) + θf( )) f(( ) θ f( )). The ca be obtai after simplifyig: f(( ) + θf( )) f(( ) θ f( )) 3 4 3 3 3 4 = θ F e + 6 θ c F e + (8θ c F + θ c F + 4 θ c F ) e + (0θc F 3 4 3 4 4 5 cc3f c4f cc3f e O e + 6θ + 8θ + 0 θ ) + ( ) (5) Cosiderig a geometric series ad usig equatio () ad equatio (5) computig ca be obtai after simplifyig: θf( ), f(( ) + θf( )) f(( ) + θf( )) θf( ) = e c e + c c θ c F e + θ c c F f(( ) + θf( )) f(( ) + θf( )) 3 ( 3 3 ) ( 3 3 4 5 c4f cc3 c4 c e O e 4θ + 7 3 4 ) + ( ). (6) The, substitutig equatio (6) ito equatio (8) ad = + e, ca obtai as followig: 3 3 = + + ( 3 + θ 3 ) + (3 4 + 4θ 4 + 4 θ 3 y c e c c F c e c c F c c c F Applyig Taylor epasio of f( y ) ad usig equatio (7), ca be obtai as followig: 3 3 3 3 = + θ 3 + 3 + θ 4 θ 3 + 4 f( y ) c Fe ( c F c F c F) e (4 c F c c F 3c F 3 4 5 7 c c ) e + O( e ) (7) 3 4 5 3 + 5c F 7 c c F) e + O( e ) (8) From equatio (9), by doig simple calculatio for equatio () ad equatio (8), ed up with: ( ) 3 3 4 3 3 e + = c c c θ F c c + β c e (9) From equatio (9) ca get that the ew iterative method free derivative (PFDM) has four order of covergece. Thiseds the proof.
4 Alyauma Hajjah: Solvig a Noliear Equatio Usig a New Two-Step Derivative Free Iterative Methods 4. Numerical Eperimets I this sectio will preset five umerical eamples, umerical simulatio are performed i order to compare the umber of iteratio DM by equatio (8) ad equatio (9), HM by equatio (0) ad equatio (), KM by equatio () ad equatio (3) ad PFDM by equatio (8) ad equatio (9) for the solutio of the oliear equatio. I this compariso use the followig test five fuctios: f = cos( ), = 0.739085335606 [0] f = si( ) 0.5, = 0.53598775598989[4, 5] f3 = si ( ) +, =.40449648534 [0] 3 4 3 f = +, =.676998865760 [4, 5] f5 = ta( ) cos( ) 0.5, = 0.85705677647869 [4, 5] All the computatio is doe i Maple 3, use tolerace is ε =.0 0 5, ad the maimum umber of iteratio allowed is00 iteratio. Stop the iteratio process by the followig criteria:. + < ε,. f( + ) < ε. Table. Comparisos of the discussed methods. Fuctios 0 The umber of iteratio of DM HM KM PFDM -.5 5 4 36 4 f -.0 5 4 4 4-0.5 6 3 3 3 0.0 4 3 3 3-0. 3 3 3 3 f 0.0 3 3 3 0.5 3.0 3 3 3 0.3 7 5 9 4 f 3 0.5 5 5 6 3.3 3 3.5 3 3 -.7 3 3 3 f 4-0.9 4 4 90-0.5 5 5 3 0.0 6 8 55 3 0.5 3 4 3 3 f 5 0.7 3 3 3 3. 7 7 3.5 3 9 3 3 0.739085335606 0.53598775598989.40449648534.676998865760 0.85705677647869 I Table, sig o the umber of iteratios states that the methods coverge to aother root (ot the same as the root). Based o umerical computatios i Table, i geeral for all fuctios after compariso show that the PFDM is superior to the iteratio methods such as DM, HM, ad MK. This is evidet from the umber of iteratio required to obtai a approimatio of the root or solvig a oliear equatio. 5. Coclusio Suggest ad aalyze a ew two step iterative methods free derivative to solve a oliear equatio. The ew iterative method free derivative has four order of covergece ad four evaluatios of the fuctio per iteratio, so it possesses.44 as the efficiecy ide. Although the efficiecy ide of our method is worse tha that of DM, KM ad HM but umerical eperimet show that it is comparable to the eitig method i terms of the umber of iteratio. So it ca be state that the proposed method is superior i fidig the success of roots from a oliear equatio. Ackowledgemets This work was supported by the Departmet of Iformatics Techical STIKOM Pelita Idoesia. Also authors thaks to Mr. M. Imra ad M. D. H. Gamal the Departmet of Mathematics, Uiversity of Riau, Pekabaru, Idoesia for their costat ecouragemet. Refereces [] K. E. Atkiso. Elemetary Numerical Aalysis, d Ed. Joh Wiley, New York, 993. [] M. S. M. Bahgat, ad M. A. Hafiz, 04. Three-Step Iterative Method with eighteeth order covergece for solvig oliear equatios, Iteratioal Joural of Pure ad Applied Mathematics. Vol.93, No., 85-94. [3] W. Bi, H. Re, ad Q. Wu (009). Three-step Iterative Methods with Eight-order Covergece for Solvig Noliear Equatios, Joural of Computatio ad Applied Mathmatics, 5, 05-. [4] A. Cordero, J. L. Hueso, E. Martiez, ad J. R. Torregrosa, 0. Steffese type methods for solvig oliear equatios, Joural of Computatioal ad Applied Mathematics, doi: 0.06/j.cam. 00.08.043. [5] M. Dehgha ad M. Hajaria. Some Derivative Free Quadratic ad Cubic Covergece Iterative Formulas for Solvig Noliear Equatio, J. Comput. Appl. Math, 9 (00), 9-3.
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