Three-Step Iterative Methods with Sixth-Order Convergence for Solving Nonlinear Equations

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1 Article Three-Step Iteratie Methods with Sith-Order Coergece or Solig Noliear Eqatios Departmet o Mathematics, Kermashah Uiersity o Techology, Kermashah, Ira (Correspodig athor; bghabary@yahoocom Receied: Jaary, Reised: Febrary, Accepted: 6 March Abstract I this paper, we deelop ew amilies o sith-order methods or solig simple zeros o o-liear eqatios These methods are costrcted sch that the coergece is o order si Each member o the amilies reqires two ealatios o the gie ctio ad two o its deriatie per iteratio These methods hae more adatages tha Newto s method ad other methods with the same coergece order, as show i the illstratio eamples Keywords: Iteratie methods, Simple-zero o oliear eqatios, Newto s method Itrodctio Newto s method or the calclatio o the simple root α ie, ( α ad ( α, o a oliear eqatio ( gie by, + ( is probably the most widely sed iteratie method o qadratic order, : is a cotiosly dieretiable real aled ctio Drig the last seeral years meros papers [-7], deoted to iteratie schemes o dieret order o coergece, hae appeared i arios jorals I this paper, we cosider the iteratie methods to id a simple zero i sch a way that ew approimatios to a zero o ( is calclated by calclatig (, ad possibly its deriaties or a mber o ales o the idepedet ariable, at each step I the literatre, some higher order methods hae bee deeloped by cosiderig three-step methods i which we se three dieret poits per iteratio Cosider the ollowig ith-order iteratie scheme deied i [], as ( + + y y ( + 7 ( y 5 ( ( ( ( ( y ( Ch ad Ham [] proposed a sith-order amily as ollows + z ( y ( ( y z ( z + + μ + γ, ( ( ( ( ( ( y z y (5 ( ( y ( ad μ, γ Throghot the rest o this paper y is deied by Eq (, ad z by Eq (5 Gra ad Barrero [] deeloped a sith-order method which is gie by Walailak J Sci & Tech ; 9(: 9 5

2 Three Step Iteratie Methods with Sith Order + + ( β ( y z + β ( y ( (6 ad β Ko ad Li [5] preseted a sith-order method as ollows + z ( z (7 + L ( +, ( ( z ( ( L ( [ ( ] (8 Aother sith-order amily o modiied Ostrowski s method was cosidered by Sharma ad Gha [6] + ( y + z (9 + ( y ( Weihog et al [7] deeloped a oeparameter amily o eighth-order method, which is ( + ( + α ( t ( w + z ( + α ( t F( ( ( y ( y t y ( 5 ( y ( ( ( y w z ( 5 ( y F( ad F ( [ z, y ] + [ z,, ]( z y A sith-order method deeloped by Parhi ad Gpta [8], is gie by: ( y +, ( ( + (, ( + ( y ( y + ( ( ( + ( y ( Some more recet methods are also sed i the literatre [9-] Motiated by the recet actiities i this directio, i this paper we propose a class o sithorder methods to id a simple zero The rest o this paper is orgaized as ollows I Sectio, we cosider a geeral iteratie scheme, aalyze it to preset a amily o ith-order methods the seeral kow special cases o this amily are listed Sectio, the ormlae are tested ad their perormace is compared with some kow methods Fially, coclsios are draw i the last sectio Mai reslt I this sectio, by the idea o costrctig iteratio methods with kow ctios, we hae geeralized ad aalyzed a iteratie scheme i three steps i the ollowig orm ( y +, ( ( t, ( ( + ( y ( t + H ( (, ( y, ( I Theorem, we will impose the coditio o H (, i sch a way that the iteratie methods deied by Eq ( hae a sith-order o coergece, bt irst let s poit ot the deiitio o order o coergece or ied poit iteratio method as ollows Deiitio o order o coergece: The seqece prodced by the iteratio ctio F, as + F( is said to be coerget to the ied poit α o F, ie, F ( α α, o order p i ad oly i we hae p F ( α + + Ce p, C p ade α Theorem Let α I be a simple zero o a sicietly dieretiable ctio : I o a ope iteral which cotais as a close iitial approimatio toα I H (, satisies the coditios H (,, H (,, H (, H (,, the the method deied by Eq (, is o at least sith-order 5 Walailak J Sci & Tech ; 9(

3 Three Step Iteratie Methods with Sith Order Proo Let α be a simple zero o Cosider the ied-poit iteratio ctio F to be deied as ( ( ( F( H( (, ( y( t, ( ( Where ( y(, ( t( ( ( + ( y( Here, de to tedios ealatio o deriaties o F, we employ the symbolic comptatio o the Maple package to compte the Taylor epasio o F ( arod α We id that F( α + K e + K e + + K e + O( e, Where K ( + H (,( c + 6 c It ca be easily eriied that i we take H (,, ( the we hae K Cosiderig Eq (, K leads to ( ( K H (, + c + H (, + c c This also ca be aished, by lettig H (,, ( Sbstittig Eq ( ad Eq ( ito K 5, we get ( (, + (, ( H (, H (, c c K H H c + 5 So the proo is complete i we hae H (, H (, It is see that, Eq ( with + H (,, gies a sith-order method + which has bee itrodced by Parhi ad Gpta [8] ad Eq ( Moreoer some ew sith-order methods ca be itrodced by certai choices or H (, i Eq ( For eample, i we take H (, i Eq (, we obtai the ollowig sith-order method y+,, + ( y ( y ( ( + ( 7 ( y + ( ( The mai poit that shold be metioed here is that the order o Newto's method is improed or its with additioal ealatios o the two ctios Ideed per iteratio i the methods deied by Eq ( reqires two ctios ad two irst deriatie ealatios I we cosider the deiitio o the eiciecy ide [] as r p, p is the order o the method ad r is the mber o ctioal ealatios per iteratio reqired by the method, ths the iteratio scheme deied by Eq ( has the eiciecy ide eqal to 6 565, which is better tha that o Newto s method So the order o coergece ad comptatioal eiciecy o Newto s method are greatly improed Nmerical eamples I this sectio, Method (BGM, deied by Eq (, is employed to sole some oliear eqatios ad compared with Newto's method (NM, Eq (, Ko ad Li s method [5] (KM, Sharma ad Gha s method deied by Eq (9 (SM or solig the ollowig test ctios ( si ( +, ( +, ( (, + 7 (, 5 (, 6 7 ( ( e ( e +, Tables ad respectiely show the mber o eeded iteratios ad ctio comptatios or all methods carried ot i the Maple with 8 digit 5 loatig poit reqired sch that ( < Walailak J Sci & Tech ; 9( 5

4 Three Step Iteratie Methods with Sith Order Table Comparisos o mber o iteratio o arios sith-order coerget iteratie methods NM KM, 5 SM BGM, 5, 6, 6, 5 6, 5 7, 5 8, 5 5 5, 8 Di 8 5, 6, 5 6, 6 7, 5 Di 6 6 7, Table Comparisos o mber o ctioal comptatios eeded o arios sith-order coerget iteratie methods NM KM, 8 SM 8 BGM 8, 8 8 8, 8 8 8, 8, , 5, , , , 6, 6 6 6, 7, 5-8, Walailak J Sci & Tech ; 9(

5 Three Step Iteratie Methods with Sith Order The reslts i Tables ad show that the ew method reqires a lower mber o ctioal comptatios tha other methods, especially Newto s method Thereore, it is o practical iterest ad ca work better tha them Coclsios We hae obtaied a class o three-step methods o order sith, which prodces may sith-order iteratie schemes Per iteratio the preset method reqires three ealatios o the ctio ad oe ealatio o its irst It ca be see that the merical reslts, agree well with the coergece aalysis deeloped i Theorem Ackowledgemets The athor wold like to thak the reeree or his/her costrctie commets ad sggestios which sbstatially improed the qality o this paper Reereces [] JF Trab Iteratie Methods or the Soltio o Eqatios Pretice Hall, Eglewood Clis, New Jersey, 96 [] L Fag, L S ad G He A eiciet Newto-type method with ith-order coergece or solig oliear eqatios Comp Appl Math 8; 7, 69-7 [] C Ch ad YM Ham Some sith-order ariats o Ostrowski root-idig methods Appl Math Compt 7; 9, 89-9 [] M Gra ad JLD Barrero A improemet to Ostrowski root-idig method J Math Aal Appl 6; 7, 5-6 [5] J Ko ad Y Li A ariat o Chebyshe s method with sith-order coergece Nmer Algorithms 6;, 7-8 [6] JR Sharma ad RK Gha A amily o modiied Ostrowski methods with accelerated sith-order coergece Appl Math Compt 7; 9, -5 [7] B Weihog, R Hogmi ad W Qigbiao Three-step iteratie methods with eighthorder coergece or solig oliear eqatios J Compt Appl Math 9; 5, 5- [8] SK Parhi ad DK Gpta A sith order method or oliear eqatios Appl Math Compt 8;, 5-5 [9] C Ch ad B Neta A ew sith-order scheme or oliear eqatios Appl Math Lett ; 5, 85-9 [] L Che ad Y Ma A ew modiied Kig- Werer method or solig oliear eqatios Compt Math Appl ; 6, 7-5 [] MS Petkoić, J Džić ad B Neta Iterpolatory mltipoit methods with memory or solig oliear eqatios Appl Math Compt ; 8, 5- [] X Zho, X Che ad Y Sog Costrctig higher-order methods or obtaiig the mltiple roots o oliear eqatios J Compt Appl Math ; 5, 99-6 [] L Fag ad G He Some modiicatios o Newto's method with higher-order coergece or solig oliear eqatios J Compt Appl Math 9; 8, 96- [] W Gatschi Nmerical Aalysis: A Itrodctio Birkhaiser, Qi-Woodbie, Woodbie, New Jersey, 997 [5] J Ko ad Y Li A improemet o Jarratt method Appl Math Compt 7; 89, 86- Walailak J Sci & Tech ; 9( 5

Some Variants of Newton's Method with Fifth-Order and Fourth-Order Convergence for Solving Nonlinear Equations

Some Variants of Newton's Method with Fifth-Order and Fourth-Order Convergence for Solving Nonlinear Equations Copyright, Darbose Iteratioal Joural o Applied Mathematics ad Computatio Volume (), pp -6, 9 http//: ijamc.darbose.com Some Variats o Newto's Method with Fith-Order ad Fourth-Order Covergece or Solvig