New Twelfth Order Iterative Methods for Finding Multiple Roots of Nonlinear Equations

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1 Aerica Joural o Coputatioal ad Applied Matheatics 05 5(: -4 DOI: 059/jajca New Twelth Order Iterative Methods or Fidig Multiple Roots o Noliear Equatios R Thukral Padé Research Cetre 9 Deaswood Hill Leeds West Yorkshire LS7 5JS Eglad Abstract The ais o this paper are irstly to preset our ew twelth order iterative ethods or solvig oliear equatios ad secodly to itroduce ew orulas or approiatig the ultiplicity o the iterative ethod It is proved that these ethods have the covergece order o twelve requirig si uctio evaluatios per iteratio Nuerical coparisos are icluded to deostrate eceptioal covergece speed o the proposed ethods Keywords Modiied Newto-type ethod Root-idig Noliear equatios Multiple roots Order o covergece Eiciecy ide Itroductio Fidig the root o oliear equatios is oe o iportat proble i sciece ad egieerig [-8] I this paper we preset our ew ultipoit higher-order iterative ethods to id ultiple roots o the oliear equatio ( = 0 : I R R or a ope iterval I is a scalar uctio The ultipoit root-solvers is o great practical iportace sice it overcoes theoretical liits o oe-poit ethods cocerig the covergece order ad coputatioal eiciecy I recet years ay odiicatios o the Newto-type ethods or siple roots have bee proposed ad aalysed [] ad little work has bee doe o ultiple roots Thereore the prie otive o this study is to develop a ew class o ulti-step ethods or idig ultiple roots o oliear equatios o a higher tha the eistig iterative ethods I order to costruct the ew twelth order ethod or idig ultiple roots we use the well-established ourth order ethod give i [ ] The purpose o this paper is to show urther developet o the ith order ethods ad itroduce ew orulas or approiatig the ultiplicity o the iterative ethods This paper is actually a cotiuatio o the previous study [] The etesio o this ivestigatio is based o the iproveet o the ith order ethod I additio the ew iterative ethods have a better eiciecy ide tha the eight to te covergece order ethods [0 ] Hece the proposed twelth order ethods are sigiicatly better whe copared with these established ethods * Correspodig author: rthukral@hotailcouk (R Thukral Published olie at Copyright 05 Scietiic & Acadeic Publishig All Rights Reserved The structure o this paper is as ollows Soe basic deiitios relevat to the preset work are preseted i the sectio I sectio the ew ulti-poit ethods are deied ad proved I sectio 4 the ew orulas or approiatig the ultiplicity o the iterative ethods are described I sectio 5 two well-established ethods are stated it will deostrate the eectiveess o the ew twelth order iterative ethods Fially i sectio 6 uerical coparisos are ade to deostrate the perorace o the preseted ethods Preliiaries I order to establish the order o covergece o the ew twelth order ethods we state soe deiitios: be a real-valued uctio with a Deiitio Let root α ad let { } be a sequece o real ubers that coverge towards α The order o covergece p is give by li α = ζ 0 ( p ( α p R ad ζ is the asyptotic error costat [6 7] Deiitio Let ek = k α be the error i the kth iteratio the the relatio p p k ζ k k e = e Ο e ( is the error equatio I the error equatio eists the p is the order o covergece o the iterative ethod [6 7] Deiitio Let r be the uber o uctio evaluatios

2 4 R Thukral: New Twelth Order Iterative Methods or Fidig Multiple Roots o Noliear Equatios o the ethod The eiciecy o the ethod is easured by the cocept o eiciecy ide ad deied as r p ( p is the order o covergece o the ethod [6] Deiitio 4 Suppose that ad are three successive iteratios closer to the root α The the coputatioal order o covergece ay be approiated by l δ δ COC (4 l δ δ ( ( δ = [] i i i Costructio o the Methods ad Covergece Aalysis I this sectio we deie ew twelth order iterative ethods or idig ultiple roots o a oliear equatio I order to costruct ew twelth order ethods we use well kow ourth order iterative ethods preseted by Thukral Shara et al Shegguo et al ad Soleyai et al [ ] Method It is well established that the irst two step is the Thukral ourth order ethod [0] ad the ew third step is i the or o the Osada third order ethod [] Cosequetly we obtai a ew twelth order ethod based o these two well-established ethod The ew schee is give as ( y = z = y i (5 i / ( y ( (6 i= ( ( ( z = z ( ( N 0 is the iitial guess ad provided that the deoiator o (7 is ot equal to zero Theore Let α I be a ultiple zero o a suicietly dieretiable uctio : I R R or a ope iterval I with ultiplicity which icludes 0 as a (7 iitial guess o α The the iterative ethod deied by schee (7 has twelth order covergece Proo Let α be a ultiple root o ultiplicity o a suicietly dieretiable uctio ( ad ( α = 0 We deote the errors give by each step as e= α e = yα ad eˆ = z α Usig the Taylor series epasio o ( ( α y y about α we have ( = e ce ce ce (8! ( ( α ( = e (! (9 ce ce ( ( α ( y = e ce ce ce (0! ( ( α ( y = e (! ( ce ce N ad c k = ( k! ( α ( ( k! ( α ( Moreover by (5 we have ( y = e = e ( ( c c c e e e The epasio o ( y about α ad sipliyig yields ( α c ( y = e! c ( c e c

3 Aerica Joural o Coputatioal ad Applied Matheatics 05 5(: -4 5 u c 4 u c c 4 c 6 c c e c (4 u = u = (5 Dividig (4 by (8 we have c ω ω e e e ( y = ( c c Furtherore we have ( ω ( y c ω e e e c (6 ω = (7 ω = c c 4 ω = ( ( c ( ( cc 4 ( c 6 cc Substitutig appropriate epressios i (6 we obtai z α = y α i we obtai the asyptotic error costat (8 i / ( y ( (9 i= ( ( ( 4 z α = c c c e (0 about α We progress to epad ( z ( z ( z we have ( ( α = eˆ ce ˆ ceˆ ce ˆ (! ( ( α (! = eˆ ( ce ˆ ceˆ ( ( α (! = eˆ ( ( ( ce ˆ ceˆ Substitutig appropriate epressios i (7 ( z ( e = z α ( (4 ( Sipliyig (4 we obtai the asyptotic error costat 4 e = c ( c c c c c ( c c c e (5 The epressio (5 establishes the asyptotic error costat or the twelth order o covergece or the ew Newto-type ethod deied by (7 Method Aother twelth order iterative ethod is costructed by usig a ourth order ethod preseted by Shegguo et al [5] As beore the irst two steps is the Shegguo et al ethod ad third step is i the or o Osada third order ethod The ew twelth-order iterative ethod is give as y ( = (6 z ( ( y ( ( = y (7

4 6 R Thukral: New Twelth Order Iterative Methods or Fidig Multiple Roots o Noliear Equatios Theore ( z = z ( (8 ( α I be a ultiple zero o a suicietly Let dieretiable uctio : I R R or a ope iterval I with ultiplicity which icludes 0 as a iitial guess o α The the iterative ethod deied by schee (8 has twelth order covergece Proo Usig appropriate epressios i the proo o the theore ad substitutig the ito (8 we obtai the asyptotic error costat 4 6 e = 54 ( ( c c c c c (9 5 ( sc s cc c e s = (0 s = ( The epressio (9 establishes the asyptotic error costat or the twelth order o covergece or the ew Newto-type ethod deied by (8 Method The third twelth order iterative ethod is based o the Shara et al ourth order ethod preseted i [7] Here also the irst two steps is the Shara et al ethod ad third step is i the or o Osada third order ethod The ew twelth order iterative ethod is give as y ( = ( ( z = k 8 ( ( k ( k ( y ( y ( ( ( z = z ( (4 ( k = 4 8 (5 k = ( (6 ( ( Theore Let α I be a ultiple zero o a suicietly dieretiable uctio : I R R or a ope iterval I with ultiplicity which icludes 0 as a iitial guess o α The the iterative ethod deied by schee (4 has twelth order covergece Proo Usig appropriate epressios i the proo o the theore ad substitutig the ito (4 we obtai the asyptotic error costat 4 6 e = 54 ( c c c c c 5 ( sc s4 cc c e 5 4 (7 s = (8 s4 = (9 The epressio (7 establishes the asyptotic error costat or the twelth order o covergece or the ew Newto-type ethod deied by (4 4 Method 4 The ourth twelth order iterative ethod is based o the Soleyai et al ourth order ethod preseted i [6] Here also the irst two steps is the Soleyai et al ethod ad third step is i the or o Osada third order ethod The ew twelth order iterative ethod is give as

5 Aerica Joural o Coputatioal ad Applied Matheatics 05 5(: -4 7 y ( = 4q t z = q t q q t qt ( z (40 (4 = z ( (4 ( q Theore 4 ( = q = 6 p t ( ( y = t = (4 (44 α I be a ultiple zero o a suicietly Let dieretiable uctio : I R R or a ope iterval I with ultiplicity which icludes 0 as a iitial guess o α The the iterative ethod deied by schee (4 has twelth order covergece Proo Usig appropriate epressios i the proo o the theore ad substitutig the ito (4 we obtai the asyptotic error costat 7 6 e = 54 ( ( c c c c c (45 6 ( sc 5 s6 cc c e 4 s5 = 8 (46 4 s6 = (47 The epressio (45 establishes the asyptotic error costat or the twelth order o covergece or the ew Newto-type ethod deied by (4 4 New Forulas or Approiatig the Multiplicity I this sectio we derive soe ew orulas to approiate the ultiplicity o the ethod I [6] Thukral preseted a ew orula or approiatig the ultiplicity as ( (48 ( ( ( I act this orula was discovered by Lagouaelle i [9] ad apparetly Thukral rediscovered this orula However epirically we have oud that the orula should be epressed as ( (49 The approiatios obtaied by the ew ad old orulas are based o the Schroder secod order ethod [4] give as ( ( = (50 Deiitio 5 Suppose that ad are three successive iteratios closer to the root α The the coputatioal order o covergece ay be approiated by the ollowig; l ( l (5 This orula was actually preseted by Traub [7] ad the ollowig ew orulas are actually the iproveets o the above orulas; (5 r 4 r rr r 5 r rr 6 r r r r r r r r r 7 r r r = = = (5 (54 (55 (56 r r r (57

6 8 R Thukral: New Twelth Order Iterative Methods or Fidig Multiple Roots o Noliear Equatios The errors o the above approiats are give by e = (58 The perorace o these orulas are displayed i the table 4 5 The Established Methods For the purpose o copariso two iterative ethods preseted i [0 ] are cosidered Sice these ethods are well established the essetial orulas are used to calculate the approiate solutio o the give oliear equatios ad thus copare the eectiveess o the ew twelth order ethod The irst o the ethod is i act a eighth order ethod preseted i [] ad is epressed as y ( = (59 z ( y = y (60 y = z (6 ( z The secod ethod is by Mir et al ad is preseted i [0] This ethod is actually a teth order ad is epressed as y ( = (6 / ( y ( y z = y ( ( y / = z ( y 6 Nuerical Results (6 (64 I this sectio we shall preset the uerical results obtaied by eployig the iterative ethods (7 (8 (4 (4 (6 ad (64 to solve soe oliear equatios with kow ultiplicity To deostrate the perorace o the ew higher order iterative ethods we use te particular oliear equatios We shall deterie the cosistecy ad stability o results by eaiig the covergece o the ew iterative ethods The idigs are geeralised by illustratig the eectiveess o the higher order ethods or deteriig the ultiple root o a oliear equatio Cosequetly we give estiates o the approiate solutio produced by the ethods cosidered ad list the errors obtaied by each o the ethods The uerical coputatios listed i the tables were perored o a algebraic syste called Maple I act the errors displayed are o absolute value ad isigiicat approiatios by the various ethods have bee oitted i the ollowig tables The ew twelth order ethod requires si uctio evaluatios ad has the order o covergece twelve To deterie the eiciecy ide o the ew ethods we shall use the deiitio Hece the eiciecy ide o the ew ethods give by (7 (8 (4 ad (4 is 6 as the eiciecy ide o the eighth ad teth order ethods (6 ad (64 is give by 6 8 ad 6 0 respectively We ca see that the eiciecy ide o the ew twelth order ethod has better eiciecy ide tha the eighth ad teth order ethod The test uctios with kow ultiplicities ad their eact root α are displayed i table The dierece betwee the root α ad the approiatio or test uctios with iitial guess 0 are displayed i table Table shows the absolute errors obtaied by each o iterative ethods described we see that the ew twelth order ethods are producig better results tha the established ethods Furtherore the coputatioal order o covergece (COC are displayed i table Fro the table we observe that the COC perectly coicides with the theoretical result I additio the dierece betwee the ultiplicity ad the approiatio with iitial guess 0 are displayed i table 4 I table 4 we observe that there is o sigiicat dierece betwee the Lagouaelle orula (48 ad the recetly itroduced orulas (5-(56 as the Traub s ethod (5 is perorig poorly I act is calculated by usig the sae total uber o uctio evaluatios (TNFE or all ethods which is ater three iteratios 7 Coclusios I this paper our ew twelth order iterative ethods or solvig oliear equatios with ultiple roots have bee itroduced Covergece aalysis proves that the ew ethods preserve their order o covergece Siply cobiig the two well-established ethods we have achieved a twelth order o covergece The prie otive o presetig these ew ethods was to establish a higher order o covergece ethod tha the eistig ethods [-8] The eectiveess o the ew twelth order ethods is eaied by showig the accuracy o the ultiple roots o several oliear equatios Ater a etesive eperietatio it ca be cocluded that the covergece o the tested ultipoit ethods o the twelth order is rearkably ast The ai purpose o deostratig the ew ethods or dieret types o oliear equatios was purely to illustrate the accuracy o the approiate solutio the stability o the covergece the cosistecy o the results ad to deterie the eiciecy o the ew iterative ethods

7 Aerica Joural o Coputatioal ad Applied Matheatics 05 5(: -4 9 We have show uerically ad veriied that the ew iterative ethods coverge to the order twelth Epirically we have oud i ay cases that the ew orulas or approiatig the ultiplicity are perorig better tha Table Test uctios ultiplicity root α ad iitial guess 0 the established ethods Fially we have costructed ew higher order iterative ethods but uortuately these ew ethods are ot o optial order hece urther ivestigatio is essetial Fuctios Roots Iitial guess = ( 4 0 = 50 α = = 7 ( = e si ( cos( 5 = 0 α = = 5 ( = ( = α = 0 = 4 = ep 70 = 50 α = 0 = ( 5 = cos = 99 α = = 08 6 si = = 0 α = = ( = e e ( ( ta ( e 0 ( l ( 5 7 = 5 α = = 4 8 = = 00 α = = 8 = = 7 α = = 5 = = 000 α = = 59 Table Copariso o ultipoit iterative ethods i (6 (64 (8 (4 (4 (7 086e e e e-4 098e-98 08e-84 06e-0 00e-7 058e e e e-59 0e e e-54 09e e e e e e-45 09e-45 0e e e e e e e e e e-56 05e e e-84 06e e e e-74 04e e-7 098e e- 089e e e e e e-86 0e e-79 07e-5 060e e e-6 0e e e e e-484

8 40 R Thukral: New Twelth Order Iterative Methods or Fidig Multiple Roots o Noliear Equatios Table Perorace o COC i (6 (64 (8 (4 (4 ( Table 4 Perorace o ew orulas or approiatig ultiplicity i (56 (49 (55 (5 (54 (5 (5 09e-49 09e e e e-5 09e e-4 085e-4 089e-4 089e-4 070e-4 085e e-6 046e-6 0e-6 0e-6 04e-6 046e e e e e e e e-8 04e-8 074e-8 074e-8 069e-8 04e e e e e-40 0e e e- 08e- 044e- 044e- 045e- 08e e-0 040e-0 07e-0 07e-0 07e-0 040e e-0 07e-0 048e-0 048e-0 049e-0 07e e-49 0e e e e-49 0e REFERENCES [] Biazar B Ghabari A ew third-order aily o oliear solvers or ultiple roots Coput Math Appl 59 ( [] C Chu B Neta A third-order odiicatio o Newto ethod or ultiple roots Appl Math Coput ( [] C Chu H Bae B Neta New ailies o oliear third-order solvers or ultiple roots Coput Math Appl 57 ( [4] C Dog A basic theore o costructig a iterative orula o the higher order o coputig ultiple roots o a equatio Math Nuber Siica ( [5] C Dog A aily o ultipoit iterative uctios or idig ultiple roots o equatio It J Coput Math ( [6] W Gautschi Nuerical Aalysis: a Itroductio

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