Tekle Gemechu Department of mathematics, Adama Science and Technology University, Ethiopia

Transcription

1 ISSN (Paper) ISSN 5-05 (Olie) Vl7, N0, 07 Se Multiple ad Siple Real Rt Fidig Methds Tele Geechu Departet atheatics, Adaa Sciece ad Techlgy Uiversity, Ethipia Abstract Slvig liear equatis with rt idig is very c i sciece ad egieerig dels I particular, e applies it i atheatics, physics, electrical egieerig ad echaical egieerig It is a researchable area i uerical aalysis This preset wr cuses se iterative ethds higher rder r ultiple rts New ad eistig vel ultiple ad siple rt idig techiques are discussed Methds idepedet a ultiplicity a rt r, which ucti very well r bth siple ad ultiple rts, are als preseted Errr-crrecti ad variatal techique with se ucti estiatis are used r the cstructis Fr the aalysis rders cvergece, se basic theres are applied Aple test eaples are prvided (i C++) r test eiciecies with suitable iitial guesses Ad cvergece se ethds t a rt is shw graphically usig atlab applicatis Keywrds:Iterative algriths, errr-crrecti, variatial ethds, ultiple rts, applicatis Itrducti Nliear equatis appear i st sciece ad egieerig prbles Fr eaple, i cliate dels, electric circuit aalysis, aalysis state equatis r the geeral gas law, echaical tis ad i ay ther prcesses (Ali et al, 000) A practical prble applicati ultiple rts is idig ultiple Eige values square atrices (see test equati i uit-6) The rt idig algriths are usually cstructed by the ethds iterplatis (liear, quadratic, cubic, etc) (Tele, 07& 04), perturbati ethd (Padeirli, 008), variatial techiques (Muhaad et al; 0), the Newt s ethd apprach, ied pit techiques, etc (Gerud ad Ae, 008; Jurge,994; Athy ad Philip, 978) Se iterative ethds r ultiple rts ca be reerred t the wrs (Muhaad et al;0;; YI Ji ad Baha,00 ;Jaa ad Raji, 0;JSter ad Bulirsch, 99), where the diied Newt ethd ad se higher rder cverget ethds are accessible Slvig ultiple rts with iterative ethds ay have se diiculties The basic rt idig algriths such as the Secat ethd, Bisecti ethd, Regula Falsi ethd, Newt s ethd, ied pit iterati, Muller s ethd ad ay thers are t eective r ultiple rts Se the eed diicatis t apply r ultiple rts Fr eaple, Richard ( 977) used a prcedure that ivlves the divided dierece t apply the Secat ethd r ultiple rts The Mdiied Newt ethd r ultiple rts is btaied assuig that r is a actr ( with a ultiplicity ad the applyig liearizati r Newt s ethd r siple rt r ( (Ali et al, 000; J Ster ad R Bulirsch, 99) Chebyshev s ethd, Euler s ethd, Halley s ethd, Osada s ethd ad Chu-Neta (CNM) ethd are se well w r ultiple rts (Muhaad et al, 0;Jaa ad Raji, 0; Chugbu&Bey Neta,009) The variatial techique was used t geerate vel third rder algriths r ultiple rts based diied Newt s ethd (Muhaad et al, 0) I this paper, ur preset study, we start with liearizati techique t id liear cverget ethds r ultiple rts (Ali et al,000; JSter ad RBulirsch,99) Ad derivig r the [errr-crrecti] hr (r bth siple ad ultiple rts) r the derivative (s) the assued epressi, we btai se ew diied ethds rder at least tw r ultiple rts We als btai ethds that are idepedet ultiplicity a rt r ( We develp third rder ethds r ultiple rts based se ideas i variatial ethds (Muhaad et al,0) with additial ccepts With a apprpriate iitial iput, st the ew algriths are better cpetet (i ters speed cvergece) with se eistig ethds r ultiple rts The tivati ces r the authr s previus wr siple rts (Tele, 07) Fr related ccepts reer t (Faretal, 06; Far & Muhaad,04; Chugbu&Bey Neta, 009) Fr applicatis ultiple rts, see: (PNG 50: Phase Relatis i Reservir Egieerig, i a cubic equati state) Discussis se Eistig Multiple Rt Fidig Csider the basic diied Newt s iterative rula ; 0,,,, ( ) + ()

2 ISSN (Paper) ISSN 5-05 (Olie) Vl7, N0, 07 t uerically appriate a rt r ultiplicity a liear equati ( 0 (Ali et al, 000; J Ster & RBulirsch, 99) Where ( ( r) r se Ad ( r) /( r) + ( r) g, g ( r) 0 () Assuig g' ( 0 ad applyig Newt s ethd t slve ( 0, we have () which is quadratic cverget (Ath &Philip, 978) Se basic cubic cverget ethds are Chu-Neta ethd (), Halley s ethd (4), Osada s ethd (5), Euler-Chebyshev s ethd (5b) ad the urth rder (6) are as listed belw (Muhaad et al,0; YI Ji & Baha,00;Jaa&Raji,0) See als (Tele, 04) [ ] () ( + ( ) [ ( ] ( + ( )[ ( )] ) ) ( φ () φ (4) ( ( φ ( + 05( ) + 05( ) (5) ( ) ) [ ] ' ( ( + 05() 05 ( ) [ ( )] φ (5b) (+ )[ ( ] φ ( ( + )[ ( )] (+ ) ) ( ) ) + [ )] ' ) (6) Oe the vel third rder ethds develped usig the variatial techique by Nr, Far& Nr is ψ( / [( ) + ( /[ ( ] )] (6b) ( ( β Nte: i, the () ad (5) bece Newt s ethd r a sigle rt r Ad (4) ad (5b) bece Halley s ethd ad Chebyshev s ethd r siple rts respectively Oe has t tice that () is btaied usig ' ( ) [ ] i (4), (Chugbu&Bey Neta, 009) With, (6) gives a urth rder '' (6c) belw r siple rts Methds i (6) &(6c) were preseted by YI JIN & BAHMAN (00) [ ( ] ( 6[ ( )] 6 ) ( ) ) + [ )] ' ) φ (6c) A quadratic cverget (6d), a cubic e (6e) &4 th rder ethds therei by Muish et al (05) are ) ( ) φ ( ) + 05[ () ] (6d) ( ) ) ) ( ) ψ ( ) + 05[ ( ) () ] (6e) ( ) ) See als higher rder ethds by Rajider Thural (0) Cstructi se Iterative Methds r Multiple Rts I r is a rt ultiplicity 0, the usig () abve, the Newt s rula r ultiple rts beces (7b) as shw belw With liear estiati arud r, h ( r) / ( ( r), (7) [( r) g + ] φ ( ( r) (7b) [( r) g + ]

3 ISSN (Paper) ISSN 5-05 (Olie) Vl7, N0, 07 (7b) is liear cverget with rate r) / > 0 φ ( r) ad it is quadratic cverget (Ali et al, 000) Se id replaceets ay be used as llws Suppse hr ad we put g v, the r (7b) we btai ( + φ ( (8) [ ( ] But i hr ad we use g v i (7b), the we ca get ( + φ ( (9) [ ( )] ) ) φ r all > Ad i, the ' / 0 I i (8), the we als get (9), a eistig ethd Eq (9) is idepedet ad hlds well r bth siple ad ultiple rts It ca be btaied applyig Newt s ethd r slvig u( / ( (Gerud&Ae, 008; Athy & Philip,978) Assuig that the rder (7b) ay be p > ad slvig the cditi φ 0 i it hlds, we ca btai () g h (9b) hg'' + g' Puttig Puttig g b ad h / i the right part (9b), we have ( ) ( + φ ( (9c) [ ( )] ) ) g c ad h / i the right part (9b), we have I '' ( ) / ( ) ( + φ ( (0) [[ ( )] ) )] i (0) ad settig t be we have ( ) + φ ( (b) [ ] ( Replacig ()/() by ()/() i (b) gives the secd rder, ( ) + φ ( (c) [ ] ( A iterestig aspect a algrith i (c) is that it ca be btaied r eteded Halley s ethd i eq(4) with '' /( )( ) / Ad usig '' ( )/( )( ) / i eq(4) yields diied Newt s ethd i eq() Eaples belw shw that Newt s Methd (N) r siple rts ad rulae (8), (0) ad (b) whe applied here r ultiple rts are slw cverget Cpare their eiciecies with (9) ad (c) which are relatively uch better, based the uber iteratis required r cvergece as i table- belw We tae ( with 5, ad rt r i [, ), ( [cs] 0 with 0, ad r i (0, ), See table results belw

4 ISSN (Paper) ISSN 5-05 (Olie) Vl7, N0, 07 Table-: Cpariss se ethds r ultiple rts (tl 0 - ) ( N (8) (9) (0) (b) (c) ( 5, 6 90, 95, 9, 4 5, 6, ( ), 9 9,, 6, 5, 9, Se Secd Order Methds r Multiple Rts By derivig r the errr-crrecti h r eq() abve i dieret ways, we develp se iterative ethds rder tw The ai is t shw ather way t diy the slw cverget ethds i secti- abve ad btai better ethds r ultiple rts Csider agai the equati, ( r) /( r) + ( r) g () ' /( r) + g / () φ ( (a) g / Eq(a) ca als be btaied taig F( ϕ ( + λ[ / ] ad applyig variatial techique [as i eq(b) belw] I g' g i (a) [there ca be ay cases], the e gets a ultiple rt idig φ ( (b) ( g ' c / g I i (a), the we get ( ( ψ ( ( c (c) The ethd i (b) was btaied als by (Muhaad et al, 0) Let us assue that g 0 That is g ( is t a cstat Ad suppse ' /( r) + c g' ( (d) Ad usig, h g r g / h, hr (4) (d) hr cg' (5) hg' Or () h + (6) g ' ( / c) I i (5), the we btai the llwig iterative ucti r ultiple rts ( φ ( (7) ( '[ ] Taig 0 i (7) gives the diied Newt s rula Ad i 0 but, the we get the Newt s rula r siple rts ' Ad isertig ' ad i (7) yields the Halley s r Chebyshev s ethd r sigle rts Fr (-)/() we btai the ethd belw r slvig a ultiple rt ( ) ( ) [ + φ (8) Ipsig the cditi < i (8), we have ( ) ) )] 4

5 ISSN (Paper) ISSN 5-05 (Olie) Vl7, N0, 07 Usig [ ] ( ) [+ ] ( ) ( ) ' h ad g ' i (6), we ca btai + φ ( [ ] (0) ( ) [ ( )] + φ (9) I -, the (0) beces Chebyshev s ethd r siple rts I 0, the (0) is diied Newt s ethd Ad i 0 ad, it is the Newt ethd r siple rts ( ) Taig '' [ ] / ad i (0), we btai + φ ( [+ ] () ( ) Nw with ad g ' ( '/[ ] ) g ad h r used i () r i (d) with c, h g r g / h we ca get h + h[[( ( ] ] () ( [ ( ] 4 [ ( ] I, h / the we have + φ ( [ ([ ( ] ) ] () ( Usig (), we ca epress h as [ ( ] h [[ ( ] ([ ( ] ) ] (4) Fr which we derive a iterati ucti [ ( ] φ ( (5) [[ ( ] [[ ( ] ] ] I ' i (5), the we get the ethd belw ( + φ( (6) [[ ( ] [[ ( ] ] ] I g' g i () r i (d) with c, the puttig g we get [ ] φ (, tae / (7) [ ( ( ] I g' g i () the, we btai ( φ (, tae / (8) ([ ( ] [ ] ) Als r () r () assue 5

6 ISSN (Paper) ISSN 5-05 (Olie) Vl7, N0, 07 ' + h g' / h+ h g' ', (9) h h Let g' g, g'' ( + ) g, h r, put g /h i (9) t btai h, (0) h' h( + ) (Use h / i the right part) This gives, ( φ ( [ [ ( + ][ ] + [ ( ] + [ ] ( () I g' ' 0i (9) r i (0) ad h i the right part (9) r (0), the with se (sig) chages, we get () ( φ ( () [[ ( ] + [ ] ( Fr a prper value, () hlds bth r sigle ad ultiple rts idepedet ultiplicity I 0, the () is (9) ad r we btai a duble purpse algrith (b) belw 4 Oe ca als tae i () ( φ( (b) [[ ] + / 4[ ( ( ' Nte als that the geeral class ultiplicity idepedet ethds ca be btaied r (7b) ad is give i the r φ (, g' ( + but it is t easily ccluded r which g it hlds 4 Cstructi Se Third Order Methds r Multiple Rts We w develp third r higher rder ethds based variatial techique The variatial techique used by (Muhaad et al, 0) is that i ϕ( is a iterative ethd rder p > with ultiplicity >, the vel ultiple rt idig ethds rder at least p+ are give by ϕ' ( ψ ( ϕ( () p[ ( + g' ( ] r se auiliary ucti g The pr by (Muhaad et al, 0) was idig the cstat λ usig the p/ ptial cditi the iterati ucti F( ϕ ( + λ[ ] t slve 0 Mdiied Newt s ethd () was used i () t geerate higher rder ethds p/ Here, siilar pr is used but w taig ather F( ϕ ( + λ[ / ] t btai ϕ' ( ψ ( ϕ( (b) p[ ( g' ( ] I this paper, i φ( ad ϕ( are tw secd rder ethds r ultiple rts, the with little chage i (b) several ethds ψ( rder p > ca be btaied by ϕ' ( ψ ( φ( (c) [ ( g' ( ] Se the ay ethds btaied r (c) are shw belw I we tae φ( the diied Newt s 6

7 ISSN (Paper) ISSN 5-05 (Olie) Vl7, N0, 07 ethd ad r eq(c), we use ad taig Ad whe g ' g ψ( i (c),the we get ( ) ϕ (, ( ) ( ( ) / [ ( ( ( ) [ ( 05 ( ( ) φ ( ϕ(, withg ' g ( ) ( /[ ( ] )] ( ' /[, we have ] )] (4) ( + ) ( + ) ψ ( /[ ( /[ ( ] )] ( ) ( ( ( ) [ ( /[ ( ] )] ( ) 05 (4b) ( ) ( ( ( ) ( ( ) ( ( ( ( I φ, ϕ, the with g g we btai ( ) [( ψ( 05 ( ) ( /[ ( ] ( )] (5) ) ( ) ( ) ) I φ ) 05[ () ], e has ( + ψ( ) + ( ) ϕ (, withg g ( ) ( ) ( ) ( + ) 05[ () ] /[ ( /[ ( ])] ( ) ) ( (6) Nte that e ca chse - r i (4), (4b) r (5) ad (6) but there ay be ay cases It is als pssible t use ay secd rder r ultiple rts i (c) Fr urther irati variatial techiques r rts, reer t (Far et al, 06; Far & Muhaad, 04; Muhaad et al, 0) ', 5 Cvergece Aalysis There 5 Let φ ( ϕ( be a arbitrary iterati ucti rder p at a ultiple rt r with ultiplicity Suppse als that φ( is ctiuusly dieretiable at r, the ψ ( i () has rder at least p+(muhaad et al, 0) See als prs by (Muhaad et al, 0) There 5 (Order Cvergece) Assue that φ ( has suicietly ay derivatives at a rt r ( The rder ay e-pit iterati ucti φ ( is a psitive iteger p, re especially φ ( has rder p i ad ly i ( j) ( p) φ ( r ) r ad φ ( r) 0 r 0 < j < p, ( r) 0 φ (Athy &Philip, 978; Ali et al, 000) All the algriths i this paper eed a apprpriate chice ly e iitial guess i a iterval I [a, b] 7

8 ISSN (Paper) ISSN 5-05 (Olie) Vl7, N0, 07 Ad rad chices ay lead us t uecessary wrs S we eed t d that Fr there 6 abve ad cvergece ied pit iterati ethd φ(, < φ' ( ) < The case φ ( ) > is divergece Ad ) ' φ r alli [a, b] Fr which h φ eeds especial treatet (rerulatis r eed r alterative ethds; Gerud & Ae, 008) ) Pr rder cvergece p (9) ad (b) Oe has tw cases sice bth algriths are used r slvig siple ad ultiple rts I (r is siple rt ( ), the (9) ca be writte as φ( ( θ( )/ T(, where T( /[ ( ] ad θ( is Newt s iterati ucti which has rder p We see that φ (, θ r) 0, θ' r) 0 Applyig there 6, dieretiatig (9) r φ (, we have φ' ( r) 0ad φ ' r) 0 O the ther had, i p, the we easily shw that φ' ( r) 0but φ ' r) 0 S p r (9) whe I case > ( r is ultiple rt), (9) ca be writte as φ( / ( θ( )/ T(, where θ( is diied Newt s iterati ucti rder p as prved by (Athy&Philip,978) Applyig the sae pr θ ( here r φ( ad addig ccepts i there 5, e btais p ) I, the (b) ca be writte as φ( ( θ( )/ B(, with ( B( + / 4 ( ' /[ ] ad θ( is Newt s iterati ucti Applyig there 6 as i ) abve, we get p r Ad i >, the (b) ca be epressed as φ( / ( θ( )/ B(, with θ( is diied Newt s ethd Applyig siilar prs as i the secd case ), agai we get p ) Pr rder cvergece p (7) (7) ca be epressed as φ( ( θ( )/ H(, where H( [ ] / (, ad ( θ ( w r ( θ is the diied Newt s iterati ucti p Applyig the pr r rder ) φ here, r by there 6 e btais p Hece (7) ad (8) are secd rder cverget Siilar prs ca be used t shw that (c), (), (6), (8) ad (b) are all rder p Nte als that (c) hlds very well r siple rts Fr (c) ( ) ( ( ) ( ϕ Ad r siple rts S this gives agai a quadratic cverget ethd 4 ( ( ϕ Nte that bth (c) ad () culd be btaied r (4) 4) By there 5, p ad the (4b) bece rder p+ + Siilarly (4), (5) ad (6) are cubic cverget Nte that there 5 hlds als r (b) ad (c) The prs r st the irst rder ethds i secti- ca be r the iterati ucti i (7b) which is liear Oe ca als use ther way prs i the literatures 6 Test Equatis ad Nuerical Results The llwig equatis were selected r test cvergece ( + 0, with 0,,, ad r i [, ), ( [cs] 0 8

9 ISSN (Paper) ISSN 5-05 (Olie) Vl7, N0, 07 with 0,, ad r i (0,), ( e + e 0, with 0,, ad r i (0,), ( + 4, 05, 5, ad rt i [, ), + 4 5( , -, 0,, ad rt i [-, 0), 4 6 ( [ e / ] 0, 05, 5,,, ad r i [0, ], ( [l + 9) ] 0, 7 05, 5, ad r i [0, ], Cpariss were ade relative t diied Newt ethd (MN), Halley s ethd (HM), the algriths i equatis (9), (8), (), (6), (7), (8), (b)(4), (4b) C++ ipleetati was de r each algriths ad the uber iteratis tae t cverge t a rt r was recrded ad writte i the bdy the et table- uder each ethd The stppig criteria were usig the errr Ei i + i such that E i ε, r chse ε 0 Nte: The ubers i each cells table- the et page crrespd t the uber iteratis eeded r cvergece at each the three iitial guesses a rt r ( 0, i,,,7 Fr the table Halley s ethd (HM) r slvig ( e + e 0 cverges at steps,, r the iitial guesses tae at 05,, i respectively ad beig ε give I the irst clu, Fuctis () reers t the uber ucti evaluatis up t derivatives ad Eiciecy (e) represets the cputatial eiciecy ide calculated by p / A algrith with the least average uber iteratis (Nar) t cverge t a rt r wuld be raed the astest cverget Taig re iitial guesses r re eaples gives gd average raig easure Nte that average uber iterati was als used by Chagbu Chu & Bey Neta (009) I the table, the highest value eiciecy ide is 44 (r third rder ethds (4), (4b) ad (HM) ) ad the lwest value is 60 (r secd rder) Ay urth rder with ur ucti evaluatis [p 4, 4] ad ay secd rder with [ p ] have equal cputatial eiciecy idices e 44 It ay t be true that the higher the rder is the better the eiciecy r the ethd i geeral We ca bserve that all ethds preseted i the table are better cpetet with t average uber iteratis t cverge r bth directis at whe a apprpriate iitial guess is tae, especially ear a uit legth iterval ctaiig the rt ri is t suitably chse, the e ca epect slw cvergece ad eve divergece cases As a eaple, we have checed that algrith (9) ad (b) ail t cverge t the desired rt whe is t prperly chse r slvig siple ad ultiple rts se equatis, but are ast cverget See the table results belw Nte: i the table belw use the llwig letters as llws Nar average uber iteratis uber ucti evaluatis p rder cvergece e cputatial eiciecy idices 9

10 ISSN (Paper) ISSN 5-05 (Olie) Vl7, N0, 07 Table- Suary uerical results The uber iteratis eeded t cverge t a rt r r every triplets iitial guesses ( MN (9) (8) HM (8) 4 4b (7) b (,,,,,,,,,,,,,,,,,, ( :,,,,,,,,,,,,,,,,,, ( :,, 4,, 4,, 4,,,, 4,, 4,, 4,, 4,, 4( ) 4,, 4 4,, 4 4,, 4,, 4,, 4,,,, 4,, 4 4,, 4 ( ):,,,,,,,,,,,,,,,,,, 5 ( ) 4,, 4 4,, 4 4,, 4,, 4,, 4,,,, 4,, 4 4,, 4 6 ( ):,, 4,,,, 4,, 4,, 4,, 4,,,, 4,, 7 Nar Fu () Ord(p) Ei(e) Mdiied Newt r rts Fig : cvergece diied Newt s ethd r slvig [r ] Fied pit (4) r 4 Rts Fig : Cvergece ethd i (4) r slvig [r ] 0

11 ISSN (Paper) ISSN 5-05 (Olie) Vl7, N0, y Fig : Graph r rt lcati [r ] Nte that ^-*^+*- is the plyial a idetity atri rder, with all Eige values r 0 0 I Graph I y Fig 4: Tps (r abve) the graph shw idetical Eige values 7 Cclusis I this article, we have preseted iterative ethds rder,,, 4 bth r ultiple ad siple rts By derivig r the errr-crrecti h r the derivative (s) a epressi the equati t be slved, we btaied iterative ethds rder tw r estiatig ultiple rts scalar liear equatis We have develped third rder ethds based variatial techiques with additial ccepts We have als discussed se basic rt idig ethds such as the Newt ethd, Chu-Neta ethd, Halley s ethd, Osada s ethd, Chebyshev s ethd ad (6) r ultiple rts We have see that there are diied ethds i this paper that are secd rder (see (c), (8), (), (6), (7), (8), (b)) ad third rder cverget (see (4), (4b), (5), (6)) They are better cpetet with se eistig ethds We shwed that there are diied ethds rder tw with ly tw ucti evaluatis up t derivatives (see (8), (7), (8)) We preseted tw secd rder cverget algriths which are idepedet ultiplicity (see eq(9) a eistig ethd ad a ew e eq (b))they are ast cverget bth r siple ad ultiple rts We have als de cvergece aalysis with prs ad graphs I the uture we will preset urther aalyses the tpic ad ther higher rder iterative algriths We hpe that this result will be re cedable ad cece e t perr urther research i the area Acwledgeets I wuld lie t tha the sta atheatics i ASTU

12 ISSN (Paper) ISSN 5-05 (Olie) Vl7, N0, 07 REFERENCES Ali Quarteri, Riccard Sacc, Faust Saleri (000), Nuerical atheatics (Tets i applied atheatics; 7), Spriger-Verlag New Yr, Ic, USA Athy Ralst, Philip Rabiwitz (978), A irst curse i uerical aalysis (d ed), McGraw-Hill B Cpay, New Yr Chagbu Chu ad Bey Neta(009), A third-rder diicati Newt s ethd r ultiple rts, Applied Matheatics ad Cputati, Farq Ahed Shah, Muhaad Asla Nr ad Muhaad Wasee(06), Se secd-derivative-ree sith-rder cverget iterative ethds r -liear equatis, Maej It J Sci Techl, 0(0), 79-87; Farq Ahed Shah ad Muhaad Asla Nr(04), Variatial Iterati Techique ad Se Methds r the Appriate Sluti Nliear Equatis, Appl Math I Sci Lett, N, 85-9 Gerud Dahlquist, Ae Bjrc(008), Nuerical ethds i scietiic cputig Vlue I, Sia - Sciety r idustrial ad applied atheatics Philadelphia, USA Jaa Raj Shara ad Raji Shara (0), Mdiied Chebyshev-Halley type ethd ad its variats r cputig ultiple rts, Spriger sciece + Busiess Media, LLC Jurge Gerlach (994), Accelerated Cvergece i Newt s Methd, Sciety r idustrial ad applied atheatics,sia Review 6, 7-76 JSter, RBulirsch (99), Tets i Applied atheatics, Itrducti t Nuerical Aalysis (d ed), spriger-verlag New Yr, Ic, USA Malvi RSpecer (994), PhD Dissertati: Plyial Real Rt Fidig i Berstei Fr, Departet egieerig Brigha Yug Uiversity MM Hussei (009), A te e-step iterati ethds r slvig liear equatis, Wrld Applied Scieces Jural 7 (special issue r applied ath), 90-95, IDOSI Publicatis Muhaad Asla Nr, Farq Ahed Shah, Khalida Iayat Nr ad Eisa Al-Said (0), variati iterati techique r idig ultiple rts liear equatis, Scietiic Research ad Essays Vl6(6),pp44-50 Muish Kasal, V Kawar_, ad Saurabh Bhatia (05), O Se Optial Multiple Rt-Fidig Methds ad their Dyaics, A Iteratial Jural(AAM) : ISSN: , Vl 0, Issue, pp Padeirli, H Byacl ad H A Yurtsever 9008), A rt idig algrith with ith rder derivatives, Matheatical ad Cputatial Applicatis,Vl, -8 Rajider Thural (0), Itrducti t Higher-Order Iterative Methds r Fidig Multiple Rts Nliear Equatis, Hidawi Jural atheatics, Article ID 40465: Richard F Kig (977), A Secat Methd r Multiple Rts, BIT 7, -8 Tele Geechu (07), Rt Fidig With Etesis t Higher Diesi, Matheatical Thery ad Mdelig (IISTE, USA),Vl 7 N 4, ISSN 5-05 Tele Geechu (Dia)-(Dissertati 04), Perturbati eect scalar plyial ceiciets ad real rt idig algriths with applicatis, CCNU YI JIN AND BAHMAN KALANTARI (00) (cuicated by Peter A Clars), A cbiatrial cstructi high rder algriths r idig plyial rts w ultiplicity, Aerica Matheatical Sciety 8,