New Families of Fourth-Order Derivative-Free Methods for Solving Nonlinear Equations with Multiple Roots

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1 Arica Joural o Coputatioal ad Applid Mathatics (4: 7- DOI:.59/j.ajca.4. Nw Failis o Fourth-Ordr Drivativ-Fr Mthods or Solvig Noliar Equatios with Multipl Roots R. Thukral Padé Rsarch Ctr 9 Daswood Hill Lds Wst Yorkshir LS7 5JS Eglad Abstract I this papr two w ourth-ordr drivativ-r thods or idig ultipl zros o oliar quatios ar prstd. I trs o coputatioal cost th aily rquirs thr valuatios o uctios pr itratio. It is provd that th ach o th thods has a covrgc o ordr our. I this way it is dostratd that th proposd class o thods supports th Kug-Traub hypothsis (974 o th uppr boud o th ordr o ultipoit thods basd o + uctio valuatios. Nurical apls suggst that th w thods ar coptitiv to othr ourth-ordr thods or u ltip l roots. Kywords Modiid Nwto s Mthod Root-idig Noliar Equatios Multipl Roots Ordr o Covrgc. Itroductio I rct yars so odiicatios o Nwto s thod or ultipl roots hav b proposd. Thror idig th roots o oliar quatios is vry iportat i urical aalysis ad has ay applicatios i girig ad othr applid scics. I this papr w cosidr drivativ-r thods to id a ultipl root α o ultip licity i.. ( j ( α j... ad a oliar quatio ( ( α o (. ( : B is a scalar uctio o a op itrval B ad it is suicitly sooth i a ighbourhood o α. I rct yars so odiicatios o th Nwto thod or ultipl roots hav b proposd ad aalysd[-5-68]. Howvr thr ar ot ay drivativ-r thods kow to hadl th cas o ultipl roots. Hc w prst two ourth-ordr thods or idig ultipl zros o a oliar quatio ad oly us thr valuatios o th uctio pr itratio. I act w hav obtaid th optial ordr o covrgc which supports th Kug ad Traub cojctur[6]. Kug ad Traub cojcturd that th ultipoit itratio thods without ory basd o valuatios could achiv optial covrgc ordr. I additio th w ourth-ordr thod has a quivalt icicy id to th stablishd ourth-ordr thods prstd i[948]. Furthror * Corrspodig author: rthukral@hotail.co.uk (R. Thukral Publishd oli at Copyright Scitiic & Acadic Publishig. All Rights Rsrvd th w thod has a bttr icicy id tha th third-ordr thods giv i[-7-]. I viw o this act th w thod is sipl r wh copard with th stablishd thods. Cosqutly w hav oud that th w drivativ-r thods ar icit ad robust. Cotts o th papr ar suarizd as ollows: So basic diitios rlvat to th prst work ar prstd i th sctio. I sctio w dscrib th ourth-ordr thods that ar r ro drivativs ad prov th iportat act that th thods obtaid prsrv thir covrgc ordr. I sctio 4 w shall brily stat th stablishd thods i ordr to copar th ctivss o th w thods. Fially i sctio 5 w dostrat th prorac o ach o th thods dscribd. I th drivativ o th uctio is diicult to coput or is psiv to obtai th a drivativ-r thod is rquird. I this study th w drivativ-r itrativ thods ar basd o a classical Sts s thod[9] which actually rplacs th drivativ i th classical Nwto s thod with suitabl approiatios basd o iit dirc w + ( ( w ( w Thror th odiid Nwto s thod ( + bcos th odiid Sts s thod ( + ( w ( (4 (5

2 8 R. Thukral: Nw Failis o Fourth-Ordr Drivativ-Fr Mthods or Solvig Noliar Equatios with Multipl Roots I act it is wll kow that th odiid Nwto s thod (4 ad th odiid Sts s thod (5 hav a covrgc ordr o two[9].. Basic Diitios I ordr to stablish th ordr o covrgc o th w drivativ-r thods w stat so o th diitios: b a ral uctio with a sipl Diiti o Lt ( root α ad lt { } b a squc o ral ubrs that covrgs towards α. Th ordr o covrgc p is giv by α + li ζ p α + ζ is th asyptotic rror costat ad p. α b th rror i th kth Diiti o Lt k k itratio th th rlatio p+ ( p k+ k k (6 ζ +Ο (7 is th rror quatio. I th rror quatio ists th p is th ordr o covrgc o th itrativ thod. Diitio Lt r b th ubr o uctio valuatios o th w thod. Th icicy o th w thod is asurd by th cocpt o icicy id[4] ad did as r p (8 p is th ordr o th thod. Diitio 4 Suppos that ad + ar thr succssiv itratios closr to th root α o (. Th th coputatioal ordr o covrgc[7] ay b approiatd by l COC l + α α ( α( α. (9. Dvlopt o th Mthods ad Aalysis o Covrgc I this sctio w di two w ourth-ordr drivativ-r thods. I act th w thods ar basd o th scod-ordr Sts s thod ad ar iprovd by itroducig a wight-uctio i th scod stp o th itratio... Th Mthod M Th irst w ourth-ordr drivativ-r thod or idig u ltip l root o a oliar quatio is basd o th rctly itroducd thod giv i[5] ad is prssd as ( w + ( ( ( y ( w + abc ( ( + y bc + ( a bc ( w ( y b c a y ( ( is th iitial poit ad providd that th doiator o ( ad ( ar ot qual to zro. I ordr to ai th covrgc proprty o th w thod ( w prov th ollowig thor. Thor Lt α b a ultipl root o ultiplicity o a suicitly dirtiabl uctio : B or a op itrval B. I th i itial poit is suicitly clos to α th th covrgc ordr o itrativ thod did by (7 is our. Proo Lt α b a ultipl root o ultiplicity o a suicitly sooth uctio ( ê w α ad y α. α Usig th Taylor pasio o ( ad ( w about α w hav ( ( α + c + c +! ( ( α ( ( w ˆ ˆ ˆ + c + c +!. (4 ad c k Morovr by ( w hav y ( + k! ( α ( ( + k! ( α ( ( + c c c + Th pasio o ( y about α is giv as ( ( α ( y + c + c +.! Sipliyig (7 w gt. (5 (6 (7

3 Arica Joural o Coputatioal ad Applid Mathatics (4: 7-9 ( ( α c ( y! (8 c ( + c + + c Furthror w hav ( ( y ( α c! (9 c ( + c + + c Sic ro ( w hav + abc ( + bc ( + ( a bc ( w Substitutig appropriat prssios i ( ad atr sipliicatio w obtai th rror quatio c c + ac + c + c ( ( Th rror quatio ( stablishs th ourth-ordr covrgc o th w drivativ-r thod did by (... Th Mthod M I this sub-sctio w di aothr ourth-ordr drivativ-r thod basd o[4]. I this cas w us a cocpt rctly itroducd i[4] ad apply th drivativ-r lt i th scod stp. Th w ourth-ordr thod or idig ultipl roots o a oliar quatio is prssd as ( w + ( ( ( y w ( ( + ab + y b + ( a b w ( y ( b / ( (4 a is th iitial poit ad providd that th doiator o ( ad (4 ar ot qual to zro. Thor Lt α b a ultipl root o ultiplicity o a suicitly dirtiabl uctio : B or a op itrval B. I th i itial poit is suicitly clos to α th th covrgc ordr o itrativ thod did by (4 is our. Proo Substitutig appropriat prssios i (4 ad atr sipliicatio w obtai th rror quatio ( c a (5 Th rror quatio (5 stablishs th ourth-ordr covrgc o th w thod did by (4. 4. Th Establishd Mthods For th purpos o copariso w cosidr thr ourth-ordr thods prstd rctly i[98]. Sic ths thods ar wll stablishd w stat th sstial prssios usd i ordr to calculat th approiat solutio o th giv oliar quatios ad thus copar th ctivss o th w ourth-ordr thod or ultipl roots. 4.. Th Wu l at. Mthod I[8] Wu t al. dvlopd a ourth-ordr thod or idig ultipl roots o oliar quatios sic this thod is wll-stablishd w stat th sstial prssios usd i th thod ( y ( y ( + y y (6. (7 providd that th doiator o (6 ad (7 ar ot qual to zro. 4.. Th Shara t al. M thod I[9] Shara t al. dvlopd a ourth-ordr o covrgc thod as bor w stat th sstial prssios usd i th thod y (8 + ( k ( + k + 8 (9 ( k ( y ( y k 4+ 8 k ( + + is th iitial valu ad providd that th doiators o (8 ad (9 ar ot qual to zro.

4 R. Thukral: Nw Failis o Fourth-Ordr Drivativ-Fr Mthods or Solvig Noliar Equatios with Multipl Roots 4.. Th Shgguo t al. Mthod I[] Shgguo t al. dvlopd a ourth-ordr o covrgc thod th particular prssios o th thod is giv as y ( + ( ( ( ( ( ( ss y s + s y s ( s s + ( is th iitial poit ad providd that th doiators o ( ad ( ar ot qual to zro. 5 ( ( Ta bl. Tst uctios ad thir roots 5. Applicatio o th Nw Fourth-ordr Drivativ-r Itrativ Mthods Th prst ourth-ordr drivativ-r thods ar giv by ( ad (4 ar ployd to solv oliar quatios ad copar with th odiid Sts s thod th Wu t al. th Shara t al. ad th Shgguo t al. thods (5 (7 (9 ad ( rspctivly. To dostrat th prorac o th w ourth-ordr thods w us t particular oliar quatios. W shall dtri th cosistcy ad stability o rsults by aiig th covrgc o th w itrativ thods. Th idigs ar gralisd by illustratig th ctivss o th ourth-ordr drivativ-r thods or dtriig th ultipl roots o a oliar quatio. Cosqutly w giv stiats o th approiat solutios producd by th ourth-ordr drivativ-r thods ad list th rrors obtaid by ach o th thods. Th urical coputatios listd i th tabls wr prord o a algbraic syst calld Mapl. I act th rrors displayd ar o absolut valu ad isigiicat approiatios by th various thods hav b oittd i th ollowig tabls. Fuctios Roots Iitial Poit α ( ( ( si( cos( α ( ( 6 α. ( ( ( 4 p + α ( ( ( 5 cos + 9 α ( 8 ( ( 6 ( si( + α α ( ( α ( ( ( 9 ta α ( ( ( l α Th w ourth-ordr drivativ-r thod rquirs thr uctio valuatios ad has th ordr o covrgc our. To dtri th icicy id o th w thod w shall us th diitio. Hc th icicy id o th ourth-ordr thod giv is which is idtical to othr stablishd thods giv i sctio 4. Th tst uctios ad thir act root ar displayd i tabl. Th dirc btw th root ad th approiatio α α

5 Arica Joural o Coputatioal ad Applid Mathatics (4: 7- or tst uctios with iitial poit ar displayd i Tabl ad 4. I act is calculatd by usig th sa total ubr o uctio valuatios (TNFE or all thods. I th calculatios TNFE ar usd by ach thod. Furthror th coputatioal ordr o covrgc (COC is displayd i Tabl ad 5. Ta bl. Copariso o w itrativ thods ( i (5 a a a ( Ta bl. COC o various itrativ thods ( i (5 a a a ( Ta bl 4. Copariso o w itrativ thods (4 i a a a (9 (

6 R. Thukral: Nw Failis o Fourth-Ordr Drivativ-Fr Mthods or Solvig Noliar Equatios with Multipl Roots Ta bl 5. COC o various itrativ thods (4 i a a a (9 ( Rarks ad Coclusios I this papr w hav itroducd two w ailis o ourth-ordr drivativ-r thods or solvig oliar quatios with ultipl roots. Covrgc aalysis provs that th w drivativ-r thods prsrv th ordr o covrgc. By siply itroducig w paratrs i th w thods w hav achivd ourth-ordr covrgc. Th pri otiv or prstig ths w drivativ-r thods was to stablish a dirt approach to obtai ourth-ordr covrgc thod. W hav aid th ctivss o th w drivativ-r thods by showig th accuracy o th ultipl roots o svral oliar quatios. Atr a tsiv pritatio it ca b cocludd that th covrgc o th tstd ultipoit thods o th ourth-ordr is rarkably ast. Furthror i ost o th tst apls pirically w hav oud that th bst rsults o th w thods ar obtaid wh a ad th w thod giv by ( is producig bttr approiatio tha th othr siilar thods. Thr ar two ajor advatags o ths w drivativ-r thods. Firstly w do ot hav to valuat th drivativ o th uctios; thror thy ar spcially icit th coputatioal cost o th drivativ is psiv ad scodly w hav stablishd a w highr ordr o covrgc thod which is sipl to costruct. W hav aid th ctivss o th w drivativ-r thods by showig th accuracy o th u ltip l root o a oliar quatio. Th ai purpos o dostratig th w highr ordr drivativ-r thods or ay dirt typs o oliar quatios was purly to illustrat th accuracy o th approiat solutio th stability o th covrgc th cosistcy o th rsults ad to dtri th icicy o th w drivativ-r thods. Fially w cojctur that ths w thods ca b iprovd to obtai highr ordr thods. ACKNOWLEDGEMENTS I a gratul to th rviwr or his hlpul cots o this papr. REFERENCES [] C. Chu B. Nta A third-ordr odiicatio o Nwto s thod or ultipl roots Appl. Math. Coput. ( [] B. Ghabari B. Rahii M. G. Porshokouhi A w class o third-ordr thods or ultipl zros It. J. Pur Appl. Sci. Tch. ( [] G. Fradz-Torrs J. Vasquz-Aquio Thr w optial ourth-ordr itrativ thods to solv oliar quatios Adv. Nur. Aal. doi..55// [4] W. Gautschi Nurical Aalysis: a Itroductio Birkhausr 997. [5] S. Kuar V. Kawar S. Sigh O so odiid ailis o ultipoit itrativ thods or ultipl roots o solvig oliar quatios Appl. Math. Coput. 8 ( [6] H. Kug J. F. Traub Optial ordr o o-poit ad ultipoit itratio J. Assoc. Coput. Math. ( [7] N. Osada A optial ultipl root-idig thod o ordr thr J. Coput. Appl. Math. 5 ( [8] E. Schrodr Ubr udich vil Algorith zur Aulosug dr Glichug Math. A. ( [9] J. R. Shara R. Shara Modiid Jarratt thod or coputig ultipl roots Appl. Math. Coput. 7 (

7 Arica Joural o Coputatioal ad Applid Mathatics (4: 7- [] J. R. Shara R. Shara Modiid Chbyshv-Hally typ thod ad its variats or coputig ultipl roots Nur. Algor. 6 ( [] L. Shgguo L. Xiagk C. Lizhi A w ourth-ordr itrativ thod or idig ultipl roots o oliar quatios Appl. Math. Coput. 5 ( [] J. F. Sts Rark o itratio Skad. Aktuar Tidsr. 6 ( [] R. Thukral A w third-ordr itrativ thod or solvig oliar quatios with ultipl roots J. Math. Coput. 6 ( [4] R. Thukral Itroductio to highr ordr itrativ thods or idig ultipl roots o solvig oliar quatios J. Math. Vol. doi:.55//4465. [5] R. Thukral A w aily o ourth-ordr itrativ thods or solvig oliar quatios with ultipl roots subittd to J. Nur. Math. Stoch.. [6] J. F. Traub Itrativ Mthods or solutio o quatios Chlsa publishig copay Nw York 977. [7] S. Wrakoo T. G. I. Frado A variat o Nwto s thod with acclratd third-ordr covrgc Appl. M ath. Ltt. ( [8] Z. Wu X. Li A ourth-ordr odiicatio o Nwtos thod or ultipl roots IJRRAS ( [9] Q. Zhg P. Zhao L. Zhag W. Ma Variats o Sts-scat thod ad applicatios Appl. Math. Co[put. 6 (

New Sixteenth-Order Derivative-Free Methods for Solving Nonlinear Equations

New Sixteenth-Order Derivative-Free Methods for Solving Nonlinear Equations Amrica Joural o Computatioal ad Applid Mathmatics 0 (: -8 DOI: 0.59/j.ajcam.000.08 Nw Sixtth-Ordr Drivativ-Fr Mthods or Solvig Noliar Equatios R. Thukral Padé Rsarch Ctr 9 Daswood Hill Lds Wst Yorkshir