Several New Families of Jarratt s Method for Solving Systems of Nonlinear Equations

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1 Avlble t htt://vm.ed/m Al. Al. Mth. ISSN: 9-9 Vol. 8 Ie December. 7 7 Alcto d Aled Mthemtc: A Itertol Jorl AAM Severl New mle o Jrrtt Method or Solvg Stem o Noler Eqto V. Kwr* Uvert Ittte o Egeerg d echolog Pb Uvert Chdgrh- Id vmthl@hoo.co. Seev Kmr d Rmdee Behl School o Mthemtc & Comter Alcto hr Uvert Ptl-7 Pb Id eev.mc@thr.ed rmbehl87@hoo. Receved: Jl ; Acceted: November *Correodg Athor Abtrct I th td we gget d lze ew d wde geerl cl o Jrrtt method or olvg tem o oler eqto. hee method hve orth-order covergece d do ot reqre the evlto o ecod or hgher-order réchet dervtve. I term o comttol cot ll thee method reqre evlto o oe cto d two rt-order réchet dervtve. he erormce o rooed method comred wth ther cloet comettor ere o mercl eermet. It worth metog tht ll the method codered here re od to be eectve d comrble to the robt method vlble the ltertre. Keword: Nmercl l; tem o oler eqto; tertve method; order o covergece; Jrrtt method AMS-MSC No: A H 7

2 7 V. Kwr et l.. Itrodcto h er ddree the roblem o dg rel root o oler tem o the orm where d : R R cetl deretble vector cto. M roblem bot dg root o hve emerged m cece d egeerg lcto. he zero o oler tem c ot geerl be ereed cloed orm th tertve method or romtg olto o tem o oler eqto re the mot reqetl ed techqe. hereore dg root o h become oe o the mot mortt d chllegg roblem comttol mthemtc. M robt d ecet method or olvg re lred egged. Oe o the mot bc rocedre or romtg olto o the oler tem the qdrtcll coverget Newto method rb 9 d gve b } where } the vere o rt réchet dervtve o the cto. I order to mrove the locl order o covergece o Newto method mber o method hve bee rooed the ltertre. or tem o eqto ow the rt-order réchet dervtve mtr wth evlto whle the ecod-order réchet dervtve h evlto. h mle tht hge mot o comttol wor reqred to evlte ever terto Amt et l.. hrd order tertve method le Hlle method Amt et l. Gterrez d Herdez 997 d Chebhev method Amt et l. Gterrez d Herdez 997 re cloe reltve o Newto method. hee method reqre the evlto o the ecod-order réchet dervtve er terto. hereore dete ther cbc covergece the re codered le rctcl rom the comttol ot o vew. Mltot tertve method or olvg oler tem l gct role the eld o tertve rocee ce the crcmvet the drwbc o oe-ot terto ch Newto method. Sch cotrcto occoll oe better order o covergece d ecec de or olvg the tem o oler eqto. I recet er ome ew hgher order tertve method hve bee develoed d lzed to olve the oler tem wthot g the ecod-order réchet dervtve c. Homeer Gr-S chez et l. Shrm et l. Cordero et l. 9 Drvh d Brt 7 d Nedzhbov 8.

3 AAM: Iter. J. Vol. 8 Ie December 7 I th er or m obectve to develo wde geerl cl o orth-order Jrrtt method Jrrtt 9 or olvg oler tem wthot g the ecod or hgher-order réchet dervtve. or th roe we eted the cheme o Behl et l. to the - dmeol ce mle w. We lo erorm deret mercl tet tht corm the theoretcl relt d llow to comre the method wth ome other recetl blhed method.. Decrto o New Geerl Cl o Jrrtt Method More recetl Behl et l. hve rooed ew otml ml o Jrrtt method or olvg clr oler eqto. h gve b ] ][ [ ] 7 [ where R ch tht ether or otherwe thee mle o method hve thrd -order o covergece. I th ecto we ted to develo tertve cheme o hgher order or olvg tem o oler eqto wthot g ecod-order r e chet dervtve. or th roe we trodce the ollowg modcto over the ml Error! Reerece orce ot od. or mltdmeol ce: } h h where ] 7 [ B I ] ][ [ I B B I h H B } d H H h d where I deote the dett mtr d R where R ch tht ether or.

4 7 V. Kwr et l.. Covergece Al I order to elore the covergece roerte o cheme we recll the ollowg relt o lor ere ereo o vector cto [ee Örteg d Rheboldt 97] d lemm roved b Nedzhbov [ee Nedzhbov 8]. Lemm.. Let : V R R be C cto deed o V : < r} ; the or v r the ollowg ereo hold: v v vv vvv v v R!!! where v R. V! Lemm.. Let : V R R C cto d h locll coverget qe root V. rther oe tht the Jcob vertble eghborhood o the the ollowg ereo hold: h h h B B B heorem.. I H H H I H H H H H. 9 Let : V R R C cto oe cove et V R. Ame tht there et V ch tht d et. he there et > ch tht or ever tl ge U the eqece o terte geerted b well deed coverge to d h orth-order covergece whe. Proo: rom eqto we get

5 AAM: Iter. J. Vol. 8 Ie December 7 } h. 7 We trodce the ollowg otto 8. 9 Now eqto 7 c be rewrtte } [ e h ]. Ug Lemm. we c rereet h b g the ollowg lor ere eo h h h e h ee h eee O e. Sce root o tem. thereore h H d h I b Lemm.. hereore eqto rther gve h e h ee h eee O e. Smlrl we ere B B B e B ee O e. Ug Lemm. d eqto we obt I h e B ee O e. Ug d we hve h eh [ ] ee [ h 9 B 9 h ] eeeo e. 8

6 7 V. Kwr et l. Ug we obt I h e [ B ] h eeo e. 9 Sbtttg Error! Reerece orce ot od. d Error! Reerece orce ot od. we hve } h ee [9B h 9h ] eee O e }. or eqto become } 9B h h } eee O e. Accordg to Lemm. d btttg the ereo o h h d B we hve } H H H } eee O e. Let derette twce the eqto H I ; we thereb obt H H H. 7 Ug eqto 7 Error! Reerece orce ot od. ll we get O e. 8 h comlete the roo o the theorem... Secl Ce ll b g deret ecc vle o d whch re deed heorem. we get the vro method rom orml ollow:

7 AAM: Iter. J. Vol. 8 Ie December 77 or ml red. 7 } h ew orth-order ml o method or olvg tem o oler eqto. Sb ecl ce o ml Error! Reerece orce ot od. or ml Error! Reerece orce ot od. red. } } h the well-ow Jrrtt method [Nedzhbov 8] or olvg tem o oler eqto. b or ml Error! Reerece orce ot od. red. } 7 h ew orth-order method or olvg tem o oler eqto. or ml red. 7 } 9 h other ew orth-order ml o method or olvg tem o oler eqto.

8 78 V. Kwr et l. Sb ecl ce o ml 9 or ml 9 red. 9 } h modcto over the well-ow Jrrtt method Jrrtt 9 or olvg tem o oler eqto. b or ml 9 red. 8 9 } h ew orth-order method or olvg tem o oler eqto. or d ml red } h g ew orth-order method or olvg tem o oler eqto. Note tht ml c rodce everl ew mltot mle o Jrrtt method wthot g ecod-order réchet dervtve or mle root o oler tem b g oe o the doble rmeter mel or.. Comttol Ecec he trdtol w to obt emet o the ecec de Otrow 97 o tertve method gve b C E where the order o covergece d C the comttol cot er terto. or the tem o o-ler eqto ow the comttol cot er terto gve b [ee Gr-Sáchez et l. ]

9 AAM: Iter. J. Vol. 8 Ie December 79 C P where d rereet the mber o evlto o d reectvel P the mber o rodct er terto d d re the rto betwee rodct d vlto reqred to ere the vle o C term o rodct. Now let comre the ecec de o the rooed method mel Error! Reerece orce ot od. MJM d Error! Reerece orce ot od. MJM wth tht o Newto method NM thrd order method b Homeer HM [Homeer ] d hrmoc me Newto method HMNM [Gr-S chez et l. ] orth-order method b Shrm et l. WNM [Shrm et l. ] Cordero et l. CM 7 [ Cordero et l. 9] d Drvh DM 8 [Drvh d Brt 7]. he HM gve b } }. he HMNM gve b } } }. he WNM gve b } 9 } } I }. he CM gve b

10 7 V. Kwr et l. } 7 } } }. he DM gve b } } 8. I the tertve method tht Newto method ted o comtg the vere oertor we olve ler tem where we hve / rodct d / qotet the LU decomoto d rodct d qotet the reolto o two trglr ler tem. I we oe tht qotet eqvlet to rodct the P. 7 I geerl we deote b the mber o clr rodct er terto b d the mber o comlete reolto o ler tem LU decomoto d reolto o two trglr tem b. We cll the mber o reolto o two trglr tem whe LU decomoto comted other te the me terto the totl mber o rodct ee Behl et l. P 8 I ble we reet the vle o dc or ech tertve method lzed th er 8. ble. Coecet ed d 8 locl order o covergece d comttol cot o tertve method 8 Method C / 7 / /

11 AAM: Iter. J. Vol. 8 Ie December 7 / / / 7 / 8 /.. Comro Betwee the Ecece Let deote the ecece o 8 to b. M Coder the rto log log log log C C M M G 9 where. C It cler tht > G the tertve method more ecet th. g to ccot tht border betwee two comttol ecece gve b G th bodr g 9 c be ereed b eqto whch wrtte [ee Gr-S chez et l. ] where log log log log log log log log log log log log log log log log. log log d

12 7 V. Kwr et l. I order to comre the ecec de o the tertve method tht M wth the ecec dce o the other method g. the le or reectvel or we reet the bodr G betwee M d M b dotted le S the bodr G betwee M d M b dhed le S the bodr G 7 betwee M d M 7 b dot-dhed le S the bodr G 8 betwee M d M 8 b old le S. he ot tht we c t comre the ecec dce o the tertve method & d & ce the codto log log volted here. Le S dvde the mmm ecec rego betwee & beg tht E > E bove S. Smlrl le S dvde the mmm ecec rego betwee & beg tht E > E bove S le S dvde mmm ecec rego betwee & 7 beg tht E > E7 bove S le S dvde mmm ecec rego betwee & 8 beg tht E > E8 bove S. gre.. Bodr Le - le or reectvel or he bove relt cocerg ecec dce re mmrzed the ollowg theorem: heorem.. or ll we hve M > M or > b M > M or 8log > log c M > M 7 or > d M > M8 or >.

13 AAM: Iter. J. Vol. 8 Ie December 7 O mlr le we c lo comre the ecec de o the rooed method Error! Reerece orce ot od. MJM wth tht o Newto method NM the thrd order method b Homeer HM Homeer d hrmoc me Newto method HMNM Gr-S chez et l. the orth-order method b Shrm et l. WNM Shrm et l. Cordero et l. CM 7 Cordero et l. 9 d Drvh DM 8 Drvh d Brt 7.. Nmercl Eermet Now we reet ome mercl emle to lltrte the comro o the erormce o the ewl develoed method mel method Error! Reerece orce ot od. MJM method MJM wth tht o clcl Error! Reerece orce ot od. MJM d method Newto method NM thrd order method b Homeer HM Error! Reerece orce ot od. d the orth-order method b Cordero et l. CM Cordero et l. 9 Drvh DM Drvh d Brt 7 Shrm et l. WNM Shrm et l. d Jrrtt method JM Nedzhbov 8 reectvel or olvg tem o oler eqto gve ble. Comtto hve bee erormed g MALAB vero 7. R 7b doble reco rthmetc. We e tolerce error. ollowg tog crter re te or comter rogrm: < <. We lze the mber o terto eeded to coverge to the reqred olto. Coder the ollowg tem o oler eqto Emle.... Solto Emle.. e co. Solto.

14 7 V. Kwr et l. Emle.. log. Solto Emle.. e. Solto. Emle... Solto Emle.. 9. Solto Emle.7. co e. Solto

15 AAM: Iter. J. Vol. 8 Ie December 7 Emle o.. ble. otl mber o terto Itl ge NM HM CM DM WNM JM MJM MJM MJM 7. Coclo Evdetl we hve rooed d lzed wde geerl cl o Jrrtt method or olvg oler eqto the mltvrte ce. h cl geerlzto over the ml o Jrrtt method rooed b Behl et l. d deed o two doble rmeter. hee method hve orth-order covergece d do ot reqre the ecod-order réchet dervtve. I term o comttol cot ll thee method reqre evlto o oe cto d two rt-order réchet dervtve. ll the comttol relt ver tht the ml o method re ecet d ehbt eql or better erormce comred to other well-ow method vlble ltertre. Acowledgemet he thor re gretl debted to the omo revewer or ther vlble ggeto o the rt vero o th er. REERENCES Amt S. Bqer S. d Gterrez J.M.. Geometrc cotrcto o tertve cto to olve oler eqto J. Comt. Al. Mth. Vol Bbee D.K.R. Dhoo M.Z. Drvh M.. Krmb A. d Brt A.. Al o two Chebhev-le thrd order method ree rom ecod dervtve or olvg tem o oler eqto J. Comt. Al. Mth. Vol.. -. Behl R. Kwr V. d Shrm K.K.. Otml eq-cled mle o Jrrtt method It. J. Comt. Mth Vol Cordero A Mrtíez E. d orregro J.R. 9. Itertve method or order or d ve or tem o o-ler eqto Al. Mth. Comt. Vol.. -.

16 7 V. Kwr et l. Cordero A. d orregro J.R.. Vrt o Newto method or cto o everl cto Al. Mth. Comt. Vol Drvh M.. d Brt A. 7. A orth-order method rom qdrtre orml to olve tem o o-ler eqto Al. Mth. Comt. Vol Ezqerro J.A. d Herdez M.A.. A rmetrc Hlle-te terto wth ree ecod dervtve It. J. Pre Al. Mth. Vol.. -. Ezqerro J.A. d Herdez M.A.. O Hlle-te terto wth ree ecod dervtve J. Comt. Al. Mth. Vol rot M. d Sorm E.. hrd-order method rom qdrtre ormle or olvg tem o oler eqto Al. Mth. Comt. Vol Gr-Sáchez M. Gr A. d Noger M.. O the comttol ecec de d ome tertve method or olvg tem o o-ler eqto J. Comt. Al. Mth. Vol Gterrez J.M. d Herdez M.A A ml o Chebhev-Hlle te method Bch ce Bll. At. Mth. Soc. Vol.. -. Herdez M.A.. Secod-dervtve-ree vrt o the Chebhev method or oler eqto J. Ot. heor. Al. Vol.. -. Homeer H.H.H.. Moded Newto method wth cbc covergece: he mltvrte ce J. Comt. Al. Mth. Vol Jrrtt P. 9. Some orth-order mltot tertve method or olvg eqto Mth. Comt. Vol Nedzhbov G.H. 8. A ml o mlt-ot tertve method or olvg tem o oler eqto J. Comt. Al. Mth. Vol.. -. Örteg J.M. d Rheboldt W.C. 97. Itertve Solto o Noler Eqto Severl Vrble Acdemc Pre New Yor. Otrow A.M. 97. Solto o Eqto Eclde d Bch Sce Acdemc Pre New Yor. Shrm J.R. Gh R.K. d Shrm R.. A ecet orth-order weghted-newto method or tem o o-ler eqto Nmer. Algor. Vol rb J.. 9. Itertve Method or the Solto o Eqto Pretce-Hll Eglewood Cl NJ.