New Efficient Optimal Derivative-Free Method for Solving Nonlinear Equations

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1 Itratioal Joral o Mathmatis ad Comptatioal Si Vol No 05 pp Nw Eiit Optimal Drivativ-Fr Mthod or Solvig Noliar Eqatios Q W Go Y H Qia * Dpartmt o Mathmatis Zhjiag Normal Uivrsit Jiha Zhjiag Chia Abstrat I this papr w sggst a w thiq whih ss Lagrag polomials to gt drivativ-r itrativ mthods or solvig oliar qatios With th s o th proposd thiq ad Sts o-lik mthods a w optimal orth-ordr mthod is drivd B sig thr-dgr Lagrag polomials with othr two-stp mthods whih ar iit optimal mthods ighth-ordr mthods a b ahivd Bsids w a gt sitth-ordr mthods i w s othr thr-stp mthods ad highr-ordr dgr Lagrag polomials Th rror qatios ad asmptoti ovrg ostats ar obtaid or th proposd mthods Som mrial ampls ar illstratd to vri th ara o th proposd omptatioal shm Kwords Lagrag Polomials Sts o-lik Mthod Drivativ-Fr Covrg Ordr Eii Id Rivd: Marh 7 05 / Aptd: April 6 05 / Pblishd oli: Ma 05 Th Athors Pblishd b Amria Istitt o Si This Op Ass artil is dr th CC BY-NC lis Itrodtio O o th most basi ad ompl problms i mrial aalsis is to solv oliar qatios i trms o ( = 0 Howvr it is somwhat diilt to id th roots o oliar qatios aratl Itrativ mthods ar th most ommol sd thiqs to obtai approimat soltios Lt * ab [ b a root o th qatio ( = 0 ( W hag th qatio ( as qivalt orm = φ ( Takig 0 [ ab ostrt th rrsiv ormla = k φ k k = 0 ( W gt a sq { k } k = 0 throgh th rrsiv ormla ( Takig limit o both sids o th ormla ( w a gt ɶ = φ ɶ ( whrɶ is a limit o th sq{ k } k = 0 wh k Obviosl ɶis a root o th qatio ( throgh th ormla ( Bas o qatio ( is qal to ( w wold obtai = ɶ lim k = ɶ (5 * k ad th qatio ( is alld itrativ ormla Mawhil th mthod sig qatio ( is amd itrativ mthod Nwto s mthod [ is th lassial approah or solvig oliar qatios th omptatioal algorithm is writt as ollows: whr = = (6 This is a ampl o a o-poit itratio shm [0 I som appliatios ( is mor diilt to valat tha ( Rtl mh rsarh ort * Corrspodig athor addrss: qh00@zjd (Y H Qia

2 Itratioal Joral o Mathmatis ad Comptatioal Si Vol No 05 pp [ lightd th idas o rmovig drivativs rom itratio tios i ordr to avoid th ssit o diig w tio valatios sh as th irst drivativ or sod drivativ For ampl Sts [ mad s o th orward dir approimatio to rpla ( to propos th ollowig omptatioal shm ( ( = (7 Rtl Dhgha ad Hajaria [6 prstd a third ordr itrativ mthod (DM ad iorporatd or tio valatios pr ah itratio stp as ollows whr ( ( z ( ( ( = z ( ( ( = ( Bsids R t al [5 proposd aothr optimal orth-ordr mthod i trms o th ollowig shm (RMa a R ad a is a paramtr ( = z = z (9 ( = [ [ z [ z a( z I Eq (9 th tio [ is rprstd b ( a ( b [ a b = Atrward Li t al [ prstd a a b orth-ordr mthod (LM as ollows ( = z = z [ [ z [ z = [ (0 Th aormtiod mthods [6 5 ar optimal i aorda with Kg-Trab ojtr to obtai a drivativ-r omptatioal shm It is ot limitd to th abov omptatioal shm [6 5 ma variats o Sts s mthods wr proposd i th past dad Sharma [6 itrodd a Nwto-Sts mthod to ahiv th third ordr ovrg Cordro t al [ proposd a w orth-ordr Sts tp mthod I additio othr rsarhrs also itrodd a varit o w mthods to solv oliar qatios basd o Sts s tp mthod (s th litratr pblishd i [ 5 Mawhil thr ar varios mthods sig liar tios to allat ( For ampl Gstavo t al i [ prstd thr w optimal orth-ordr mthods to osidr th polomial Cordro t al [5 ad Solmai [9 proposd Padé approimats to stimat ( I this papr a w optimal thiq is prstd to adopt Lagrag s itrpolatio ad Sts-lik mthods to ld drivativ tios i th omptatioal itratio shm Th papr is orgaizd as ollows th mthodolog ad priipl o th w optimal thiq sig Lagrag s itrpolatio or solvig oliar qatios is itrodd i Stio I th t stio th proposd thiq is gralizd or gttig ighth-ordr mthods whih ar both drivativ-r ad optimal I Stio w a driv th sitth-ordr mthods wh th proposd thiq ad othr thr-stp optimal mthods ar sd Illstrativ ampls ar giv to vri th ara o th proposd mthod i Stio 5 Fiall th papr ds with oldig rmarks i Stio 6 Forth-Ordr Optimal Mthod W osidr a gral oliar qatio ( = 0 Assmig that α is a simpl root o th qatio ( = 0 ad γ is a giv iitial gss whih is los ogh to α Aordig to Talor s sris pasio arod γ givs ( = ( γ ( γ ( γ γ γ ( Th irst-ordr approimatig qatio is ( γ ( γ ( γ 0 = ( Th rom Eq ( w a ir th ollowig itrativ mthod = ( This is th lassial Nwto s mthod with a sod-ordr ovrg Usig th ollowig orward-dir to rpla ( ilds ( ( = ( W a gt Sts s mthod b makig s o Eq ( Th w obtai a mthod ormd b th ompositio o Sts ad Nwto mthods as ollows:

3 0 Q W Go ad Y H Qia: Nw Eiit Optimal Drivativ-Fr Mthod or Solvig Noliar Eqatios ( ( ( ( = z z Whr = (5 = this ss or tio valatios whih ar I ordr to avoid th valatio o th irst drivativ ad rd th tio valatios it is sggstd approimatig it b th drivativ p ( z o th ollowig Lagrag polomial o dgr ( t ( t z ( ( z ( t ( t ( z ( z ( z ( t ( t z ( ( z p t = (6 Th th drivativ o as p = valatd i a b prssd ( z ( z ( ( ( z ( z ( z ( ( ( z ( ( z ( z (7 Sbstittig Eq (7 i th sod qatio o Eq (5 w gt a w two-stp itrativ mthod (GM sig th Lagrag polomial whos prssio is writt as ollows: p( t ( ( z ( = ( = ( ( z ( z ( ( z ( ( z ( z ( ( ( z Thorm Lt α Db a simpl zro o a siitl dirtiabl tio : D R R i a itrvald I 0 is siitl los to α th th mthod ( has optimal orth-ordr ovrg Proo Lt b th rror i whih is Talor padig at = α w hav = 5 ( ( O α B = (9 ( k whr = / k! k = k α O th basis o z ( = w gt = ( ( ( ( ( ( ( ( 5 O( z (0 Sbstittig Eq (9 ad (0 i th irst qatio o Eq ( w hav ( ( α = ( (( 5 Th w gt ( ( 6 O 5 ( ( ( = ( ( (( 5 7 ( ( 6 O 5 ( ( with Eq (9 (0 ad ( w a dd that th rror qatio o th GM mthod is ( ( 5 = O ( showig that th mthod (GM has optimal orth-ordr ovrg

4 Itratioal Joral o Mathmatis ad Comptatioal Si Vol No 05 pp Eighth-Ordr Optimal Mthods Th thiq a b rthr applid to highr ordr mthods b sig highr dgr Lagrag polomials To obtai drivativ-r mthods havig optimal ighth-ordr ovrg w osidr a thr-stp l i whih th irst two stps ar a drivativ-r optimal orth-ordr mthods ad ϕ is a tio that dis a optimal drivativ r mthod or = th ombiatio o ad ϕdis a optimal drivativ-r itrativ mthod or = = [ z [ z ( ( ( Th th drivativ p ( o th ollowig Lagrag polomial o dgr is sd to rpla ( Aordig to th Lagrag polomial w gt ( t ( t z ( t = ( ( z ( ( t ( t z ( t ( ( ( z ( ( t ( t ( t z ( z ( z ( z ( z ( t ( t ( t z ( ( ( ( z p t (5 Th th drivativ o p ( t valatd i a b prssd as p ( = z ( ( z ( ( ( ( z ( ( z ( ( ( ( z ( ( ( ( z ( z ( z Usig p ( to rpla ( thr-stp lass ϕ (6 obtais th ollowig [ z [ z ( ( = D (7 whr D ( is th drivativ o p( t i i othr words D ( = p ( W valat or tios whih ar ( ( ( z ( i ah itratio Th omptatioal mthod statd i Eq (7 will b optimal i w prov that it has ighth-ordr ovrg Thorm Lt α Db a simpl zro o a siitl dirtiabl tio : D R R i a itrval D I 0 is siitl los to α th th mthod (7 has optimal ighth-ordr ovrg Proo Lt = α B Talor padig at = α w hav ( = 6 6 ( k 7 7 whr = / k! k = k α O th basis o z ( 9 O = w obtai 5 5 ( z = ( ( ( ( ( ( 5 9 ( O ( (9 Sbstittig Eq ( ad (9 i th irst qatio o Eq (7 w hav α = ( ( ( ( ( 5 ( ( 6 9 O (0 Th w pad ( arodα with Eq (0 to driv th ollowig prssio

5 06 Q W Go ad Y H Qia: Nw Eiit Optimal Drivativ-Fr Mthod or Solvig Noliar Eqatios ( = ( ( ( ( 5 7 ( ( ( 6 9 O ( W di that th irst two stps ar optimal orth-ordr mthod ad assm = h h h h h O( 9 ( with Eq ( (9 ad ( w a ths dd th rror qatio o th mthod as ( h 9 O( h = ( Th abov mthod shows th optimal ighth-ordr ovrg For ampl applig RM (assm a = i (7 w gt th mthod (GRM: ( ( z ( = z = ( ( = [ [ z [ z z = D Th abov allatio mthod satisis th ollowig rror qatio: ( ( ( = 7 ( ( ( ( ( 9 O 7 (5 I additio i w appl th mthod o LM i Eq (7 w a gt aothr mthod (GLM i th orm o: ( = z = z [ [ z [ z = [ = D This mthod satisis th ollowig rror qatio: ( ( 9 O 7 (6 (7 Sitth-Ordr Optimal Mthods To rthr obtai drivativ-r mthods o optimal sitth-ordr ovrg w osidr a or-stp l i whih th irst thr stps ar a drivativ-r optimal ighth-ordr mthod as ollows: v = φ [ z [ z [ z = φ = φ = v ( v ( v ( Whr ϕ is a itrativ tio o optimal ighth ordr drivativ-r mthod omposd with ϕ ϕ Th drivativ p ( v o th ollowig Lagrag polomial o dgr is adoptd to rpla ( v ( t ( t z ( t ( t v = ( ( z ( ( v ( t ( t z ( t ( t v ( ( ( z ( ( v ( t ( t ( t ( t v ( z ( z ( z ( z ( z v ( t ( t ( t z ( t v ( ( ( ( z ( v ( t ( t ( t z ( t ( v ( v ( v ( v z ( v p t ( ( ( ( ( = 7 (9 Th th drivativ o p( t valatd iv a b prssd as

6 Itratioal Joral o Mathmatis ad Comptatioal Si Vol No 05 pp v v v v p ( = v v v z v ( v ( v ( z ( ( v ( ( ( z ( v ( v ( v z ( ( ( v ( ( z ( v ( v ( v z ( ( ( v ( ( z ( v ( v ( v ( z ( z ( v z ( z ( z Usig p ( v rplas ( v thr-stp lass othrwords ( v v [ z [ z [ z (0 w gt th ollowig ( v ( v = v D ( whr v is th drivativ o pt i v i D = p W valat iv tios i ah itratio Th mthod will b optimal i w prov that it has sitth-ordr ovrg Thorm Lt α Db a simpl zro o a siitl dirtiabl tio : D R R i a itrval D I 0 is siitl los to α th th mthod ( has optimal sitth-ordr ovrg Pr oo hav D Lt α B Talor padig at = α w 7 ( ( O = ( ( k whr k = α / k! k = O th basis o = z w obtai = ( ( ( ( (( ( 5 z ( ( = 7 O v ( Sbstittig Eq ( ad ( i th irst qatio o Eq ( w hav α = ( ( 6 7 O ( Th w pad ( arod α with Eq ( to ahiv th ollowig qatio ( = ( ( ( ( 5 7 ( ( ( 6 7 O (5 W di that th irst thr stps ar optimal ighth-ordr mthod w ths gt v = h h h = j h h ( ( ( ( 5 h 6 6 O( 7 h h 0 0 O( h h (6 (7 Fiall th rror qatio o th mthod a b drivd as ollows: ( ( 5h j 6 7 O( j = ( Th abov qatio shows that th mthod ( has a optimal orth-ordr ovrg 5 Nmrial Eampls ( Th proposd mthods dsribd i Stio sig th w thiq ar mplod to solv som illstrativ oliar problms Th obtaid rslts ar ompard with Sts s mthod (SM (7 Dhgha ad Hajaria s mthod (DM ( Usig Lagrag polomials w gt a w optimal orth-ordr mthod (GM ad two w optimal

7 0 Q W Go ad Y H Qia: Nw Eiit Optimal Drivativ-Fr Mthod or Solvig Noliar Eqatios ighth-ordr mthods (GRM GLM All th omptatios wr odtd b sig MATLAB 0 w sltd a approimat soltio rathr tha th at root omptd with 500 digits Th stoppig ritrio sd is 50 k k ( k < 0 th itrativ sssio ovrg was hkd to vri th ara o th approimat soltios Tabl shows th mbr o th itratios dd to rah th aptabl tolra or ah mthod ad stimats th omptatioal ordr o ovrg ρ (sall alld ACOC did b Cordro ad Torrgrosa [ : l( k k / k k ρ = (9 l( / k k k k I Tabl w show th ompariso o ii idis [ or dirt mthods o varios ordrs Th ii idis is did as p / whr p is th ordr o ovrg is th mbr o tio valatios pr stp W s th ollowig tios i mrial ompariso: ( = α = = os α = 7769 = l α = 5590 = α = 05 5 = 0 α = 55 6 = si α = 09 7 = 0 α = 650 = os α = 50 It is od that th ii id o th mthods (GM GRM GLM ar bttr tha othr mthods rom Tabl From Tabl w a id that th mthod (GM ovrgs mor rapidl tha Sts s mthod Dhgha ad Hajaria s mthod Th ighth-ordr mthods (GRM GLM rqir lss itrativ stps to rah th stoppig ritrio All thortial ordr o ovrg Tabl Ordrdr ad ii idis o som mthods Mthods SM DM GM GRM GLM Ordr Eii id Tabl Nmrial rslts or oliar qatio 0 SM DM GM GRM GLM Itr Error Itr Error Itr Error Itr Error Itr Error ρ ρ ρ

8 Itratioal Joral o Mathmatis ad Comptatioal Si Vol No 05 pp SM DM GM GRM GLM ρ ρ ρ ρ ρ Colsios W proposd a w drivativ-r orth-ordr mthod (GM whih is optimal ordr o ovrg i th ss o Kg-Trab ojtr Makig s o th w thiq w obtai othr two ighth-ordr mthods whih ar a optimal mthod withot drivativ tios Highr ordr mthods a also b ostrtd sig th proposd mthod sh as sitth-ordr mthods Th ovrg o th mthod is provd i th papr Illstrativ mrial ampls ar show i this papr to dmostrat th ii o th w mthods i trms o its ii o id ad omptatioal ii id Th proposd mthods ar apabl o solvig varios oliar qatios ad a ahiv good ovrg Akowldgmts Th athor YH Qia gratll akowldg th spport o th Natioal Natral Si Fodatios o Chia (NNSFC throgh grat No 09 ad th iaial spport o Chia Sholarship Coil (CSC throgh grat No Th athor QW Go gratll akowldg th spport o th Natioal Traiig Programs o Iovatio ad Etrprrship or Udrgradats Th athors wish to akowldg th valabl partiipatio o Dr SK Lai i th proo-radig o this papr Rrs [ Argros I K: Covrg ad Appliatio o Nwto-Tp Itratios Sprir Nw York 00 [ Cordro A Torrgrosa JR: Variats o Nwto s mthod sig ith-ordr qadratr ormlas ApplMath Compt 90 ( [ Cordro A Torrgrosa JR: A lass o Sts tp mthods with optimal ordr o ovrg Appl Math Compt 7( [ Cordro A Hso J L Martíz E Torrgrosa J R: Sts tp mthods or solvig o-liar qatios J Compt Appl Math 6( (0 [5 Cordro A Hso J L Martíz E Torrgrosa JR: A w thiq to obtai drivativ-r optimal itrativ mthods or solvig oliar qatios J Compt Appl Math

9 0 Q W Go ad Y H Qia: Nw Eiit Optimal Drivativ-Fr Mthod or Solvig Noliar Eqatios [6 Dhgha M Hajaria M: Som drivativ r qadrati ad bi ovrg itrativ ormlas or solvig oliar qatios J Compt Appl Math 9( [7 Ehbst N Shvrdt M L Viga R P: Two drivativ-r mthods or solvig drdtrmid oliar sstms o qatios Compt Appl Math 0( [ Fradz-TorrsG Vasqz-Aqio J: Thr Nw Optimal Forth-Ordr Itaiv Mthods to Solv Noliar Eqatios Adv Nmr Aal0Artil ID [9 Fradz-TorrsG: Drivativ r itrativ mthods with mmor o arbitrarhigh ovrg ordr Nmr Algor [0 Kg HT Trab JF: Optimal ordr o o-poit ad mlti-poit itratio J Asso Compt Math [ Li ZL Zhg Q Zhao P: A variat o Sts s mthod o orth-ordr ovrg ad its appliatios Appl Math Compt [ Ortga JM Rhiboldt WG: Itrativ Soltios o Noliar Eqatios i Svral Variabls Aadmi Prss Nw York 970 [ Ostrowski AM: Soltios o Eqatios ad Sstms o Eqatios Aadmi [ Păvăloi I Cătiaş E: O a Nwto-Sts tp mthod Appl Math Ltt 6( [5 R HM W QB Bi WH: A lass o two-stp Sts tp mthods with orth-ordr ovrg Appl Math Compt 09( [6 Sharma JR: A omposit third ordr Nwto-Sts mthod or solvig olia qatios Appl Math Compt 69( [7 Sharma JR Arora H: A iit drivativ r itrativ mthod or solvig sstms o oliar qatios Appl Aal Disrt Math [ Solmai F Karimi Vaai S: Optimal Sts-tp mthods with ighth ordr o ovrg Compt Math Appl 6( [9 Solmai F Karimi Vaai S Jamali Paghalh M: A lass o thr-stp drivativ-r root solvrs with optimal ovrg ordr J Appl Math 0 Artil ID [0 Thkral R: Nw Highr Ordr Drivativ-Fr Mthods or Solvig Noliar Eqatios J Nmr Math Stoh ( [ Wag H Li SB: A amil o drivativ-r mthods or oliar qatiorv Mat Complt