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

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1 Amrica Joural o Computatioal ad Applid Mathmatics 0 (: -8 DOI: 0.59/j.ajcam Nw Sixtth-Ordr Drivativ-Fr Mthods or Solvig Noliar Equatios R. Thukral Padé Rsarch Ctr 9 Daswood Hill Lds Wst Yorkshir LS7 5JS Eglad Abstract Nw our-poit drivativ-r sixtth-ordr itrativ mthods or solvig oliar quatios ar costructd. It is provd that ths mthods hav th covrgc ordr o sixt rquirig oly iv uctio valuatios pr itratio. I act w hav obtaid th optimal ordr o covrgc which supports th Kug ad Traub cojctur. Kug ad Traub cojcturd that th multipoit itratio mthods without mmory basd o valuatios could achiv optimal covrgc ordr. Thus w prst w drivativ-r mthods which agr with th Kug ad Traub cojctur or 5. Numrical comparisos ar mad with othr xistig mthods to show th prormac o th prstd mthods. Kywords Drivativ-Fr Mthods Noliar Equatios Optimal Ordr o Covrgc Computatioal Eicicy Kug-Traub Cojctur. Itroductio Multipoit itrativ mthods or solvig oliar quatios ar o grat practical importac sic thy ovrcom thortical limits o o-poit mthods cocrig th covrgc ordr ad computatioal icicy. Th w itrativ mthods ar applid to id a simpl root α o th oliar quatio ( x 0 ( whr : D is a scalar uctio o a op itrval D ad it is suicitly smooth i a ighbourhood o α. I this papr a w amily o our-poit drivativ-r mthods o th optimal ordr ight ar costructd by combiig optimal two-stp ourth-ordr mthods ad thr-stp ighth-ordr mthods. I ordr to obtai ths w sixt ordr drivativ-r mthods w rplac drivativs with suitabl approximatios basd o dividd dirc. I act it is wll kow that th various mthods hav b usd i ordr to approximat th drivativs[67]. Th prim motiv o this study is to dvlop a class o vry icit our-stp drivativ-r mthods or solvig oliar quatios. Th sixtth-ordr mthods prstd i this papr is drivativ-r ad oly uss iv valuatios o th uctio pr itratio. I act w hav obtaid th optimal ordr o covrgc which supports th Kug ad Traub cojctur. Kug ad Traub cojcturd that th multipoit itratio mthods without mmory basd o valuatios could achiv optimal covrgc ordr - * Corrspodig author: rthukral@hotmail.co.uk (R. Thukral Publishd oli at Copyright 0 Scitiic & Acadmic Publishig. All Rights Rsrvd Thus w prst w drivativ-r mthods which agr with th Kug ad Traub cojctur or 5. I additio ths w sixtth-ordr drivativ-r mthods hav a quivalt icicy idx to th rctly stablishd mthods prstd i[56]. Furthrmor th w sixtth-ordr drivativ-r mthods hav a bttr icicy idx tha xistig th two-stp ad thr-stp ordr drivativ-r mthods prstd i[ ] ad i viw o this act th w mthods ar sigiicatly bttr wh compard with th stablishd mthods. Also w hav oud that thr is a typo rror i[6] hc w shall show ad us ths ight ordr drivativ-r mthods to costruct sixt ordr mthods. It should b otd that th ighth-ordr drivativ-r mthods prstd i[6] ar o optimal ordr ad thir ordr o covrgc hav b provd ad show i may xampls. Howvr th typo rrors actually occur i th wight uctios o th ighth-ordr drivativ-r mthods i (.7 ad (.[6]. Sic ths ighth-ordr drivativ-r mthods hav b provd to covrg o th ordr ight w shall thror us ad simpliy various xprssios giv i[6]. Cosqutly w hav oud that th w sixtth-ordr drivativ-r mthods ar icit ad robust. Th papr is orgaizd as ollow. A w amily o our-poit drivativ-r mthods o optimal ordr sixtth ar costructd i th xt sctio by combiig two-poit ourth-ordr mthods ad thr-poit ighth-ordr mthods. Th purpos o this papr is to obtai a suitabl approximatio o th drivativs o a uctio i ordr to rduc th umbr o uctio valuatios. Th total umbr o uctio valuatios o th proposd our-poit drivativ-r mthods is iv ad accordig to th Kug-Traub cojctur is o th optimal ordr[98]. I sctio w shall compar

2 Amrica Joural o Computatioal ad Applid Mathmatics 0 (: -8 th ctivss o th w mthods with th rctly itroducd sixtth-ordr mthods[56]. Fially i sctio 4 som computatioal aspcts o th proposd our-poit drivativ-r mthods ad compariso with xistig mthods ar also giv.. Drivatio o th Mthods ad Covrgc Aalysis I this sctio w shall di w sixtth-ordr drivativ mthods. I ordr to stablish th ordr o covrgc o ths w drivativ-r mthods w stat th thr sstial diitios. Diitio Lt ( x b a ral uctio with a simpl root α ad lt { x } b a squc o ral umbrs that covrg towards α. Th ordr o covrgc m is giv by α lim ρ 0 ( m x α whr ρ is th asymptotic rror costat ad m. Diitio Lt λ b th umbr o uctio valuatios o th w mthod. Th icicy o th w mthod is masurd by th cocpt o icicy idx[48] ad did as /λ µ ( whr µ is th ordr o th mthod. Diitio Suppos that x x ad x ar thr succssiv itratios closr to th root α o (. Th th computatioal ordr o covrgc[9] may b approximatd by whr. l COC l ( x α( x α ( x α( x α.. Th our-poit drivativ-r mthods Cosidr th ollowig itratio schm ( x y x ( x ( y z y ( y (5 ( z a z ( z ( a a ( a This schm cosists o our stps i which th Nwto mthod is rpatd. It is clar that th ormula (5 rquirs ight valuatios pr itratio ad usig ( th icicy idx o (5 is which is sam as th classical Nwto mthod. I act schm (5 dos ot icras th (4 computatioal icicy. Th purpos o this papr is to stablish w drivativ-r mthods with optimal ordr; hc w rduc th umbr o valuatios to iv by usig som suitabl approximatio o th drivativs. To driv a highr icicy idx w cosidr approximatig th drivativs by dividd dirc mthod. Thror th drivativs i (5 ar rplacd by ( w ( x ( x [ w x] (6 w x ( x ( y ( y [ x y] (7 x y ( z [ y z] [ x y] [ x z] (8 [ y a] [ z a] ( a. (9 [ y z] Substitutig (6-(9 ito (5 w gt w x ( x ( x y x [ w x ] ( y z y (0 [ x y] ( z a z [ y z] [ x y] [ x z] [ y z] ( a a [ y a] [ z a] Th irst stp o th ormula (0 is th classical Sts scod-ordr mthod[5] ad th scod stp is th improvmt o th irst stp. Furthrmor w hav oud that th scod third ad ourth stp dos ot produc a optimal ordr our ight ad sixtth rspctivly. Thror w hav itroducd wight uctios i th scod third ad ourth stp i ordr to achiv th dsird optimal ordr o covrgc. First w shall dot th ollowig uctios as ( z ( z ( y u u u ( x ( w ( x ( ( y ( a ( a u4 u5 u6. ( w ( x ( w I ordr to achiv th ourth-ordr covrgc i th scod stp w hav oud that thr ar thr dirt orms o wight uctios which ar xprssd as φ u4 ( φ u u ( 4 4 φ [ ] [ ] x w y w (4 ad to achiv th ighth-ordr covrgc i th third stp th wight uctios ar xprssd as η ( uu 4 ( u (5 Furthrmor to achiv th sixtth-ordr o covrgc i th ourth stp th wight uctio is xprssd as uu uuu 4 u5 u6 uu4 4 4 σ [ ] u u uu u u x y. (6

3 4 R. Thukral: Nw Sixtth-Ordr Drivativ-Fr Mthods or Solvig Noliar Equatios Th th itratio schm (5 i its ial orm is giv as w x ( x ( x y x [ w x ] ( y z y φ (7 [ x y] ( z a z η [ y z] [ x y] [ x z] [ y z] ( a z σ [ y a] [ z a] whr k φ ησ ar giv by (4 (5 (6 rspctivly ad providd that th domiators i (7 ar ot qual to zro. Thus th schm (7 dis a w multipoit mthod with suitabl wight uctios. To obtai th solutio o ( by th w sixtth ordr drivativ-r mthods w must st a particular iitial approximatio x 0 idally clos to th simpl root. I umrical mathmatics it is vry usul ad sstial to kow th bhaviour o a approximat mthod. Thror w shall prov th ordr o covrgc o th w sixtth-ordr mthod. Thorm Lt α D b a simpl root o a suicitly dirtiabl uctio : D i a op itrval D. I x 0 is suicitly clos to α th th ordr o covrgc o th w drivativ-r mthod did by (7 is sixt. Proo Lt α b a simpl root o ( x i.. ( α 0 ad ( α 0 ad th rror is xprssd as x α (8 Usig th Taylor xpasio w hav ( x ( α ( α ( α (9 iv 4 6 ( α 4 ( α Takig ( α 0 ad simpliyig xprssio (9 bcoms x c c c c (0 whr ad c 4 k 4 ( k ( α ( k! or k 4 ( Expadig th Taylor sris o ( w ad substitutig ( x giv by (0 w hav ( ( w c c c c c c c. ( Substitutig (0 ad ( i th irst stp o th xprssio (7 w obtai ( x y α x α [ w x] ( c ( c c Th xpasio o ( y about α is giv as ( y c c c c c c c cc c c cc c. (4 Th xpasio o th particular trms usd i (7 ar giv as ( w ( x [ w x] c ( c cc (5 w x ( c c c c c c w y [ w y] c ( c cc w y c c c c c c c c c c x y [ x y] c x y cc c cc c c [ w x] [ x y] [ w y] c cc c cc cc c (6 (7 (8 Basd o th particular wight uctio φ ad substitutig appropriat xprssios i th scod stp o (7 w obtai [ w x] ( y z α y α. (9 as [ ] [ ] z about α is giv x y w y Th Taylor sris xpasio o ( ( 4 z c c c c c cc 4 cc ccc ccc. (0 I ordr to valuat th sstial trms o (7 w xpad trm by trm ( y ( z [ y z] y z ( cc c c c ( x ( z [ x z] x z ( c c c c 4 Collctig th abov trms [ ] [ ] [ ] cc ( c ψ y z x y x z c c ( u ω cc ccc cc c c c ( (4 cc c ξ ( uu 4 (5 c

4 Amrica Joural o Computatioal ad Applid Mathmatics 0 (: -8 5 Substitutig appropriat xprssios i (7 w obtai a α z α ψωξ ( z (6 Simpliyig (6 w obtai th rror quatio o ighthordr covrgc 7 ( ( ( a α c c c cc c 8 c c c c c c c cc c cc c (7 Th rror quatio (7 is a simpliid vrsio o th ighth-ordr drivativ-r mthod stablishd i[6]. Th Taylor sris xpasio o ( z about α is giv as 6 ( ( cc 4 ccc cc 4 8 cc c cc 4c a c c c cc c (8 W progrss to xpad th trms usd i th ourth-stp o (7 ( z ( a [ z a] z a (9 ( ( c c c cc c 4 c [ ] y z y z y z ( c c c c ( y ( a [ y a] y a ( c c c c ( ( ( c c cc c cc c 6 6 c (40 (4 σ (4 [ y z] [ ] [ ] ν y a z a ( ( ( c c cc c cc c 6 7 c c a α σν ( a (4 (44 Collctig ad Simpliyig th appropriat trms usd i (7 w obtai th rror quatio c c( c ( cc c ( cc4 ccc cc4 cc c cc 4c 4 4 ( 5ccc 4 cc 0ccc 4 cc 4ccc 4 4 5ccc cc 7ccc 6cc ccc cc 9c (45 Th xprssio (45 stablishs th asymptotic rror costat or th sixt ordr o covrgc or th w drivativ-r mthod did by (7.. Mthod : (Liu Rctly w hav itroducd thr-stp ighth-ordr drivativ-r mthods[6] ad o o thm was costructd by usig th two-stp ourth-ordr mthod prstd by Liu t al.[7] ad th third-stp was dvlopd to achiv th ighth-ordr. It appars that th ormula o th ighth-ordr drivativ-r mthod basd o Liu t al. mthod giv by (.7 i[6] has a typo rror. This rror actually occurs i th wight uctio o th third stp o th ighth-ordr drivativ-r mthod. W obsrv that th third stp is corrctd ad simpliid i th ollowig sixtth-ordr mthod giv blow ( y z y (46 ( [ x y] [ w y] [ w x] whr a z [ u] uu 4 ( z ( [ y z] [ x y] [ x z] (47 x z lν z (48 l uu u5 u6 uu4 uu uu u u ( ( [ ] [ ] 4 4 zu x y a y x (49 w y ar giv i (7 ν is giv by (4 x 0 is th iitial approximatio ad providd that th domiators o (46-(49 ar ot qual to zro. Thorm Lt α D b a simpl root o a suicitly dirtiabl uctio : D i a op itrval D. I x 0 is suicitly clos to α th th ordr o covrgc o th w drivativ-r mthod did by (48 is sixt. Proo Usig appropriat xprssios i th proo o th thorm ad substitutig thm ito (48 w obtai th asymptotic rror costat 5 ( c c c cc c c c c c c cc4 cc c cc c ( ccc ccc 4cc 0ccc 4 cc ccc 5ccc 4cc 4 4 7ccc 4cc 4cc cc c cc c 6 60 (50 Th xprssio (50 stablishs th asymptotic rror costat or th sixt ordr o covrgc or th w drivativ-r mthod did by (48... Mthod : (Liu Aothr thr-stp ighth-ordr drivativ-r mthod rctly itroducd i[6] ad o o thm was costructd by usig th two-stp ourth-ordr mthod prstd by Liu t al.[0] ad th third-stp was dvlopd to achiv th ighth-ordr. It appars that th ormula o th ighth-ordr

5 6 R. Thukral: Nw Sixtth-Ordr Drivativ-Fr Mthods or Solvig Noliar Equatios drivativ-r mthod basd o Liu t al. mthod giv by (. i[6] has a typo rror. Hr also th rror actually occurs i th wight uctio o th third stp o th ighth-ordr drivativ-r mthod. W obsrv that th third stp giv by (5 is corrctd ad simpliid i th ollowig sixtth-ordr mthod giv blow ( y ( [ x y] [ w y] [ w x] z y ( [ x ] y (5 a z [ u ] u 4 uu 4 (5 ( z ( [ y z] [ x y] [ x z] z lν ( z (5 whr l uu uu 4 u5 u6 uu 4 uu uu 4 ( u u (54 zu [ xy ] [ a y] x 4 w y is giv i (7 ν is giv by (4 x 0 is th iitial approximatio ad providd that th domiators o (5-(54 ar ot qual to zro. It is wll stablishd that th ighth-ordr drivativ-r mthod giv by (5 covrgs to ordr ight s[6]. Howvr w progrss to stablish th sixtth-ordr mthod giv by (5. Thorm Lt α D b a simpl root o a suicitly dirtiabl uctio : D i a op itrval D. I x 0 is suicitly clos to α th th ordr o covrgc o th w drivativ-r mthod did by (5 is sixt. Proo Usig appropriat xprssios i th proo o th thorm ad substitutig thm ito (5 w obtai th asymptotic rror costat 5 ( ( ( c c c cc cc c c c c c 4 ( ccc ccc 4cc 0ccc 4 cc ccc 5ccc 4cc 4 4 4ccc 4cc 8cc cc c4cc 5 c. c c c c c cc c cc 4c (55 Th xprssio (55 stablishs th asymptotic rror costat or th sixtth-ordr o covrgc or th w drivativ-r mthod did by (5.. Kim ad Gum Sixtth Ordr Mthods For th purpos o compariso w cosidr two sixtth-ordr mthods prstd rctly i[56]. Sic ths mthods ar wll stablishd w shall stat th sstial xprssios usd i ordr to calculat th approximat solutio o th giv oliar quatios ad thus compar th ctivss o th w sixt ordr drivativ-r mthods. Th irst o th Kim t al. mthod[6] is giv as x whr ( ( x ( y ( x ( z ( ( ( s ( ( x y x z y K u s z H u v w x x s W u v w t K ( u 5β βu ( u β ( β u ( 9 4 ( σ u u w H ( u v w v σ w u ( σ vw W ( u v w t v w t ( σ vw G( u w ( y ( z u v ( x ( y ( z ( s w t. ( x ( z Thr ar may vrsios o ( (56 (57 (58 (59 (60 G u w s[6] or th purpos o this papr w shall cosidr th ollowig Gu ( w uw { 6 u u ( 4 β (6 a b uφ 4σ} φw a φ β 66β 6 (6 b φ u ( σ σ 9 4σ 6 (6 β σ. (64 Aothr sixtth-ordr mthods itroducd by Kim t al. is giv i[5] ad is xprssd as x whr ( ( x ( y ( x ( z ( ( ( s ( ( x y x z y K u s z H u v w x x s W u v w t K ( u 5β βu ( u β ( β u ( 9 4 ( σ u u w H ( u v w v σ w (65 (66 (67

6 Amrica Joural o Computatioal ad Applid Mathmatics 0 (: -8 7 u ( σ vw W ( u v w t (68 v w t ( σ vw G( u v w ( y ( z u v (69 x y ( z ( s w t ( x ( z As bor w shall cosidr th particular wight uctio G( u w usd i (68 which is giv as Gu v w uv uw w ( 4 u v v w w (70 4 β σ. (7 whr x 0 is th iitial approximatio ad providd that th domiators o (65 ar ot qual to zro. I additio urthr dtails ad xprssios o th rctly itroducd sixtth-ordr mthod ar giv[5]. 4. Applicatio o th Nw Sixtth- Ordr Drivativ-Fr Itrativ Mthods Th prst sixtth-ordr mthods giv by (7 (48 (5 ar mployd to solv oliar quatios ad compar with th Kim t al typ ad typ mthods (56 ad (65. To dmostrat th prormac o th w sixtth-ordr mthods w us t particular oliar quatios. W shall dtrmi th cosistcy ad stability o rsults by xamiig th covrgc o th w drivativ-r itrativ mthods. Th idigs ar gralisd by illustratig th ctivss o th sixtth-ordr mthods or dtrmiig th simpl root o a oliar quatio. Cosqutly w shall giv stimats o th approximat solutio producd by th sixtth-ordr mthods ad list th rrors obtaid by ach o th mthods. Th umrical computatios listd i th tabls wr prormd o a algbraic systm calld Mapl. I act th rrors displayd ar o absolut valu ad isigiicat approximatios by th various mthods hav b omittd i th ollowig tabls. Rmark Th w our-stp drivativ-r mthods rquir iv uctio valuatios ad hav th ordr o covrgc sixt. Thror ths w mthods ar o optimal ordr ad support th Kug-Traub cojctur[9]. To dtrmi th icicy idx o ths w drivativ-r mthods w shall us th diitio. Hc th icicy idx o th sixtth-ordr drivativ-r mthods giv is which ar bttr tha th othr two ad thr poit drivativ-r mthods giv i[ ]. Rmark Th tst uctios ad thir xact root α ar displayd i tabl. Th dirc btw th root α ad th approximatio x or tst uctios with iitial approximatio x 0 ar displayd i Tabl. I act x is calculatd by usig th sam total umbr o uctio valuatios (TNFE or all mthods. I th calculatios 5 TNFE ar usd by ach mthod. Furthrmor th computatioal ordr o covrgc (COC is displayd i Tabl. Tabl. Tst uctios ad thir roots Fuctios Roots x xp x si x l x α 0 x x α 0 x x x x xp x α 4 ( x ( x xp si ( x x xp cos x α 0 ( 5 ( x si( x x α x xp x cos x α x l x x x α ( x x x x α x cos x 5 x α ( x ( x x α 0 0 si Tabl. Compariso o various our-poit itrativ mthods Mthods (7 (48 (5 (56 (65 ( x x x x x x x x x x x x x x x x x x x x

7 8 R. Thukral: Nw Sixtth-Ordr Drivativ-Fr Mthods or Solvig Noliar Equatios Tabl. COC o various itrativ mthods Mthods (7 (48 (5 (56 (65 ( x x ( x x ( x x ( x x ( x x ( x x ( x x ( x x ( x x ( x x Coclusios I this study w hav costructd w sixtth-ordr drivativ-r mthods or solvig oliar quatios. Covrgc aalysis provs that ths w drivativ-r mthods prsrv thir ordr o covrgc. From th rsults i th tabls basd o a umbr o umrical xprimts it ca b cocludd that th covrgc o th w multipoit mthod o th sixtth-ordr is rmarkably ast. Furthrmor w hav oud that th w our-poit mthods (7 producd idtical rsults with all thr typs o wight uctio giv by (-(4 usd i th scod stp o (7. Atr a xtsiv xprimtatio w wr ot abl to dsigat a spciic itrativ mthod which always producs th bst rsults or all tstd oliar quatios. Thr ar two major advatags o th highr ordr drivativ-r mthods. Firstly w do ot hav to valuat th drivativ o th uctios; thror thy ar spcially icit whr th computatioal cost o th drivativ is xpsiv ad scodly w hav stablishd a highr ordr o covrgc mthod tha th xistig drivativ-r mthods[ ]. W hav xamid th ctivss o th w drivativ-r mthods by showig th accuracy o th simpl root o a oliar quatio. Th mai purpos o dmostratig th w sixtth-ordr drivativ-r mthods or may dirt typs o oliar quatios was purly to illustrat th accuracy o th approximat solutio th stability o th covrgc th cosistcy o th rsults ad to dtrmi th icicy o th w itrativ mthods. Fially w coclud that th w our-poit mthods may b cosidrd a vry good altrativ to th classical mthods. REFERENCES [] S. D. Cot Carl d Boor Elmtary Numrical Aalysis A Algorithmic Approach McGraw-Hill 98 [] A. Cordro J. L. Huso E. Martiz J. R. Torrgrosa Sts typ mthods or solvig oliar quatios Comput. Appl. Math. (00 doi:0.06/jcam [] M. Dhgha M. Hajaria Som drivativ r quadratic ad cubic covrgc itrativ ormulas or solvig oliar quatios J. Comput. Appl. Math. 9 ( [4] W. Gautschi Numrical Aalysis: a Itroductio Birkhausr 997. [5] Y. H. Gum Y. I. Kim A amily o optimal sixtth-ordr multipoit mthods with a liar ractio plus a trivariat polyomial as th ourth-stp wightig uctio Comp. Math Appl. 6 ( [6] Y. H. Gum Y. I. Kim A biparamtric amily o optimally covrgt sixtth-ordr multipoit mthods with thir ourth-stp wightig uctio as a sum o a ratioal ad a gric two-variabl uctio Comput. Appl. Math. 5 ( [7] S. K. Khattri R. P. Agarwal Drivativ-r optimal itrativ mthods Comput. Mt. Appl. Math. 0 ( [8] S. K. Khattri I. K. Argyros Sixth ordr drivativ r amily o itrativ mthods Appl. Math. Comput. (0 doi:0.06/jamc [9] H. Kug J. F. Traub Optimal ordr o o-poit ad multipoit itratio J. Assoc. Comput. Math. ( [0] Z. Liu Q. Zhg P. Zho A variat o Sts s mthod o ourth-ordr covrgc ad its applicatios Appl. Math. Comput. 6 ( [] Y. Pg H. Fg Q. Li X. Zhag A ourth-ordr drivativ-r algorithm or oliar quatios J. Comput. Appl. Math. 5 ( [] M. S. Ptkovic S Ilic J. Dzuic Drivativ r two-poit mthods with ad without mmory or solvig oliar quatios Appl. Math. Comput. 7 ( [] F. Solymai S. K. Vaai Optimal Sts-typ mthods with ighth ordr covrgc Comp. Math. Appl. 6 ( [4] F. Solymai S. K. Vaai M. J. Paghalh A class o thr-stp drivativ-r root solvrs with optimal covrgc ordr ISRN Appl. Math. I prss (0. [5] J. F. Sts Rmark o itratio Skad. Aktuar Tidsr. 6 ( [6] R. Thukral Eighth-ordr itrativ mthods without drivativs or solvig oliar quatios J. Appl. Math. (0 -. [7] R. Thukral Nw amily o highr ordr Sts-typ mthods or solvig oliar quatios J. Mod. Mth. Numr. Math. (0-0. [8] J. F. Traub Itrativ Mthods or solutio o quatios Chlsa publishig compay Nw York 977. [9] S. Wrakoo T. G. I. Frado A variat o Nwto s mthod with acclratd third-ordr covrgc Appl. Math. Ltt. ( [0] Q. Zhg J. Li F. Huag A optimal Sts-typ amily or solvig oliar quatios Appl. Math. Comput. 7 (