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1 Docunt downloadd fro: This papr ust b citd as: Cordro Barbro, A.; Fardi, M.; Ghasi, M.; Torrgrosa Sánchz, JR. (. Acclratd itrativ thods for finding solutions of nonlinar quations and thir dynaical bhavior. Calcolo. 5(:7-3. doi:.7/s Th final publication is availabl at Copyright Springr Vrlag (Grany Additional Inforation

2 Nona anuscript No. (will b insrtd by th ditor Acclratd itrativ thods for finding solutions of nonlinar quations and thir dynaical bhavior A. Cordro M. Fardi M. Ghasi J.R. Torrgrosa Rcivd: dat / Accptd: dat Abstract In this papr, w prsnt a faily of optial, in th sns of Kung-Traub s conjctur, itrativ thods for solving nonlinar quations with ighth-ordr convrgnc. Our thods ar basd on Chun s fourth-ordr thod. W us th Ostrowski s fficincy indx and svral nurical tsts in ordr to copar th nw thods with othr known ighth-ordr ons. W also xtnd this coparison to th dynaical study of th diffrnt thods. Kywords Convrgnc ordr Efficincy indx Basin of attraction Priodic orbit Dynaical plan Nonlinar quations Itrativ thods Mathatics Subjct Classification ( 5B99 5H5 5P99 37F This rsarch was supportd by Ministrio d Cincia y Tcnología MTM-83-C- and by th Cntr of Excllnc for Mathatics, Univrsity of Shahrkord, Iran. A. Cordro Instituto d Matática Multidisciplinar, Univrsitat Politècnica d València, Spain E-ail: acordro@at.upv.s M. Fardi Dpartnt of Mathatics, Islaic Azad Univrsity, Najafabad Branch, Najafabad, Iran. E-ail: fardiojtaba@yahoo.co M. Ghasi Dpartnt of Applid Mathatics, Faculty of Mathatical Scinc, Shahrkord Univrsity, Shahrkord, Iran E-ail: h ghasi@yahoo.co J.R. Torrgrosa Instituto d Matática Multidisciplinar, Univrsitat Politècnica d València, Spain E-ail: jrtorr@at.upv.s

3 A. Cordro t al. Introduction On of th ost iportant probls in nurical analysis is solving nonlinar quations. In rcnt yars, uch attntion hav bn givn to dvlop a nubr of itrativ thods for solving th nonlinar quations, paying attntion to th ffctivnss of th schs, usually analyzd by ans of th fficincy indx introducd by Ostrowski in []. This indx is dfind as I = p /d, whr p is th ordr of convrgnc and d is th total nubr of functional valuations pr stp. In this sns, Kung and Traub conjcturd in [8] that a ultistp thod without ory prforing n + functional valuations pr itration can hav at ost convrgnc ordr n, in which cas it is said to b optial. Rcntly, diffrnt optial itrativ thods of ordr of convrgnc ight hav bn publishd. For instanc, optial ighth ordr thods can b found in [,,3,9,,,], all of th with fficincy indx.8. So of th will b usd in th nurical and dynaical sctions, in ordr to copar th with th nw schs introducd in this papr. In particular, Liu and Wang in [9] prsnt a thr-stp itrativ sch whos xprssion is: y = x f(x f (x, z = y f(x f(x 9f(y [ x + = z f(z f (x f (x, 8 f(x f(y + ( + 3 3f(y +β f(z + f(x +β f(z that will b dnotd by LW. Morovr, Cordro t. al. dsignd in [] th optial ighth-ordr thod y = x f(x f (x, z = x f(x f(x f(x f(y f (x, ( [ ] x + = z f(z f(y f(x f(y f (x f(y 3f(z f(x f(z f(y f(y, basd on Ostrowski s fourth-ordr sch, that will b dnotd by CTV. Finally, Solyani t. al. in [] dsign th following sch by using wight functions: y = x f(x f (x, z = y f(x f(x 5 x + = z dnotd by SK. 3 f[z,x ] f (x ( f(y f(x f (x, [ + 3 ( f(y f (x ( f(z f(x + f (x ], f(y + ( f(z f(y (3 ](, Now, lt us rcall so basic concpts on coplx dynaics (s [] and [7], for xapl. Givn a rational function R : Ĉ Ĉ, whr Ĉ is th Riann

4 Mthods for nonlinar quations and thir dynaical bhavior 3 sphr, th orbit of a point z Ĉ is dfind as: z, R (z, R (z,..., R n (z,... W ar intrstd in th study of th asyptotic bhavior of th orbits dpnding on th initial condition z, that is, w ar going to analyz th phas plan of th ap R dfind by th diffrnt itrativ thods. To obtain ths phas spacs, th first of all is to classify th starting points fro th asyptotic bhavior of thir orbits. A z Ĉ is calld a fixd point if it satisfis R (z = z. A priodic point z of priod p > is a point such that R p (z = z and R k (z z, k < p. A pr-priodic point is a point z that is not priodic but thr xists a k > such that R k (z is priodic. A critical point z is a point whr th drivativ of rational function vanishs, R (z =. On th othr hand, a fixd point z is calld attractor if R (z <, suprattractor if R (z =, rpulsor if R (z > and parabolic if R (z =. Th basin of attraction of an attractor α is dfind as th st of pr-iags of any ordr: A (α = {z Ĉ : Rn (z α, n }. Th st of points z Ĉ such that thir failis {Rn (z} n N ar noral in so nighborhood U (z, is th Fatou st, F (R, that is, th Fatou st is coposd by th st of points whos orbits tnd to an attractor (fixd point, priodic orbit or infinity. Its coplnt in Ĉ is th Julia st, J (R ; thrfor, th Julia st includs all rplling fixd points, priodic orbits and thir pr-iags. That ans that th basin of attraction of any fixd point blongs to th Fatou st. On th contrary, th boundaris of th basins of attraction blong to th Julia st. In this papr w prsnt an optial faily of itrativ thods which ar fr of scond drivativs and ar of ighth-ordr of convrgnc. Th rst of this papr is organizd as follows. Th proposd thods ar dscribd in Sction and th convrgnc analysis is carrid out to stablish th ordr of convrgnc. In Sction 3, so nurical xapls confir th thortical rsults and allow us to copar th proposd thods with othr known thods ntiond in th Introduction. Sction is dvotd to th coplx dynaical analysis of th dsignd thods on quadratic and cubic polynoials. In Sction 5, w nd this papr with so conclusions. Dvlopnt of th ighth-ordr faily and convrgnc analysis In this sction, w driv a faily of ighth-ordr thods using an approxiation for th last drivativ. Lt us considr th faily of fourth-ordr thods

5 A. Cordro t al. proposd by Chun in [3] y = x f(x, f (x z = y f 3 (x f 3 (x f (x f(y +αf(x f (y α f 3 (y, f (x whr α R. If w copos this sch with Nwton s thod, it is known that th rsulting algorith is of ighth-ordr of convrgnc, but it is not optial, sinc it uss two additional functional valuations. In ordr to iprov th fficincy, w ar going to approxiat f (z trying to hold th ordr of convrgnc. By using th Taylor xpansion, and f (z can b approxiatd by + f (y (z y + f (y (z y, ( f (z f (y + f (y (z y. (5 In ordr to avoid th coputation of th scond drivativ, w can xprss f (y as follows f (y f[z, x, x ] = (f[z, x ] f (x z x, ( whr f[, ] dnots th dividd diffrnc of first ordr. Fro (, (5 and (, w hav f (z f[z, y ] + f[z, x, x ](z y. It can b provd that by using this approxiation, th coposd sch only rachs svnth-ordr of convrgnc, for any valu of α. So, w propos to us a wight function to attain th optial ordr. W considr th following thr-stp itration sch y = x f(x, f (x f z = y 3 (x f 3 (x f (x +αf(x f (y α f 3 (y, (7 f (x x + = z Af(x +B Cf(x +Df(z f[z,,y ]+f[z,x,x ](z y whr A, B, C, D and α ar paratrs to b dtrind such that th itrativ thod dfind by (7 has th ordr of convrgnc ight. Thor Assu that function f : D R R is sufficintly diffrntiabl and f has a sipl zro x D. If th initial point x is sufficintly clos to x D, thn th thods dfind by (7 convrg to x with ighth-ordr undr th conditions A = C, C B + D = and α =, and with rror quation + = c ( c 3 3c 3 + c c 3 c 8 + O( 9.

6 Mthods for nonlinar quations and thir dynaical bhavior 5 Proof: Lt = x x b th rror at th th itration and c k = f (k (x k! f (x, k =, 3,.... By using Taylor xpansions, w hav: f(x = f (x [ + c + c c + c c +c c 8 8 ] + O( 9, f (x = f (x [ + c + 3c 3 + c 3 + 5c 5 + c 5 +7c 7 + 8c c 9 8 ] + O( 9. Now, fro (8 and (9, w hav (8 (9 y x = c + (c 3 c 3 + (3c 3c c 3 (c 3 c c +(c 5 c c c 3 + c 3 c 8c 5 ( Fro (, w gt +(7c c 3 + 8c c 3c c c c 3 + 5c 5c 3 c 3 + c 5 ] + O( 7, = f (x [c + (c 3 c 3 + (3c 7c c 3 + 5c 3 +( c 3 + c 3 c c c + c 5 c 5 ( +(7c c 3 + 3c c 3c c 5 + 5c + 37c c 3 73c 3 c 3 + 8c 5 ] + O( 7. Cobining (8, (9, ( and (, w hav z x = ( ( + αc 3 c c 3 (( + α + α c ( + 3αc c 3 + c 3 + c c 5 ( Fro (, w gt + ( ( 5 + α + 7α c 5 ( 5 + α + 8α c 3 c 3 +( + 3αc c 7c 3 c + 3c ( ( + 8αc 3 c 5 + O( 7. = f (x [ ( ( + αc 3 c c 3 ( (( + α + α c ( + 3αc c 3 + c 3 + c c 5 (3 + ( ( 5 + α + 7α c 5 ( 5 + α + 8α c 3 c 3 +( + 3αc c 7c 3 c + 3c ( ( + 8αc 3 c 5 ] + O( 7. By using th Taylor xpansions (8 and (3, w gt Af(x + B Cf(x + D = A C + (BC ADc ( ( + αc c 3 3 C (BC AD (( 5 + α + α c 3(3 + αc c 3 + c 3 + c c C Morovr, fro (8, (9, (, (, ( and (3 w obtain f[z, y ] + f[z, x, x ](z y = ( ( + αc 3 c c 3 (( + a + a C ( + 3aC C 3 + C 3 + C C 5 + ( ( 5 + α + 7α c 5 ( 5 + α + 8α c 3 c 3 + O( 5. +( + 3αc c 7c 3 c + 3c ( ( + 8αc 3 c 5 + O( 7,

7 A. Cordro t al. and, thn th rror quation is + = (A Cc ( ( + αc c 3 C + (A C (( + α + α c ( + 3αc c 3 + c 3 + c c 5 C (A C [ ( 5 + α + 7α c 5 ( 5 + α + 8α c 3 C c 3 +( + 3αc ( ] c 7c 3 c + 3c ( + 8αc 3 c 5 + O( 7, which shows that th convrgnc ordr of any thod of th faily (7 is at lst svn if A = C, and th rror quation is ( ( + = c ( + αc c 3 ( + α(b Dc + ( B + C + Dc 3 7 C + c ( (( ( + α 9 + α + α B + C + αc ( 9 + α + α D C C + ( ( 3 + α + α B + ( + 3α + α C + ( 3 + α + α D c c 3 +(( + 3αB 8(3 + αc ( + 3αDc c 3 +( B + C + Dc ( + α(b 3C Dc 3 c +( B + 7C + Dc c 3 c 8 + O( 9. Finally, if α = and B + C + D =, thn th rror quation is + = c c 3 ( 3c 3 + c c 3 c 8 + O( 9, and th proof is copltd. Thrfor, th thods of th nw faily ar optial in th sns of Kung- Traub s conjctur, so thy hav fficincy indics 8 =.8, as wll as othr ighth-ordr schs dscribd in [,,3,9,,,]. In what follows, w giv so concrt itrativ thods of (7, that w ar going to us in th following sction. FA. If A = C =, D =, B = 3, y = x f(x, f (x z = y x + = z f(x +3 f(x +f(z FA. If A = C =, D =, B =, f 3 (x, f 3 (x f (x f(y f(x f (y f 3 (y f (x f[z.,y ]+f[z,x,x ](z y y = x f(x, f (x z = y x + = z f(x + f 3 (x, f 3 (x f (x f(x f (y f 3 (y f (x f(x f(z f[z.,y ]+f[z,x,x ](z y

8 Mthods for nonlinar quations and thir dynaical bhavior 7 FA3. If A = C =, D = 3, B =, y = x f(x, f (x z = y x + = z f(x f 3 (x, f 3 (x f (x f(x f (y f 3 (y f (x f(x 3f(z f[z.,y ]+f[z,x,x ](z y 3 Nurical rsults W prsnt so xapls to illustrat th fficincy of th proposd thods coparing th with th ighth-ordr thods dscribd in th introduction, spcifically, w copar our thods with LW sch (xprssion ( with β = β =, CTV thod dfind in ( and SK thod showd in (3. Th tst functions usd ar: f (x = sin (x x + ; x.9853; f (x = sin(x x ; x ; f 3 (x = cos(x x; x ; f (x = x x ; x.79385; f 5 (x = x x 3x + ; x ; f, x = LW CTV SK F A F A F A3 f(x n x n+ x n itr ρ f, x = LW CTV SK F A F A F A3 f(x n x n+ x n itr ρ f 3, x = LW CTV SK F A F A F A3 f(x n x n+ x n itr ρ f, x = LW CTV SK F A F A F A3 f(x n x n+ x n itr ρ f 5, x = LW CTV SK F A F A F A3 f(x n x n+ x n itr ρ Tabl : Coparison of various itrativ thods.

9 8 A. Cordro t al. For ach thod and ach tst function w show in Tabl th valu of function in th last itration, th distanc btwn th two last itrations, coputd with 3 significant digits, th nubr of itrations such that x n+ x n 8 or f(x n+ 8 and th coputational ordr of convrgnc ρ, approxiatd using th forula (s [5] ρ ln( x + x / x x ln( x x / x x. All coputations wr don by using MAPLE. Dynaical analysis In this sction, w study th dynaics of fixd point oprators whn thy ar applid on quadratic and cubic polynoials. As w will obsrv in th following, th dynaics of th proposd thods is vry rich, svral priodic orbits appar with infinit pr-iags, Julia st is connctd and th connctd coponnts of Fatou st ar also infinit. In our calculations, w usually considr th rgion [, ] [, ] of th coplx plan, with points and w apply th corrsponding itrativ thod starting in vry x in this ara. If th squnc gnratd by itrativ thod rachs a zro x of th polynoial with a tolranc x k x < and a axiu of itrations, w dcid that x is in th basin of attraction of ths zro and w paint this point in a color prviously slctd for this root. In th sa basin of attraction, th nubr of itrations ndd to achiv th solution is showd in darkr or brightr colors (th lss itrations, th brightr color. Black color dnots lack of convrgnc to any of th roots (with th axiu of itrations stablishd or convrgnc to th infinity. Whn w apply th fixd point oprator of th nw thods (s Figurs a to for FA, FA and FA3, rspctivly on th scond-dgr polynoial p(z = z, a rational function is obtaind, with polynoials of dgr 3 and 9 in th nurator and dnoinator, rspctivly. Analyzing th fixd points, w found that th roots of p(z ar, obviously, suprattractiv and thr xist othr 8 rpulsiv strang fixd points. That is, all strang fixd points ar on th Julia st. Obsrving Figur, w not that th thod FA nvr fails, anwhil two priodic orbits of priod appar in th dynaical plan of FA (th orbits ar {.9885i,.938i} and {.9885i,.938i} and also of FA3 (th first orbit is {.9i,.537i} and th scond {.9i,.537i}. In cas of doubl roots, all th dynaical plans ar th sa as in Nwton s cas, as can b sn in Figur f. Th dynaical plans of th nw thods for cubic polynoials ar showd in Figur. W obsrv that th dynaical bhavior of th thr schs is siilar. W found again that th roots of th polynoial ar th only suprattractiv fixd points. If w ar nar th origin (s Figur a, th connctd coponnts of th basin of attraction containing th roots s to xpand

10 Mthods for nonlinar quations and thir dynaical bhavior 9 z= i.9885 z=.5 9+i II{z} II{z} IR{z}.5.5 IR{z} (a Orbit in FA on z (b Orbit in FA on z z= 3. 8+i.537 z=.93 8+i II{z} II{z} IR{z} IR{z} (c Orbit in FA3 on z (d Orbit in FA3 on z II{z} II{z} IR{z} IR{z} ( FA on z (f Nw thods on z Fig. : Dynaical plans of th nw thods on quadratic polynoials filling all th plan. Howvr, w can s in Figurs b to d that th structur around th origin is infinitly rplicatd. Indd, th black rgions btwn th copis contain thr priodic orbits of priod in ach nw thod. Two of th ar on th diagonals of th plan (conjugatd btwn th and th third on is on th ral axis. Th orbits of FA ar: {.89.79i, i, i, i} { i, i, i, i} { 9.8,.88,.9795,.359} In Figur 3 w can s two of ths priodic orbits.

11 A. Cordro t al..5.5 II{z} II{z} IR{z} IR{z} (a Dtail of FA on z 3 (b FA on z 3 II{z} II{z} IR{z} IR{z} (c FA on z 3 (d FA3 on z 3 Fig. : Dynaical plans of th nw thods on cubic polynoials z=.59+i z=.359+i II{z} II{z} IR{z} IR{z} (a Diagonal orbit (b Horizontal orbit Fig. 3: So orbits of priod of FA Finally, w can copar th rsults obtaind for th prsntd thods with th sa schs that hav bn usd in nurical tsts. All th thods hav a good bhavior on quadratic polynoials, but in cas of cubic ons, th diffrncs btwn th ar clar. It can b obsrvd (s Figur that CTV is th ost stabl thod on quadratic and cubic polynoials, anwhil LW sch shows a siilar bhavior than th thods proposd in this papr. Howvr, SK procdur has big rgions of convrgnc to th infinity (in black

12 Mthods for nonlinar quations and thir dynaical bhavior in Figurs and f in quadratic and cubic polynoials. In this cas, no priodic orbits appar but starting points in ths black rgions lad to infinity aftr fw itrations II{z} II{z} IR{z} IR{z} (a LW sch on z (b LW sch on z II{z} II{z} IR{z} IR{z} (c CTV sch on z (d CTV sch on z II{z} II{z} IR{z} IR{z} ( SK sch on z (f SK sch on z 3 Fig. : Dynaical plans of known thods on polynoials 5 Conclusions W hav dsignd and studid a faily of optial itrativ thod (in th sns of Kung-Traub s conjctur of ighth-ordr. W hav tstd so lnts of th faily and copard th with othr known schs. Finally,

13 A. Cordro t al. a dynaical analysis of th particular thods has bn ad on quadratic and cubic polynoials, showing th dynaical richnss of th faily. Acknowldgnts Th authors would lik to thank Mr. Francisco Chicharro for his valuabl hlp for drawing th dynaical plans and to th anonyous rfrs, whos suggstions hav iprovd th radability of this papr. Rfrncs. W. Bi, H. Rn, Q. Wu, Thr-stp itrativ thods with ighth-ordr convrgnc for solving nonlinar quations, Journal of Coputational and Applid Mathatics, 55, 5- (9. P. Blanchard, Coplx Analytic Dynaics on th Riann Sphr, Bull. of th Arican Mathatical Socity, (, 85- (98 3. C. Chun, So variants of Kings fourth-ordr faily of thods for nonlinar quations, Appl. Math. Coput, 9, 57- (7. A. Cordro, J.L. Huso, E. Martínz, J.R. Torrgrosa, Nw odifications of Potra-Pták s thod with optial fourth and ighth ordr of convrgnc, Journal of Coputational and Applid Mathatics, 3, ( 5. A. Cordro, J.R. Torrgrosa, Variants of Nwton s thod using fifth-ordr quadratur forulas, Applid Mathatics and Coputation, 9, 8-98 (7. A. Cordro, J.R. Torrgrosa, M.P. Vassilva, A faily of odifid Ostrowski s thod with optial ighth ordr of convrgnc, Applid Mathatics Lttrs, (, 8-8 ( 7. A. Douady and J.H.Hubbard, On th dynaics of polynoials-lik appings, Ann. Sci. Ec. Nor. Sup. (Paris, 8, ( H.T. Kung, J.F. Traub, Optial ordr of on-point and ulti-point itration, Journal of th Association for Coputing Machinry,, 3-5 (97 9. L. Liu, X. Wang Eighth-ordr thods with high fficincy indx for solving nonlinar quations, Applid Mathatics and Coputation, 5, (. A.M. Ostrowski, Solutions of quations and systs of quations. Acadic Prss, Nw York-London (9. J.R. Shara, R. Shara, A faily of odifid Ostrowski s thods with acclratd ighth ordr convrgnc, Nurical Algorits, 5, 5-58 (. F. Solyani, S. Karii Banani, M. Khan, M. Sharifi, So odifications of King s faily with optial ighth ordr of convrgnc, Mathatical and Coputr Modlling, 55, ( 3. R. Thukral, M.S. Ptkovic, A faily of thr-point thods of optial ordr for solving nonlinar quations, Journal of Coputational and Applid Mathatics, 33, 78-8 (