NEW NEWTON-TYPE METHOD WITH k -ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION R. Thukral Padé Research Cetre, 39 Deaswood Hill, Leeds West Yorkshire, LS7 JS, ENGLAND ABSTRACT The objective of this paper is to defie a ew Newto-type method for fidig simple root of a polyomial. It is proved that the ew oe-poit method has the covergece order of k requirig fuctio evaluatios per iteratio, where k is the umber of terms i the geeratig series. Kug ad Traub cojectured that the multipoit iteratio methods, without memory based o evaluatios, could achieve maximum covergece order, but the ew method produces covergece order of k, which is better tha the expected maximum covergece order of eight. Therefore, we show that the cojecture fails for a particular set of polyomial equatios. We will demostrate that the ew method is very simple to costruct. Keywords: Newto-type method; Polyomial equatio; Kug-Traub s cojecture; Efficiecy idex; Optimal order of covergece. Subject Classificatios: AMS (MOS): 6H0. INTRODUCTION I this paper, we preset a ew oe-poit k of a polyomial equatio. Let -order iterative method to fid a simple root p( x ) a a x a x a x a x be a polyomial of 3 q 0 3 degree q ad a i are real costats. Fidig zeros of a polyomial has bee iterest i theoretical ad may areas of applied mathematics. It is well established that may higher order multi-poit variats of the Newto-type method have bee developed based o the Kug ad Traub cojecture [4]. Here we preset a ew iterative method which has a better efficiecy idex tha the classical Newto method [3,,6,7,0]. This paper is actually a cotiuatio form the previous studies [8,9], hece the ew oe-poit method is applied to higher order polyomial equatio. For the purpose of this paper, we costruct a ew Newto-type iterative method of k - order for fidig simple root of polyomial equatios. The ew oe-poit method preseted i this paper oly uses evaluatios of the fuctio per iteratio. Kug ad Traub cojectured that the multipoit iteratio methods, without memory based o evaluatios, could achieve optimal covergece order. I fact, we have obtaied a higher order of covergece tha the maximum order of covergece suggested by Kug ad Traub cojecture [4]. We demostrate that the Kug ad Traub cojecture fails for a particular case. Progressive Academic Publishig, UK Page www.idpublicatios.org
Prelimiaries I order to establish the order of covergece of the iterative method [3,,6,0], some of the defiitios are stated: Defiitio Let f x be a real fuctio with a simple root ad let real umbers that coverge towards. The order of covergece p is give by x lim 0 p x x be a sequece of where p ad is the asymptotic error costat (AEC). Let e x be the error i the th iteratio, the the relatio p p e e e, () is the error equatio. If the error equatio exists, the p is the order of covergece of the iterative method, [3,,6,0]. Defiitio Let be the umber of fuctio evaluatios of the iterative method. The efficiecy of the iterative method is measured by the cocept of efficiecy idex ad defied as E, p p (3) where p is the order of covergece of the method, [6]. Defiitio 3 (Kug ad Traub cojecture) Let x g x () defie as a iterative fuctio without memory with -evaluatios. The pg p opt, (4) where p opt is the maximum order, [4]. Covergece Aalysis I this sectio we defie a ew class of oe-poit k -order method for fidig simple root of a polyomial equatio. I fact, the ew iterative method is a extesio of the Thukral s quadratic ad cubic methods, give i [8,9]. Hece, we will demostrate that the ew oe-poit method ca be costructed to fid a simple root of ay order of polyomial equatio. The order of covergece the ew iterative method is determied by the k, that is umber of terms i the geeratig series, hece depedig o k we ca costruct ay desired order of covergece. The Method The ew oe-poit k -order Newto-type method is expressed by x x rhk r, t, t, t3, tp () where k,,, 3,,,, 3, ; H r t t t t AEC r t t t t i r i k p p (6) i Progressive Academic Publishig, UK Page www.idpublicatios.org
f x f x f x r, t, t, f x! f x 3! f x f x f x f x t, t, t, iv v vi 3 4 4! f x! f x 6! f x vii viii ix f x f x f x t6, t7, t8, 7! f x 8! f x 9! f x where x is the iitial guess ad provided that deomiators of (7) are ot equal to zero. 0 Now, we shall verify the covergece property of the ew oe-poit k -order iterative method (). Theorem Let D be a simple zero of a sufficietly smooth fuctio f : D for a ope iterval D. If the iitial guess x 0 is sufficietly close to, the the covergece order of the ew oe-poit iterative method defied by () is k. Proof Let be a simple root of e x. f x, i.e. f 0 ad f 0 The Taylor series expasio ad takig ito accout f 0 (7), ad the error is expressed as, we have 3 4 6 7 8 3 4 6 7 8 3 4 6 7 3 4 6 7 8 3 4 6 3 4 6 7 8 f ( x ) f e c e c e c e c e c e c e c e. (8) f ( x ) f c e 3c e 4c e c e 6c e 7c e 8c e. (9) f ( x ) f c 6c e c e 0c e 30c e 4c e 6c e. (0) f ( x ) f 6c 4c e 3 4 60c e 0c e 0c e 336c e. 3 4 6 7 8 () iv f ( x ) f 4c 0c e 3 4 360c e 840c e 680c e. 4 6 7 8 () v f ( x ) f 0c 70c e 3 0c e 670c e. 6 7 8 (3) vi f ( x ) f 70c 0c e 060c e. 6 7 8 (4) vii f ( x ) f 0c 4030c e. where () 7 8 Progressive Academic Publishig, UK Page 3 www.idpublicatios.org
iv v vi f f f f f c, c3, c4, c, c6, f f f f f (6) Dividig (8) by (9), we have f x e 3 ce c c3 e. f x (7) ad f x c c 3c3 e c 4c 9c3 6 c4 e. fx (8) f x 6c3 c4 cc3 e. f x (9) iv f x 4c4 4c cc4 e. fx (0) iv f x 0c 403 c6 cc e. fx () vi f x 70c6 707c7 cc6 e. fx () vii f x 040c7 4030c8 0080 cc7 e. fx (3) viii f x 4030c8 36880c 9 80640 cc8 e. fx (4) Substitutig (9) i (), we obtai f x e 3 e ce c c3 e, f x () Therefore, the asymptotic error costat give by () is expressed as AEC 0 ce. (6) Progressive Academic Publishig, UK Page 4 www.idpublicatios.org
It is well kow that (6) is the asymptotic error costat for the classical Newto method. Therefore, we take error equatio (6) as our ext term of the geeratig series give i (). Furthermore, we obtai a family of higher iterative method by icreasig the terms of summatio series of (6). Hece, we show the asymptotic error costat AEC k for the k -order Newto-type method. I order to obtai the followig terms of the geeratig series, we must substitute the essetial parts i AEC k, thus c t, c3 t, c4 t3, c t4, c6 t, (7) e r. (8) where r ad t i are give by (7). We use this priciple to obtai the followig oe-poit k -order iterative method; Secod-order Polyomial equatio This method has bee recetly preseted i [8] ad we review the basic expasio,. k : Oe-poit third-order iterative method is give by k i x x r AEC i r x r tr i (9) ad the error equatio 3 AEC ce (30). k : Oe-poit fourth-order iterative method is give by x x r tr t r (3) ad the error equatio 3 4 AEC ce. (3) 3. k 3: Oe-poit fifth-order iterative method is give by 3 3 x x r tr t r t r (33) ad the error equatio 4 AEC 3 4 ce. (34) 4. k 4: Oe-poit sixth-order iterative method is give by 3 3 4 4 x x r tr t r t r 4t r (3) ad the error equatio 6 AEC 4 4 ce. (36). k : Oe-poit seveth-order iterative method is give by Progressive Academic Publishig, UK Page www.idpublicatios.org
3 3 4 4 x x r tr t r t r 4t r 4t r (37) ad the error equatio 6 7 AEC 3 ce. (38) 6. k 6 : Oe-poit eighth-order iterative method is give by 3 3 4 4 6 6 x x r tr t r t r 4t r 4t r 3t r (39) ad the error equatio 7 8 AEC 6 49 ce. (40) It is well established that the maximum order of covergece of optimal methods with three fuctios evaluatios is four. As illustrated above we have obtaied order of covergece greater tha four, hece the Kug ad Traub cojecture fails for k 3. To produce the ext oe-poit higher order of covergece with oly three fuctio evaluatios, we use the followig priciple. The process is very simple, usig the coefficiet of the error equatio AEC k as our coefficiet of the ext term of the geeratig series, thus we ca calculate the ext higher order of covergece method. Hece, k -order Newto-type method. AEC k is the error equatio of the Third-order Polyomial equatio This cubic equatio method has bee recetly preseted i [9] ad review some of the results of the method.. k : Oe-poit third-order iterative method is give by x x r tr (4) ad the error equatio AEC c 3 c3 e (4). k : Oe-poit fourth-order iterative method is give by x x r tr t t r (43) ad the error equatio AEC c 4 c c3 e (44) 3. k 3: Oe-poit fifth-order iterative method is give by 3 x x r tr t t r t t t r (4) ad the error equatio AEC 3 4c 4 3c3 c c3 e (46) 4. k 4: Oe-poit sixth-order iterative method is give by Progressive Academic Publishig, UK Page 6 www.idpublicatios.org
3 4 4 x x r tr t t r t t t r 4t 3t t t r (47) ad the error equatio AEC 4 4c 4 6 3c c3 6c c3 e (48). k : Oe-poit seveth-order iterative method is give by 3 4 4 4 x x r tr t t r t t t r 4t 3t t t r 4t 3t t 6t t r (49) ad the error equatio AEC 6c 6 3 4 7 c3 30c c3 c c3 e (0) 6. k 6 : Oe-poit eighth-order iterative method is give by 3 4 4 x x r tr t t r t t t r 4t 3t t t r 4 6 3 4 6 4t 3t t 6t t r 6t t 30t t t t r () ad the error equatio AEC 6 3c 6 4 8 43c c3 330c c3 49c c3 e () Due to Kug ad Traub cojecture the maximum order of covergece of optimal methods with four fuctios evaluatios is eight. As illustrated above whe k 7 we have obtaied order of covergece greater tha eight, hece the Kug ad Traub cojecture fails. Fourth-order Polyomial equatio. k : Oe-poit third-order iterative method is give by x x r tr (3) ad the error equatio AEC c 3 c3 e (4). k : Oe-poit fourth-order iterative method is give by x x r tr t t r () ad the error equatio AEC c 3 4 cc3 c4 e (6) 3. k 3: Oe-poit fifth-order iterative method is give by 3 3 x x r tr t t r t tt t3 r (7) ad the error equatio AEC 3 4c 4 3c3 c c3 6cc4 e (8) 4. k 4: Oe-poit sixth-order iterative method is give by Progressive Academic Publishig, UK Page 7 www.idpublicatios.org
3 3 4 4 x x r tr t t r t tt t3 r 4t 3t t t 6tt 3 r (9) ad the error equatio AEC 4 4c 3 6 8c c3 84c c3 7c3c4 8c c4 e (60). k : Oe-poit seveth-order iterative method is give by 3 3 x x r tr t t r t tt t3 r 4t 4 4 3 3t t t 6tt 3 r 4t 8tt 84t t 7tt3 8t t3 r (6) ad the error equatio AEC 3c 6 3 4 3 7 c3 80c c3 330c c3 4c4 0c c4 7cc3 c4 e (6) 6. k 6 : Oe-poit eighth-order iterative method is give by 3 3 4 4 x x r tr t t r t tt t3 r 4t 3t t t 6tt 3 r 4 8 84 7 3 8 3 3 80 330 4 3 0 3 7 3 t t t t 3 t t t t t r t 6 t 3 t t t 4 t t t 3 t t t t r 6 (63) ad the error equatio AEC 6 49c 7 3 3 4 8 6c c3 990c c3 87c c3 49c c4 4c c4 4c 3c4 49c c3c4 e (64) The maximum order of covergece of optimal methods with five fuctios evaluatios is sixtee. To establish the order of covergece greater tha sixtee we have to display ext fiftee tedious terms, hece we have omitted them ad it is clear that the Kug ad Traub cojecture fails whe k. Fifth-order Polyomial equatio. k : Oe-poit third-order iterative method is give by x x r tr (6) ad the error equatio AEC c 3 c3 e (66). k : Oe-poit fourth-order iterative method is give by x x r tr t t r (67) ad the error equatio AEC c 3 4 cc3 c4 e (68) 3. k 3: Oe-poit fifth-order iterative method is give by Progressive Academic Publishig, UK Page 8 www.idpublicatios.org
3 3 x x r tr t t r t tt t3 r (69) ad the error equatio AEC 3 4c 4 3c3 c c3 6cc4 c e (70) 4. k 4: Oe-poit sixth-order iterative method is give by 3 3 4 4 x x r tr t t r t tt t3 r 4t 3t t t 6tt 3 t4 r (7) ad the error equatio AEC 4 4c 3 6 8c c3 84c c3 7c3c4 8c c4 7cc e (7). k : Oe-poit seveth-order iterative method is give by 3 3 x x r tr t t r t tt t3 r 4t 4 4 3 3t t t 6tt 3 t4 r 4t 8tt 84t t 7tt3 8t t3 7tt 4 r (73) ad the error equatio AEC 3c 6 3 4 3 7 c3 80c c3 330c c3 4c4 0c c4 7cc3 c4 8c3c 36c c e (74) 6. k 6 : Oe-poit eighth-order iterative method is give by x x r tr t t r t tt t3 r 4t 3t t t 6tt 3 t4 r 3 3 4 4 4t 8t t 84t t 7t t 8t t 7t t r 3 3 3 4 6 3 4 3 6 3t t 80t t 330t t 4t3 0t t3 7ttt3 8tt4 36t t4 r (7) ad the error equatio AEC 6 49c 6c c 990c c 87c c 49c c 4c c 7 3 3 4 3 3 3 4 4 3 8 4c 3c4 49c c3c4 9c4c 6c c 90cc 3c e (76) The maximum order of covergece of optimal methods with six fuctios evaluatios is thirty-two. To establish the order of covergece greater tha thirty-two we have to display ext thirty-oe tedious terms, hece we have omitted them ad it is clear that the Kug ad Traub cojecture fails whe k 3. Sixth-order Polyomial equatio. k : Oe-poit third-order iterative method is give by x x r tr (77) ad the error equatio AEC c 3 c3 e (78). k : Oe-poit fourth-order iterative method is give by Progressive Academic Publishig, UK Page 9 www.idpublicatios.org
x x r tr t t r (79) ad the error equatio AEC c 3 4 cc3 c4 e (80) 3. k 3: Oe-poit fifth-order iterative method is give by 3 3 x x r tr t t r t tt t3 r (8) ad the error equatio AEC 3 4c 4 3c3 c c3 6cc4 c e (8) 4. k 4: Oe-poit sixth-order iterative method is give by 3 3 4 4 x x r tr t t r t tt t3 r 4t 3t t t 6tt 3 t4 r (83) ad the error equatio AEC 4 4c 3 6 8c c3 84c c3 7c3c4 8c c4 7cc c6 e (84). k : Oe-poit seveth-order iterative method is give by 3 3 x x r tr t t r t tt t3 r 4t 4 4 3 3t t t 6tt 3 t4 r 4t 8tt 84t t 7tt3 8t t3 7tt 4 t r (8) ad the error equatio AEC 3c 6 3 4 3 7 c3 80c c3 330c c3 4c4 0c c4 7cc3 c4 8c3c 36c c 40cc 6 e (86) 6. k 6 : Oe-poit eighth-order iterative method is give by x x r tr t t r t tt t3 r 4t 3t t t 6tt 3 t4 r 3 3 4 4 4t 8t t 84t t 7t t 8t t 7t t t r 3 3 3 4 6 3 4 3 6 3t t 80t t 330t t 4t3 0t t3 7ttt3 8tt4 36t t4 40tt r (87) ad the error equatio AEC 6 49c 6c c 990c c 87c c 49c c 4c c 7 3 3 4 3 3 3 4 4 3 8 4c 3c4 49c c3c4 9c4c 6c c 90cc 3c 4c 3c6 c c6 e (88) The maximum order of covergece of optimal methods with seve fuctios evaluatios is sixty-four. To establish the order of covergece greater tha sixty-four we have to display ext sixty-three tedious terms, hece we have omitted them ad it is clear that the Kug ad Traub cojecture fails whe k 63. Seveth-order Polyomial equatio. k : Oe-poit third-order iterative method is give by Progressive Academic Publishig, UK Page 0 www.idpublicatios.org
x x r tr (89) ad the error equatio AEC c 3 c3 e (90). k : Oe-poit fourth-order iterative method is give by x x r tr t t r (9) ad the error equatio AEC c 3 4 cc3 c4 e (9) 3. k 3: Oe-poit fifth-order iterative method is give by 3 3 x x r tr t t r t tt t3 r (93) ad the error equatio AEC 3 4c 4 3c3 c c3 6cc4 c e (94) 4. k 4: Oe-poit sixth-order iterative method is give by 3 3 4 4 x x r tr t t r t tt t3 r 4t 3t t t 6tt 3 t4 r (9) ad the error equatio AEC 4 4c 3 6 8c c3 84c c3 7c3c4 8c c4 7cc c6 e (96). k : Oe-poit seveth-order iterative method is give by 3 3 x x r tr t t r t tt t3 r 4t 4 4 3 3t t t 6tt 3 t4 r 4t 8tt 84t t 7tt3 8t t3 7tt 4 t r (97) ad the error equatio AEC 3c 6 3 4 3 7 c3 80c c3 330c c3 4c4 0c c4 7cc3 c4 8c3c 36c c 40cc 6 e (98) 6. k 6 : Oe-poit eighth-order iterative method is give by x x r tr t t r t tt t3 r 4t 3t t t 6tt 3 t4 r 3 3 4 4 4t 8t t 84t t 7t t 8t t 7t t t r 3 3 3 4 6 3 4 3 6 3t t 80t t 330t t 4t3 0t t3 7ttt3 8tt4 36t t4 40tt r (99) ad the error equatio AEC 6 49c 6c c 990c c 87c c 49c c 4c c 7 3 3 4 3 3 3 4 4 3 8 4c 3c4 49c c3c4 9c4c 6c c 90cc 3c 4c 3c6 c c6 80c c7 e (00) Progressive Academic Publishig, UK Page www.idpublicatios.org
The maximum order of covergece of optimal methods with eight fuctios evaluatios is 8. To establish the order of covergece greater tha 8 we have to display ext 7 tedious term, hece we have omitted them ad it is clear that the Kug ad Traub cojecture fails whe k 7. The above procedure may be repeated for higher-order of polyomial equatio. Remark The ew oe-poit iterative method requires fuctio evaluatios ad has the order of covergece k. To determie the efficiecy idex of the ew method, defiitio shall be used. Hece, the efficiecy idex of the ew iterative method give by () is k ad the efficiecy idex of the classical Newto method is. For particular set of values of k, we have show that the efficiecy idex of the ew oe-poit method is much better tha the other similar methods which are based o the Kug ad Traub cojecture. Remark Kug ad Traub cojecture fails whe the total umber of terms required i the geeratig series (6) is give by d d (0) where d is the degree of the polyomial equatio. CONCLUSION I this work, a ew oe-poit k -order Newto-type method has bee preseted. The prime motive for presetig the ew class of iterative method was to exted ad demostrate the use of the Thukral s quadratic ad cubic equatio methods recetly itroduced i [8,9]. Furthermore, demostrate that the Kug ad Traub cojecture fails. It is very simple to evaluate the terms of the geeratig series (6), hece we ca costruct ay desire order of covergece. Recetly we have established the essetial advatages of the ew oe-poit method are: very high computatioal efficiecy; the ew method is ot limited to the Kug ad Traub cojecture; better efficiecy idex tha the classical Newto method; simple oestep iteratio, ot limited to quadratic ad cubic equatios [,,8,9], simple to costruct, efficiet ad robust. Fially, further ivestigatio is eeded to improve ad reduce the umber of terms required for the Kug ad Traub cojecture to fail. REFERENCES [] F. Ahmad, Higher order iterative methods for solvig matrix vector equatios (0), Researchgate, doi: 0.340/RG...9.487. [] D. K. R. Babajee, O the Kug-Traub cojecture for iterative methods for solvig quadratic equatios, Algorithms 06, 9,, doi 0.3390/a90000. [3] W. Gautschi, Numerical Aalysis: a Itroductio, Birkhauser, 997. [4] H. Kug, J. F. Traub, Optimal order of two-poit ad multipoit iteratio, J. Assoc. Comput. Math. (974) 643-6. [] M. S. Petkovic, B. Neta, L. D. Petkovic, J. Dzuic, Multipoit methods for solvig oliear equatios, Elsevier 0. Progressive Academic Publishig, UK Page www.idpublicatios.org
[6] A. M. Ostrowski, Solutios of equatios ad system of equatios, Academic Press, New York, 960. k -order covergece for solvig [7] R. Thukral, New improved Newto method with quadratic equatios, Euro. J. math. Computer. Sci. 3 () (06) 3-8. [8] R. Thukral, Aother Newto-type method with (k+) order covergece for solvig quadratic equatios, J. Adv. Math. Vol (9) (06) 678-68. k -order covergece for [9] R. Thukral, New improved Newto-type method with solvig cubic equatios, Researchgate (06). [0] J. F. Traub, Iterative Methods for solutio of equatios, Chelsea publishig compay, New York 977. Progressive Academic Publishig, UK Page 3 www.idpublicatios.org