A New Family of Multipoint Iterative Methods for Finding Multiple Roots of Nonlinear Equations

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Horatio Riley
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1 Aerica Joural o Coputatioal ad Applied Matheatics 3, 3(3: DOI:.593/j.ajca A New Faily o Multipoit Iterative Methods or Fidig Multiple Roots o Noliear Equatios R. Thukral Padé Research Cetre, 39 Deaswood Hill, Leeds, West Yorkshire, LS7 5JS, Eglad Abstract I this paper, we preset a ew aily o ultipoit iterative ethods or idig ultiple zeros o oliear equatios. Per iteratio the ew ethod requires three evaluatios o uctios ad oe o its irst derivative. We have aalysed ad proved the order o covergece o the ew ethods. Fially, the uerical eaples deostrate that the proposed ethods are superior to the eistig ethods. Keywords Modiied Newto Method, Root-Fidig, Noliear Equatios, Multiple Roots, Order o Covergece, Eiciecy Ide. Itroductio Fidig the roots o oliear equatios is very iportat i uerical aalysis ad has ay applicatios i egieerig ad other applied scieces. I this paper, we cosider iterative ethods to id a ultiple root o u ltip licity, i.e., ( ( α, α ( j ( α =, j =,,..., o a oliear equatio ( =. ad ( where : I is a scalar uctio o a ope iterval I ad it is suicietly sooth i the eighbourhood o α. I recet years, soe odiicatios o the Newto ethod or ultiple roots have bee proposed ad aalysed [-]. However, there are ot ay ethods kow to hadle the case o ultiple roots. Hece we preset two ew iterative ethods o higher order or idig u ltip le zeros o a oliear equatio ad oly use our evaluatios o the uctio per iteratio. I additio, the ew ethods have a better eiciecy ide tha the third ad ourth order ethods give i [,3-]. I view o this act, the ew ethods are sigiicatly better whe copared with the established ethods. Cosequetly, we have oud that the ew ethods are eiciet ad robust. The well-kow Newto s ethod or idig ultiple roots is give by ( = +, * Correspodig author: rthukral@hotail.co.uk (R. Thukral Published olie at Copyright 3 Scietiic & Acadeic Publishig. All Rights Reserved ( which coverges quadratically [,]. For the purpose o this paper, we use ( to costruct ew ultipoit iterative ethods or idig ultiple roots o a oliear equatio.. Costructio o the Methods ad Covergece Aalysis I this sectio we deie two ew iterative ethods. I order to establish the order o covergece o the ew ethods we state the three essetial deiitios. Deiitio Let be a real uctio with a sip le ( α { } root ad let be a sequece o real ubers that coverge towards α. The order o covergece p is give by α + li = ζ p ( α where is the asyptotic error costat ad Deiitio Let k be the uber o uctio evaluatios o the ew ethod. The eiciecy o the ew ethod is easured by the cocept o eiciecy ide [,] ad deied as /k p (4 where p is the order o the ethod. Deiitio 3 Suppose that ad are three successive iteratios closer to the root o (. The the coputatioal order o covergece [] ay be approiated by (3 + ζ p. l COC l, + α ( + α( α ( α( α. (5

2 Aerica Joural o Coputatioal ad Applied Matheatics 3, 3(3: where Recetly Thukral [7] preseted a ourth-order ethod or idig ultiple roots o a oliear equatio, which is give as y, = + abc ( + = y bc, ( a bc ( + ( y where b =, ( c, ( y a, (6 (7 =, is the iitial value ad provided that the deoiator o (6 ad (7 are ot equal to zero. I act, the ew iterative ethods preseted i this paper are the etesio o the above schee. To develop the higher order iterative ethods we use the above two steps ad itroduce the third step with a ew paraeter. First we deie sith-order iterative ethod ad the ollowed by ith-order iterative ethod... The New Sith-order Iterative Method The ew sith-order ethod or idig ultiple root o a oliear equatio is epressed as + abc z, (9 = y bc ( a bc ( + 5 a ( 8a + a bc + ( + 4a( bc + = z st 5 a ( a + a bc ( ( ( where y c b = = y z t z s = =, (, y = (8 ( ad, a is the i itial value. Theore Let : D be a suicietly sooth uctio deied o a ope iterval D, eclosig a ultiple zero o (, with ult iplicity >. The the aily o iterative ethods deied by schee ( has sith-order covergece. Proo Let α be a u ltiple root o ultip licity o a suicietly sooth uctio, ad e = y α, where y is deied i (8. Usig Taylor epasio o ad about, we have where ad e= α ( ( y α ( ( α ( = e + ce + ce +! ( ( α ( e + = + ce + (! c k = ( + k! ( α ( ( + k! ( α ( (3 (4 Moreover by (8, we have ( c e = e = e +. (5 ( ( y α ( ( α The epasio o about is give as ( y = e + ce + ce +.! (6 Sipliyig (6, we get ( ( α c ( y =! (7 c ( + c e + e + c Furtherore, we have c c + c b= e + e + (8 c Sice ro (9 ad (, we have + abc ( z = e bc, (9 + ( a bc ( a ( a + a bc 5 a 8a + a bc + + 4a bc e+ = z st 5 (. ( ( Substitutig appropriate epressios i ( ad ater sipliicatio we obtai the error equatio..

3 7 R. Thukral: A New Faily o Multipoit Iterative Methods or Fidig Multiple Roots o Noliear Equatios e+ = c + a e + ( The error equatio ( establishes the ourth-order covergece o the ew ethod deied by (... The New Fith-Order Iterative Method The ew ith-order ethod or idig ultiple root o a oliear equatio is epressed as (, y = (, (3. (4 where are give i (,, a is the iitial value ad provided that the deoiator o (-(4 are ot equal to zero. Theore Let : D be a suicietly sooth uctio deied o a ope iterval D, eclosig a ultiple zero o (, with ult iplicity >. The the aily o iterative ethods deied by schee (4 has ith-order covergece. Proo Substitutig appropriate epressios i (4 ad ater sipliicatio we obtai the error equatio e+ = c ( + a e + (5 The error equatio (5 establishes the ith-order covergece o the ew ethod deied by (4. 3. The Established Methods For the purpose o copariso, we cosider oe sith-order ethods ad two ith-order ethods preseted recetly i [8,9]. Sice these ethods are well established, we shall state the essetial epressios used i order to calculate the approiate solutio o the give oliear equatios ad thus copare the eectiveess o the ew ultipoit order ethods or idig ultiple roots. I [9], Thukral developed a sith ad ith order o covergece ethods, sice these ethods are well established we state the essetial epressios used i each o the ethods. The s ith-order ethod or idig ultiple roots o a oliear equatio is epressed as ( + abc z = y bc ( a bc + ( + = z st ( bcst,,,, (, y = (6 i/ 3 ( y ( z = i i= ( ( i/ 3 ( y + = z i i= (,(7, (8 where, is the iitial value ad provided that the deoiators o (6 ad (7 are ot equal to zero. The secod o the ethod give i[9] is the ith-order ethod or idig ultiple root o a oliear equatio is epressed as (9,(3, (3 where, is the iitial value ad provided that the deoiators o (- ( are ot equal to zero. I [8] Thukral developed aother ith-order ethod or idig ultiple root o a oliear equatio is epressed as (3, (33 where, is the iitia l value ad provided that the deoiators o (6 ad (7 are ot equal to zero. 4. Applicatio o the New Multipoit Iterative Methods The preset ultipoit ethods give by ( ad (4 are eployed to solve oliear equatios with ultiple roots ad copare with the Thukral ethods, (8, (3 ad (33. To deostrate the perorace o the ew u ltipoit z (, y = i/ 3 y ( z = i i= ( ( = z + z (, y = ( y ( y / + = y ( + z z = ( y (,

4 Aerica Joural o Coputatioal ad Applied Matheatics 3, 3(3: ethods, we use te particular oliear equatios. We deterie the cosistecy ad stability o results by e a iig the covergece o the ew iterative ethods. The idigs are geeralised by illustratig the eectiveess o the ew ultipoit ethods or deteriig the ultiple root o a oliear equatio. Cosequetly, we give estiates o the approiate solutio produced by the ethods cosidered ad list the errors obtaied by each o the ethods. The uerical coputatios listed i the tables were perored o a algebraic syste called Maple. I act, the errors displayed are o absolute value ad isigiicat approiatios by the various ethods have bee oitted i the ollowig tables. The ew ultipoit ethod requires our uctio evaluatios. To deterie the eiciecy ide o the ew ethods, we shall use the deiitio. Hece, the eiciecy Ta ble. Test uctios ad their roots ide o the ultipoit ethod give by ( is whereas the eiciecy ide o the ith-order ethod is give as. We ca see that the eiciecy ide o the ew sith-order ethod is better tha the other siilar ethods. The test uctios ad their eact root α are displayed i table. The dierece betwee the root α ad the approiatio or test uctios with iitial estiate, are displayed i Table. I act, is calculated by usig the sae total uber o uctio evaluatios (TNFE or all ethods. I the calculatios, TNFE are used by each ethod. Furtherore, the coputatioal order o covergece (COC is displayed i Table 3. Fuctios Roots Iitial Est iate 3 = + + = 7 α = =.9 3 ( (( si 3cos 5 = e + + = 4 α = =. = = 9 α = =. 4 = ep + = 95 α = = 3 5 = cos + = 5 α = = 6 ( si 8 = + = 5 α = =.7 = e e + = 3 α = = ( ( ta ( e ( ( l( = = 55 α = = = = α = =.4 = = 3 α = = 6

5 7 R. Thukral: A New Faily o Multipoit Iterative Methods or Fidig Multiple Roots o Noliear Equatios Ta ble. Copariso o various iterative ethods i (33 (3 (4 (8 (.39e-84.39e-8.6e-96.5e-59.84e-87.3e-3.69e-8.43e-6.94e-4.7e-5 3.4e-56.87e-55.57e-7.54e-97.69e e-3.649e-8.37e-48.e-5.468e e e e-4.84e-35.3e e-76.6e-73.e-8.53e-37.7e-53.8e-94.8e-9.76e-9.8e-8.853e-3.44e-5.983e-3.45e-3.33e-46.4e e-9.e e-9.9e-8.66e-48.86e-44.4e e-46.36e e-4 Ta ble 3. COC o various iterative ethods i 3 4 (33 (3 (4 (8 (

6 Aerica Joural o Coputatioal ad Applied Matheatics 3, 3(3: Rearks ad Coclusios I this paper, we have itroduced two ew ultipoit iterative ethods or solvig oliear equatios with u ltip le roots. Covergece aalysis proves that the ew ethods preserve their order o covergece. Sip ly itroducig ew paraeter i ( we have achieved sith-order o covergece. The prie otive o presetig these ew ethods was to establish a higher order o covergece ethod tha the eistig third, ourth ad ith order ethods [-]. We have eaied the eectiveess o the ew iterative ethods by showig the accuracy o the u ltiple roots o several oliear equatios. Ater a etesive eperietatio, it ca be cocluded that the covergece o the tested ultipoit ethods o the sith order is rearkably ast. Furtherore, i ost o the test eaples, epirically we have oud that the best results o the ew iterative ethods ( are obtaied whe a =. The a i purpose o deostratig the ew ethods or dieret types o oliear equatios was purely to illustrate the accuracy o the approiate solutio, the stability o the covergece, the cosistecy o the results ad to deterie the eiciecy o the ew iterative ethod. The ajor advatages o the ew ultipoit iterative ethods are; irst, able to evaluate siple ad ultiple roots, secodly, siple orula whe copared to eistig ethods cotaiig log epressios o (see[3], thirdly, produces better estiate tha the siilar ethods. Fially, we cojecture that ew cocept itroduced i this paper by Thukral ay be applied to trasor the iterative ethods used or idig siple root o oliear equatios. REFERENCES [] W. Gautschi, Nuerical Aalysis: a Itroductio, Birkhauser, 997. [] S. Kuar, V. Kawar, S. Sigh, O soe odiied ailies o ultipoit iterative ethods or ultiple roots o solvig oliear equatios, Appl. Math. Coput. 8 ( [3] S. G. Li, L. Z. Cheg, B. Neta, Soe ourth-order oliear solvers with closed orulae or ultiple roots, Cop. Math. Appl. 58 ( [4] J. Peg, Z. Zeg, A approach idig ultiple roots o oliear equatios or polyoials, Adv. Elec. Co. Web Appl. Co. 49 ( [5] E. Schroder, Uber uedich viele Algorithe zur Aulosug der Gleichuge, Math. A. ( [6] R. Thukral, A ew third-order iterative ethod or solvig oliear equatios with ultiple roots, J. Math. Coput. 6 ( [7] R. Thukral, A ew aily o ourth-order iterative ethods or solvig oliear equatios with ultiple roots, subitted to J. Nuer. Math. Stoch.. [8] R. Thukral, A ew ith-order iterative ethod or idig ultiple roots o oliear equatios, Aer. J. Coput. Math. 6 ( [9] R. Thukral, Itroductio to higher order iterative ethods or idig ultiple roots o solvig oliear equatios, J. Math. Vol. 3, doi:.55/3/ [] J. F. Traub, Iterative Methods or solutio o equatios, Chelsea publishig copay, New York 977. [] S. Weerakoo, T. G. I. Ferado, A variat o Newto s ethod with accelerated third-order covergece, Appl. Math. Lett. 3 ( [] Z. Wu, X. Li, A ourth-order odiicatio o Newto s ethod or ultiple roots, IJRRAS ( ( 66-7.

New Twelfth Order Iterative Methods for Finding Multiple Roots of Nonlinear Equations

New Twelfth Order Iterative Methods for Finding Multiple Roots of Nonlinear Equations Aerica Joural o Coputatioal ad Applied Matheatics 05 5(: -4 DOI: 059/jajca050500 New Twelth Order Iterative Methods or Fidig Multiple Roots o Noliear Equatios R Thukral Padé Research Cetre 9 Deaswood Hill