Some Variants of Newton's Method with Fifth-Order and Fourth-Order Convergence for Solving Nonlinear Equations

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1 Copyright, Darbose Iteratioal Joural o Applied Mathematics ad Computatio Volume (), pp -6, 9 http//: ijamc.darbose.com Some Variats o Newto's Method with Fith-Order ad Fourth-Order Covergece or Solvig Noliear Equatios Yao-tag Li *, Ai-qua Jiao Departmet o Mathematics, School o Mathematics ad Statistics, Yua Uiversity, Kumig Yua, 659, P. R. Chia, liyaotag@yu.edu.c Abstract Chu's two-step predictor corrector type iterative method or solvig oliear equatios (Iterative Methods Improvig Newto's Method by the Decompositio Method, J. Comput. Math. Appl.5, ) is improved ad a kid o ew iterative method is preseted. Some preset methods derived rom Adomia Decompositio method are uiied i a orm ad a series o ew methods with high covergece order ad the value o EFF are obtaied by itroducig parameter also. Those methods ca be cosidered as a sigiicat improvemet o the Newto's method ad its variat orms. Keywords: Noliear equatio, Iterative method, Newto's method, Adomia decompositio method, Predictor corrector type iterative method.. Itroductio It is oe o the oldest ad most basic problems i mathematics to solve oliear equatios () =. Iterative methods are usually used to solvig these equatios. Newto's method is oe o the most powerul ad well-kow iterative methods kow to coverge quadratic. Recetly, there has bee some progress o iterative methods with higher order o covergece that do require the computatio o as lower-order derivatives as possible [-6]. Correspodig author : Yao-tag Li This work is supported by the Natioal Natural Sciece Foudatio o Chia (No.775) ad the Natural Scieces Foudatio o Yua Provice (Nos. 7AM ad 5FM).

2 Yao-tag Li, Ai-qua Jiao O the other had, there has bee aother approach based o the Adomia decompositio method [, 7 ad 8]. The Adomia decompositio method is the method which cosiders the solutio as a iiite series usually covergig to a accurate solutio, ad has bee successully applied to a wide class o uctioal equatio over the last years [9-]. The covergece o the decompositio series have bee ivestigated by several authors [7, 9, ad ]. Abbaoui ad Cherruault [7] applied the method to solve the equatio () =. ad proved the covergece o the series solutio. The Adomia method has bee modiied also so as to costruct umerical schemes [, 7]. I this paper, we costruct a kid o higher-order iterative methods based o the Adomia decompositio method. I Sectio, two relative deiitios are give. I Sectio, we obtai a ew kid o iterative methods, which uiy may preset methods i a orm by itroducig parameters, ad a series o ew methods with high covergece order ad with the value o EFF are obtaied by takig these parameters as dieret values. I Sectio, we give a detailed covergece aalysis o the proposed methods. I Sectio 4, we give the comparisos o the iormatio eiciecies EFF o these algorithms. I Sectio 5, several umerical eamples are give to illustrate the eiciecy ad perormace o those ew methods. Deiitio. (Order o covergece [6]). I the sequece { } teds to a limit * i such a way ad * lim k C ( * ) q k k or some C ad q, the the order o covergece o the sequece is said to be q, ad C is called the asymptotic error costat. I q=, q= or q=, the covergece is said to be liear, quadratic or cubic, respectively * Let e be the error i the k th k k iterate o the method which produces the sequece. { } The, the relatio q k k q q k k e C e O e O e is called the error equatio. The value o q is called the order o covergece o this method. Deiitio.. The iormatio eiciecy EFF [6] is deied: EFF = q, where q is the order o the method ad d is the umber o uctioal evaluatios per iteratio required by the method. /d IJAMC

3 Some variats o Newto's method with ith-order.... Algorithms costructio Cosider oliear equatio () = () Assume that á is a simple root o equatio () ad ã is a iitial guess suicietly close to á. We rewrite the oliear equatio () as a equivalet coupled system by usig the Taylor epasio o () at ã, ( ) ( )( ) g( ) () g( ) ( ) ( ) ( )( ) () Rewrite equatio () i the ollowig orm : or where ( ) g ( ) ( ) ( ) c N( ) ( ) g ( ) c N ( ) ( ) ( ) The Adomia decompositio is to look or a solutio (4) ad the oliear operator N () ca be decomposed as where Ad A are called the Adomia's Polyomials. The irst ew polyomials are give by N ( ) A, d i A N( i),,..., (7)! d i,, A N ( ) A N ( ) A N( ) N ( ) (5) (6) darbose

4 4 Yao-tag Li, Ai-qua Jiao Upo substitutig (5) ad (6) ito (4) yields c A. It ollows rom (8) that Note that ca be approimated by c A =,,... (8) X A A A m m m that is lim. A elemetary calculatio shows that, m X m g ( A N( ) ) N( ) ( ) ( ) ( ) ( ) A N( ) ( ) ( ) N ( ) ( ) ( ) A N( ) N ( ) A N( ) A N ( ) ( ) ( ( ) ( ) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) So X A A A c A A A = = is a approimatio o. Now, by itroducig some parameters, we geeralize X to the ollowig orm X 4 (9) where ad á, á, á, á4 are parameters. Thus some ew ad high order IJAMC

5 methods ca be obtaied by takig dieret values o parameters á, á,á, ad á. 4 (i) I, á = á =á = á =,the the equatio (9) has the ollowig orm : 4 X This allows us to costruct the ollowig oe-step iterative method or solvig Equatio (). Algorithm.. For a give, compute the approimate solutio iterative scheme, + ( ), ( ),,,,. ( ) o equatio () by the which is the well-kow Newto's method ad Algorithm. has a secod-order covergece. (ii) I á = -, á =á = á =, the the equatio (9) has the ollowig orm : 4 Some variats o Newto's method with ith-order... 5 X Algorithm.. For a give, compute the approimate solutio iterative scheme. Predictor-step : + y o equatio () by the ( ), () ( ) ( ) Corrector-step : ( y ) y =,,.... ( ) We would like to metio that Algorithm. has cubic covergece []. (iii) I, á = -, á = á = á =, the the equatio (9) has the ollowig orm: 4 X Usig this orm, we ca costruct the ollowig two-step iterative method or solvig Equatio () as : Algorithm.. For a give, compute the approimate solutio iterative scheme. + o equatio () by the darbose

6 6 Yao-tag Li, Ai-qua Jiao Predictor-step : Corrector-step : y ( ), ( ) ( ) ( y ) ( ) ( ) y y y, =,,.... ( ) ( ) We would like to metio that Algorithm. has cubic covergece []. (iii) I, á = -, á =, á = -, á = - 4, the the equatio (9) has the ollowig orm: X Usig this orm, we ca costruct the ollowig two-step iterative method or solvig Equatio () as : Algorithm.4. For a give, compute the approimate solutio iterative scheme. + o equatio () by the Predictor-step : Corrector-step : ( ) y, ( ) ; ( ) ( y ) ( y ) ( y ) ( y ) ( y ) ( y ) ( y ) y ( ) ( ) ( ) ( ) We would like to metio that Algorithm. has cubic covergece []. =,,.... (iv) I á = -, á =, á =, á = - 5 4, the the equatio (9) has the ollowig orm: 5 X Usig this orm, we ca costruct the ollowig two-step iterative method or solvig Equatio () as : Algorithm.5. For a give, compute the approimate solutio iterative scheme. + o equatio () by the IJAMC

7 Some variats o Newto's method with ith-order... 7 Predictor-step : Corrector-step : ( ) y, ( ) ; ( ) ( ) ( ) ( ) ( ) ( ) y,,,,. y y y 5 y y ( ) ( ) ( ) I sectio, we will prove that Algorithm.5 has ive-order covergece. (vi) From equatio (6)(see Sectio ) ad equatio (7) (see Sectio ), we ca get the ollowig approimatio ( ) ( ) ( ) ( ) ( ) Upo substitutig () ito (9), ad takig á = -, á =, á =, á = - 5, the the 4 equatio (9) has the ollowig orm : X 5 Usig this orm, we ca costruct the ollowig two-step iterative method or solvig Equatio () as : () Algorithm.6. For a give, compute the approimate solutio iterative scheme. + o equatio () by the Predictor-step : Corrector-step : ( ) y, ( ) ; ( ) ( y) ( y) ( y) ( y) ( y) ( ) ( ) ( ) ( ) y 5,,,,. I sectio, we will prove that Algorithm.6 has ive-order covergece. (vii) Replace derivate by dieret, that is ( ) ( ) where, we get the ollowig approimatio : darbose

8 8 Yao-tag Li, Ai-qua Jiao Upo substitutig () ito (9), the the equatio (9) has the ollowig orm : X ( ( ) ( ) ( ) ( ) ) ( 4 ) ( ) ) ( () () Set 4 where â is a costat. The the equatio (9) has the ollowig orm : X Usig this orm, we ca costruct the ollowig two-step iterative method or solvig Equatio () as : Algorithm.7. For a give, compute the approimate solutio iterative scheme. + o equatio () by the Predictor-step : Corrector-step : ( ) y, ( ) ; ( ) ( y ) ( y ) ( y ) y,,,, ( ) ( ) ( ) ( ) ( ) where â is a costat. The sectio, we will prove that Algotirhm.7 is a amily method with our-order covergece.. Covergece Aalysis or Algorithms Now, we aalysis the order o covergece o those methods deied by Algorithm.5, Algorithm.6 ad Algorithm.7. IJAMC

9 Some variats o Newto's method with ith-order... 9 Theorem.. Let I e a simple root o the oliear equatio () =, : I R R be suicietly dieretiable or a ope iterval I. I is suicietly close to á, the the orders o covergece o Algorithm.5 is ive, ad the error equatio o Algorithm.5 is e c c 4 c e O ( e ) (4) ( k ) ( ) where ck k=,,, e k! ( ) Proo : Let á be a simple zero o. Sice is suicietly dieretiable, by epadig ( ) ad ' ( ) at á, we get ( ) ( )[ e c e c e c e c e O( e )] (5) ( ) ( )[ c e c e 4c e 5 c e O( e )] (6) where c k ( k ) ( ) k=,,, e k! ( ) Dividig (5) by (6) gives us From () ad (7), we obtai, Now, epadig (y ) at á ad usig (8), we obtai ( y ) ( )[ c e ( c c ) e (c 5c 7 c c ) e - ( ) e c e ( c c ) e (7c c c 4 c) e4 ( ) 4 ( c c c c 4 c 6 c 8 c ) e O ( e ) y c e ( c c ) e (c 4c 7 c c ) e 4 4 ( c c 4 c c c 6 c 8 c ) e O ( e ) (4 c c c c 4 c 6 c c ) e O ( e )] Agai, epadig (y ) about á ad usig (8), we get y y ( ) y ( ) ( ) ( )( ) ( )...! ( )[ c e (4c c 4 c ) e (6c c 8c c c ) e (8c c 8c c c c 6 c ) e O( e )] (7) (8) (9) () darbose

10 Yao-tag Li, Ai-qua Jiao Epadig (y ) at á ad usig (8) too, we have (4) ( y ) ( ) ( )( y ) ( ) ( y )...! ( )[ c 6 c c e ( c c c ) e (8 c c c c 4 c c 4 c c ) e ( c c 4c c c c c 6 c c c 48 c c ) e O ( e )] Ater a elemetary calculatio, we get ( y) c e (c c) e (8c c c c ) e O( e4) 4 ( ) () ( y) c e ( c c ) e (c c 4 c c ) e ( ) 4 4 (64 c c 4 c c c c 8 c ) e O ( e ) ( y) c e (6c c ) e (6c c 6c 4 c ) e 4 ( ) ( c c 4 c 6 c c 5 c 9 c ) e O ( e ) ( y ) c 4ce 8 ce (c 8c c 6 c4) e O( e4), 4 ( ) () () (4) ( y ) ( y ) c e (c 6 c ) e (c 7c c c ) e ( ) 4 4 ( c c 4 c c c 8 c 4 c ) e O ( e ) (5) ( y) ( y) ( ) c e (8c c c ) e O( e ) ( y) ( y) ( ) c e (8c c c ) e O( e ) (6) IJAMC

11 Some variats o Newto's method with ith-order... ( y) ( y) ( ) ( ) c e (6c c c ) e O( e ) (7) From equatio (8), (), (5) ad (6), we obtai ( y ) ( ) ( ) 5 ( ) ( ) y y y y y ( ) ( ) ( ) (4 c c 4 c ) e O ( e ) (8) Thus, or Algorithm.5, rom ad equatio (8), that is ( ) ( ) ( ) ( ) ( ) y 5 y y y y y ( ) ( ) ( ) e (4 c c 4 c ) e O ( e ) e (4 c c 4 c ) e O ( e ) This meas that the order o covergece o Algorithm.5 is ive. I c =, the order o covergece o Algorithm.5 is si. Theorem.. Let I be a simple root o the oliear equatio () =, : I R R be suicietly dieretiable or a ope iterval I. I is suicietly close to á, the the orders o covergece o Algorithm.6 is ive, ad the error equatio o Algorithm.6 is e 4 c 4 6 c c 5 6 e O ( e ) (9) ( k ) ( ) where ck k=,,, e k! ( ) Proo : Let á be a simple zero o. Sice is suicietly dieretiable, rom equatio (8), (), (5) ad (7), we obtai ( y ) ( y ) ( y ) ( y ) ( y ) y 5 ( ) ( ) ( ) ( ) (4 c 6 c c ) e O ( e ) () darbose

12 Yao-tag Li, Ai-qua Jiao For Algorithm.6, rom ad equatio (), ( y ) ( ) ( ) ( ) ( ) y y y 5 y y ( ) ( ) ( ) ( ) e (4 c 6 c c ) e O ( e ) that is e (4c 6 c c ) e O ( e ) This meas that the order o covergece o Algorithm.6 is ive. I c =, the order o covergece o Algorithm.5 is si. Theorem.. Let I be a simple root o the oliear equatio () =, : I R R be suicietly dieretiable or a ope iterval I. I is suicietly close to á, the the orders o covergece o Algorithm.7 is our, ad the error equatio o Algorithm.7 is ( k ) where ( ) c k=,,, k e k! ( ) Proo : Let á be a simple zero o. Sice is suicietly dieretiable, rom equatio () ad (), we obtai e 5 c c c e O( e ), ( y) ce (4c c 7 c) e4 O ( e5) ( ) ( ) () () From Equatio (8), (), () ad (), we obtai ( y ) ( ) ( ) c e O ( e ) 4 5 () y ( y ) ( y ) ( y ) ( ) ( ) ( ) ( ) ( ) (5 ) c c c e O ( e ) 4 5 (4) IJAMC

13 Some variats o Newto's method with ith-order... For Algorithm.7, rom ad equatio (4), that is, ( y) ( y ) ( ) y y ( ) ( ) ( ) ( ) ( ) e (5 ) c c c e O( e ) 4 5 e (5 ) c c c e O ( e ) 4 5 This meas that the order o covergece o Algorithm.7 is ive. I c =, the order o covergece o Algorithm.7 is si. (5) (6) 4. Compariso o The iormatio eiciecy EFF o Algorithms Table 4. : Compariso o algorithms iormatio eiciecies EFF Iterative methods q Asymptotic errors d EFF Algorithm. c.44 Algorithm. c Algorithm. 4 5c 4 Algorithm.4 5 4c4 5 Algorithm.5 5 4cc 4c 4 5 Algorithm.6 5 4c4 6 cc 4 Algorithm.7 4 (5 )c c c ( k ) ( ) k! ( ) where c or j =,. k 5. Numerical eamples I this sectio, we preset some eamples to illustrate the eiciecy o the iterative methods i this paper. darbose

14 Table 5.: Compariso o Newto's method ad those methods proposed i this paper. (Algorithm.7, â= ; Algorithm.8, â= ). IJAMC

15 4 Yao-tag Li, Ai-qua Jiao As or the covergece criteria, the distace o two cosecutive approimatios ä is required. Also the umbers o iteratios to approimate the zero (IT) ad the value are displayed. (see Table 5.). The ollowig stoppig criteria are used or computer programs : (i) (ii) The eamples are the same as i [], ( ) 4 e ( ) ( ) ( ) 5 ( ) si cos 5 7 e ( ) si ( ) cos 4 ( ) 6 7 ( ) e 8 6. Coclusio I this paper, may eistig methods or solvig oliear equatios are uiied i a orm by itroducig parameter based o Chu's method. We also get a series o ew algorithms with high covergece order ad the value o EFF. From Table 4., we ca easily see that the last ree algorithms ot oly with high covergece order, but also required little o uctioal evaluatios per iteratio. The last two algorithms do ot ivolve the higher derivatio o uctio. From Table 5., we also ca see that the ew algorithms, compared to the eistig algorithms, ca easily get the root i more little steps ad more eactly error. I a word, it is easily to see the eiciecy o our ew algorithms. Ackowledgmets: The authors sicerely thak the reerees or valuable commets ad poitig out some errors i the origial versio o this paper, which led to a substatial improvemet i cotets o this paper. Reereces []. Chagbum Chu, Iterative Methods Improvig Newto's Method by the Decompositio Method, J. Comput. Math. Appl.5, , (5). ]. ]. E. Babolia ad J. Biazar, Solutio o oliear equatios by modiied Adomia decompositio method, Appl. Math. Corput., 6 7, (). M. Grau ad M. Noguera, A variat o Cauchy's method with accelerated ith-order covergece, Appl. Math. Lett. 7 (5), 59-57, (4). 4]. H.H.H. Homeier, O Newto-type methods with cubic covergece, J. Corput. Appl.Math. 76, 45-4,(5). 5]. 6]. X.Wu ad D. Fu, New high-order covergece iteratio methods without employig derivatives or solvig oliear equatios, Computers Math. Applic. 4 (/4), , (). S. Weerakoo ad G.I. Ferado, A variat o Newto's method with accelerated third-order covergece, Appl. Math. Lett. 7 (8), 87-9, (). darbose

16 6 Yao-tag Li, Ai-qua Jiao 7]. 8]. 9]. ]. ]. ]. ]. K. Abbaoui ad Y. Cherruault, Covergece o Adomia's method applied to oliear equatios, Mathl.Corput. Modellig (9), 69-7, (994). S. Abbasbady, Improvig Newto-Raphso method or oliear equatios by modiied Adomia decompositio method, Appl. Math. Corput. 45,887-89, (). G. Adomia, Noliear Stochastic Systems ad Applicatios to Physics, KluwerAcademic Publishers, Dordrecht, (989). G. Adomia ad R. Rach, O the solutio o algebraic equatios by the decompositio method, Math. Aal. Appl. 5, 4-66, (985). G. Adomia, Solvig Frotier Problems o Physics: The Decompositio Method, Kluwer Academic Publishers, Dordrecht, (994). K. Abbaoui ad Y. Cherruault, New idea or provig covergece o decompositio methods, Computers Math. Applic. 9 (7), -8, (995). Y. Cherruault ad G. Adomia, Decompositio methods: A ew proo o covergece, Math. Comput. Modellig 8 (), -6, (99). 4]. W. Gautschi, Numerical Aalysis, Birkh~iuser, Bosto, MA, (997). 5]. M. Frotii ad E. Sormai, Some variat o Newto's method with third-order covergece, J. Comput.Appl. Math. 4, 49-46, (). [6]. D.K.R. Babajee, M.Z. Dauhoo, A aalysis o the properties o the variats o Newto's method with third order covergece, Appl. Math. Comput. 8, , (6). IJAMC