Modification of Weerakoon-Fernando s Method with Fourth-Order of Convergence for Solving Nonlinear Equation

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1 ISSN: Iteratioal Joural o AdvacedResearch i Sciece, Egieerig ad Techology Vol. 5, Issue 8, August 018 Modiicatio o Weerakoo-Ferado s Method with Fourth-Order o Covergece or Solvig Noliear Equatio Wartoo,Desjebrialdi, Rahmawati, Irma Suryai Departmet o Mathematics, Faculty o Scieces ad Techology, Uiversitas Islam Negeri Sulta Syari Kasim Riau, Pekabaru, Idoesia ABSTRACT: Weerakoo-Ferado s method is a iterative method to solve a oliear equatio with third order o covergece that ivolve three evaluatio o uctios. I this paper, the author developed the method by usig cetroidal meas ad secod order Taylor s polyomial, the its secod derivative is reduced by usig approimatio o sum o orward ad backward dierece. Based o the covergece aalysis, we show that the ew iterative method coverges quartically ad ivolves three evaluatios o uctio with eiciecy ide equal to 4 1/ Numerical simulatio preset that the proposed method is comparable to other methods discussed ad gives better results. KEYWORDS: cetroidal meas, oliear equatio, order o covergece, Taylor s polyomial, Weerakoo- Ferado s method. I. INTRODUCTION Noliear equatio is mathematical represetatio rom various pheomeo o scietiic ad egieerig ield. The problem o the oliear equatio is how to solve the equatio by usig aalytical techique. Thereore, may oliear equatios are ot able to solve aalytical techique, the umerical solvig is a alterative solutio that use loopig calculated. It is kow widely as iterative method. I this paper, we cosider iterative method to solve a simple root rom a oliear equatio i orm ( = 0, (1 where : D or ope iterval I is a scalar uctio. The classical iterative method that kow widely as basic approimatio or solvig oliear equatio is writte as ollow ( 1. ( ( The method o ( coverges quadratically with eiciecy ide equal to 1/ I order to improve the local o order o covergece i (, may techical approimatios have bee proposed to costruc a ew iterative method which coverges cubically, such as quadratic polyomial [1, 1], Adomia decompositio method [, 4], circle o curvature [7], ad modiicatio o Newto s method [, 4, 6, 8, 11,1, 14, 16, 19]. The last techical approimatio appear a iterative method that is kow widely as two poit iterative method. Oe o the two poit iterative method is a Weerakoo-Ferado s method [19] writte i ( 1. ( ( '( y Iterative method ( is a third-order o covergece ivolvig three uctioal evaluatios with eiciecy ide equal to 1/ Based o Traub [17], the order o covergece o a iterative method will be optimal i the eiciecy ide o the iterative method satisies 1 where is umber o poit o the iterative method. For two poit iterative Copyright to IJARSET 649

2 ISSN: Iteratioal Joural o AdvacedResearch i Sciece, Egieerig ad Techology Vol. 5, Issue 8, August 018 method with three uctioal evaluatios, order o covergece o the method will be optimal i its order o covegece is our. I this paper, we developed the Weerakoo-Ferado s method usig secod order Taylor series epasio ad reduced its secod derivative by usig a equality o Newto-Steese method with oe real parameter [15] ad Weerakoo-Ferado s method [19]. Furthermore, a umerical simulatio is give to compare the perormace o the proposed method ad the compare its perormace with other methods. II. THE SCHEMATIC OF WEERAKOON-FERNANDO S ITERATIVE METHOD Weerakoo-Ferado s iterative method has bee developed by usig Newto s theorem ad trapezoidal rule, which is writte as ollowig, respectively ad ( ( '( t dt (4 '( '( '( t dt ( (5 Substitute (5 ito (4, we get '( '( ( ( ( I we take = +1, the (6 ca be writte as '( 1 '( ( 1 ( ( 1 Suppose, +1 is a approimatio root o oliear equatio at (+1th iteratio which coverges closely to,so that( Thereore, we ca wrtie (7 as orm '( 1 '( ( ( 1 0 or ( 1 * '( '( where * 1 is Newto s method deied i (. Equatio (9 is third-order iterative method ivolvig three evaluatio uctios with eiciecy ide equal to 1/ The iterative method is widely kow as Weerakoo-Ferado s method which is commoly writte as (. 1 III. THE PROPOSED METHOD AND CONVERGENCE ANALYSIS To costruct a ew iterative method, we cosider Weerakoo-Ferado s method i ( as arithmetic meas orm ( 1, (10 ( '( y wherey is give by (. Furthermore, suppose a = '( ad b = '(y, the by usig cetroidal meas, the Equatio o (10 ca be writte as ( ( '( y 1. (11 ( ( ( y '( y (6 (7 (8 (9 Copyright to IJARSET 649

3 ISSN: Iteratioal Joural o AdvacedResearch i Sciece, Egieerig ad Techology Vol. 5, Issue 8, August 018 The equatio o (11 is variat o Weerakoo-Ferado s method with third-order o covergece. Thereore, to improve the method, we cosider a secod order Taylor series epasio about as orm 1 ( ( '( ( ''( (. (1 Suppose +1 is a root aproimatio or oliear equatio that closely to a its eact root, the ( Thereore, the secod order Taylor series epasio at = +1 i (1 ca be ormed as where * 1 1 * 1 ( ( ''(, (1 ( '( is give by equatio o (11. The, the equatio (11 substitutes ito (1, we ca write (1 as ollow ( 9 '( '( y ( ''( 1 ( y y. (14 8 '( '( '( '( '( Equatio (14 is a two-poit iterative method with our evaluatios o uctios ad ivolves secod dervative o (. To avoid the secod derivative, we reduce it by usig sum o orward ad backward dierece as orm '( '( '( y ''(. (15 ( Substitute (15 ito (14, we get a iterative method without secod derivative ( 9 ( '( '( y '( '( y 1 ( 8 '( '( '( y '( y. (16 Furthermore, '(y i (16 is approimated by usig equality o Newto-Steese method with oe real parameter [7] ad variat o Newto method [8] as orm '( ( ( ( y '( y. (17 ( ( y Subtitute '(y i (16 by usig (17 the we will get a ew iterative method ( y '(, (18 9 ( y ( ( 1 ( ( y ( 1 ( y ( ( y ( 1 1. (19 ( ( 6( 1 ( ( 6 4 ( y y Equatio (18 (19 is a two poit iterative method that ivolve two evaluatio o uctios ad oe its derivative. Based o Traub [17], the propose method will be a optimal iterative method i its order o covergece is our. Theorem. Let : D be a dieretiable uctio i ope iterval D. The assume that is a simple root o ( = 0. Suppose 0 is a give value that is suicietly close to, the iterative method (18 (19 has ourth-order o covergece or 0 with error e1 cc c e O( e, (0 where c k ( k (, k,,4,5,.... k! '( Copyright to IJARSET

4 ISSN: Iteratioal Joural o AdvacedResearch i Sciece, Egieerig ad Techology Vol. 5, Issue 8, August 018 Proo. Letis a eact root o (, the ( 0. Furthermore, assume ( 0 ad = e +, the Taylor series epasio or ( ad ' ( about, we have ad ( '( e c e c e c e O( e, ( '( '( 1 c e c e 4 c e ( e. ( Usig (1 ad (, we have ( e 5 ce (c c e O( e, ( ad hece, y ce ( c c e O( e5. ( Agai, use epasio series Taylor or (y about, we have 5 ( y ( ( ce (c c e O( e. (4 Based o (1, ( ad (4, we have ( ( e c e ( c 4 9 c e O( e, (5 ad ( y ( c e (c c e O( e. (6 Furthermore, use (1, (, (4, (5 ad (6, we oud ( ( y ( e (1 c e ((1 c c e (5 c 7 c c (1 c e4 O( e5, (7 4 y y e c e c c e O e, (8 ( ( 1 ( ( ( 1 ( ( ( ( 1 4 ( 5 ( 6( 1 ( ( y ( 6 4 ( y ( e 6 c e ( 1 1 c 6( 1 c e ad 4 ( 5, (9 Oe ( ( y ( ( c e ( c c e O( e. (0 4 5 Use (5, (6, (7, (8, (9, (0 ad +1 = e +1 +, we will get 1 e ce (4 1 cc 9 8 c e O( e. (1 Equatio (1 give a iormatio that the covergece order o (18 (19 will icrease i we take = 0. So, by subtitutig agai= 0 ito (1, we get e1 cc c e O( e. ( The proposed method (18 (19 requires three evaluatios o uctio per iteratio. Based o deiitio o eiciecy ide, i.e I.E = p 1/m, where p is the order o the iterative method ad m is the umber o uctioal evaluatios per iteratio, the propose method i (18 (19 have the eeciecy ide equals to 4 1/ , which is better tha the Newto s method 1/ [17], Halley s method 1/ 1.44 [1], Weerakoo-Ferado s method 1/ 1.44 [19]. Theproposed iterative method i (18 (19 with oe real parameter will appear oe ourth-order ad some third-order iterative methods. For = 0, we get ourth-order iterative method that is give by y y 1 1 ( ( ( 4 ( y y 9 ( ( ( ( (. ( ( For= 1, the equatio (19 preseted a third-order iterative method Copyright to IJARSET

5 ISSN: Iteratioal Joural o AdvacedResearch i Sciece, Egieerig ad Techology Vol. 5, Issue 8, August ( ( y ( ( 1 1. (4 ( ( ( y For=, the equatio(19 preseted a third-order iterative method 9 ( y ( ( y ( ( y (. (5 ( 1 1 ( ( ( 4 ( y y IV. NUMERICAL SIMULATION I this sectio, we preseted umerical simulatio to show the perormace o the proposed method. The idicators o the perormace ivolved the umber o iteratio, computatioal o order o covergece (COC, absolute value o uctio, value o absolute error. The result o umerical simulatio o the proposed method both o 0 (MT1 ad = 0 (MT are compared with Newto s method (MN [17], Halley s method (MH [1], ad Weerakoo-Ferado s method (MWF [19]. I this umerical simulatio, all computig are perormed by usig MAPLE 1.0 with 850 digits loatig poit arithmetic s ad the ollowig stoppig criterio or computer program 1. (6 I additio, whe the stoppig criterio is satisied, the +1 is take as the eact root. Furthermore, COC may be approimated as l ( 1 (. (7 l ( ( We used the ollowig eight test uctio while its eact roots displayed the computed approimate zeros roud up to 15th decimal places 1 ( e 0,1, , ( e 4, , ( cos(, , 4( ( 1 1, = , 5( 4 10, , 1 6( e cos( 1 1, = , 7( si ( 1, , 8(, Table 1 shows the umber o iteratio (IT requirig such thatsatiies (6 where = ad the computatioal order o covergece (COC i the paretheses by usig ormulas (7. I the Table 1, irst colmu shows some test uctios deoted by, secod colum shows iitial value deoted by 0, ad third colum to seveth colum shows umber o iteratio ad COC or ive compared iterative methods. Table 1. The umber o iteratio adcocor several iterative methods with ( 0 MN MH MWF MT1 MT ( 0 = 0 1 ( 0. 8( ( ( ( ( ( ( ( ( ( ( 4.0 8( ( ( ( ( ( ( ( ( ( ( 0.1 8( ( ( ( ( Copyright to IJARSET

6 ISSN: Iteratioal Joural o AdvacedResearch i Sciece, Egieerig ad Techology Vol. 5, Issue 8, August ( 5 ( 6 ( 7 ( 8 ( 1.5 7( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Based o the results i the Table 1, we ca see that order o covergece o the proposed methods is three or 0 ad our or = 0.From computatioal order o covergece (COC o each iterative method, MT has a covergecy higher order tha others. Thereore, the MT move aster i reachig the root o oliear equatio. Beside, Table 1 coirms the order o covergece o the proposed method theoretically. The accuracy o the proposed method ad several methods by usig the same total umber o uctioal evaluatios(tnfe as compariso are preseted at Table. Table. Value o absolute error or TNFE = 1 ( 0 MN MH MWF MT1 ( 0 MT ( = 0 1 ( ( ( 4 ( 5 ( 6 ( 7 ( 8 ( (e (e (e4 8.0 (e (e7.471 (e (e (e (e (e (e67.10 (e (e (e.0985 (e4.199 (e (e (e (e (e (e (e (e61.10 (e (e61.80 (e5.547 (e (e (e65.47 (e (e (e66.086(e (e (e (e (e57.65 (e (e49.90 (e (e (e (e (e6 1.07(e (e (e (e59 1.1(e4.18 (e (e (e (e (e (e (e (e8.80 (e (e (e (e (e (e (e (e (e (e (e (e (e (e (e (e (e (e (e (e (e (e (e15 Table preseted that all o compariso methods coverge to the root o oliear equatio or all test uctios. The accuracy o the proposed method (18 (19 or = 0 is better tha other iterative methods. V.CONCLUSION This study discovers a ew variat o Weerakoo-Ferado s method as a alterative iterative method to id the roots o the oliear equatio that require evaluatio o two uctio ad oe irst derivative per iteratio with eiciecy ide equal to 4 1/ Numerical simulatio preset that the proposed method is better tha Weerakoo-Ferado s method ad other discussed methods. Copyright to IJARSET

7 ISSN: Iteratioal Joural o AdvacedResearch i Sciece, Egieerig ad Techology Vol. 5, Issue 8, August 018 REFERENCES [1] Amat, S., Busquier, S., ad Gutierrez, J. M., Geometric costructios o iterative uctios to solve oliear equatios, Joural o Computatioal ad Applied Mathematics, 157, 00, [] Ababeh, O. Y., New Newto s method with third-order covergece or solvig oliear equatios, Iteratioal Joural o Mathematical, Computaioal, Physical, Eletrical ad Copute Egieerig, 6(1, 01, [] Abbasbady, S., Improvig Newto-Raphso method or oliear equatios by modied Adomia decompositio method, Applied Mathematics as Computatio, 145, 00, [4] Chu, C., Iterative methods improvig Newto s method by the decompositio method, Computers as Mathematics with Applicatios, 50, 005, [5] Chu, C., A geometric costructio o iterative uctios o order three to solve oliear equatio, Computers ad Mathematics with Applicatios, 5, 007, [6] Chu, C., A simply costructed third-order modiicatios o Newto s method, Joural o Coputatioal ad Applied Mathematics, 19, 008, [7] Chu, C. ad Kim, Y., Several ew third-order iterative methods or solvig oliear equatios, Acta Appl. Math, 109, 010, [8] Homeier, H. H. H., O Newto-type methods with cubic covergece, Joural o Computatioal ad Applied Mathematics, 176, 005, [9] Kasal, M., Kawar, V. ad Bhatia, S., Optimized mea base secod derivative-ree amilies o Chebyshev-Halley type methods, Numerical Aalysis ad Applicatio, 9(, 016, [10] Kasal, M., Kawar, V., ad Bhatia, S., New modiicatios o Hase-Patrick s amily with optimal ourth ad eighth orders o covrgece, Applied Mathematics ad Computatio, 69, 015, [11] Kou, J., Li, Y., ad Wag, X., A modiicatio o Newto method with third-order covergece:, Applied Mathematics ad Computatio, 181, 006, [1] Kou, J., To improvemets o modiied Newto s method, Applied Mathematics as Computatio, 189, 007, [1] Melma, A., Geometry ad covergece o Euler s ad Halley s methods, SIAM Review, 9(4, 1997, [14] Ozba, A. Y., Some ew variat o Newto s method, Applied Mathematics Letters, 17, 004, [15]Sharma, J. R., A composite third order Newto-Steese method or solvig oliear equatios, Applied Mathematics as Computatio, 169, 005, [16] Thrukal, R., New modiicatio o Newto mehod with third-order covergece or solvig oliear equatio o type (0 = 0, America Joural o Computatioal ad Applied Mathematics, 6(1, 016, [17] Traub, J. F., Iterative method or the solutio o equatios, Pretice-Hall: New-York, [18] Wartoo, Soleh, M.., Suryai, I., Zulakmal ad Muhaza, A ew variat o Chebyshev-Halley s method without secod derivative with covergece o optimal order, Asia Joural o Scietiic Research, 11(, 018, [19] Weerakoo, S., ad Ferado, T. G. I., A variat o Newto s method with accelerated third-order covergece, Applied Mathematics Letters, 1, 000, Copyright to IJARSET