Some New Iterative Methods for Solving Nonlinear Equations


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1 World Applied Scieces Joural 0 (6): , 01 ISSN IDOSI Publicatios, 01 DOI: /idosi.wasj Some New Iterative Methods for Solvig Noliear Equatios Muhammad Aslam Noor, Khalida Iayat Noor ad Kshif Aftab Departmet of Mathematics, COMSATS Istitute of Iformatio Techology, Park Road, Islamabad, Pakista Abstract: I this paper, we suggest ad aalyze some ew iterative methods for solvig oliear equatios by usig a ew series expasio of the oliear fuctio. Some special cases are also discussed. These ew methods ca be viewed as sigificat modificatio ad improvemet of the Newto method. Several examples are give to illustrate the efficiecy ad robustess of these methods. Kew words: Householder method Iterative method Covergece Noliear equatio INTRODUCTION f(x) = 0 (1) It is well kow a wide class of liear ad oliear Various umerical methods have bee developed problems which arise i differet braches of usig the Taylor series ad other techiques. I this mathematical such as physical, biomedical, regioal, paper, we use aother series of the oliear fuctio f(x) optimizatio, ecology, ecoomics ad egieerig which ca be obtaied by usig the trapezoidal rule ad scieces ca be formulated i terms of oliear the Fudametal Theorem of Calculus. To be more equatios. Iterative methods for fidig the approximate precise, we assume that is a simple root of (1) ad is solutios of the oliear equatio f(x) = 0 are beig a iitial guess sufficietly close to Now usig the developed usig several differet techiques icludig trapezoidal rule ad fudametal theorem of calculus, oe Taylor series, quadrature formulas, homotopy ad ca show that the fuctio f(x) ca be approximated by decompositio techiques, see [15, 911, 1317, 19, 1] the series ad the refereces therei. Ispired ad motivated by the ogoig research activities i this area, we x f( x) = f + [ ( x) + ] suggest ad aalyze a ew iterative method for () solvig oliear equatios. To derive these iterative methods, we show that the oliear fuctio ca where f'(x) is the differetial of f. be approximated by a ew series which ca be obtaied by usig the trapezoidal rule for From (1) ad (), we have approximatig the itegral i cojuctio with the fudametal theorem of calculus. This ew expasio is x = x. used to suggest these ew iterative methods for (3) solvig oliear equatios.. We also cosider the covergece aalysis of these methods. Several examples Usig (3), oe ca suggest the followig iterative are give to illustrate the efficiecy ad compariso with method for solvig the oliear equatios (1). other methods. Algorithm 1: For a give iitial choice x 0, fid the Iterative Methods: It is well kow that a wide class of approximate solutio x +1 by the iterative scheme problems, which arise i various fiekds of pure ad applied scieces ca be formulated i terms of oliear f( x) ( x+ 1) x+ 1 = x ( x+ 1 x), equatios of the type. = 0,1,,... Correspodig Author: M. Aslam Noor, Mathematics Departmet, COMSATS Istitute of Iformatio Techology, Park Road, Islamabad, Pakista. 870
2 Algorithm 1 is a implicit iterative method. Algorithm 5: For a give iitial choice x 0, fid To implemet Algorithm 1, we use the predictorcorrector the approximate solutio x +1 by the iterative techique. Usig the Newto method as a predictor ad schemes: Algorithm 1, as a corrector, we suggest ad aalyze the followig twostep iterative method for solvig the f( x) ( x) x+ 1 = x = 0,1,,... oliear equatio (1) ad this is the mai motivatio of [ ( x)] f( x) ( x) this ote. which is kow as the Halley method. It ca Algorithm : For a give iitial choice x 0, fid the easily show that Halley method has cubic approximate solutio x +1 by the iterative schemes: covergece. I a similar way, oe ca obtai several kow ad ew iterative methods form these y f( x) = x f f x = x ( y x ), = 0,1,, f( x) f( x) ( y) x+ 1 = y +, = 0,1,,... ( x) ( x) ( x) x = f. + ( x) algorithms. We ow cosider the covergece aalysis of Algorithm. I a similar way, oe ca prove the covergece of Algorithm 3 ad algorithm 4. From Algorithm, we ca deduce the followig Theorem.1: Let I be a simple zero of iterative method for solvig the oliear equatios sufficietly differetiable fuctio f : I R R for a ope f(x) = 0 which appears to be ew oe. iterval I. If x 0 is sufficietly close to the the iterative method defied by Algorithm has secodorder Algorithm 3: For a give iitial choice x 0, fid the covergece. approximate solutio x +1 by the iterative scheme Proof: Let be a simple zero of f. The by expadig f(x) Algorithm 3 is called the modified Householder method for solvig the oliear equatios (1), which does ot ivolve the secod derives. From (1) ad (), we ca have ad f'(x ) about we have f( x) = e + ce + ce 3 + ce 4 + O( e), ad ( x) = 1+ ce + 3ce 3 + 4ce 4 + 5ce 5 + O( e), (4) (5) This fixed poit formulatio eables us to suggest the followig iterative method for solvig the oliear equatio. Algorithm 4: For a give iitial choice x, fid the 0 approximate solutio x by the iterative schemes: +1 y f( x) = x f f x 1 + = x = 0,1,,... + Usig the Taylor series expasio of f'(y ), oe obtai the followig iterative method for solvig the oliear equatio f(x) = 0. 1 f ( k) ck =, k,3, k! = where ad e = x. Now, from (4) ad (5), we have f( x) = e ce + ( c c3) e+ (7c3 c 4c 3 c4) e + O( e). ( x) (6) From (6), we have = y ce ( c c ) e (7cc 4c 3 c) e O( e). (7) From (7), we have 871
3 ( y) = 1 + ce + 4( cc 3 c) e + ( 11cc 3+ 6cc 4+ 8 c) e + O( e). (8) From (5) ad (8), we have ( y) = ce + ( 3c3+ 6 c) e + ( 16c 4c4+ 16 cc 3) e + O( e). (9) From (7), we have = y x e c e ( c c ) e (7c c 4c 3 c ) e O( e ). (10) From (9) ad (10), we have ( y x) e 3 ce (5c 10 c ) e ( 30cc 30c 7 c) e O( e). = ( x) (11) Thus, from (6) ad (11), we have = x ce ( c 6 c ) e O( e ), (1) which implies that = e ce ( c 6 c ) e O( e). (10) This shows that Algorithm is secodorder coverget. Numerical Results: We preset some examples to illustrate the efficiecy of the ew developed twostep iterative methods, see Table 1. We compare the Newto method (NM), 15 Algorithm (NR1) ad Algorithm 3 (NR).. We used = 10. The followig stoppig criteria is used for computer programs: () i x x <, ( ii) f < The examples are the same as i Chu []. 1 f ( x) = si x x + 1, f ( x) = x e 3x + f ( x) = cos x x, f ( x) = ( x 1) x 5 6 f ( x) = x 10, f ( x) = xe si x + 3cos x x + 7x 30 f ( x) = e 1. 3 x As for the covergece criteria, it was From the Table 1, we see that our method is required that the distace of two cosecutive comparable with the Newto Method. I fact, our approximatios 15 for the zero was les tha 10. methods ca be cosidered as sigificat improvemet of Also displayed is the umber of iteratios to the Newto Method ad ca be cosidered as alterative approximate the zero (IT), the approximate zero x ad method to other secod order coverget methods of the value f(x ) solvig oliear equatios. 87
4 Table 1: (Numerical Examples ad Compariso) Method IT x f(x ) f 1, x 0 = 1 NM e e6 NR e e8 NR e e3 f, x 0 = NM e e8 NR e3.11e16 NR e e19 f 3, x 0 = 1.7 NM e3.34e16 NR e e7 NR e e7 f 4, x 0 = 3.5 NM 8.06e4 8.8e NR e e30 NR e e8 f 5, x 0 = 1.5 NM e e8 NR e45.30e3 NR e e19 f 6, x 0 = NM e40.73e1 NR e3 5.8e17 NR e e f 7, x 0 = 3.5 NM e e5 NR e e19 NR e e16 CONCLUSION Islamabad, Pakista (CIIT), for providig excellet research facilities. Authors are also grateful to Prof. Dr. I this paper, we have used a ew series of the Syed Tauseeh Mohyud Di, EditorChief for valueable fuctio f(x), which is obtaied by usig the trapezoidal suggestios ad commets. rule ad fudamet theorem of calculus. This series is used to suggest ad aalyzed a ew iterative method for REFERENCES solvig the oliear equatios. It is a iterestig problem to use this expasio of the fuctio to 1. Richard L. Burde ad J. Douglas Faires, 001. suggest ad cosider some ew iterative method for Numerical Aalysis, PWS publishig compay solvig the variatioal iequalities ad related problems, Bosta. see [69, 170] ad the referece therei. I our other. Chu, C., 005. Iterative methods improvig papers, we will the homotopy perturbatio method ad Newto s method by the decompositio method, some decompositios method to derive several iterative Computers Math. Appl., 50: methods for solvig the oliear equatios. It is a 3. Householder, A.S., The Numerical Treatmet of iterestig problem to derive the iterative methods for a Sigle Noliear Equatio, McGrawHill, New York. solvig system of oliear equatios. 4. Aslam Noor, M., 007. New family of iterative methods for oliear equatios, Appl. Math. ACKNOWLEDGEMENT Computatio, 190: Noor, M.A., New classes of iterative methods for The authors would like to thak Dr. S. M. Juai Zaidi, oliear equatios, Appl. Math. Computatio. Rector, COMSATS Istitute of Iformatio Techology, 191(007),
5 6. Noor, M.A., Geeral variatioal iequalities, 15. Noor, M.A. ad W.A. Kha, 01. Fourthorder Appl. Math. Letters, 1: iterative method free from secod derivative for 7. Noor, M.A., 004. Some developmets i geeral solvig oliear equatios, Appl. Math. Sci., variatioal iequalities, Appl. Math. Comput., 6(93): : Noor, M.A., W.A. Kha ad S. Youus, Noor, M.A., 009. Exteded geeral variatioal Homotopy perturbatio techiques for the solutio iequalities, Appl. Math. Letters, : of certai oliear equatios, Appl. Math. Sci., 9. Noor, M.A., 011. O iterative methods for oliear 6(130): equatios usig homotopy perturbatio techique, 17. Noor, M.A. ad K.I. Noor, 013. Auxiliary priciple Appl. Math. Iform Sci., 4: techique for solvig split feasibility problems, Appl. 10. Noor, M.A., 010. Some iterative methods for solvig Math. Iform. Sci., 7(1): 17. oliear equatios usig homotopy perturbatio 18. Noor, M.A., K.I. Noor ad Th. M. Rassias, method, Iter. J. Computer Math., 87: Some aspects of variatioal iequalities, J. Comput. 11. Noor, M.A. ad K.I. Noor, 006. Iterative schemes for Appl. Math., 47: solvig oliear equatios, Appl. Math. 19. Noor, M.A., K.I. Noor, E. AlSaid ad M. Waseem, Computatio, 183: Some ew iterative methods for oliear 1. Noor, K.I. ad M.A. Noor, 007. Predicotcorrector equatios, Math. Prob. Eg. 010(010), Article ID, Halley method for oliear equatios, Appl. Math : 1. Comput., 188: Noor, M.A., K.I. Noor ad E. AlSaid, 01. Iterative 13. Noor, K.I., M.A. Noor ad S. Momai, 007. Modified methods for solvig ocovex equilibrium Householder iterative method for oliear variatioal iequalities, Appl. Math. Iform. Sci., equatios, Appl. Math. Computatio, 190: (): Noor, M.A. ad W.A. Kha, 01. New iterative 1. Traub, J.F., Iterative Methods for Solutio of methods for solvig oliear equatios by usig Equatios, PreticeHall, Eglewood Cliffs, NJ, homotopy perturbatio method, Appl. Math. Comput., 19:
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