# Some New Iterative Methods for Solving Nonlinear Equations

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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 secod-order coverget. Numerical Results: We preset some examples to illustrate the efficiecy of the ew developed two-step 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 e-6 NR e e-8 NR e e-3 f, x 0 = NM e e-8 NR e-3.11e-16 NR e e-19 f 3, x 0 = 1.7 NM e-3.34e-16 NR e e-7 NR e e-7 f 4, x 0 = 3.5 NM 8.06e-4 8.8e- NR e e-30 NR e e-8 f 5, x 0 = 1.5 NM e e-8 NR e-45.30e-3 NR e e-19 f 6, x 0 = NM e-40.73e-1 NR e-3 5.8e-17 NR e e- f 7, x 0 = 3.5 NM e e-5 NR e e-19 NR e e-16 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, Editor--Chief 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 [6-9, 17-0] 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, McGraw-Hill, 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. Fourth-order 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): 1-7. 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. Al-Said ad M. Waseem, Computatio, 183: Some ew iterative methods for oliear 1. Noor, K.I. ad M.A. Noor, 007. Predicot-corrector 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. Al-Said, 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, Pretice-Hall, Eglewood Cliffs, NJ, homotopy perturbatio method, Appl. Math. Comput., 19:

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