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A Modification on Newton's Method for Solving Systems of Nonlinear Equations

Authors: Jafar Biazar, Behzad Ghanbari


In this paper, we are concerned with the further study for system of nonlinear equations. Since systems with inaccurate function values or problems with high computational cost arise frequently in science and engineering, recently such systems have attracted researcher-s interest. In this work we present a new method which is independent of function evolutions and has a quadratic convergence. This method can be viewed as a extension of some recent methods for solving mentioned systems of nonlinear equations. Numerical results of applying this method to some test problems show the efficiently and reliability of method.


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[1] R.L. Burden, J.D. Faires, Numerical Analysis, 7th ed., PWS Publishing Company, Boston, 2001.
[2] J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several variables, Academic Press, 1970.
[3] E. Babolian, J. Biazar, A.R. Vahidi, Solution of a system of nonlinear equations by Adimian decomposition method, Appl. Math. Comput. Vol. 150, 2004, pp. 847-854.
[4] Frontini, E. Sormani, Third-order methods from quadrature formulae for solving systems of nonlinear equations, Appl. Math. Comput, Vol. 149, 2004, pp. 771-782.
[5] A. Cordero, J.R. Torregrosa, Variants of Newton's method for functions of several variables, Appl. Math. Comput., Vol. 183, 2006, pp.199-208.
[6] F. Freudensten, B. Roth, Numerical solution of systems of nonlinear equations, J. ACM, Vol. 10 , 1963, pp. 550-556.
[7] M. Grau-S├ínchez, J.M. Peris, J.M. Gutiérrez, Accelerated iterative methods for finding solutions of a system of nonlinear equations, Appl. Math. Comput. Vol. 190 , 2007, pp. 1815-1823.
[8] H.H.H. Homeier, A modified Newton method with cubic convergence: The multivariate case, J. Comput. Appl. Math. 169 , 2004), pp 161-169.
[9] J. Kou, A third-order modification of Newton method for systems of nonlinear equations, Appl. Math. Comput, Vol. 191 , 2007, pp. 117-121.
[10] L.F. Shampine, R.C. Allen, S. Pruess, Fundamentals of Numerical Computing, John Wiley and Sons, New York, 1997.
[11] M. Kupferschmid, J.G. Ecker, A note on solution of nonlinear programming problems with imprecise function and gradient values , Math. Program. Study, Vol. 31 , 1987, pp. 129-138.
[12] M.N. Vrahatis, T.N. Grapsa, O. Ragos, F.A. Zafiropoulos, On the localization and computation of zeros of Bessel functions, Z. Angew. Math. Mech, Vol 77 (6, pp, 1997, pp. 467-475.
[13] M.N. Vrahatis, G.D. Magoulas, V.P. Plagianakos, From linear to nonlinear iterative methods, Appl. Numer. Math, Vol. 45 , No.1, 2003, pp. 59-77.
[14] M.N. Vrahatis, O. Ragos, F.A. Zafiropoulos, T.N. Grapsa, Locating and computing zeros of Airy functions, Z. Angew. Math. Mech, Vol. 76, No.7, 1996, pp. 419-422.
[15] W. Chen, A Newton method without evaluation of nonlinear function values, CoRR cs.CE/9906011, 1999.
[16] T.N. Grapsa, E.N. Malihoutsaki, Newton's method without direct function evaluations, in: E. Lipitakis (Ed.), Proceedings of 8th Hellenic European Research on Computer Mathematics & its Applications-Conference, HERCMA 2007, 2007.
[17] E.N. Malihoutsaki, I.A. Nikas, T.N. Grapsa, Improved Newton's method without direct function evaluations, Journal of Computational and Applied Mathematics, Vol. 227 , 2009, pp. 206-212.
[18] T.N. Grapsa, Implementing the initialization-dependence and the singularity difficulties in Newton's method, Tech. Rep. 07-03, Division of Computational Mathematics and Informatics, Department of Mathematics, University of Patras, 2007.
[19] T.N. Grapsa, M.N. Vrahatis, A dimension-reducing method for solving systems of nonlinear equations in ¶Çü£n , Int. J. Comput. Math, Vol. 32 , 1990, pp. 205-216.
[20] T.N. Grapsa, M.N. Vrahatis, A dimension-reducing method for unconstrained optimization, J. Comput. Appl. Math, Vol. 66, No.1-2, 1996, pp. 239-253.
[21] D.G. Sotiropoulos, J.A. Nikas, T.N. Grapsa, Improving the efficiency of a polynomial system solver via a reordering technique, in: D.T. Tsahalis (Ed.), Proceedings of 4th GRACM Congress on Computational Mechanics, Vol. III, 2002.