{"title":"A Modification on Newton's Method for Solving Systems of Nonlinear Equations","authors":"Jafar Biazar, Behzad Ghanbari","volume":34,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":841,"pagesEnd":846,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/13521","abstract":"
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.<\/p>\r\n","references":"[1] R.L. Burden, J.D. Faires, Numerical Analysis, 7th ed., PWS Publishing\r\nCompany, Boston, 2001.\r\n[2] J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations\r\nin Several variables, Academic Press, 1970.\r\n[3] E. Babolian, J. Biazar, A.R. Vahidi, Solution of a system of\r\nnonlinear equations by Adimian decomposition method, Appl. Math.\r\nComput. Vol. 150, 2004, pp. 847-854.\r\n[4] Frontini, E. Sormani, Third-order methods from quadrature formulae for\r\nsolving systems of nonlinear equations, Appl. Math. Comput, Vol. 149,\r\n2004, pp. 771-782.\r\n[5] A. Cordero, J.R. Torregrosa, Variants of Newton's method for functions\r\nof several variables, Appl. Math. Comput., Vol. 183, 2006, pp.199-208.\r\n[6] F. Freudensten, B. Roth, Numerical solution of systems of nonlinear\r\nequations, J. ACM, Vol. 10 , 1963, pp. 550-556.\r\n[7] M. Grau-S\u251c\u00ednchez, J.M. Peris, J.M. Guti\u00e9rrez, Accelerated iterative\r\nmethods for finding solutions of a system of nonlinear equations, Appl.\r\nMath. Comput. Vol. 190 , 2007, pp. 1815-1823.\r\n[8] H.H.H. Homeier, A modified Newton method with cubic convergence:\r\nThe multivariate case, J. Comput. Appl. Math. 169 , 2004), pp 161-169.\r\n[9] J. Kou, A third-order modification of Newton method for systems of\r\nnonlinear equations, Appl. Math. Comput, Vol. 191 , 2007, pp. 117-121.\r\n[10] L.F. Shampine, R.C. Allen, S. Pruess, Fundamentals of Numerical\r\nComputing, John Wiley and Sons, New York, 1997.\r\n[11] M. Kupferschmid, J.G. Ecker, A note on solution of nonlinear\r\nprogramming problems with imprecise function and gradient values ,\r\nMath. Program. Study, Vol. 31 , 1987, pp. 129-138.\r\n[12] M.N. Vrahatis, T.N. Grapsa, O. Ragos, F.A. Zafiropoulos, On the\r\nlocalization and computation of zeros of Bessel functions, Z. Angew. Math. Mech, Vol 77 (6, pp, 1997, pp. 467-475.\r\n[13] M.N. Vrahatis, G.D. Magoulas, V.P. Plagianakos, From linear to\r\nnonlinear iterative methods, Appl. Numer. Math, Vol. 45 , No.1, 2003, pp. 59-77.\r\n[14] M.N. Vrahatis, O. Ragos, F.A. Zafiropoulos, T.N. Grapsa, Locating and\r\ncomputing zeros of Airy functions, Z. Angew. Math. Mech, Vol. 76, No.7, 1996, pp. 419-422.\r\n[15] W. Chen, A Newton method without evaluation of nonlinear function\r\nvalues, CoRR cs.CE\/9906011, 1999.\r\n[16] T.N. Grapsa, E.N. Malihoutsaki, Newton's method without direct\r\nfunction evaluations, in: E. Lipitakis (Ed.), Proceedings of 8th Hellenic European Research on Computer Mathematics & its Applications-Conference, HERCMA 2007, 2007.\r\n[17] E.N. Malihoutsaki, I.A. Nikas, T.N. Grapsa, Improved Newton's method\r\nwithout direct function evaluations, Journal of Computational and\r\nApplied Mathematics, Vol. 227 , 2009, pp. 206-212.\r\n[18] T.N. Grapsa, Implementing the initialization-dependence and the\r\nsingularity difficulties in Newton's method, Tech. Rep. 07-03, Division\r\nof Computational Mathematics and Informatics, Department of\r\nMathematics, University of Patras, 2007.\r\n[19] T.N. Grapsa, M.N. Vrahatis, A dimension-reducing method for solving\r\nsystems of nonlinear equations in \u00b6\u00c7\u00fc\u00a3n , Int. J. Comput. Math, Vol. 32 ,\r\n1990, pp. 205-216.\r\n[20] T.N. Grapsa, M.N. Vrahatis, A dimension-reducing method for\r\nunconstrained optimization, J. Comput. Appl. Math, Vol. 66, No.1-2,\r\n1996, pp. 239-253.\r\n[21] D.G. Sotiropoulos, J.A. Nikas, T.N. Grapsa, Improving the efficiency of\r\na polynomial system solver via a reordering technique, in: D.T. Tsahalis (Ed.), Proceedings of 4th GRACM Congress on Computational\r\nMechanics, Vol. III, 2002.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 34, 2009"}