Local Convergence and Radius of Convergence for Modified Newton Method

Open access


We investigate the local convergence of modified Newton method, i.e., the classical Newton method in which the derivative is periodically re-evaluated. Based on the convergence properties of Picard iteration for demicontractive mappings, we give an algorithm to estimate the local radius of convergence for considered method. Numerical experiments show that the proposed algorithm gives estimated radii which are very close to or even equal with the best ones.

[1] I.K. Argyros and S. George, Convergence analysis of a three step Newton-like method for nonlinear equations in Banach space under weak conditions, Analele Universitatii de Vest, Timisoara Seria Matematica si Informatica, LIV, (2016), 37–46.

[2] J.A. Ezquerro and M.A. Hernandez, An optimization of Chebyshev’s method, Journal of Complexity, 25, (2009), 343–361.

[3] M.A. Hernndez-Veron and N. Romero, On the Local Convergence of a Third Order Family of Iterative Processes, Algorithms, 8, (2015), 1121–1128.

[4] T.L. Hicks and J.D. Kubicek, On the Mann iteration process in a Hilbert spaces, J. Math. Anal. Appl., 59, (1977), 489–504.

[5] St. Maruster, The solution by iteration of nonlinear equations in Hilbert spaces, Proc. Amer. Math.Soc., 63, (1977), 69–73.

[6] St. Maruster and I.A. Rus, Kannan contractions and strongly demicontractive mappings, Creative Mathematics and Informatics, 24, (2015), 171 – 180.

[7] St. Maruster, Estimating local radius of convergence for Picard iteration, Algorithms, 10, (2017), doi:10.3390/a10010010.

[8] J.M. Ortega and W.C. Rheinboldt, Iterative solution of nonlinear equation in several variables, Acad. Press, New York, 1970.

[9] C.L. Outlaw, Mean value iteration for nonexpansive mappings in a Banach space, Pacific J. Math., 30, (1969), 747–759.

[10] F.A. Potra and V. Ptak, Nondiscrete induction and iterative proccesses, 1984.

[11] Y. Qing and B.E. Rhoades, T-stability of Picard iteration in metric spaces, Fixed Point Theory and Applications, (2008), Article ID418971.

[12] W.C. Rheinboldt, An adaptive continuation process for solving systems of nonlinear equations, Polish Acad. Sci. Banach Center Publ., 3, (1975), 129–140.

[13] H.F. Senter and W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44, (1974), 375–308.

[14] J.R. Sharma and P.K. Gupta, An efficient fifth order method for solving systems of nonlinear equations, Comput. Math. Appl., 67, (2014), 591–601.

[15] F. Soleymani, Optimal eighth-order simple root-finders free from derivative, WSEAS Transactions on Information Science and Applications, 8, (2011), 293–299.

[16] R. Thukral, New modification of Newton with third order of convergence for solving nonlinear equation of type f(0) = 0, Amer. J. Comput. Appl. Math., 6, (2016), 14–18.

[17] J.F.Traub, Iterative methods for the solution of equations, Chelsea Publishing Company, New York, 1982.

Journal Information

Mathematical Citation Quotient (MCQ) 2016: 0.01

Target Group

researchers in all branches of mathematics and computer science


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 160 160 28
PDF Downloads 74 74 10