Local Convergence and Radius of Convergence for Modified Newton Method

Măruşter, Ştefan

doi:10.1515/awutm-2017-0020

Abstract

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.

References

[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), 37–46.10.1515/awutm-2016-0013
Search in Google Scholar Back to article
[2] J.A. Ezquerro and M.A. Hernandez, An optimization of Chebyshev’s method, Journal of Complexity, 25, (2009), 343–361.10.1016/j.jco.2009.04.001
Open DOI Search in Google Scholar Back to article
[3] M.A. Hernndez-Veron and N. Romero, On the Local Convergence of a Third Order Family of Iterative Processes, Algorithms, 8, (2015), 1121–1128.10.3390/a8041121
Search in Google Scholar Back to article
[4] T.L. Hicks and J.D. Kubicek, On the Mann iteration process in a Hilbert spaces, J. Math. Anal. Appl., 59, (1977), 489–504.10.1016/0022-247X(77)90076-2
Search in Google Scholar Back to article
[5] St. Maruster, The solution by iteration of nonlinear equations in Hilbert spaces, Proc. Amer. Math.Soc., 63, (1977), 69–73.10.1090/S0002-9939-1977-0636944-2
Open DOI Search in Google Scholar Back to article
[6] St. Maruster and I.A. Rus, Kannan contractions and strongly demicontractive mappings, Creative Mathematics and Informatics, 24, (2015), 171 – 180.10.37193/CMI.2015.02.10
Search in Google Scholar Back to article
[7] St. Maruster, Estimating local radius of convergence for Picard iteration, Algorithms, 10, (2017), doi:10.3390/a10010010.10.3390/a10010010
Search in Google Scholar Back to article
[8] J.M. Ortega and W.C. Rheinboldt, Iterative solution of nonlinear equation in several variables, Acad. Press, New York, 1970.
Search in Google Scholar Back to article
[9] C.L. Outlaw, Mean value iteration for nonexpansive mappings in a Banach space, Pacific J. Math., 30, (1969), 747–759.10.2140/pjm.1969.30.747
Search in Google Scholar Back to article
[10] F.A. Potra and V. Ptak, Nondiscrete induction and iterative proccesses, 1984.
Search in Google Scholar Back to article
[11] Y. Qing and B.E. Rhoades, T-stability of Picard iteration in metric spaces, Fixed Point Theory and Applications, (2008), Article ID418971.10.1155/2008/418971
Open DOI Search in Google Scholar Back to article
[12] W.C. Rheinboldt, An adaptive continuation process for solving systems of nonlinear equations, Polish Acad. Sci. Banach Center Publ., 3, (1975), 129–140.10.4064/-3-1-129-142
Search in Google Scholar Back to article
[13] H.F. Senter and W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44, (1974), 375–308.10.1090/S0002-9939-1974-0346608-8
Open DOI Search in Google Scholar Back to article
[14] J.R. Sharma and P.K. Gupta, An efficient fifth order method for solving systems of nonlinear equations, Comput. Math. Appl., 67, (2014), 591–601.10.1016/j.camwa.2013.12.004
Search in Google Scholar Back to article
[15] F. Soleymani, Optimal eighth-order simple root-finders free from derivative, WSEAS Transactions on Information Science and Applications, 8, (2011), 293–299.
Search in Google Scholar Back to article
[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.
Search in Google Scholar Back to article
[17] J.F.Traub, Iterative methods for the solution of equations, Chelsea Publishing Company, New York, 1982.
Search in Google Scholar Back to article

Local Convergence and Radius of Convergence for Modified Newton Method

Abstract

Paradigm

My account