Convergence analysis for the two-step Newton method of order four

Ioannis K. Argyros; Sanjay K. Khattri

doi:10.33993/jnaat431-993

Authors

Ioannis K. Argyros Cameron University, USA
Sanjay K. Khattri Stord Haugesund University College, Norway

DOI:

https://doi.org/10.33993/jnaat431-993

Keywords:

two-step Newton method, Newton's method, Banach space, Kantorovich hypothesis, majorizing sequence, Lipschitz/center-Lipschitz condition

Abstract views: 286

Abstract

We provide a tighter than before convergence analysis for the two-step Newton method of order four using recurrent functions. Numerical examples are also provided in this study.

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References

S. Amat, S. Busqouier and J.M. Gutierrez, On the local convergence of secant-type methods, Int. J. Comput. Math., 81 (2004), no.9, pp. 1153-1161, https://doi.org/10.1080/00207160412331284123 DOI: https://doi.org/10.1080/00207160412331284123

J.Appell, E. De Pascale, N.A. Wvkhuta and P.P. Zabrejko, On the two-stup Newton method for the solution of nonlinear operator equations, Math. Nachr, 172, (1995), pp. 5-14, https://doi.org/10.1002/mana.19951720102 DOI: https://doi.org/10.1002/mana.19951720102

I.K. Argyro, On a multistep Newton method in Banach spaces and the Ptak error estimates, Adv. N onlinear Var. Inequal., 6, (2003), no.2, pp. 121-135.

I.K. Argyros, A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach spaces, J. Math. Anal. Appl., 298 (2004), no.2, pp. 374-397, https://doi.org/10.1016/j.jmaa.2004.04.008 DOI: https://doi.org/10.1016/j.jmaa.2004.04.008

I.K. Argyros, J.Y. Cho and S. Hilout, Numerical emthods for equations and its applications, CRC Press Taylor & Francis Group 2012, New York. DOI: https://doi.org/10.1201/b12297

R.P. Brent, Algorithms for m inimisation qithout derivatives, Prentice Hall, Englewood Cliffs, New Jersey, 1973.

E. Catinas, On some iterative methods for solving nonlinear equations, Rev. Anal. Numer. Theor., Approx., 23 (1994), no. 1, pp. 47-53.

J.A. Ezquerro and M.A. Hernandez, Multipoint super-Halley type approximation algorithms in Banach spaces, Numer. Funct. Anal. Optim., 21 (2000), no.7-8, pp.845-858, https://doi.org/10.1080/01630560008816989 DOI: https://doi.org/10.1080/01630560008816989

J.A.Ezquerro, M.A., Hernandez and M.A. Salanova, A Newton-like method for solving some boundary value problems, Numer. Funct. Anal. Optim., 23 (2002), no. 7-8, pp. 791-805, https://doi.org/10.1081/nfa-120016270 DOI: https://doi.org/10.1081/NFA-120016270

J.A. Ezquerro, M.a. Hernandez and M.A. Salanova, A discretization scheme for some conservative problems, J. Comput. Appl. Math., 115 (2000), no.1-2, pp. 181-192, https://doi.org/10.1016/s0377-0427(99)00115-6 DOI: https://doi.org/10.1016/S0377-0427(99)00115-6

M.A. Hernandez, M.J. Rubio and J.A. Ezquerro, Secant-like methods for solving nonlinear integral equaitons of the Hammerstein type, J. Comput. Appl. Math., 115 (2000), no.1-2, pp. 245-254, https://doi.org/10.1016/s0377-0427(99)00116-8 DOI: https://doi.org/10.1016/S0377-0427(99)00116-8

M.A. Hernandez and M.J. Rubio, Semilocal convergence of the secant method under mild convergence conditions of differentiab ility. Comput. Math. Appl., 44 (2002), no. 3-4, pp. 277-285, https://doi.org/10.1016/s0898-1221(02)00147-5 DOI: https://doi.org/10.1016/S0898-1221(02)00147-5

L.V. Kantorovich and G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.

A.M. Ostrowski, Solutions of equations in Euclidean and Banach spaces, A Series of Monographs and Textbooks, Academic Press, New York, 1973.

J.M. Ortega and W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970.

I. Pavaloiu, A convergence theorem concerning the method of Chord, Rev. Anal. Numer. Theor. Approx., 21 (1972), no.1, pp. 59-65.

F.A. potra and V. Ptak, Nondiscrete induction and iterative processes, Research Notes in Mathematics, 103, Pitman Avanced Publ. Program, Boston, 1974.