Extending the Convergence Domain of Newton’s Method

Abstract

We present a local as well a semilocal convergence analysis for Newton’s method in a Banach space setting. Using the same Lipschitz constants as in earlier studies (S. Amat, S. Busquier, J. Math. Anal. Appl. 336:243–261, 2007, [2], I.K. Argyros, J. Comput. Math. 169:315–332, 2004, [4], I.K. Argyros, Math. Comput. 80:327–343, 2011, [5], I.K. Argyros, D. González, Appl. Math. Comput. 234:167–178, 2014, [6], I.K. Argyros, S. Hilout, J. Complex., AMS, 28:364–387, 2012, [7], I.K. Argyros, S. Hilout, Numerical methods in Nonlinear Analysis, 2013, [8], I.K. Argyros, S. Hilout, Appl. Math. Comput. 225:372–386, 2013, [9], J.A. Ezquerro, M.A. Hernández, How to improve the domain of parameters for Newton’s method, to appear in Appl. Math. Lett, [11], J.M. Gutiérrez, Á.A. Magreñán, N. Romero, Appl. Math. Comput. 221:79–88, 2013, [13], L.V. Kantorovich, G.P. Akilov, Functional Analysis, 1982, [14], J.F. Traub, Iterative Methods for the Solution of Equations, 964, [15]) we extend the applicability of, Newton’s method as follows: Local case: A larger radius is given as well as more precise error estimates on the, distances involved. Semilocal case: the convergence domain is extended; the error estimates are tighter and the information on the location of the solution is at least as precise as before. Numerical examples further justify the theoretical results. It follows (I.K. Argyros, Á.A. Magreñán, Extending the applicability of the local and semilocal convergence of Newton’s method, submitted for publication, 2015, [10]).