Finding good starting points for solving equations by Newton's method

Authors

  • Ioannis K. Argyros Cameron University, USA

DOI:

https://doi.org/10.33993/jnaat381-897

Keywords:

Newton's method, Banach space, Newton-Kantorovich theorem/hypothesis, Fréchet-derivative, Lipschitz, center-Lipschitz conditions, good starting points for Newton's method
Abstract views: 252

Abstract

We study the problem of finding good starting points for the semilocal convergence of Newton's method to a locally unique solution of an operator equation in a Banach space setting. Using a weakened version of the Newton-Kantorovich theorem we show that the procedure suggested by Kung [6] is improved in the sense that the number of Newton-steps required to compute a good starting point can be significantly reduced (under the same computational cost required in the Newton--Kantorovich theorem [3], [5]).

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References

Argyros, I.K., A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl., 298, 2, pp. 374-397, 2004, https://doi.org/10.1016/j.jmaa.2004.04.008 DOI: https://doi.org/10.1016/j.jmaa.2004.04.008

Argyros, I.K., An improved approach of obtaining good starting points for solving equations by Newton's method, Adv. Nonlinear Var. Ineq., 8, pp. 133-142, 2005.

Argyros, I.K., Computational theory of iterative methods, Studies in Computational Mathematics, 15, Elsevier, 2007, New York, U.S.A., https://doi.org/10.1016/s1570-579x(13)60006-3 DOI: https://doi.org/10.1016/S1570-579X(13)60006-3

Avila, J., Continuation method for nonlinear equations, Technical Report TR-142, Computer Science Center, University of Maryland, January, 1971.

Kantorovich, L.V. and Akilov, G.P. Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1982. DOI: https://doi.org/10.1016/B978-0-08-023036-8.50010-2

Kung, H.T., The complexity of obtaining starting points for solving operator equations by Newton's method, Technical Report NR 044-422, Department of Computer Science, Carnegie-Mellon University, Pittsburgh, PA, 15213, October, 1975.

Traub, J.F. (editor), Analytic Computational Complexity, Academic Press, 1975.

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Published

2009-02-01

How to Cite

Argyros, I. K. (2009). Finding good starting points for solving equations by Newton’s method. Rev. Anal. Numér. Théor. Approx., 38(1), 1–8. https://doi.org/10.33993/jnaat381-897

Issue

Vol. 38 No. 1 (2009)

Section

Articles

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Copyright (c) 2015 Journal of Numerical Analysis and Approximation Theory

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