Local convergence of Newton's method using Kantorovich convex majorants

Ioannis K. Argyros

doi:10.33993/jnaat392-1029

Authors

Ioannis K. Argyros Cameron University, USA

DOI:

https://doi.org/10.33993/jnaat392-1029

Keywords:

Newton's method, Banach space, Kantorovich's majorants, convex function, local/semilocal convergence, Fréchet-derivative, radius of convergence

Abstract views: 269

Abstract

We are concerned with the problem of approximating a solution of an operator equation using Newton's method. Recently in the elegant work by Ferreira and Svaiter [6] a semilocal convergence analysis was provided which makes clear the relationship of the majorant function with the operator involved. However these results cannot provide information about the local convergence of Newton's method in their present form. Here we have rectified this problem by using two flexible majorant functions. The radius of convergence is also found. Finally, under the same computational cost, we show that our radius of convergence is larger, and the error estimates on the distances involved is finer than the corresponding ones [1], [11]-[13].

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References

J. Appel, E. De Pascale, J.V. Lysenko and P.P. Zabrejko, New results on Newton--Kantorovich approximations with applications to nonlinear integral equations, Numer. Funct. Anal. and Optimiz., 18, pp. 1-17, 1997, https://doi.org/10.1080/01630569708816744 DOI: https://doi.org/10.1080/01630569708816744

I.K. Argyros, On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math., 169, pp. 315-332, 2004, https://doi.org/10.1016/j.cam.2004.01.029 DOI: https://doi.org/10.1016/j.cam.2004.01.029

I.K. Argyros, A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. and Appl., 298, 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

I.K. Argyros, Computational theory of iterative methods, Studies in Computational Mathematics, 15, Elsevier, 2007, New York, U.S.A.

I.K. Argyros, On the convergence of the Secant method under the gamma condition, Cent. Eur. J. Math., 5, pp. 205-214, 2007, https://doi.org/10.2478/s11533-007-0007-3 DOI: https://doi.org/10.2478/s11533-007-0007-3

O.P. Ferreira and B.F. Svaiter , Kantorovich's majorants principle for Newton's method, Comput. Optim. and Appl., 2007, in press. DOI: https://doi.org/10.1007/s10589-007-9082-4

J.M. Gutiérrez, M.A. Hernánadez and M.A. Salanova, Accessibility of solutions by Newton's method, Inter. J. Comput. Math., 57, pp. 239-247, 1995, https://doi.org/10.1080/00207169508804427 DOI: https://doi.org/10.1080/00207169508804427

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

Y. Nesterov and A. Nemirovskii, Interior-point polynomial algorithms in convex programming, SIAM Studies in Appl. Math., 13, Philadelphia, PA, 1994. DOI: https://doi.org/10.1137/1.9781611970791

F.A. Potra, The Kantorovich theorem and interior point methods, Mathematical Programming, 102, pp. 47-50, 2005, https://doi.org/10.1007/s10107-003-0501-8 DOI: https://doi.org/10.1007/s10107-003-0501-8

W.C. Rheinboldt, An adaptive continuation process for solving systems of nonlinear equations, Banach Center Publ., 3, pp. 129-142, 1975, https://doi.org/10.4064/-3-1-129-142 DOI: https://doi.org/10.4064/-3-1-129-142

S. Smale, Newton method estimates from data at one point. The merging of disciplines: new directions in pure, applied and computational mathematics, (R. Ewing, K. Gross, C. Martin, eds), Springer-Verlag, New York, pp. 185-196, 1986, https://doi.org/10.1007/978-1-4612-4984-9_13 DOI: https://doi.org/10.1007/978-1-4612-4984-9_13

X. Wang, Convergence on Newton's method and inverse function theorem in Banach space, Math. Comput., 68, pp. 169-186, 1999, https://doi.org/10.1090/s0025-5718-99-00999-0 DOI: https://doi.org/10.1090/S0025-5718-99-00999-0

T.J. Ypma, Local convergence of inexact Newton Methods, SIAM J. Numer. Anal., 21, pp. 583-590, 1984, https://doi.org/10.1137/0721040 DOI: https://doi.org/10.1137/0721040