A note on the quadratic convergence of the inexact Newton methods
Keywords:nonlinear systems of equations in \(R^n\), inexact Newton methods, inexact and perturbed Newton methods, convergence orders, backward errors
We show that a new sufficient condition for the convergence with \(q\)-order two of the inexact Newton iterates may be obtained by considering the normwise backward error of the approximate steps and a result on perturbed Newton methods.
This condition is in fact equivalent to the characterization given by Dembo, Eisenstat and Steihaug.
E. Catinas, Inexact perturbed Newton methods, and some applications for a class of Krylov solvers, J. Optim. Theory Appl., 108 (2001) no. 3, pp. 543–570, https://doi.org/10.1023/a:1017583307974
E. Catinas, On the high convergence orders of the Newton-GMBACK methods, Rev. Anal. Numer. Theor. Approx., 28 (1999) no. 2, pp. 125–132.
E. Catinas, Newton and Newton-Krylov methods for solving nonlinear systems in Rn, Ph.D. thesis, ”Babes-Bolyai” University of Cluj–Napoca, Romania, 1999.
E. Catinas, The relationship between three practical models of Newton methods with high convergence orders , submitted.
E. Catinas, Inexact perturbed Newton methods for nonlinear systems with singular Jacobians, submitted.
E. Catinas, The high convergence orders of the successive approximations, submitted.
E. Catinas, Finite difference approximations in the Newton-Krylov methods, manuscript.
D. Cores and R.A. Tapia, Perturbation Lemma for the Newton Method with Application to the SQP Newton Method, J. Optim. Theory Appl., 97 (1998), pp. 271–280, https://doi.org/10.1023/a:1022622532499
R. S. Dembo, S. C. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), pp. 400–408, https://doi.org/10.1137/0719025
J. E. Dennis, Jr. and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice–Hall Series in Computational Mathematics, Englewood Cliffs, NJ, 1983.
N. J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, PA, 1996.
St. Maruster, Numerical Methods for Solving Nonlinear Equations, Ed. Tehnica, Bucharest, 1981 (in Romanian).
J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
I. Pavaloiu, Introduction to the Approximation of the Solutions of Equations, Ed. Dacia, Cluj-Napoca, 1976 (in Romanian).
F. A. Potra, On Q-order and R-order of convergence, J. Optim. Theory Appl., 63 (1989), pp. 415–431, https://doi.org/10.1007/bf00939805
W. C. Rheinboldt, Methods for Solving Systems of Nonlinear Equations, Second ed., SIAM, Philadelphia, 1998.
J. L. Rigal and J. Gaches, On the compatibility of a given solution with the data of a linear system, J. ACM, 14 (1967), pp. 543–548, https://doi.org/10.1145/321406.321416
H. F. Walker, An approach to continuation using Krylov subspace methods, Research Report 1/97/89, Dept. of Math., Utah State University, appeared in Computational Science in the 21st Century, M.–O. Bristeau, G. Etgen, W. Fitzgibbon, J. L. Lions, J. Periaux and M. F. Wheeler, eds., John Wiley and Sons, Ltd., 1997, pp. 72–82
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