No q-superlinear convergence to a fixed point \(x^\ast\) of a nonlinear mapping \(G\) may be attained by the successive approximations when \(G^\prime(x^\ast)\) has no eigenvalue equal to 0, as shown by us in .
However, high q-convergence orders may be attained if one considers perturbed successive approximations.
We characterize the correction terms which must be added at each step in order to obtain convergence with q-order 2 of the resulted iterates.
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)
fixed point problems; acceleration of convergence; nonlinear system of equations in Rn; inexact Newton method; linear systems of equation in Rn; residual; local convergence; q-convergence order.
E. Cătinaş, On accelerating the convergence of the successive approximations method, Rev. Anal. Numér. Théor. Approx., 30 (2001) no. 1, pp. 3-8.
 I. Argyros, F. Szidarovszky, The Theory and Applications of Iteration Methods, CRC Press, Boca Raton, 1993.
 I. Argyros, On the convergence of the modified contractions, J. Comp. Appl. Math., 55(1994), 183–189.
 E. Catinas, Newton and Newton-Krylov methods for solving nonlinear systems in Rn, PhD Thesis, Babes-Bolyai University of Cluj-Napoca, Cluj-Napoca, Romania, 1999.
 E. Catinas, On the high convergence orders of the Newton-GMBACK methods, Rev. Anal. Numer. Theor. Approx., 28 (1999) no. 2, 125-132.
 E. Catinas, A note on the quadratic convergence of the inexact Newton methods, Rev. Anal. Numer. Theor. Approx. 29 (2000) no. 2, 129-133.
 E. Catinas, Inexact perturbed Newton methods and applications to a class of Krylov solvers, J. Optim. Theory Appl., 108 (2001) no. 3, 543-570.
 E. Catinas, The inexact, inexact perturbed and quasi-Newton methods are equivalent models, submitted.
 E. Catinas, On the superlinear convergence of the successive approximations method, submitted.
 R.S. Dembo, S.C. Eisenstat, T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), 400-408.
 J.E. Dennis, Jr., J. J. More, A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Comp., 28 (1974), 549-560.
 J.E. Dennis, Jr., R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall Series in Computational Mathematics, Engle-wood Cliffs, 1983.
 P. Deuflhard, F. A. Potra, Asymptotic mesh independence of Newton-Galerkin methods via a refined Mysovskii theorem, SIAM J. Numer. Anal.,29 (1992), 1395-1412.
 S.C. Eisenstat, H.F. Walker, Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput.,17(1996), 16-32.
 Emil Catinas, N.J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia,1996.
 V.I. Istratescu, Introduction to the Fixed Points Theory, Editura Academiei RSR, Bucharest, Romania, 1973 (in Romanian).
 C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM, Philadelphia, Pennsylvania, 1995.
 St. Maruster, Quasi-nonexpansivity and two classical methods for solving nonlinear equations, Proc. AMS, 62 (1977), 119-123
 St. Maruser, Numerical Methods for Solving Nonlinear Equations, Editura Tehnica, Bucharest, Romania, 1981 (in Romanian).
 J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.
 A.M. Ostrowski, Solution of Equations and Systems of Equations, Academic Press, New York, 1966.
 I.Pavaloiu, Introduction to the Theory of Approximating the Solutions of Equations, Editura Dacia, Cluj-Napoca, Romania, 1976 (in Romanian).
 F.A. Potra, V. Ptak, Nondiscrete Induction and Iterative Processes, Pitman, London,1984.
 F.A. Potra, On Q-order and R-order of convergence, J. Optim. Theory Appl., 63 (1989), 415–431.
 F.A. Potra, Q-superlinear convergence of the iterates in primal-dual interior-point methods, Math. Progr., to appear.
 W.C. Rheinboldt, Methods for Solving Systems of Nonlinear Equations, SIAM, Philadelphia, 1998.
 H.F. Walker, An approach to continuation using Krylov subspace methods, Computational Science in the 21st Century, M.-O. Bristeau, G. Etgen, W. Fitzgibbon, J. L.Lions, J. Periaux and M. F. Wheeler, editors, John Wiley and Sons, Ltd., 72-82, 1997