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 [8].

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.


Emil Cătinaş
(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.

Cite this paper as:

E. Cătinaş, On accelerating the convergence of the successive approximations method, Rev. Anal. Numér. Théor. Approx., 30 (2001) no. 1, pp. 3-8.


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[1] I.  Argyros,  F.  Szidarovszky, The  Theory  and  Applications  of  Iteration  Methods, CRC Press, Boca Raton, 1993.

[2] I. Argyros, On the convergence of the modified contractions, J. Comp. Appl. Math., 55(1994), 183–189.

[3] 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.

[4] E. Catinas, On the high convergence orders of the Newton-GMBACK methods, Rev. Anal. Numer. Theor. Approx., 28 (1999) no. 2, 125-132.

[5] E. Catinas, A note on the quadratic convergence of the inexact Newton methods, Rev. Anal. Numer. Theor. Approx. 29 (2000) no. 2, 129-133.

[6] E. Catinas, Inexact  perturbed  Newton  methods  and  applications  to  a  class  of  Krylov solvers, J. Optim. Theory Appl., 108 (2001) no. 3, 543-570.

[7] E. Catinas, The inexact, inexact perturbed and quasi-Newton methods are equivalent models, submitted.

[8] E. Catinas, On the superlinear convergence of the successive approximations method, submitted.

[9] R.S. Dembo, S.C. Eisenstat, T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), 400-408.

[10] 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.

[11] 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.

[12] 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.

[13] S.C.  Eisenstat,  H.F.  Walker, Choosing  the  forcing  terms  in  an  inexact  Newton method, SIAM J. Sci. Comput.,17(1996), 16-32.

[14] Emil Catinas, N.J.  Higham, Accuracy  and  Stability  of  Numerical  Algorithms,  SIAM,  Philadelphia,1996.

[15] V.I. Istratescu, Introduction  to  the  Fixed  Points  Theory,  Editura  Academiei  RSR, Bucharest, Romania, 1973 (in Romanian).

[16] C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM, Philadelphia, Pennsylvania, 1995.

[17] St. Maruster, Quasi-nonexpansivity and two classical methods for solving nonlinear equations, Proc. AMS, 62 (1977), 119-123

[18] St. Maruser, Numerical  Methods  for  Solving  Nonlinear  Equations, Editura Tehnica, Bucharest, Romania, 1981 (in Romanian).

[19] J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.

[20] A.M. Ostrowski, Solution  of  Equations  and  Systems  of  Equations, Academic Press, New York, 1966.

[21] I.Pavaloiu, Introduction to the Theory of Approximating the Solutions of Equations, Editura Dacia, Cluj-Napoca, Romania, 1976 (in Romanian).

[22] F.A. Potra, V. Ptak, Nondiscrete Induction and Iterative Processes, Pitman, London,1984.

[23] F.A. Potra, On Q-order  and R-order  of  convergence,  J. Optim. Theory Appl., 63 (1989), 415–431.

[24] F.A. Potra, Q-superlinear  convergence  of  the  iterates  in  primal-dual  interior-point methods, Math. Progr., to appear.

[25] W.C.  Rheinboldt, Methods  for  Solving  Systems  of  Nonlinear  Equations,  SIAM, Philadelphia, 1998.

[26] 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

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