GMBACK is a Krylov solver for linear systems in \(\mathbb{R}^n\).

We analyze here the high convergence orders (superlinear convergence) of the Newton-GMBACK methods, which can be characterized applying three different existing results.

In this note we show by some direct calculations that these characterizations are equivalent.


Emil Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)


nonlinear system of equations in Rn; inexact Newton method; Krylov methods; linear systems of equation in Rn; residual; local convergence; superlinear convergence.

Cite this paper as:

E. Cătinaş, On the high convergence orders of the Newton-GMBACK methods, Rev. Anal. Numér. Théor. Approx., 28 (1999) no. 2, pp. 125-132.


Scanned paper.

Latex-pdf version of the paper.

About this paper

Print ISSN


Online ISSN






Google Scholar citations


Paper in html format


[1] P.N. Brown, A theoretical comparison of the Arnoldi and GMRES algorithms, SIAM, J. Sci. Stat. Comput., 12 (1991) no. 1, pp. 58–78.

[2] P.N.  Brown and H.F.  Walker, GMRES on (nearly) singular systems,  SIAM  J. Matrix Anal. Appl., 18 (1997) no. 1, pp. 37–51.

[3] E. Catinas, A note on inexact secant methods, Rev. Anal. Numer. Theor. Approx., 25 (1996) nos. 1–2, pp. 33–41.

[4] E. Catinas, Newton and Newton-Krylov methods for solving nonlinear systems in Rn, Ph.D. Thesis, Cluj-Napoca, Romania, (to be defended).

[5] E. Catinas, Inexact perturbed Newton methods and some applications for a class of Krylov solvers, J. Optim. Theory Appl., submitted.

[6] R.S. Dembo, S.C. Eisenstat, and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982) no. 2, pp. 400–408.

[7] J.E. Dennis, Jr. and J.J. More, A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Comput., 28 (1974) no. 126, pp. 549–560.

[8] J.E.  Dennis,  Jr. and J.J.  More, Quasi-Newton methods, motivation and theory, SIAM Review, 19 (1977) no. 1, pp. 46–89.

[9] 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.

[10] S.C. Eisenstat and H.F. Walker, Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17 (1996) no.1, pp. 16–32.

[11] E.M.  Kasenally, GMBACK: a generalised minimum backward error algorithm for nonsymmetric linear systems, SIAM J. Sci. Comput., 16 (1995) no. 3, pp. 698–719.

[12] E.M. Kasenally and V. Simoncini, Analysis of a minimum perturbation algorithm for nonsymmetric linear systems, SIAM J. Numer. Anal, 34 (1997) no. 1, pp. 48–66. 132

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

[14] H.J. Martinez, Z. Parada and R.A. Tapia, On the characterization of Q-superlinear convergence of quasi-Newton interior-point methods for nonlinear programming,  Bol.Soc. Mat. Mexicana, 1 (1995) no. 3, pp. 137–148.

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

[16] I. Pavaloiu, Introduction in the Approximation of the Solutions of Equations Theory, Ed. Dacia, Cluj-Napoca, 1976 (in Romanian).

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

[18] F.A. Potra, On Q-order and R-order of convergence,  J.  Optim.  Theory  Appl.,  63 (1989) no. 3, pp. 415–431.

[19] W.C.  Rheinboldt, Methods for Solving Systems of Nonlinear Equations,  SIAM, Philadelphia, 1996.

[20] Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, Math.Comp., 37 (1981) no. 155, pp. 105–126.

[21] Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Pub. Co., Boston, 1996.

[22] Y. Saad and M.H. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986) no. 3, pp.856–869.

[23] G.W.  Stewart and J.-G.  Sun, Matrix Perturbation Theory,  Academic  Press,  New York, 1990.

[24] H.F. Walker, An approach to continuation using Krylov subspace methods, Research Report 1/97/89, Dept. of Math., Utah State University, submitted to Computational Science in the 21st Century.

Related Posts