On the high convergence orders of the Newton-GMBACK methods
Keywords:nonlinear systems of equations, Newton-Krylov methods, linear systems of equations in R^n, Krylov methods, inexact Newton methods, quasi-Newton methods, convergence orders
The high convergence orders of the Newton-GMBACK methods can be
characterized applying three different existing results. In this note we
show by some direct computations that these characterizations are equivalent.
P.N. Brown, A theoretical comparison of the Arnoldi and GMRES algorithms, SIAM J. Sci. Stat. Comput., 12 (1991) no. 1, pp. 58–78,https://doi.org/10.1137/0912003
P.N. Brown and H.F. Walker, GMRES on (nearly) singular systems, SIAM J. Matrix Anal. Appl., 18 (1997) no. 1, pp. 37–51https://doi.org/10.1137/s0895479894262339
E. Catinas, A note on inexact secant methods, Rev. Anal. Numer. Theor. Approx., 25(1996) nos. 1–2, pp. 33–41.
E. Catinas, Newton and Newton-Krylov methods for solving nonlinear systems in Rn, Ph.D. Thesis, Cluj-Napoca, Romania, (to be defended).
E. Catinas, Inexact perturbed Newton methods and some applications for a class of Krylov solvers, J. Optim. Theory Appl., submitted, https://doi.org/10.1023/a:1017583307974
R.S. Dembo, S.C. Eisenstat, and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982) no. 2, pp. 400–408, https://doi.org/10.1137/0719025
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,https://doi.org/10.1090/s0025-5718-1974-0343581-1
J.E. Dennis, Jr. and J.J. More, Quasi-Newton methods, motivation and theory, SIAM Review, 19 (1977) no. 1, pp. 46–89, https://doi.org/10.1137/1019005
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.
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, https://doi.org/10.1137/0917003
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, https://doi.org/10.1137/0916042
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, https://doi.org/10.1137/s0036142994266844
C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM, Philadelphia, 1995, https://doi.org/10.1137/1.9781611970944
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.
J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
I. Pavaloiu, Introduction in the Approximation of the Solutions of Equations Theory, Ed. Dacia, Cluj-Napoca, 1976 (in Romanian).
F.A. Potra and 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) no. 3, pp. 415–431, https://doi.org/10.1007/bf00939805
W.C. Rheinboldt, Methods for Solving Systems of Nonlinear Equations, SIAM, Philadelphia, 1996.
Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, Math. Comp., 37 (1981) no. 155, pp. 105–126, https://doi.org/10.1090/s0025-5718-1981-0616364-6
Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Pub. Co., Boston, 1996.
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, https://doi.org/10.1137/0907058
G.W. Stewart and J.-G. Sun, Matrix Perturbation Theory, Academic Press, New York, 1990.
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.
How to Cite
Copyright (c) 2015 Journal of Numerical Analysis and Approximation Theory
This work is licensed under a Creative Commons Attribution 4.0 International License.
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.