Abstract
We review and extend results on the local convergence of the classical Newton-Kantorovich method. Then we discuss globally convergent damped and inexact Newton methods and point out advantages of using a minimal error conjugate gradient method for the linear systems arising at each Newton step.
Finally application on a nonlinear elliptic problem is considered. A combination of nested iterations, damped inexact Newton method and two-level grid finite element methods for the solution of the linear boundary value problems encountered at each step are discussed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
O. Axelsson, Conjugate gradient type methods for unsymmetric and inconsistent systems of linear equations, Linear Algebra and its Applications, 29 (1980), 1–16.
O. Axelsson, On multigrid methods of the two-level type. In Proceedings, Conference on multigrid methods, DFVLR, Köln-Porz, November 23–27th, 1981, Springer Verlag, to appear.
R.E. Bank and D.J. Rose, Global approximate Newton methods, Numer. Math. 37 (1981), 279–295.
R.S. Dembo, S.C. Eisenstat, and T. Steihaug, Inexact Newton Methods. Series # 47, School of Organization and Management, Yale University, 1980.
J.E. Dennis and J.J. Moré, A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Comp. 28 (1974), 549–560.
L.V. Kantorovich, Functional analysis and applied mathematics, Uspekhi Mat. Nauk. 3 (1948), 89–185; English transl., Rep. 1509, National Bureau of Standards, Washington, D.C., 1952.
L. Kronsjö and G. Dahlquist, On the design of nested iterations for elliptic difference equations, BIT 11 (1971), 63–71.
L. Mansfield, On the solution of nonlinear finite element systems, SIAM J. Numer. Anal. 17 (1980), 752–765.
J.M. Ortega and W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970.
L.B. Rall, Computational solution of nonlinear operator equations, Wiley, New York, 1969.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1982 Springer-Verlag
About this paper
Cite this paper
Axelsson, O. (1982). On global convergence of iterative methods. In: Ansorge, R., Meis, T., Törnig, W. (eds) Iterative Solution of Nonlinear Systems of Equations. Lecture Notes in Mathematics, vol 953. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069371
Download citation
DOI: https://doi.org/10.1007/BFb0069371
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-11602-8
Online ISBN: 978-3-540-39379-5
eBook Packages: Springer Book Archive