Summary
Newton-like methods in which the intermediate systems of linear equations are solved by iterative techniques are examined. By applying the theory of inexact Newton methods radius of convergence and rate of convergence results are easily obtained. The analysis is carried out in affine invariant terms. The results are applicable to cases where the underlying Newton-like method is, for example, a difference Newton-like or update-Newton method.
Similar content being viewed by others
References
Axelsson, O.: On global convergence of iterative methods. In: Iterative solution of nonlinear systems of equations Ansorge, R., Meis, T., Törnig, W., (eds.). Lecture Notes in Mathematics, Vol. 953, pp. 1–19. Berlin: Springer 1982
Bank, R.E., Rose, D.J.: Global approximate Newton methods. Numer. Math.37, 279–295 (1981)
Broyden, C.G., Dennis, J.E., Moré, J.J.: On the local and superlinear convergence of quasi-Newton methods. J. Inst. Maths. Applics.12, 223–245 (1973)
Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SINUM19, 400–408 (1982)
Dennis, J.E., Walker, H.F.: Local convergence theorems for quasi-Newton methods. Cornell Computer Science TR 79-383. Cornell: Cornell Computer Science 1979
Dennis, J.E., Walker, H.F.: Convergence theorems for least-change secant update methods. SINUM18, 949–987 (1981)
Dennis, J.E., Moré, J.J.: A characterization of superlinear convergence and its application to quasi-Newton methods. Math. Comput.28, 549–560 (1974)
D'Jakonov, E.G.: On certain iterative methods for solving nonlinear differential equations. In: Proc. Conf. on the numerical solution of differential equations Morris, J.L. (ed.), Lecture Notes in Mathematics, Vol. 109, pp. 7–22. Berlin: Springer 1969
Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. New York: Academic Press 1970
Rheinboldt, W.C.: Methods for solving systems of nonlinear equations. Philadelphia: SIAM 1974
Richardson, L.F.: The approximate arithmetical solution by finite differences of physical problems involving differential equations with an application to the stresses in a masonry dam. Philos. Trans. Roy. Soc. London Ser. A.210, 307–357 (1910)
Schwetlick, H.: Numerische Lösung nichtlinearer Gleichungen. München: Oldenbourg 1979
Sherman, A.H.: On Newton-iterative methods for the solution of systems of nonlinear equations. SINUM15, 755–771 (1978)
Ypma, T.J.: Local convergence of difference Newton-like methods. Math. Comput.41, 527–536 (1983)
Ypma, T.J.: Local convergence of inexact Newton methods. SINUM21, 583 (1984)
Ypma, T.J.: Difference Newton-like methods under weak continuity conditions. Computing in press (1984)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ypma, T.J. Convergence of Newton-like-iterative methods. Numer. Math. 45, 241–251 (1984). https://doi.org/10.1007/BF01389469
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01389469