Abstract
We give a local convergence analysis of difference Newton-like methods for solving the nonlinear equationF(x)=0, without assuming Lipschitz continuity of the derivativeF′. The results are obtained by regarding difference Newton-like methods as inexact Newton methods.
Zusammenfassung
Wir befassen uns mit einer Konvergenzanalyse für Newton-ähnliche Verfahren vom Differenztyp zur Lösung der nichtlinearen GleichungF(x)=0, ohne die Voraussetzung, daß die AbleitungF′ Lipschitz-stetig ist. Die Resultate entstehen daraus, daß wir Newton-ähnliche Verfahren vom Differenztyp als approximäre (inexact) Newton-Verfahren ansehen.
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Bus, J. C. P.: Numerical solution of systems of nonlinear equations. (Mathematical Centre Tract 122.) Amsterdam: Mathematisch Centrum 1980.
Dembo, R. S., Eisenstat, S. C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal.19, 400–408 (1982).
Dennis, J. E., Moré, J. J.: A characterization of superlinear convergence and its application to quasi-Newton methods. Math. Comput.28, 549–560 (1974).
Deuflhard, P., Heindl, G.: Affine invariant convergence theorems for Newton's method and extensions to related methods. SIAM J. Numer. Anal16, 1–10 (1979).
Gill, P. E., Murray, W., Saunders, M. A., Wright, M. H.: Computing forward-difference intervals for numerical optimization. SIAM J. Sci. Stat. Comp.4, 310–321 (1983).
Griewank, A., Toint, P.: Local convergence analysis for partitioned quasi-Newton updates. Numer. Math.39, 429–448 (1982).
Jankowska, J.: Theory of multivariate secant methods. SIAM J. Numer. Anal.16, 547–562 (1979).
Mönch, W.: Secant methods for sparse systems of nonlinear equations with a special structure. Computing30, 212–223 (1983).
Ortega, J. M., Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. New York: Academic Press 1970.
Rokne, J.: Newton's method under mild differentiability conditions with error analysis. Numer. Math.18, 401–412 (1972).
Schwetlick, H.: Numerische Lösung nichtlinearer Gleichungen. München: Oldenbourg 1979.
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. SIAM J. Numer. Anal. (to appear, 1984).
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Ypma, T.J. Difference Newton-like methods under weak continuity conditions. Computing 33, 51–64 (1984). https://doi.org/10.1007/BF02243075
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DOI: https://doi.org/10.1007/BF02243075