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Asymptotic error estimate for general Newton-type methods and its application to differential equations

Kulikov, Gennady Yu; Merkulov, A.I.; Shindin, S.K.

Russian Journal of Numerical Analysis and Mathematical Modelling, 22(6) (2007), 567-590

In the paper we present a new result for evaluating the convergence rate of iterative Newton-type methods with respect to the number of iteration steps. We obtain an explicit asymptotically correct estimate that provides a fruitful basis for many practical situations. As examples of such applications, we solve four important problems arising in numerical integration of ordinary differential equations and index 1 semi-explicit differential-algebraic systems.