Majorizing sequences and error bounds for iterative methods

Given a sequence {xn}n=o in a Banach space, it is well known that if there is a sequence {tn}n=O such that llxn+ -xlx 6 tn+ tn and lim tn = t ?n (tn nd 1xn @ n1 t1 -o Representative applications to infinite series and to iterates of types xn = Gxn and Xn = H(xn, xn1) are given for i = 1. Error estimates with 0 < , < 2 are shown to be valid and optimal for Newton iterates under the hypotheses of the Kantorovich theorem. The unified convergence theory of Rheinboldt is used to derive error bounds with 0 < , < 1 for a class of Nbwton-type methods, and these bounds are shown to be optimal for a subclass of methods. Practical limitations of the error bounds are described.

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