Majorizing Sequences and Error Bounds for Iterative Methods

Given a sequence {xn}n=Q in a Banach space, it is well known that if there a sequence {r^J^—QSUch that Wxn+X — xn\\ < tn + x — tn and lim tn = t* < °°, then {xn}n=Q converges to some x* and the error bounds IIjc* — jc II < f* — tn hold. It is shown that certain stronger hypotheses imply sharper error bounds, ■**-*„« <—-—Jxn-Xn-X*11* * \'*1-'0|M" > »• Representative applications to infinite series and to iterates of types xn = Gxn_x and xn = H(xn, xn_x ) are given for u = 1. Error estimates with 0 < p < 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 < u < 1 for a class of Newton-type methods, and these bounds are shown to be optimal for a subclass of methods. Practical limitations of the error bounds are described.