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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Majorizing sequences and error bounds for iterative methods

Author: George J. Miel
Journal: Math. Comp. 34 (1980), 185-202
MSC: Primary 65J05; Secondary 47H10
MathSciNet review: 551297
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Abstract: Given a sequence $ \{ {x_n}\} _{n = 0}^\infty $ in a Banach space, it is well known that if there is a sequence $ \{ {t_n}\} _{n = 0}^\infty $ such that $ \left\Vert {{x_{n + 1}} - {x_n}} \right\Vert \leqslant {t_{n + 1}} - {t_n}$ and $ \lim {t_n} = {t^\ast} < \infty $, then $ \{ {x_n}\} _{n = 0}^\infty $ converges to some $ {x^\ast}$ and the error bounds $ \left\Vert {{x^\ast} - {x_n}} \right\Vert \leqslant {t^\ast} - {t_n}$ hold. It is shown that certain stronger hypotheses imply sharper error bounds,

$\displaystyle \left\Vert {{x^\ast} - {x_n}} \right\Vert \leqslant \frac{{{t^\as... ...})}^\mu }}}{\left\Vert {{x_1} - {x_0}} \right\Vert^\mu },\quad \mu \geqslant 0.$

Representative applications to infinite series and to iterates of types $ {x_n} = G{x_{n - 1}}$ and $ {x_n} = H({x_n},{x_{n - 1}})$ are given for $ \mu = 1$. Error estimates with $ 0 \leqslant \mu \leqslant 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 \leqslant \mu \leqslant 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.

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Keywords: Majorizing sequences, iterative methods, exit criteria
Article copyright: © Copyright 1980 American Mathematical Society

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