# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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## A quadratically convergent iteration method for computing zeros of operators satisfying autonomous differential equationsHTML articles powered by AMS MathViewer

by L. B. Rall
Math. Comp. 30 (1976), 112-114 Request permission

## Abstract:

If the Fréchet derivative P’ of the operator P in a Banach space X is Lipschitz continuous, satisfies an autonomous differential equation $P’(x) = f(P(x))$, and $f(0)$ has the bounded inverse $\Gamma$, then the iteration process ${x_{n + 1}} = {x_n} - \Gamma P({x_n}),\quad n = 0,1,2, \ldots ,$ is shown to be locally quadratically convergent to solutions $x = {x^\ast }$ of the equation $P(x) = 0$. If f is Lipschitz continuous and $\Gamma$ exists, then the global existence of ${x^\ast }$ is shown to follow if $P(x)$ is uniformly bounded by a sufficiently small constant. The replacement of the uniform boundedness of P by Lipschitz continuity gives a semilocal theorem for the existence of ${x^\ast }$ and the quadratic convergence of the sequence $\{ {x_n}\}$ to ${x^\ast }$.
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