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Mathematics of Computation

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A quadratically convergent iteration method for computing zeros of operators satisfying autonomous differential equations

Author: L. B. Rall
Journal: Math. Comp. 30 (1976), 112-114
MSC: Primary 65H05; Secondary 47H15
MathSciNet review: 0405831
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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

$\displaystyle {x_{n + 1}} = {x_n} - \Gamma P({x_n}),\quad n = 0,1,2, \ldots ,$

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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Keywords: Nonlinear operator equations, iteration methods, quadratic convergence, variants of Newton's method
Article copyright: © Copyright 1976 American Mathematical Society