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

Rall, L. B.

doi:10.1090/S0025-5718-1976-0405831-4

A quadratically convergent iteration method for computing zeros of operators satisfying autonomous differential equations
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by L. B. Rall PDF

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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