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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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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 ${x^\ast }$.
References
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Math. Comp. 30 (1976), 112-114
  • MSC: Primary 65H05; Secondary 47H15
  • DOI: https://doi.org/10.1090/S0025-5718-1976-0405831-4
  • MathSciNet review: 0405831