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

DOI:
https://doi.org/10.1090/S0025-5718-1976-0405831-4

MathSciNet review:
0405831

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Abstract | References | Similar Articles | Additional Information

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 ${x^\ast }$.

- Robert G. Bartle,
*Newton’s method in Banach spaces*, Proc. Amer. Math. Soc.**6**(1955), 827–831. MR**71730**, DOI https://doi.org/10.1090/S0002-9939-1955-0071730-1 - J. E. Dennis Jr.,
*Toward a unified convergence theory for Newton-like methods*, Nonlinear Functional Anal. and Appl. (Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1971, pp. 425–472. MR**0278556** - Louis B. Rall,
*Computational solution of nonlinear operator equations*, John Wiley & Sons, Inc., New York-London-Sydney, 1969. With an appendix by Ramon E. Moore. MR**0240944** - L. B. Rall,
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Keywords:
Nonlinear operator equations,
iteration methods,
quadratic convergence,
variants of Newton’s method

Article copyright:
© Copyright 1976
American Mathematical Society