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Nondiscrete induction and an inversion-free modification of Newton’s method. (English) Zbl 0563.65040
For the solution of an operator equation $$f(x)=0$$ in a Banach space the so-called inversion free modification of Newton’s method has the form (1) $$x_{n+1}=x_ n-G_{n+1}f(x_ n)$$, $$G_{n+1}=2G_ nf'(x_ n)$$, $$n=0,1,...$$. If G is an approximation of $$f'(x_ n)^{-1}$$ with $$| I-Gf'(x_ n)| <1$$ then $$f'(x_ n)^{-1}$$ can be expanded in powers of $$I-Gf'(x_ n)$$ and (1) is obtained by replacing the inverse in Newton’s method with the first two terms of that series. The method appears to have been proposed first by S. Ul’m [Izv. Akad. Nauk Èston. SSR, Fiz. Mat. 16, 403-411 (1967; Zbl 0189.485)] and has since then been studied by several authors. In this paper the second author’s nondiscrete induction is applied to the development of a natural rate of convergence of (1) to the solution of $$Gf(x)=0$$ when f’ is Lipschitz continuous. The approach follows that developed in earlier papers of the authors, notably [Numer. Math. 34, 63-72 (1980; Zbl 0434.65034)].
Reviewer: W.C.Rheinboldt

##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators 47J25 Iterative procedures involving nonlinear operators
##### Citations:
Zbl 0189.485; Zbl 0434.65034
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