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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

65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
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