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Local convergence of inexact Newton methods. (English) Zbl 0566.65037
Let $D\subset {\bbfR}\sp m$ and $F: D\to {\bbfR}\sp m$ be a mapping. The author studies the approximate solution of the equation $F(x)=0$ by means of the iterative method for $n=0,1,2,....:$ (*) $x\sb{n+1}:=x\sb n+s\sb n\in {\bbfR}\sp m$ with $s\sb n$ from $F'(x\sb n)s\sb n=-F(x\sb n)+r\sb n$ for some sequence $\{r\sb n\}\subset R\sp m$. He gives an affine invariant condition involving $r\sb n$ which ensures the local convergence of (*) to a solution of $F(x)=0$. Moreover he deduces a radius of convergence result for (*) which is shown to be sharp for both Newton’s method and the general difference Newton-like method. The results are applied to the latter two methods and the general Newton-like method in which the iterates are perturbed by the presense of rounding errors. No numerical example.
Reviewer: B.Döring

65H10Systems of nonlinear equations (numerical methods)
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