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Local convergence of inexact Newton methods. (English) Zbl 0566.65037
Let $D\subset {ℝ}^{m}$ and $F:D\to {ℝ}^{m}$ be a mapping. The author studies the approximate solution of the equation $F\left(x\right)=0$ by means of the iterative method for $n=0,1,2,····:$ (*) ${x}_{n+1}:={x}_{n}+{s}_{n}\in {ℝ}^{m}$ with ${s}_{n}$ from ${F}^{\text{'}}\left({x}_{n}\right){s}_{n}=-F\left({x}_{n}\right)+{r}_{n}$ for some sequence $\left\{{r}_{n}\right\}\subset {R}^{m}$. He gives an affine invariant condition involving ${r}_{n}$ which ensures the local convergence of (*) to a solution of $F\left(x\right)=0$. 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

##### MSC:
 65H10 Systems of nonlinear equations (numerical methods)