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Zbl 0566.65037
Ypma, T.J.
Local convergence of inexact Newton methods.
(English)
[J] SIAM J. Numer. Anal. 21, 583-590 (1984). ISSN 0036-1429; ISSN 1095-7170/e

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.
[B.Döring]
MSC 2000:
*65H10 Systems of nonlinear equations (numerical methods)

Keywords: Newton-like methods; local convergence; error analysis; presence of inaccuracies; iterative method

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