On the convergence of modified contractions
The convergence of modified iteration methods of the form \(x_{k+ 1}= \alpha_ k{\mathbf x}_ k+ (1-\alpha_ k) A({\mathbf x}_ k)\), where \(A: D\to D\) with \(D\subseteq \mathbb{R}^ n\) being convex is studied. Earlier results, given by \textit{D. Liu} and \textit{F. Szidarovszky} [Appl. Math. Comput. 43, No. 3, 237-240 (1991; Zbl 0733.93065)], guarantee the convergence of the process (even for nonstationary cases) when the original algorithm \({\mathbf z}_{k+ 1}= A({\mathbf z}_ k)\) also converges to the unique fixed point of the mapping \(A\). In the paper, some convergence criteria, valid also in cases when the original algorithm diverges, are proved and the optimal selection of the weights \(\alpha_ k\) is determined for such cases, using the quadratic programming approach.
scientific article; zbMATH DE number 758529