Some dynamically trivial mappings with applications to the improvement of simple iteration
The aim is the construction of algorithms \(x_{n+k}=g[\varphi(x_ n,x_{n+1},\dots,x_{n+k-1})]\), \(n=0,1,2,\dots\), with \(k\)-dimensional (arithmetic) means \(\varphi\), which converge (globally) against a fixed point \(\alpha\) of the real function \(g\) where the simple iteration \(x_{n+1}=g(x_ n)\) may even fail to be convergent. The assumptions on \(g\) and \(\alpha\) are as follows. There is an interval \(I\) with \(g(I)\subset I\), \(g\) is monotonically decreasing and convex, and \(\alpha\) satisfies a non-expansion-condition \(-1\leq g'(\alpha)<0\) (for global convergence) and \(-1\leq g'(\alpha)<1\) (for local convergence). Rates of convergence in terms of the weights of the means are given.
scientific article; zbMATH DE number 120302