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arXiv · 1801.07525

On the quasi-arithmetic Gauss-type iteration

Paweł Pasteczka
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Abstract

For a sequence of continuous, monotone functions $f_1,\dots,f_n \colon I \to \mathbb{R}$ ($I$ is an interval) we define the mapping $M \colon I^n \to I^n$ as a Cartesian product of quasi-arithmetic means generated by $f_j$-s. It is known that, for every initial vector, the iteration sequence of this mapping tends to the diagonal of $I^n$. We will prove that whenever all $f_j$-s are $\mathcal{C}^2$ with nowhere vanishing first derivative, then this convergence is quadratic. Furthermore, the limit $\frac{\text{Var}\, M^{k+1}(v)}{(\text{Var}\, M^{k}(v))^2}$ will be calculated in a nondegenerated case.

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Paweł Pasteczka. 2018-01-23. On the quasi-arithmetic Gauss-type iteration. https://doi.org/10.1007/s00010-018-0568-1

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