On the quasi-arithmetic Gauss-type iteration

TL;DR

This paper proves quadratic convergence of a class of quasi-arithmetic Gauss-type iterative methods under smoothness conditions and calculates the asymptotic ratio of variances in the nondegenerate case.

Contribution

It establishes quadratic convergence for smooth, monotone quasi-arithmetic mean iterations and derives a formula for the variance ratio in the limit.

Findings

Convergence is quadratic when functions are $ ext{C}^2$ with nonzero derivatives.

The limit of the variance ratio is explicitly calculated.

The results extend understanding of convergence behavior in mean-type iterations.

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.