On the quasi-arithmetic Gauss-type iteration

Abstract

For a sequence of continuous, monotone functions $f_1,\dots ,f_n:I\to \mathbb{R}$ ($I$ is an interval) we define the mapping $M:I^n\to I^n$ 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.

References in corpus (4)

• Computer aided solution of the invariance equation for two-variable Gini means
• Computer aided solution of the invariance equation for two-variable Stolarsky means
• Characterization of generalized quasi-arithmetic means
• Iterated Quasi-Arithmetic Mean-Type Mappings

Cited by in corpus (1)

• Quasiarithmetic-type invariant means on probability space