关于拟算术Gauss型迭代
On the quasi-arithmetic Gauss-type iteration
摘要：
对于连续单调函数序列$f_1,\dots,f_n \colon I \to \mathbb{R}$（$I$是一个区间），我们定义映射$M \colon I^n \to I^n$为由$f_j$ -s生成的拟算术平均的笛卡儿积.已知对于每个初始向量，该映射的迭代序列趋向于$I^n$的对角线。我们将证明，只要所有的$f_j$ -s是$\mathcal{C}^2$无处消失的第一阶导数，那么这个收敛是二次的。此外，将在非退化情况下计算限度$\frac{\text{Var}\, M^{k+1}(v)}{(\text{Var}\, M^{k}(v))^2}$。
英文摘要：
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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