Q-收敛阶的p步推广
A $p$-step generalization of the Q-order of convergence
AI总结：
本文提出p步Q-收敛阶以推广经典Q-收敛阶，可处理非单调序列的收敛分析，其至少α阶蕴含至少α阶的R-收敛阶，适用于多步交替迭代方法。
AI中文摘要：
Q-收敛阶的概念可说是描述收敛序列渐近行为的最重要工具，大致而言，它刻画了迭代方法的收敛“速度”。对于误差并非每一步都单调递减的序列，Q-收敛阶的概念并不总是适用。本文引入p步Q-收敛阶的概念，它通过比较间隔p次迭代的误差而非连续迭代的误差，推广了经典的Q-收敛阶概念；当p=1时，该定义退化为经典Q-收敛阶。我们证明，对于经典Q-收敛阶不存在或给出过于悲观分类的某些非单调序列，p步Q-收敛阶能提取有意义的收敛信息。我们建立了该新概念的基础理论，并通过证明至少α阶的p步Q-收敛阶蕴含至少α阶的R-收敛阶，将其纳入经典层级中。自然应用包括更新在多步间交替或循环的迭代方法。
英文摘要：
The notion of Q-order convergence is arguably the most important tool for describing the asymptotic behavior of a convergent sequence. Loosely speaking, it captures the``speed''of convergence of an iterative method. The concept of Q-order convergence is not always well suited for sequences whose errors do not decrease monotonically at every step. In this paper, we introduce the notion of $p$-step Q-order convergence. It generalizes the classical notion of Q-order convergence by comparing errors that are $p$ iterations apart rather than errors of successive iterates. This definition recovers classical Q-order convergence as the special case $p=1$. We show that it extracts meaningful convergence information from certain non-monotonic sequences for which the classical Q-order either does not exist or assigns an overly pessimistic classification. We develop the basic theory of the new notion and locate it within the classical hierarchy by proving that $p$-step Q-order at least $α$ implies R-order at least $α$. Natural applications include iterative methods whose updates alternate or cycle over multiple steps.
