A $p$-step generalization of the Q-order of convergence

Abstract

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 $\alpha$ implies R-order at least $\alpha$. Natural applications include iterative methods whose updates alternate or cycle over multiple steps.