The convergence orders (convergence rates) of convergent sequences toward their limit are fundamental notions from Mathematical Analysis and Numerical Analysis.
The convergence speed of an iterative method is high if the iterates converge superlinearly. The most popular method with high convergence rate is the Newton method, which under some standard assumptions has quadratic convergence.
In this paper we attempt to offer a comprehensive picture regarding various convergence orders: definitions, relations between them, history.
An important topic consists of the computational convergence orders (numerical convergence rates), which are rigorously defined and related to the theoretical ones.
Twenty years after the classical book of Ortega and Rheinboldt was published, five definitions for the \(Q\)-convergence orders of sequences were independently and rigorously studied (i.e., some orders characterized in terms of others), by Potra (1989), resp. Beyer, Ebanks and Qualls (1990). The relationship between all the five definitions (only partially analyzed in each of the two papers) was not subsequently followed and, moreover, the second paper slept from the readers attention.
The main aim of this paper is to provide a rigorous, selfcontained, and, as much as possible, a comprehensive picture of the theoretical aspects of this topic, as the current literature has taken away the credit from authors who obtained important results long ago.
Moreover, this paper provides rigorous support for the numerical examples recently presented in an increasing number of papers, where the authors check the convergence orders of different iterative methods for solving nonlinear (systems of) equations.