A survey on the high convergence orders and computational convergence orders of sequences


Let \(x_k\rightarrow x^\ast\in{\mathbb R}^N\); denote the errors by \( e_k: = \|x^\ast – x_k\| \) and the quotient factors by \(Q_p(k)=\frac{e_{k+1}}{{e_k}^p}\) (\(p \geq 1\)).

Consider the following definitions of Q-orders \(p_0>1\) of \(\{x_k\}\):

\begin{equation}\tag{$Q_\varepsilon$}\label{def Qe}
Q_p=0, & p \in [1,p_0),
Q_p=+\infty, & p \in (p_0,+\infty)
Q_{p_0-\varepsilon}=0, & \forall \varepsilon >0, p_0-\varepsilon>1,
Q_{p_0+\varepsilon}=+\infty, & \forall \varepsilon >0

\begin{equation}\tag{$Q$}\label{def Q}
q_l=q_u, {\rm where}
q_l=\inf \big\{p\in[1,+\infty) : \bar{Q}_p=+\infty\big\}, \quad \bar{Q}_p=\limsup\limits_{k\rightarrow\infty}Q(k)
q_u=\sup \big\{p\in[1,+\infty) : \underset{\bar{}}{Q}{}_p=0\big\}, \quad \underset{\bar{}}{Q}{}_p=\liminf\limits_{k\rightarrow\infty}Q(k)

\begin{equation}\tag{$Q_L$}\label{def Q_L}
\lim\limits_{k\rightarrow \infty}
\frac{\ln e_{k+1}}{\ln e_k}=p_0

\(\forall \varepsilon>0,  \exists A,B\geq 0, \ k_0 \geq 0\) s.t.
\begin{equation}\tag{$Q_{I,\varepsilon}$}\label{f. def Q_eps}
A\cdot e_k ^{p_{0}+\varepsilon}\leq e_{k+1} \leq B\cdot e_k^{p_{0}-\varepsilon},\  \; \; \forall k\geq k_0

\begin{equation}\tag{$Q_\Lambda$}\label{def Q_Lam}
\lim\limits_{k\rightarrow \infty}
\frac{\ln \frac{e_{k+2}}{e_{k+1}}}{\ln \frac{e_{k+1}}{e_k}}=p_0.

Twenty years after the classical book of Ortega and Rheinboldt was published, the above definitions for the Q-orders 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.


Emil Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)


Convergent sequences in \(\mathbb{R}^n\); (Q)-convergence orders; (C)-convergence orders; (R)-convergence orders; convergence rates; rates of convergence; convergence speed; speed of convergence; computational convergence orders.

Cite this paper as

E. Cătinaş, A survey on the high convergence orders and computational convergence orders of sequences, Appl. Math. Comput., 343 (2019) 1-20.
doi: 10.1016/j.amc.2018.08.006


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The sequence \(z_k\) from Example 2.8 actually does not have Q-order \(2\):
z_k = \begin{cases} 2^{-2^k/k}, & k \text{  odd}\\ 2^{-k2^k}, & k \text{  even.} \end{cases}

Post version

November 7, 2021: errata added.

November 11, 2018: post created.

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