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

This is the first paper which surveys the rigorous definitions and relations of convergence orders (the Q-orders; the C-orders; the R-orders) of convergent sequences, and the computational variants.

## Abstract

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 $\{x_k\}$:

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

\tag{$Q$}\label{def Q}
q_l=q_u, {\rm where}
\begin{cases}
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)
\end{cases}

\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.
\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

\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.

## Authors

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

## Keywords

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

Elsevier

0096-3003

## Errata

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 11, 2018: post created.

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

author address: Tiberiu Popoviciu Institute of Numerical Analysis, Romanian

version submitted to Appl. Math. Comput., published with some changes in vol.
343 (2019) 1-20.

### Abstract

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

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. Tight connections
between some asymptotic quantities defined by theoretical and
computational elements are shown to hold.

Keywords. convergent sequences in

${ℝ}^{n}$;
Q-, C-, and R-convergence orders of sequences; computational convergence
orders.

MSC 2010. 65J05.

### 1 Introduction

Consider in the beginning a sequence

$\left\{{x}_{k}$

from

$ℝ$

, which converges
to a finite limit

${x}^{\ast }$

.
Its speed of convergence is characterized by several measures, called convergence
orders, which are fundamental notions in Mathematical and in Numerical
Analysis.

The relation

 $\underset{k\to \infty }{\mathrm{lim}}$ |x∗−xk|p = Cp ∈ (0,+∞) (1)

gives the classical definition of the convergence with order

$p>1$

, and,
in a less rigorous but sufficiently intuitive statement, it can be traced back in
1818, to Fourier [1]. Even deeper traces may be found in history: in a letter of
Newton dated in 1675, it appears he was aware of the rough doubling of the
number of correct significant digits in one step (characteristic to the
quadratic convergence) of the iterative method which bears his name
[2].

The existence of the nonzero value

${C}_{p}$

is a
very restrictive condition not only in theory, but also in examples from
practice – not to mention that in the multidimensional setting it is a
norm-dependent problem. Moreover, as it requires the knowledge of both

$p$

and

${x}^{\ast }$

,
relation (1 ) is only of theoretical use.

Still, such definition (along with the

$Q$

-superlinear convergence,
which requires

${C}_{1}$

for

$p=1$

)
is extremely important in studying the local convergence of the main
iterative methods for nonlinear problems, either of Newton type (see,
e.g., [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]), or the more general
successive approximations (see, e.g., [3], [4, ch. 10], [6], [14], [15]) – which,
under some supplementary assumptions (including smoothness) may
be regarded in fact as equally general (see [14]). This definition is also
important in the field of the matrix function theory (polar decomposition,
matrix sign function, computing the matrix inverse by Schulz or different
iterative methods), in a setting where the matrix is an element of a normed
space (see, e.g., [16], [17]); we should also mention the acceleration of
the convergence of sequences (see, e.g., [18]), and perhaps other fields
too.

From a numerical standpoint, the aim is to define the so-called
computational convergence orders, that iteratively approximate the convergence
orders using only the elements of the sequence known at the step

$k$

.
To this end (or just for theoretical purposes) new definitions have been
introduced and studied. We attempt to follow the origin of each notion in the
following section; this is a difficult task, as most of the papers treat the
convergence orders as a collateral topic; no survey has been written since two
monographs containing entire chapters devoted to the rigorous analysis of the

$Q$

– and

$R$

-convergence
orders. The book of Ortega and Rheinboldt [4], though first published in 1970, is
a standard reference in the field of iterative methods and their convergence orders
(see also [6]); it was republished in 2000 in the Classics in Applied Mathematics
series of SIAM. The book of Potra and Pták (1984) [8] contains further,
complementary results.

Two important papers were subsequently published, independently and close
in time, where the equivalence of different definitions of (computational)

$C$

– and

$Q$

-convergence
orders were given. The complementary results of these two works allow us to
form a complete, simplified and rigorous picture of the topic. The first paper was
written by Potra (1989) [19]; as of November 2017, Scholar Google reported 96
citations of it, which shows that the mathematicians are aware of it. The other
paper however, written by Beyer, Ebanks and Qualls (1990) [20], is almost
unknown to the mathematical community: as of November 2017, we have found
only 6 citations in Google Scholar (out of which two papers in Chinese, two
papers in the field of Bioarchaeology, one paper solving an unsolved problem

from [20], and one paper dealing with the quadratic convergence in period
doubling for trapezoid maps).

Unfortunately, the fact that the paper of Beyer, Ebanks and
Qualls remained unknown, allowed some authors to (honestly)
reinvent such notions, e.g., in [21] and [22]. The very cited paper
[21]1
has the distinctive feature that not only the definition of the
computational convergence order proposed there is not original
(nor proved, nor in fact entirely computational – as it requires

${x}^{\ast }$

), but even
the iterative method itself (as it was given in the well known book of J.F. Traub [23,
formula (8-14), p. 164]2 ).

In numerous works, the classical

$Q$

and

$R$

-orders
defined by Ortega and Rheiboldt lead to statements expressed as

$\left\{{x}_{k}$

has

$Q$

-order (at
least)

${p}_{0}$

and

$R$

-order
(at least)

${p}_{1}$

,

${p}_{1}$

”. The

$Q$

-order
we study here is a completion to the classical one, and no
longer allows such conclusions. Instead, if having at all an order

${p}_{0}$

,
the sequences will be only in one of the cases (the

$C$

-order
is obtained by taking norms in (1)):

• $\left\{{x}_{k}$has

$R$-order

${p}_{0}$(but neither

$C$
nor

$Q$-order);

• $\left\{{x}_{k}$has the same

$R$
and

$Q$-order

${p}_{0}$(but no

$C$-order);

• $\left\{{x}_{k}$has the same

$R$-,

$Q$
and

$C$-order

${p}_{0}$.

In less words, the relation reads for

${p}_{0}$

as

with no converse implication holding in general.

Nonetheless, the comments of Tapia, Dennis Jr. and
Schäfermeyer [13, p. 49] remain true: “The distinction between

$Q$

– and

$R$

-convergence
is quite meaningful and useful and is essentially sacred to workers in the area of
computational optimization. However, for reasons not well understood,
computational scientists who are not computational optimizers seem to be at
best only tangentially aware of the distinction.”

The structure of this paper is the following: in Section 2 we
review, in chronological order, the definitions of convergence
orders. In Section 3 we analyze the high convergence orders
(

$p>1$

), giving full
proofs for

$C$

and

$Q$

-orders.
We present first the relation (implication, resp. equivalence) between the introduced
convergence orders; the results are based on the proofs from [19] and [20], which
we can simplify for some cases. Then we review the main properties of the

$R$

-orders. We analyze next
the computational

$C$

– and

$Q$

-convergence orders
based on the corrections

${x}_{k+1}$

(the approach was dealt with in [20], but the authors assumed real and monotone
sequences in treating certain cases; a result of Potra and Pták [8] and
the Dennis-Moré lemma [5] allow us to complete the proofs). We also
consider computational convergence orders based on the nonlinear residuals

$F\left({x}_{k}$

,
when solving nonlinear systems of equations having the same number
of equations and unknowns. The section ends with a summary of all
the relations obtained, illustrated by a diagram. Finally, in Section 4
we recall, without proofs, the relationships between some convergence
orders in the linear case. Some conclusions are presented in the end of the
paper.

### 2 Definitions – a brief historical review

#### 2.1 C- and Q-type (computational) convergence orders.

##### 2.1.1 Convergence orders based on the errors x*-xk

We turn now our attention to a sequence

$\left\{{x}_{k}$

from

${ℝ}^{n}$

, which converges
to a finite limit

${x}^{\ast }$

;
we prefer the common setting of a normed
space—

${ℝ}^{n}$

endowed with
a given norm

$\parallel \cdot \parallel$

—though
most of the results from this paper can be presented in a
more general setting, of a metric space. We assume that

${x}_{k}$

$k\ge 0$

.
Since this leads to dealing with positive numbers, we can simplify the notations
and consider further only the errors (called such as a short for ”norms of errors”)

 ${e}_{k}$k∥.

As we have mentioned, the oldest definition of convergence with order

${p}_{0}$

is
that

 $\underset{k\to \infty }{\mathrm{lim}}$(ek)p0 = Cp0 ∈ (0,+∞), (2)

which we call, as in [20],

$C$

-convergence with
order

${p}_{0}$

(the linear
convergence requires

${p}_{0}$

and

$0<{C}_{1}$

)

$Q$

comes from quotient. If it exists, it is uniquely defined and it implies that

$\forall >0,\phantom{\rule{0.33em}{0ex}}\exists {k}_{0}$

such
that

 $\left({C}_{{p}_{0}}$ ≤ ek+1 ≤ (Cp0 + )ekp0 ,∀ ⁡k ≥ k0, (3)

or, less sharp, the existence of

$A,B>0$

such that

 $A{e}_{k}^{{p}_{0}}$ ≤ ek+1 ≤ Bekp0 ,∀ ⁡k ≥ 0. (4)

Remark 2.1
a) For example, if

${p}_{0}$and

${C}_{2}$is not too large, relation (3 ) says that the error is approximately squared at
each step

$k$,
for

$k$sufficiently large.

b) Some authors have used (4 ) to define the exact

$Q$-order
of convergence

${p}_{0}$(see [24], [18], [25], [8], [26], [19]).

c) An inequality of the form (4 ) still holds if the limit in (2 ) does not exist, but

$\mathrm{liminf}$and

$\mathrm{limsup}$are finite and nonzero. Denoting

 $\phantom{\rule{0.33em}{0ex}}{\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{{p}_{0}}$ (5)

relation

 $0<{\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{{p}_{0}}$ (6)

implies (4 ):

$0<{\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{{p}_{0}}$attracts the
first inequality in (4 ), while

${\overline{Q}}_{{p}_{0}}$the second one (see also [4, 9.3.3]).

d) As widely known, the larger the order

${p}_{0}$, the
faster the speed of convergence.

The

$C$

-order
is a norm-dependent notion, as the following two examples show for the

$C$

-orders

$1$

and

$2$

.

Example 2.2
[27] Let

${x}^{\ast }$,
the sequence

 ${x}_{k}$(0,2−k+1),k odd. (7)

Consider the norms

 $\parallel \left(u,v\right){\parallel }_{A}$

It follows

${C}_{1}$2for
the

$A$-norm, while

${C}_{1}$does not exist
for the

$B$-norm.

Example 2.3
Let

${x}^{\ast }$and

 ${x}_{k}$(2−2k ,2−2k ),k even. (8)

One obtains

${C}_{2}$when
considering the maximum norm

$\parallel \cdot {\parallel }_{\infty }$,
while

${C}_{2}$does not
exist for the norm

$\parallel \cdot {\parallel }_{1}$.

The following definition for the convergence order seems to be the
second oldest one (we adopt the notation from [20]). The sequence

$\left\{{x}_{k}$

has

${Q}_{L}$

-convergence
order

${p}_{0}$

if:

 ${Q}_{L}$ ln ⁡ ek → p0,as k →∞. (9)

Remark 2.4
a) As noticed by Brezinski in [28] and [26],

${Q}_{L}$is a particular case of a definition used by Bourbaki [29] in order to compare
the convergence orders of two sequences.

The measure

${Q}_{L}$was then independently considered by Wall in 1956 [30] (see also [31], [4,
ch. 9], [32, ch. 2] and [25]), argued – for

$\mathrm{lg}$instead of

$\mathrm{ln}$– as the limit of the quotient of two consecutive numbers of correct decimal
places in

${x}_{k+1}$resp.

${x}_{k}$(see also [8, p. 90]); in this sense, convergence for example with

${Q}_{L}$-order

$2$means intuitively that, from a certain step, the number of correct digits are
doubled the successive elements of

$\left\{{x}_{k}$.
The logarithm base may be taken as any positive real number

$a\ne 1$;
we shall consider further the natural base

$e$for simplicity.

b) In contrast to

${Q}_{{p}_{0}}$,
the expression

${Q}_{L}$does not require the (usually unknown) value

${p}_{0}$of the order, but provides an approximation of it. This may be advantageous
in certain situations when evaluating the order, as it turned out, e.g., in the
books of Wright [33, p. 252 and Ch. 7], Ye [34, p. 210, p. 226] and in two
papers of Potra [35] and [36] (where the authors deal with

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{L}$,
a quantity which we analyze in Proposition 3.3 ).

c) Brent, Winograd and Wolfe noticed in [37] that

$C$-order

${p}_{0}$immediately implies

${Q}_{L}$-order,
of same value.

d) We shall see later, in Remark 3.8 , that this notion (cf. [38]), as well as
the other

$Q$-orders,
is norm-independent (also noted by Potra [19]).

In 1967, Feldstein and Firestone [39] introduced essentially
the following convergence order, which we denote here by

${Q}_{}$

.

The sequence

$\left\{{x}_{k}$

has

${Q}_{}$

-convergence
order

${p}_{0}$

if:

lim k</munder >Qp0</msub ></msub > (k) = 0, > 0 with p0</msub > 1, (10)
lim k</munder >Qp0</msub >+</msub > (k) = , > 0.

Condition

${p}_{0}$

will be a standing assumption from here on, in all subsequent
formulas, while in Section 3 we will implicitely assume that

${p}_{0}$

.

The convergence with

${Q}_{}$

-convergence
order

${p}_{0}$

lim k</munder >Qp</msub >(k) = 0,1 < p < p0</msub >,
lim k</munder >Qp</msub >(k) = + ,p > p0</msub >.

Two years later, the same authors [40] expressed the above order
in an equivalent form, which in fact will be the same as the

$Q$

-order
defined below. Let

λ</msub > = sup {p : lim k</munder >Qp</msub >(k) = 0 }, (11)
λ</msup > = inf {p : lim
k
</munder >Qp</msub >(k) = }.

They noticed that

$0\le {\lambda }_{\ast }$

, and
when

${p}_{0}$

, the sequence was
said to have the order

${p}_{0}$

.

In their classical book from 1970, Ortega and Rheinboldt [4,
Ch. 9] introduced and studied the quantity denoted in (5) by

${\overline{Q}}_{p}$

for
all

$p\ge 1$

:

 ${}_{}$Q¯p = limsup ⁡ k→∞Qp (k).

They shown the following behavior for

${\overline{Q}}_{p}$

as a
”function” of

$p$

.

Proposition 2.5
[4, 9.1.2] Exactly one of the following conditions holds:

a)

${\overline{Q}}_{p}$b)

${\overline{Q}}_{p}$c)

$\exists {p}_{0}$s.t.

${\overline{Q}}_{p}$ $\forall p\in \left[1,{p}_{0}$and

${\overline{Q}}_{p}$ $\forall p\in \left({p}_{0}\right).$

The quantity

 inf ⁡ {p ∈ [1,∞) : Q¯p = +∞ } (12)

(called in [4] as

$Q$

-convergence
with order at least

${q}_{l}$

and denoted by O

${}_{Q}$

)
stood there and in many subsequent works for a measure of
the convergence order. We call it, as in [20], the lower

$Q$

-order, and
denote it by

${q}_{l}$

${p}_{l}$

used there). We shall see that the lower

$Q$

-order needs to be completed
by the upper

$Q$

-order for
obtaining the ”full”

$Q$

-order;
as a matter of fact,

${q}_{l}$

,
the lower order from (11) defined in [40].

Remark 2.6
The sequence

$\left\{{x}_{k}$was said in [4, Def. 9.1.4] that converges faster than

$\left\{{y}_{k}$if for
some

${p}_{0}$we
have

${\overline{Q}}_{{p}_{0}}$(it is interesting to note that if these two upper bounds are finite and nonzero, the
inequality may revert when changing the norm [4, p. 285]). However, in the case
of the even stronger relation

 $0<{C}_{{p}_{0}}$

numerical evidence reveals that

$\left\{{x}_{k}$does not necessarily converge faster than

$\left\{{y}_{k}$,
as shown by Ypma and Igarashi [41].

On the other hand, as noticed by Brezinski [26], the sequences given by

${x}_{k}$kand

${y}_{k}$k2</msup >,

$k\ge 0$,
have the same asymptotical constant,

${Q}_{1}$,
though they converge with quite different speeds.

Some further comments on this topic will be made in Remark 2.11 .

In 1979, Schwetlick [25, B.4.2.3] defined the

$Q$

-convergence
order in the following way, which will be used throughout the paper. Let the lower

$Q$

-order

${q}_{l}$

be
given in (12 ), recall the notation from (5)

 ${}_{}$ Q p = liminf ⁡ k→∞Qp (k)

and define the upper

$Q$

-order

${q}_{u}$

by

 ${q}_{u}$

Then the

$Q$

-convergence
is with order

${p}_{0}$

if

${p}_{0}$

This definition is equivalent to the one given by Feldstein and Firestone in (11),
as

${q}_{l}$

and

${q}_{u}$

.

In 1981, Schmidt [42] introduced the following type of convergence order, denoted here,
as in [20], by

${Q}_{\Lambda }$

:

$\left\{{x}_{k}$

has

${Q}_{\Lambda }$

-convergence
order

${p}_{0}$

if

 ${Q}_{\Lambda }$ek+1 ln ⁡ ek+1 ek → p0,   as  k →∞. (13)

The author gave no result relating the

$Q$

– and

${Q}_{\Lambda }$

-orders he introduced,
and assumed that

${Q}_{\Lambda }$

is
equivalent to an

$R$

-order
(while it is in fact a

$Q$

-order,
as we shall see).

In 1982, Beyer and Stein [43] independently introduced
the measure (19 ) below, which is somehow similar to

${Q}_{\Lambda }$

, and in fact
equivalent to

${Q}_{\Lambda }^{\prime }$

defined in subsubsection 2.1.2 .

In 1985, unaware of [42] and [43], Brezinski [26] considered the

${Q}_{\Lambda }$

-convergence order

and, assuming that

$\left\{{x}_{k}$

has the exact

$Q$

-order of
convergence

${p}_{0}$

(as defined
by (4 )), he proved that

$\left\{{x}_{k}$

has the

${Q}_{\Lambda }$

-order

${p}_{0}$

(and

${Q}_{L}$

-order

${p}_{0}$

as
well).

In 1989, Potra introduced in [19] the following type of convergence order (denoted
here by

${Q}_{I,}$

,
with ”

$I$

from ”inequality”), which will be useful in obtaining some simplified
proofs.

The sequence

$\left\{{x}_{k}$

has

${Q}_{I,}$

-convergence
order

${p}_{0}$

if

$\forall >0,$

$\exists a,b>0$

such
that

 $a{e}_{k}^{{p}_{0}}$k+1 ≤ bekp0−,∀ ⁡k ≥ 0. (14)

This can be regarded as a generalization of inequalities (4 ).

Potra shown some fundamental results in our study, that

${Q}_{I,}$

-convergence
with order

${p}_{0}$

is
equivalent to

${Q}_{}$

– and
to

${Q}_{L}$

-convergence
of same order, which we denote here, as in [20], by symbols between curled
braces:

 $\left\{{Q}_{}\right\}.$

Ortega and Rheinboldt [4, N.R. 9.2.2] have previously noticed a weaker property of

${Q}_{L}$

-convergence with order

${p}_{0}$

, namely that it implies
(partial)

$R$

-convergence
with order

${p}_{0}$

(i.e., in the
notations below, that

${p}_{0}$

).

In the same paper, Potra has also considered the lower and upper

$Q$

-orders, when noting that if
a sequence has exact

$Q$

-order

${p}_{0}$

then

${q}_{l}$

.

In 1990, Beyer, Ebanks and Qualls [20], unaware of the definitions
and the results mentioned above, introduced the definitions of the

$Q$

– and

${Q}_{\Lambda }$

-orders
and shown some other fundamental results, that convergence with

$Q$

-order

${p}_{0}$

is equivalent
to

${Q}_{\Lambda }$

and to

${Q}_{L}$

-convergence
with same order:

 $\left\{Q,{Q}_{L}\right\}.$

They considered

${Q}_{\Lambda }$

inspired by the similar measure (19), as we have already mentioned.

We end the historical remarks by noting that the references [44], [45] and [46]
are cited in certain works as containing aspects referring to convergence orders,

but we were not able to consult them.

Lemma 3.1 from Section 3 shows that

${q}_{l}$

(which justifies
the terminology lower/upper); when this inequality is strict, the sequence does not have
a

$Q$

-order,
as shown below.

Example 2.7
Let

 ${}_{}$xk = { 2−2k,keven, 3−2k ,kodd.

We get

 ${}_{}$Q¯p = { 0, 1 ≤ p < log ⁡ 34, 1, p = log ⁡ 34, +∞,p > log ⁡ 34, and  Q p = { 0, 1 ≤ p < 4log ⁡ 43, 1, p = 4log ⁡ 43, +∞,p > 4log ⁡ 43.

The graphical illustration of

${\overline{Q}}_{p}$and

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}}_{}$ p</msub >for

$p\ge 1$leads to the so called convergence profile of the sequence, as termed in [20]
(see Fig. 1 ).

This sequence does not have a

$Q$-order, as

${q}_{l}$, but it converges
with (exact)

$R$-order

$2$,
as defined later. We note the classical statement in this case,

$\left\{{x}_{k}$converges
with

$Q$-order
(at least)

${\mathrm{log}}_{3}$and
with

$R$-order
(at least)

$2$”,
which no longer holds in the setting of this paper.

An elementary example shows that in case of convergence with

$Q$

-order

${p}_{0}$

apart
of finite nonzero values of one (or both) of the asymptotical constants

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}}_{}$

p0</msub ></msub > Q¯p0</msub ></msub >

, any
of the following relations may hold:

Q p0</msub ></msub > = Q¯p0</msub ></msub > = +, (15)
Q p0</msub ></msub > = Q¯p0</msub ></msub > = 0, (16)
Q p0</msub ></msub > = 0 andQ¯p0</msub ></msub > = +. (17)

Example 2.8
Let

 ${}_{}$xk = 2−2k∕k,y k = 2−k2k ,zk = { xk,kodd, yk,k even.

It is easy to verify that for

${p}_{0}$the sequences

$\left\{{x}_{k}\right\},\left\{{y}_{k}\right\},\left\{{z}_{k}\right\}$verify respectively (15), (16) and (17) (

$\left\{{x}_{k}$was also considered in [20]).

Schwetlick [25, Ü.4.2.3, p. 93] also considered an example of the type

${x}_{k}$,

${y}_{k}$,
for the errors

${e}_{k}$converging with exact

$Q$-order

$2$.

Potra [19] considered

 ${}_{}$ek = { ek+1 = 2ekek−12,keven, ek+1 = ekek−12, k odd,

which has

$Q$-order

$2$and

${\overline{Q}}_{2}$.

Jay [27] considered the sequence of the type

${x}_{k}$</msup >,

$k$even,

${x}_{k}$</msup >k,

$k$odd, which satisfies (17) for

${p}_{0}$.

One may easily find examples when

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{{p}_{0}}$and

${\overline{Q}}_{{p}_{0}}$is finite, nonzero, or, on the other hand, when

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{{p}_{0}}$is finite, nonzero and

${\overline{Q}}_{{p}_{0}}$.
Jay [27] considered the sequence

${x}_{k}$,

$k$odd,

${x}_{k}$,

$k$even, for

$q\ge 1$,

${x}_{0}$,
for which one obtains

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{q}$and

${\overline{Q}}_{q}$.

Further examples, to be analyzed either elementary or with the aid of the
results from this paper, were given by Potra and Pták [8, p. 93]:

${x}_{k+1}$k </msup >,

$k\ge 1$,

${x}_{1}$,

$0<<1$and also

${x}_{2k}$</msup >,

${x}_{2k+1}$</msup >,

$0<<1.

Remark 2.9
Despite the fact that conditions (15 )–(17 ) may seem rather
abstract (or, at least, suitable for scholar examples), they do occur in relevant
situations from theory or practice. For a nontrivial illustration, we refer
the reader to a computational optimization problem [33, p. 252 and Ch. 7],
where one has

${p}_{0}$and

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{2}$.

Remark 2.10 When

${\overline{Q}}_{1}$i.e.,

 $\underset{k\to \infty }{\mathrm{lim}}$ek = 0,

it is said that the sequence converges (at least)

$Q$-superlinearly.
The convergence is strict

$Q$-superlinearly,
when

${q}_{l}$(i.e.,

${\overline{Q}}_{p}$,

$\forall p>1$);
there are three cases in this situation:

${q}_{l}$,
i.e.,

$Q$-order

$1$(see Example 3.6 ),

$1={q}_{l}$,
resp.

$1={q}_{l}$,

${q}_{u}$(see Example 2.14 b), formula (24)).

It is worth noting that one may also obtain

${q}_{l}$,

${q}_{u}$in lack of

$Q$-superlinear
(and even linear) convergence, shown in Example 2.14 d), formula (25)), as

${\overline{Q}}_{1}$.

Remark 2.11
Returning to the comments from Remark 2.6 ,
we note that the comparison of the convergence of two sequences

$\left\{{x}_{k}$,

$\left\{{y}_{k}$having
errors

$\left\{{e}_{k}^{\prime }$resp.

$\left\{{e}_{k}^{\prime \prime }$is
important in particular in the field of the acceleration of the convergence of sequences,

when sequences with (at most) linear convergence are considered; here it is said that

$\left\{{x}_{k}$converges
faster than

$\left\{{y}_{k}$if (see, e.g., [18]):

 $\underset{k\to \infty }{\mathrm{lim}}$ek′′ = 0(also denoted by ek′ = o(e k′′), as  k →∞). (18)

Now we see that this definition makes sense not only if the

$Q$-orders
of the two sequences are different, but even if they are equal, as the above
condition is norm-independent.

To show this, we consider the terminology proposed by Jay [27] for a sequence converging
with

$Q$-order

${p}_{0}$, who
defined that the convergence is with:

• $Q$-suborder

${p}_{0}$if

${\overline{Q}}_{{p}_{0}}$,

• $Q$-superorder

${p}_{0}$if

${\overline{Q}}_{{p}_{0}}$.

The last terminology is in fact widely used – recall the superquadratic convergence
(for

${\overline{Q}}_{2}$),
supercubic, etc.).

Now we see that for sequences with

$Q$-order

${p}_{0}$we
can briefly say that:

• $Q$-superconvergence
is faster than exact

$Q$-convergence
(in the sense of (4) or (6));

• exact
$Q$-convergence
is faster than

$Q$-subconvergence;

• (consequently)
$Q$-superconvergence
is faster than

$Q$-subconvergence.

This can be illustrated by the following sequences having

$Q$-order

$2$:

$\left\{{2}^{-k{2}^{k}}$</msup >}(

$Q$-superquadratic),

$\left\{{2}^{-{2}^{k}}$</msup >}(exact

$Q$-order

$2$),

$\left\{{2}^{-{2}^{k}}$</msup >}(

$Q$-subquadratic).

##### 2.1.2 Computational convergence orders, based on the corrections xk+1-xk and the nonlinear residuals F(xk)

We consider now some quantities which do not require the limit

${x}^{\ast }$

,
for which we adopt the terminology computational convergence
orders
. As some of them require the knowledge of the order

${p}_{0}$

itself,
we keep in mind that perhaps a more proper terminology for those would be
semi-computational.

As a notation, for the corresponding convergence orders
based on corrections we shall add a prime mark (e.g.,

${C}^{\prime }$

) as in
[20], while for those based on nonlinear residuals, two prime marks (e.g.,

${C}^{\prime \prime }$

).

Corrections.

We introduce another notation, for the corrections (including their norms
too):

 ${s}_{k}$

In 1974 Dennis and Moré [5] obtained a result which shows that if a sequence converges

$Q$

-superlinearly,
then the errors and corrections converge precisely at the same rate (Lemma 3.14
from subsubsection 3.2.1 ).

In 1984, Potra and Pták [8] shown that a sequence converges

$Q$

-superlinearly iff the corrections
converge

$Q$

-superlinearly
(Proposition 3.13 ); in 1997 Walker [47] gave a different proof, also presented
later.

In 1982, Beyer and Stein [43] considered, for sequences in

$ℝ$

, the
quantity

 $\frac{log\frac{{s}_{k}}{{s}_{k+1}}}{log\frac{{s}_{k-1}}{{s}_{k}}},$ (19)

equivalent to

${Q}_{\Lambda }^{\prime }$

defined below.

Three years later, Brezinski [26] proposed the use of the corrections

${x}_{k+1}$

(unknown) errors

${x}^{\ast }$

in the definitions of the convergence orders studied there
(

${Q}_{L}$

and exact

$Q$

-order).

In 1990, Beyer, Ebanks and Qualls [20], taking the same approach, considered

 ${Q}_{{p}_{0}}^{\prime }$skp0

with the resulted definitions being denoted correspondingly by

${C}^{\prime }$

and

${Q}_{\Lambda }^{\prime }$

; we use here the same
convention for denoting

${Q}_{}^{\prime }$

.
These authors proved the following fundamental results: for

${p}_{0}$

, the
errors and corrections converge simultaneously to zero with the same
convergence order, and therefore each convergence order is equivalent to
its corresponding computational one. The following extended relation
resulted:

 $\left\{{C}_{{p}_{0}}\right\}⇒\left\{Q,{Q}_{}$L,QΛ,Q′,Q L′,Q Λ′}.

However, the considered assumptions were very strong when proving

$\left\{{C}_{{p}_{0}}\right\}$

and

${Q}^{\prime }$

”:

$\left\{{x}_{k}$

was assumed as a
monotone sequence from

$ℝ$

(see [20, Sect. 4]). We shall see that for the general setting, some results from [8]
and [5] may be used instead to obtain simplified proofs.

In 2010, Grau-Sánchez, Noguera and Gutiérrez [48], while studying connections
between

${Q}_{\Lambda }$

– and

${Q}_{\Lambda }^{\prime }$

-orders (in the case
of a sequence from

$ℝ$

with

$C$

-order

$p$

)
considered the ”extrapolated convergence order”

 $\frac{ln\frac{|{x}_{k+1}}{|{x}_{k}}}{ln\frac{|{x}_{k}}{|{x}_{k-1}}},$ (20)

where

${\stackrel{~}{\alpha }}_{k}$

Δ2</msup >xk2</msub > ,

$k\ge 2,$

$\mathrm{\Delta }{x}_{k}$

. This convergence order
was obtained from

${Q}_{\Lambda }$

by replacing the unknown limit with an extrapolation

${\stackrel{~}{\alpha }}_{k}$

given by
the Aitken

${\mathrm{\Delta }}^{2}$

method (in order to use only the terms known at the step

$k$

).
However, this convergence order is valid only in the one dimensional case, since the
Aitken

${\mathrm{\Delta }}^{2}$

acceleration process does not have a direct extension to

${ℝ}^{n}$

.

We do not pursue this definition as, for high convergence orders, it is equivalent
to

${Q}_{\Lambda }^{\prime }$

.

In 2012, Grau-Sánchez, Noguera, Grau and Herrero [49] considered another
extrapolated convergence order

 $\frac{ln|{x}_{k}|}{ln|{x}_{k-1}|},$ (21)

for a sequence from

$ℝ$

,
having

$C$

-order

$p$

. Assuming

$C$

-order,
they shown some connections to the convergence orders (22) and (23)
below.

In [48] and [49] the authors used the technique of asymptotic
expansions in order to relate different convergence orders of

$\left\{{e}_{k}$

,

$\left\{{s}_{k}$

and

$\left\{f\left({x}_{k}$

for
sequences

$\left\{{x}_{k}$

from

$ℝ$

.
However, this approach could not be extended to the case of several
dimensions.

Remark 2.12
It is important to keep in mind that the

${C}^{\prime }$-,

${Q}_{}^{\prime }$-,

${Q}_{I,}^{\prime }$-orders
are semi-computational orders, as they require the (supposed) unknown order,
while the logarithm-based

${Q}_{L}^{\prime }$
and

${Q}_{\Lambda }^{\prime }$-orders
are (fully) computational orders, i.e., they require neither the limit

${x}^{\ast }$nor the order

$p$in their expressions.

As certain numerical experiments (see [50] and the references therein)
have not revealed a clear superiority of

${Q}_{\Lambda }^{\prime }$over

${Q}_{L}^{\prime }$,
the later seems to be the most convenient to use in practice (and in theory
too: see [33, p. 252 and Ch. 7], [34, p. 210, p. 226], [35] and [36]). Moreover,
Propositin 3.3 and Theorem 3.15 will show, in terms we will define later,
that

${q}_{l}$and

${q}_{u}$.

Nonlinear residuals.

Other computational convergence orders were considered
in the context of solving nonlinear systems of equations

$F\left(x\right)=0$

,

$F:{ℝ}^{n}$

, with solution

${x}^{\ast }$

. For simplicity,
we shall denote

${F}_{k}$

(resp.

${f}_{k}$

when

$n=1$

).

In 1981, Păvăloiu [51] defined the exact convergence order

${p}_{0}$

corresponding to (4) by

 $A\parallel {F}_{k}$ ≤∥Fk+1∥≤ B∥Fk∥p0 ,∀ ⁡k ≥ 0,

while in 1995, he introduced in [52] the expression

 $\frac{ln|{f}_{k+1}}{ln|{f}_{k}}$ (22)

and shown its equivalence to

${Q}_{L}^{\prime }$

.
Also, while extending his results from 1999 [53], he introduced in an unpublished
manuscript [54] (posted on his website) the measure

 $\frac{ln|{f}_{k+1}}{ln|{f}_{k}}$ (23)

in connection with the above ones. However, since the manuscript was not
published as of 2011, the credit of first publishing this measure goes to
Petković, who mentioned in passing the above expression in a note published in
SIAM J. Numer. Anal. (2011).

Several computational aspects regarding these quantities were subsequently
analyzed (see, e.g., [50] and the references therein).

#### 2.2 R-convergence orders.

The root convergence order (

$R$

-order)
requires weaker conditions than the

$C$

and

$Q$

-orders,
but unfortunately at the price of loosing its theoretical importance and its practical
applications. As we have quoted in the Introduction, “The distinction between

$Q$

– and

$R$

-convergence is […]
essentially sacred to workers in the area of computational optimization” [13, p. 49], meaning
that the

$R$

-order
is a much less powerful notion.

In Example 3.11 is presented a sequence for which, depending on the parameters, the
lower

$Q$

-order

${q}_{l}$

is arbitrary close to

$1$

from above, while
the (exact)

$R$

-order
is arbitrary high (and even higher is the upper

$Q$

-order

${q}_{u}$

).

We shall review certain results, but not treat all of them in the same detail as we
will for the

$C$

and

$Q$

-orders.

In 1970, Ortega and Rheinboldt [4, 9.2] introduced and
studied the root convergence factors, which we denote here by

${\overline{R}}_{p}$

:

 ${}_{}$R¯p {xk} = { limsup ⁡ k→∞(ek)1 k , ifp = 1, limsup ⁡ k→∞(ek) 1 pk , ifp > 1.

Unlike the quotient factors, the root factors do not relate each two
consecutive terms, but consider some averaged asymptotic quantities.

They proved a Proposition similar to Proposition 2.5 , but here

$1$

takes
the role of

$+\infty$

for

${\overline{Q}}_{p}$

.

Proposition 2.13
[4, 9.2.3] Exactly one of the following conditions holds:

a)

${\overline{R}}_{p}$b)

${\overline{R}}_{p}$c)

$\exists {p}_{0}$s.t.

${\overline{R}}_{p}$ $\forall p\in \left[1,{p}_{0}$and

${\overline{R}}_{p}$ $\forall p\in \left({p}_{0}\right).$

The lower

$R$

-order,
which we denote here by

${r}_{l}$

${O}_{R}$

in [4, 9.2]), defined by

 ${}_{}$rl = { ∞,   if  R¯p = 0,∀ ⁡p ≥ 1, inf ⁡ {p ∈ [1,∞) : R¯p = 1 },

stood in [4] and in many subsequent works as the definition of the root
convergence order.

Remark 2.14
a) [4, p. 290] If there exists a

${p}_{0}$with

${\overline{R}}_{{p}_{0}}$then for
any

$>0$with

${\overline{R}}_{{p}_{0}}$there
exists a

${k}_{0}$such that either

 ${e}_{k}$0, ifp0 = 1,

or

 ${e}_{k}$,∀ ⁡k ≥ k0, ifp0 > 1.

b) The value

${r}_{l}$arises in the analysis of the speed of convergence of sequences with the errors
satisfying different recurrence inequalities.

The simplest ones appear in usual circumstances for the secant method (see
[3], [9], and also [37]):

 ${e}_{k+1}$

and they yield the well-known

${r}_{l}$-order

$\left(1+\sqrt{5}\right)∕2\approx 1.6$.
It is important to note that the secant method then attains the same value
also for the lower

$Q$-order

${q}_{l}$(see [3], [9]).

For the more general inequalities

 ${e}_{k+1}$i=0m(ek−i)αi,k ≥ m,ck,αk ≥ 0,

Schmidt [42], Burmeister and Schmidt [55], [56] resp. Potra and Pták
[8, p. 107] determined in a series of works the lower

$R$-order

${r}_{l}$.

It is interesting to note that for such inequalities, Herzberger and Metzner
[57] and then Potra [19] gave certain sufficient conditions for the lower bound

${q}_{l}$. Potra
[19] has also noted that the above inequalities do not necessarily attract lower

order

${q}_{l}$greater than one: take

 ekek−1,otherwise, (24)

with

${e}_{0}$2. This sequence
satisfies

${e}_{k+1}$(which
immediately attracts

${\overline{Q}}_{1}$and

${r}_{l}$), but
in fact

${\overline{Q}}_{p}$,

$\forall p>1$(and

${q}_{u}$, as
we shall see).

Certain iterative methods have lead to some different inequalities, of the type
[4, Th. 9.2.9.]

 ${e}_{k+1}$j=0mγ jek−j,∀ ⁡k ≥ m, with γ1,...,γm ≥ 0,

c) As noted in [4, N.R. 9.2.1], the

${\overline{R}}_{1}$factor has been used implicitly in much work concerning iterative processes for
linear systems of equations, see, e.g., Varga [59]. For nonlinear systems, it was
used explicitly by Ortega and Rockoff [60].

d) Some connections between the errors and their majorizing sequences were made
by Dennis Jr. in [61] (see also [13, p. 49]). Potra [19] has noticed that the lower

$R$-order

${r}_{l}$is consistent with the natural ordering of sequences, while the lower

$Q$-order

${q}_{l}$is not:
given

$\left\{{x}_{k}$with
errors

$\left\{{e}_{k}^{\prime }$,
resp.

$\left\{{e}_{k}^{\prime \prime }$, if

${e}_{k}^{\prime }$and

$\left\{{y}_{k}$has

${r}_{l}$order

${p}_{0}$then so
has

$\left\{{x}_{k}$;
however, this statement does not hold for the lower

$Q$-order

${q}_{l}$, as one may take

$\left\{{x}_{k}$from Example
2.7 and

${y}_{k}$</msup >.
A more disturbing example was given by Jay [27], for

${y}_{k}$</msup >and

 ${x}_{k}$2−3k ,kodd. (25)

At the first sight, since

${e}_{k}^{\prime }$,
the speed of

$\left\{{x}_{k}$seems higher
than of the (exact)

$Q$-quadratic

$\left\{{y}_{k}$, but though,

$\left\{{x}_{k}$does not attain even

$Q$-linear convergence, as

${\overline{Q}}_{1}$(we shall see that in this
case the upper

$Q$-order
is

${q}_{u}$).
Another example was given in (24) above.

In 1973, Brent, Winograd and Wolfe [37] considered the following expressions,
which we denote here by

R L</msub > := liminf k</munder > |ln ek</msub > |1
k
</msup >,
(26)
R¯L</msub > := limsup k</munder > |ln ek</msub > |1
k
</msup >

(in [32] Brent considered in the same year only

${\phantom{\rule{4.1994pt}{0ex}}\phantom{\rule{-4.1994pt}{0ex}}R\phantom{\rule{-4.1994pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.1994pt}{0ex}}}_{L}$

). The sequence was said
that converges with

${R}_{L}$

-order

${p}_{0}$

if

${\phantom{\rule{4.1994pt}{0ex}}\phantom{\rule{-4.1994pt}{0ex}}R\phantom{\rule{-4.1994pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.1994pt}{0ex}}}_{L}$

, i.e.,
in a single condition,

$\underset{k\to \infty }{\mathrm{lim}}$</msup >1k =p˙0. (27)

They noted that the

$C$-order

${p}_{0}$easily implies

${R}_{L}$-order

${p}_{0}$.

In 1979, Schwetlick [25] considered the upper

$R$-order

 ${}_{}$ R p {xk} = { liminf ⁡ k→∞(ek)1 k ,  ifp = 1, liminf ⁡ k→∞(ek) 1 pk , ifp > 1,

and (in the notations of this paper)

 ${}_{}$ru = { ∞,   if   R p = 0,∀ ⁡p ≥ 1, sup ⁡ {p ∈ [1,∞) : R p = 0 },

he defined the convergence with

$R$-order

${p}_{0}$if

${p}_{0}$Similarly to the case of the

$Q$-order,
the sequence is said that converges

$R$-superlinearly
if

 $\underset{k\to \infty }{\mathrm{lim}}$k = R¯1 = 0.

Remark 2.15
Analogously to Remark 2.14 , whenever exists a

${p}_{0}$with

${\phantom{\rule{4.1994pt}{0ex}}\phantom{\rule{-4.1994pt}{0ex}}R\phantom{\rule{-4.1994pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.1994pt}{0ex}}}_{{p}_{0}}$then for
any

$>0$with

$0<{\phantom{\rule{4.1994pt}{0ex}}\phantom{\rule{-4.1994pt}{0ex}}R\phantom{\rule{-4.1994pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.1994pt}{0ex}}}_{{p}_{0}}$there
is a

${k}_{0}$such that either

 ${}^{}$ ( R p0 − )k ≤ e k,∀ ⁡k ≥ k0, ifp0 = 1,

or

 ${}^{}$ ( R p0 − )p0k ≤ ek,∀ ⁡k ≥ k0, ifp0 > 1.

In 1989, Potra [19] has introduced and studied some further measures for the

$R$-convergence order,
which we denote here by

${R}_{}$,
resp.

${R}_{I,}$:

• $\left\{{x}_{k}$converges
with

${R}_{}$-order

${p}_{0}$if

$\begin{array}{}\end{array}$lim k</munder >(ek</msub >) 1
(p0</msub >)k</msup >
</msup >
= 0, > 0,
lim k</munder >(ek</msub >) 1
(p0</msub >+)k</msup >
</msup >
= 1, > 0;

• $\left\{{x}_{k}$converges
with

${R}_{I,}$-order

${p}_{0}$if

$\forall >0,$

$\exists a,b>0$and

$0<\eta ,<1$such that

 $a\phantom{\rule{0.17em}{0ex}}{\eta }^{{\left({p}_{0}\right)}^{k}}$ ≤ ek ≤ b (p0−)k ,∀ ⁡k ≥ 0.

The exact

$R$-order
of convergence was also defined here – analogously to the definition of the exact

$Q$-order (4)
– as being

${p}_{0}$if

$\exists A,B>0$,

$0<\eta ,<1$such
that

 $A\cdot {\eta }^{{\left({p}_{0}}^{k}}$ ≤ ek ≤ B ⋅ (p0)k ,∀ ⁡k ≥ 0; (28)

we see that this definition can be immediately connected to the following one,
analogous to (6):

 $0<{\phantom{\rule{4.1994pt}{0ex}}\phantom{\rule{-4.1994pt}{0ex}}R\phantom{\rule{-4.1994pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.1994pt}{0ex}}}_{{p}_{0}}$ (29)

Potra has also considered the lower and upper

$R$-orders, when noting that if
a sequence has exact

$R$-order

${p}_{0}$then

${r}_{l}$.

Comparing the list of

$Q$
and

$R$-orders, the
definition of

${R}_{\Lambda }$was not given before, so we consider it here, by denoting the expressions

 ${}_{}$ R Λ := liminf ⁡ k→∞ |ln ⁡ ek+1 ek |1 k ,R¯Λ := limsup ⁡ k→∞ |ln ⁡ ek+1 ek |1 k

and by defining the

${R}_{\Lambda }$-order

${p}_{0}$when

${\phantom{\rule{4.1994pt}{0ex}}\phantom{\rule{-4.1994pt}{0ex}}R\phantom{\rule{-4.1994pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.1994pt}{0ex}}}_{\Lambda }$, i.e.,
when

 $\underset{k\to \infty }{\mathrm{lim}}$ek |1 k = p0. (30)

### 3 Results relating the high (computational) convergence orders

In this section we consider

$p>1$and present the relationships of different definitions of the
(computational) convergence orders, with full proofs in the case of the

$C$– and

$Q$-orders.

#### 3.1 Relationships between the convergence orders

The equivalences and implications mentioned in the previous section can be
summarized as:

$\begin{array}{}\end{array}$ {Q</msub >,QI,</msub >,QL</msub >} {R</msub >,RI,</msub >,RL</msub >} , [19],
Cp0</msub ></msub > {Q,QL</msub >,QΛ</msub >} , [20].
At this point we can simply merge the equivalences from the two statements
above and write the conclusion.

However, as we intend to provide a self-contained
material, we include the proofs of the equivalence of the

$Q$-orders,
some of them much simplified.

##### 3.1.1 C- and Q-convergence orders

We shall need the three auxiliary results below.

Lemma 3.1
[20, lemma 2.1]

(i)
If

${\overline{Q}}_{p}$then

${Q}_{s}$for all

$s,
and so

$p\le {q}_{l}$

(ii)
if

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{p}$then

${Q}_{s}$for all

$pand so

${q}_{u}$

(iii)
if

$p<{q}_{l}$then

${Q}_{p}$

(iv)
if

${q}_{u}$then

${Q}_{p}$.

Relations (iii) and (iv) show that

${q}_{l}$and justify the terminology ”lower” and ”upper”.

Proof. [20] For (i), if

${\overline{Q}}_{p}$and

$sthen

 ${Q}_{s}$eks = ekp−sQ p(k) → 0,as k →∞.

For (iii), suppose

$p<{q}_{l}$and choose

$s\in \left(p,{q}_{l}\right).$By
definition of

${q}_{l}$then,

${\overline{Q}}_{s}$. Now,
by (i),

${Q}_{p}$The other statements are proved in a similar manner.  _

Remark 3.2
Ortega and Rheinboldt have previously obtained (i) and (ii)
in [4, 9.1], see Proposition 2.5 .

Potra obtained the following general result, interesting in its own, which will
help us relating some quantities to their corresponding computational
versions.

Proposition 3.3
[19, Prop. 1.1] The following relations hold:

 ${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{L}$ ln ⁡ ek = ql (31)

and

 ${}_{}$Q¯L := limsup ⁡ k→∞ln ⁡ ek+1 ln ⁡ ek = qu.

Proof. [19] Let

 $1<{\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{L}$

Then for any

$>0$with

${p}_{0}$, there
exists

${k}_{0}$such that

 $\frac{}{}$ln ⁡ ek+1 ln ⁡ ek > p0 − ,k ≥ k0.

We may assume that

$\mathrm{ln}{e}_{k}$,
and we obtain

 $\mathrm{ln}{e}_{k+1}$

so that

 ${e}_{k+1}$0.

This proves that

${\overline{Q}}_{{p}_{0}}$and, by Lemma 3.1 (i), we get

 ${p}_{0}$

Suppose that

 ${p}_{0}$

It follows by Lemma 3.1 (iii) that

${Q}_{s}$so there
is

$c>0$such
that

${Q}_{s}$ $\forall k\ge 0,$i.e.,

 ${e}_{k+1}$

But then

 $\mathrm{ln}{e}_{k+1}$

 $\underset{k\to \infty }{\mathrm{liminf}}$ln ⁡ ek ≥ s + lim ⁡ k→∞ ln ⁡ c ln ⁡ ek = s > p0,

which is a contradiction. Hence, we have proved that

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}}_{}$ L</msub > = ql</msub >.Analogously,

${\overline{Q}}_{L}$implies

${p}_{1}$ _

Lemma 3.4
[20, lemma 3.3] If

 $\mathrm{lim}\mathrm{inf}{Q}_{\Lambda }$

then

$\left\{{e}_{k}$is monotone decreasing starting from a step

$k\ge {k}_{0}$.

Proof. [20] For some some

$>0$sufficiently small,

$\exists {k}_{0}$such that

${Q}_{\Lambda }$ $\forall k\ge {k}_{0}$Hence, the
numbers

$\mathrm{ln}\frac{{e}_{k+1}}{}$ek</msub >all have
the same sign

$\forall k\ge {k}_{0}$.
If they were positive, then we would have

${e}_{k+1}$, contradictory to the
convergence of

$\left\{{x}_{k}$Therefore,
they are all negative and

${e}_{k+1}$ _

We present now the result relating the

$C$– and

$Q$-orders,
and we choose to incorporate Proposition 3.3 .

Theorem 3.5
(cf. [19], [20]) Consider a given norm

$\parallel \cdot \parallel$in

${ℝ}^{n}$and a convergent
sequence

$\left\{{x}_{k}$to
some

${x}^{\ast }$. Then,
given some

${p}_{0}$,
in the above notations we have:

 ${C}_{{p}_{0}}$ (32)

Moreover,

 (33)

Proof. We give the proof in five steps: A.

${C}_{{p}_{0}}$, B.

$Q⇔{Q}_{}$, C.

${Q}_{}$, D.

$Q⇔{Q}_{L}$and
E.

${Q}_{\Lambda }$A1 (

${C}_{{p}_{0}}$).
The simplest approach is (also mentioned in [37]) to take logarithms in (2). The
proof in [20] used an approach similar to Lemma 3.1 .

A2 (

${Q}_{L}$)
[20] Consider any of the sequences from Example 2.8 .

B. The relation is obvious.

C1 (

${Q}_{}$)
Let

$>0$such
that

${p}_{0}$Since

${Q}_{p-}$, it follows
that exists

$b$such that

${Q}_{{p}_{0}}$ $\forall k\ge 0,$i.e., the second inequality
in the definition of

${Q}_{I,}$The first inequality is obtained in the same way.

C2 (

${Q}_{I,}$) Suppose
for some

${p}_{0}$and

$>0$with

${p}_{0}$we have

 $a{\left({e}_{k}}^{{p}_{0}}$k+1 ≤ b(ek)p0−,∀ ⁡k ≥ 0.

The second inequality implies

${\overline{Q}}_{{p}_{0}}$and by Lemma 3.1 (i),

$\mathrm{lim}{Q}_{q}$s.t.

$1The first inequality (valid for all

$>0$)
implies

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{{p}_{0}}$and again
by Lemma 3.1 (ii),

$\mathrm{lim}{Q}_{s}$with

$p+The assertion follows since

can be taken arbitrarily small.

D. [19, Prop. 1.1] Using the above result, we get

${Q}_{L}$and
vice versa.

E1 (

${Q}_{\Lambda }$)
[20] We notice that, by B and D, we have that

${Q}_{L}$, so it suffices
to show that

${Q}_{\Lambda }$.

Let

${Q}_{\Lambda }$For any

$>0,$we have

${Q}_{\Lambda }$for all

$k$sufficiently large, and using Lemma 3.4 we get

 $\mathrm{ln}\frac{{e}_{k+2}}{}$ek+1 > (p0 + )ln ⁡ ek+1 ek ,

which gives

 $\frac{}{}$ek+2 (ek+1)p0+ > ek+1 (ek)p0+,

i.e.,

${Q}_{{p}_{0}}$ $\forall k\ge {k}_{0}$Considering now

$\frac{}{}$2instead of

$,$we obtain similarly that

 ${Q}_{{p}_{0}}$2 (k + 1) > Qp0+ 2 (k),∀ ⁡k ≥ k1. (34)

We prove that the monotone sequence

$\left\{{Q}_{{p}_{0}}\right\}$is
unbounded:

${Q}_{{p}_{0}}$Otherwise, if

$\exists M>0$s.t.

$\frac{{e}_{k+2}}{}$(ek+1</msub >)p0</msub >+</msup > < M,for
all

$k,$i.e.,

${\overline{Q}}_{{p}_{0}}$then according to Lemma
3.1 (i), this implies

${\overline{Q}}_{{p}_{0}}$2 </msub > = 0in contradiction to the monotony from (34 ).

Similarly, one can prove that for any

$>0$with

${p}_{0}$we have

${Q}_{\Lambda }$for all

$k$sufficiently large
and we get

${Q}_{{p}_{0}}$E2 (

${Q}_{L}$) [20]
Assuming

${Q}_{L}$we have

 $\underset{k\to \infty }{\mathrm{lim}}$QL (k)−1 = p0p0−1 p0−1 = p0.

_

It is interesting to notice that

$Q$-convergence with
order

${p}_{0}$implies

${\overline{Q}}_{1}$However, the reverse is not true
(i.e.,

$Q$-superlinear convergence
does not imply

$Q$-convergence
with order

${p}_{0}$),
as the following examples show.

Example 3.6
a) [4, E 10.1.4] (we follow here [8, p. 94]) Let

$0and
consider the sequence

 ${x}_{k+1}$ln ⁡ xk,k ≥ 0,x0 = c,

which converges superlinearly to

$0.$Suppose there exists

${p}_{0}$such that

$\left\{{x}_{k}\right\}$converges
with

$Q$-order (and
therefore with

${Q}_{I,}$order)

${p}_{0}$Then

$\forall >0$with

${p}_{0}$s.t.

${x}_{k+1}$Hence,

 $-\frac{{x}_{k}}{}$ln ⁡ xk ≤ bxkp0−or ( 1 xk )p0−−1 ln ⁡ 1 xk ≤ b,

for all

$k$sufficiently large, which contradicts the well known limit

$\underset{t\to \infty }{\mathrm{lim}}$ln t = ,
for any

$\alpha >0$.
As

${Q}_{L}$,
we get

${q}_{l}$.

b) Some simpler examples can be found either as explicitly given by

${x}_{k}$[32, p. 22],

${x}_{k}$</msup >[26], [20],

${x}_{k}$k![27], or in the list of exercises, by

${x}_{k}$</msup >,

$c>1$[4, E.9.2.1.j]. To show these assertions we can also use that

${Q}_{L}$or

${Q}_{\Lambda }$.
Expression (24) of Potra gives yet another example.

Remark 3.7
We notice that the orders

${Q}_{}$do not have elements to define the strict

$Q$-superlinear
convergence, when

${\overline{Q}}_{1}$and

${\overline{Q}}_{p}$ $\forall p>1$.

The

$Q$-superlinear
convergence implies

$R$-superlinear
convergence, as in any norm [4, 9.3.1]

 ${}_{}$R¯1 ≤Q¯1.

In fact,

${\overline{R}}_{1}$is a
lower bound for

${\overline{Q}}_{1}$in all norms from

${ℝ}^{n}$.

It is interesting to see that sequences with infinite

$Q$-order exist too,
as one can take

${x}_{k}$</msup >(from the list of exercises in [4, E.9.2.1]).

Remark 3.8
An important aspect we address now is the problem of changing
the norms, as we have seen in Example 2.3 that for a given sequence in

${ℝ}^{n}$the existence of the

$C$-order
is norm-dependent. Ortega and Rheinboldt [4, 9.1.6] proved that the three
relations

${\overline{Q}}_{p}$,

${\overline{Q}}_{p}$finite,

${\overline{Q}}_{p}$are norm-independent. The same hold for

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{p}$,
as one can easily verify (using also Lemma 3.1 ), which shows the norm-independence
of the

$Q$-order.
By the equivalence from Theorem 3.5 , we conclude that the rest of the

$Q$-orders
are norm independent too. This assertion will be further completed by the
computational convergence orders, as we shall see in the following subsection.

It is important to recall that for sequences with

$Q$-order

${p}_{0}$,

while this

$Q$-order
is norm-independent, the values of the quantities

${\overline{Q}}_{{p}_{0}}$and

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{{p}_{0}}$however, if finite, are norm-dependent [4, E.9.1-1].

Remark 3.9
One can easily show that the sequence

$\left\{{x}_{k}$converges with

$Q$-order

${p}_{0}$iff one of the the following sequences converges with the same order:

$\left\{\frac{{e}_{k+1}}{}$ek</msub > },

$\left\{{e}_{k+1}$or

$\left\{{\left({e}_{k}}^{\alpha }$(

$\alpha >0$arbitrary, given).

##### 3.1.2 R-convergence orders

In 1972, Brent [32] stated, while in 1984, Potra and Pták [8] shown
that

 ${r}_{l}$k . (35)

In 1989 Potra [19] stated the equivalence

$\left\{{R}_{}$,
which can be easily proved.

In fact we may extend Theorem 3.5 and the above relations. For brevity, we choose to write

$\left\{Q\right\}$as a generic form for all
the five equivalent

$Q$-type
orders,

$\left\{Q,{Q}_{}$. The same will be
written in the sequel for

$\left\{{Q}^{\prime }$,

$\left\{{Q}^{\prime \prime }$,

$\left\{R\right\}$,

$\left\{{R}^{\prime }$and

$\left\{{R}^{\prime \prime }$.

Theorem 3.10
(cf. [19]) In the assumptions of Theorem 3.5 , we have:

 ${C}_{{p}_{0}}$ (36)

(with the equivalence of

${R}_{\Lambda }$requiring

$1<{q}_{l}$).

Moreover,

 ${q}_{l}$ (37)

Proof. (cf. [19]) We notice first that

${Q}_{I,}$-order

${p}_{0}$implies

${R}_{I,}$-order

${p}_{0}$(and
therefore

$\left\{Q\right\}⇒\left\{{R}_{}$,
as mentioned above); this is obtained from the proofs of Propositions
9.3.2 and 9.3.3 in [4] and taking into account Proposition 2.13
above. A simpler approach results by considering directly (37), with

${q}_{l}$.

The converse ”

$Q⇐R$
is false, even if

$\left\{{x}_{k}$has
exact

$R$-convergence
order

${p}_{0}$(i.e., (28)
or (29) hold), as shown in Examples 2.7 and 3.11 . One can easily show however that if

$\eta =$in (28), this attracts
convergence with

$Q$-order

${p}_{0}$.

The equivalence of the

${R}_{\Lambda }$-order

${p}_{0}$to the

${R}_{L}$, and therefore all
remaining

$R$-orders,
is based on the relation

 $\mathrm{ln}\frac{{e}_{k+1}}{}$ek = (ln ⁡ ek) (ln ⁡ ek+1 ln ⁡ ek − 1 ).

We note that there exist sequences (see, e.g., (25)), for which

${q}_{l}$, and which can therefore
be studied for the

$R$-orders.

Relation (37) – which is a completion of the one stated and proved by Potra [19] for the
lower orders

${q}_{l}$– can be obtained again from the well known inequalities for positive numbers

 $\underset{k\to \infty }{\mathrm{liminf}}$ak ≤ liminf ⁡ k→∞ |ak| 1 k ≤ limsup ⁡ k→∞ |ak| 1 k ≤ limsup ⁡ k→∞ak+1 ak

taking

${a}_{k}$,
and then considering (31 ) and (35 ).  _

The following example shows that one may find sequences with

$R$-order
(

${r}_{l}$) arbitrary
high and

${q}_{l}$arbitrary close to 1 from above.

Example 3.11
[19] Given any numbers

$1we construct a
sequence

$\left\{{x}_{k}\right\}$which has
the exact

$R$-convergence
order

$\tau$and
for which

${q}_{l}$Let

$0<<1$be given,
and take

$\eta ={}^{q}$with

$q>1$such
that

$qs>\tau .$Define

 ${}_{}$xk = { τk,kodd, ητk ,keven.

The sequence

$\left\{{x}_{k}\right\}$has
exact

$R$-convergence
order

$\tau .$On the
other hand, for

$k$even we obtain

 $\frac{{x}_{k+1}}{}$xks = (τ) τk (ηs) τk = ( τ qs )τk →∞,

so that

${q}_{l}$In fact, one obtains

${q}_{l}$qwhile

${q}_{u}$(

$>\tau$,
this value representing the exact

$R$-order).

This example is also suitable for justifying the mentioned comments from
[13].

In (37), any inequality may be strict. Now we see (e.g., Examples
2.7 and 3.11 ) that the inner inequality may be equality (i.e., obtain

$R$-order)
while one of the outer inequalities may be strict (i.e., no

$Q$-order).

This also shows that the sequences with

$Q$-order

$1$and

$Q$-superlinear
convergence (see Example 3.6 ) cannot have (upper)

$R$-order
greater than one, i.e., they converge strict

$R$-superlinearly. The proof
is based on relation

${\overline{R}}_{1}$(see [4]) and on (37).

Remark 3.12 We notice that the

$R$-order
is also a norm independent notion, as it can be proved by some results of
Ortega and Rheinboldt [4, 9.2.2].

#### 3.2 Results relating the computational convergence orders

In this section, the errors

${x}^{\ast }$are
replaced either by the corrections

${x}_{k+1}$or by the nonlinear residuals

$F\left({x}_{k}$,
when solving nonlinear systems having the same number of equations and unknowns. We
shall assume that

${s}_{k}$and

$F\left({x}_{k}\right)\ne 0,$ $\forall k\ge 0.$We present full proofs only for the

$C$– and

$Q$-convergence
orders.

##### 3.2.1 Corrections

$C$-,

$Q$-orders
and corrections

As we have mentioned, Beyer, Ebanks and Qualls [20] shown that, for

${p}_{0}$, the
errors and corrections converge simultaneously to zero with the same

$C$– resp.

$Q$-type
convergence order, and therefore each convergence order is equivalent to its
corresponding computational one. The following relation resulted:

 $\left\{C,{C}^{\prime }\right\}⇒\left\{Q,{Q}_{}$L,QΛ,Q′,Q L′,Q Λ′}. (38)

However, for proving

$\left\{C,{C}^{\prime }\right\}$and

${Q}^{\prime }$the considered assumptions were very strong:

$\left\{{x}_{k}\right\}$was considered as a
monotone sequence from

$ℝ$(see [20, Sect. 4]).

We shall extend relation (38 ) and show some simplified proofs. For that, we
first present two auxiliary results.

Proposition 3.13
(Potra-Pták Lemma) [8, Prop. 6.4] The sequence

$\left\{{x}_{k}$converges

$Q$-superlinearly
to

${x}^{\ast }$if and only if the sequence

$\left\{{s}_{k}$converges

$Q$-superlinearly
to zero.

Proof. Necessity. [8] Suppose

$\left\{{x}_{k}$converges

$Q$-superlinearly. Then
there exists a positive integer

${k}_{0}$such that

 ${e}_{k+1}$2ek,   for all k ≥ k0.

It follows that for

$k\ge {k}_{0}$,
we have:

 $\frac{}{}$sk+1 sk ≤ ek+1+ek+2 ek−ek+1 ≤ 3ek+1 ek ,

from which we deduce the

$Q$-superlinear convergence
of the sequence

$\left\{{s}_{k}$Sufficiency. [8] If we suppose that the sequence

$\left\{{s}_{k}$converges

$Q$-superlinearly, then there
exists a positive integer

${k}_{0}$such that

 ${s}_{k}$3sk−1,   for all k ≥ k0.

For any

$k\ge {k}_{0}$we have:

$\begin{array}{}\end{array}$ek+1</msub >
j=1</munderover >sk+j</msub > sk+2</msub >
j=0</munderover >3j</msup > = 3
2
sk+1</msub > 1
2
sk</msub >.

Finally, writing

 $\frac{}{}$ek+1 ek ≤ 3 2sk+1 sk−ek+1 ≤ 3sk+1 sk ,

we infer that

$\left\{{x}_{k}$converges

$Q$-superlinearly.
_

For necessity, an alternative proof in the one dimensional case can
be obtained using the Brezinski’s ”l’Hospital’s rule for sequences” (see
[28]).

The necessity was proved in the general case even before, by Dennis and
Moré.

Lemma 3.14
(Dennis-Moré) [5] If the sequence

$\left\{{x}_{k}$converges

$Q$-superlinearly
then

 $\underset{k\to \infty }{\mathrm{lim}}$ek = 1.

Proof. [5] The result follows immediately from the equality

 $\parallel \frac{{x}_{k+1}}{}$∥x∗−xk∥ − x∗−xk ∥x∗−xk∥ ∥ = ek+1 ek .

_

The assumption of this Lemma also implies that

$\left\{{s}_{k}$converges

$Q$-superlinearly
to zero too.

The converse of this result does not hold, as an example given by the same authors
shows [5]:

${x}_{2k-1}$k!,

${x}_{2k}$,

$k\ge 1$.

Another proof of the Potra-Pták lemma was independently given by Walker
[47], in 1997.

Proof of Potra-Pták lemma. Necessity [47]: the Dennis-Moré Lemma.

Sufficiency. [47] Suppose

${s}_{k}$,

$Q$-superlinearly.
Then

 $\frac{{e}_{k+1}}{}$ek = ∥−∑ k+1∞(xj+1 − xj)∥ ∥−∑ k∞(xj+1 − xj)∥ ≤ ∑ k+1∞sj sk −∑ k+1∞sj = ∑ k+1∞sj sk 1 −∑ k+1∞sj sk.

For each

$k$,
define

${\rho }_{k}$.
Note that

$\underset{k\to \infty }{\mathrm{lim}}$and

${s}_{k+i}$for
each

$i\ge 0$.
Then

 $\frac{{e}_{k+1}}{}$ek ≤ ρk 1−ρk 1 − ρk 1−ρk = ρk 1 − 2ρk → 0,

and it follows that

${x}_{k}$ $Q$-superlinearly.
_

We are now able to present the main result of this
subsubsection, which shows not only equivalence of

$C$-type resp.

$Q$-type orders, but also that
the lower and upper

$Q$-orders
are limits of some (truly) computational quantities.

Theorem 3.15
(cf. [20]) In the assumptions of Theorem 3.5 and notations
above, the following (generic) relations hold:

 $\left\{C,{C}^{\prime }$

Moreover,

$\begin{array}{}\end{array}$ Q L</msub > = Q L</msubsup > = q
l
</msub >and   Q¯L</msub > = Q¯L</msubsup > = q
u
</msub >,
(39)
Q qu</msub ></msub > = Q qu</msub ></msubsup >and   Q¯
q
l</msub >
</msub > = Q¯ql</msub ></msubsup >.
(40)

Proof. The proof will consist of two steps. At step A, we prove

$\left\{C,{C}^{\prime }$,
while at step B we prove the equivalence of the

$Q$-type and

${Q}^{\prime }$-type
orders. Relations (39) and (40) follow along the way.

A1 (

$C⇒{C}^{\prime }$) Suppose

$\left\{{x}_{k}\right\}$converges
with

$C$-order

${p}_{0}$Then

$\left\{{x}_{k}\right\}$converges

$Q$-superlinearly,
and by the Potra-Pták and Dennis-Moré Lemmas, we get

 $\underset{k\to \infty }{\mathrm{lim}}$(ek)p0sk+1 = 1, (41)

which shows that

$\left\{{x}_{k}\right\}$converges with

${C}^{\prime }$-order

${p}_{0}$A2 (

${C}^{\prime }$)
A2 is proved similarly to A1.

B1 (

$Q$ $⇒$ ${Q}^{\prime }$) Suppose

$\left\{{x}_{k}\right\}$converges with
some

$Q$-type
order

${p}_{0}$Then

$\left\{{x}_{k}$converges
with

$Q$-order

${p}_{0}$and therefore

$Q$-superlinearly;
by the Potra-Pták and Dennis-Moré Lemmas, it follows that

 $\underset{k\to \infty }{\mathrm{lim}}$ln ⁡ sk = lim ⁡ k→∞ln ⁡ (sk+1 ek+1ek+1 ) ln ⁡ (sk ekek ) = p0, (42)

i.e.,

$\left\{{x}_{k}\right\}$converges
with

${Q}_{L}^{\prime }$-order

${p}_{0}$One
can view the previous statement from a different viewpoint: the sequence

$\left\{{s}_{k}$converges
with

${Q}_{L}$-order

${p}_{0}$so by the results in Subsection 3.1 , it converges with any

$Q$-type
order

${p}_{0}$,
whence the conclusion.

Relation (42) and Proposition 3.3 attract (39), while (41) implies
(40).

B2 (

${Q}^{\prime }$-type

$⇒$ $Q$-type
order) The reverse implication is proved similarly.  _

Remark 3.16

a) Relation (40) shows in particular that if

$\left\{{x}_{k}$has an exact

$Q$-order
of convergence

${p}_{0}$,
then the sequence of corrections has not only the same

$Q$-order,
but even the same asymptotic constants.

b) We note that the norm-independence of the

$Q$-type
orders attract the same property for the

${Q}^{\prime }$-type
orders.

$R$-orders
and corrections

Potra and Pták [8] have obtained some fundamental results in this direction,
which can be easily completed:

Theorem 3.17
(cf. [8, Prop. 6.20]) In the assumptions of Theorem 3.5 and
notations above, the following (generic) relations hold:

 $\left\{R,{R}^{\prime }$

(with the equivalence of the

${R}_{\Lambda }$-orders

$1<{q}_{l}$).

The proof relies on the result [8, Prop. 6.20]

 ${r}_{l}$

which leads by analogy to relation

${r}_{u}$.

The following three results show that the asymptotic constants from the

$R$-convergence
orders of the (errors of a) sequence and of its corrections are
not so tightly connected as in the case of convergence with

$Q$-order
(relation (40)).

Corollary 3.18
[8, p. 111] If

$\left\{{x}_{k}\right\}$and

$\left\{{x}_{k+1}\right\}$have exact

$R$-orders
of convergence, then they must have the same exact

$R$-order
of convergence.

Proposition 3.19
[8, p. 111] If

$\left\{{x}_{k}\right\}$has an exact

$R$-order
of convergence this does not imply that

$\left\{{x}_{k+1}\right\}$has an exact

$R$-order
of convergence, and vice versa.

Example 3.20
[8, p. 111] Let

$c,r,s$be three real numbers with

$0and set

 ${}_{}$xk = { crk, k even, crk−1 + csk−1 ,k odd, yk = { −crk,k even, csk−1 ,k odd.

It is easy to see that

$r$is the exact

$R$-order
of convergence of the sequences

$\left\{{x}_{k}\right\}$and

$\left\{{y}_{k+1}\right\},$while the sequences

$\left\{{x}_{k+1}\right\}$and

$\left\{{y}_{k}\right\}$have no exact

$R$-order
of convergence.

Remark 3.21 One can easily show that the sequence

$\left\{{x}_{k}$converges with

$R$-order

${p}_{0}$iff one of the the following sequences converges with the same order:

$\left\{{e}_{k+1}$or

$\left\{{\left({e}_{k}}^{\alpha }$(

$\alpha >0$arbitrary, given). The same holds for

$\left\{\frac{{e}_{k+1}}{}$ek</msub > },
but under the supplementary assumption that

$1<{q}_{l}$.

##### 3.2.2 Nonlinear residuals $F\left({x}_{k}$

– relationships

We consider now the solving of a system of nonlinear equations

$F\left(x\right)=0$for a nonlinear
mapping

$F:D\subseteq {ℝ}^{n}$, having
a solution

${x}^{\ast }$.
The quantities

$\parallel F\left({x}_{k}$– in different formulae – were used not only to control the convergence of a given
sequence

$\left\{{x}_{k}$to

${x}^{\ast }$, but also to determine
at each step

$k$the
next element,

${x}_{k+1}$(see, e.g., the well known works of Dembo, Eisenstat and Steihaug [7],
Eisenstat and Waker [10] and some extensions we have obtained in [11],
[12]).

We assume here that

A)

$F$is differentiable
on the open domain

$D;$B)

${F}^{\prime }$continuous at

${x}^{\ast }$C)

${F}^{\prime }$is
invertible at

${x}^{\ast }$Replacing the errors by the corresponding residuals, we denote

 ${Q}_{{p}_{0}}^{\prime \prime }$∥Fk∥p0

and similarly, the resulted definitions for the

${C}^{\prime \prime }$– and all the
rest of

${Q}^{\prime \prime }$-type
orders.

$C$-,

$Q$-orders
and nonlinear residuals

The following example shows that neither of

$\left\{C,{C}^{\prime }$and

${C}^{\prime \prime }$implies the other.

Example 3.22
a) Consider

${x}_{k}$,
with

 ${}_{}$xk = { (2−2k,0), kodd, (2−2k ,2−2k−1 ),keven,

and the (linear) mapping

$F:\left({ℝ}^{2}$,

$F\left(u,v\right)=\left(u,4v\right)$.
One can see that conditions A)–C) are satisfied, and further,

$\left\{{x}_{k}$has

$C$-order

$2$,
while

$\left\{{F}_{k}$does not have

$C$-order
(in the max-norm).

b) Let

${y}_{k}$,
with

 ${}_{}$yk = { (2−2k,0), kodd, (2−2k ,2−2k+1 ),keven,

and similarly the mapping

$G\left(u,v\right)=\left(u,v∕4\right)$.
Now

$\left\{{y}_{k}$has no

$C$-order,
while

$\left\{{G}_{k}$has

$C$-order

$2$(in the max-norm).

One can also view Example 2.3 in the sense that if one takes

$F:\left({ℝ}^{2}$as the identity
operator,

$F=I$, the
sequence

$\left\{{x}_{k}$from
(8) has

$C$-order

$2$, while

$\left\{{F}_{k}$does not
have

$C$-order.

Before the result relating the orders, we recall (without proof) an
auxiliary result, which is presented in different standard monographs and
papers.

Lemma 3.23
[7] Assume the mapping

$F$is differentiable on
the open set

$D$, the
derivative

${F}^{\prime }$continuous
at the solution

${x}^{\ast }$, with the

Jacobian nonsingular at

${x}^{\ast }$:

$\exists {F}^{\prime }$. Let

$\alpha =\mathrm{max}\left\{\parallel {F}^{\prime }$2β,2β }, where

$\beta =\parallel {F}^{\prime }$. Then there
exists

$>0$such
that for any

$x\in {ℝ}^{n}$with

$\parallel x-{x}^{\ast }\parallel \le$,
one has

 $\frac{}{}$1 α ∥x − x∗∥ ≤ ∥F (x)∥ ≤ α ∥x − x∗∥.

We notice that

$\alpha >1$.
The sharpest bounds seem to be obtained by Rheinboldt [6, Th.4.2]: given

$0<<\frac{1}{}$F</msup >(x</msup >)1</msup >, there
exists

$\delta >0$s.t.

$\forall x$with

$\parallel x-{x}^{\ast }$we have

 $\left(\frac{1}{}$∥F′(x∗)−1∥ − ) ∥x − x∗∥≤ ∥F (x)∥ ≤ (∥F′ (x∗)∥ + )∥x − x∗∥.

We obtain the final result from this paper, which shows in fact the equivalence of
all

$Q$-type
orders.

Theorem 3.24
Under the assumptions of Theorem 3.5 , A)–C) and with the
above notations, the following (generic) relations hold:

$\begin{array}{}\end{array}$ {Q,Q</msup >,Q</msup >},
{C,C</msup >} C</msup >,
{C,C</msup >} C</msup >,
C</msup >{Q,Q</msup >,Q</msup >}.

Proof. Assume

$\left\{{x}_{k}\right\}$converges
with some

$Q$-type
order, say

${Q}_{L}$-order

${p}_{0}$Then, by Lemma 3.23 , we have

 $\frac{}{}$ln ⁡ ek+1−ln ⁡ α ln ⁡ ek+ln ⁡ α ≤ ln ⁡ ∥Fk+1∥ ln ⁡ ∥Fk∥ ≤ ln ⁡ ek+1+ln ⁡ α ln ⁡ ek−ln ⁡ α ,

which shows that

${Q}_{L}^{\prime \prime }$and further
the equivalence of the

${Q}^{\prime \prime }$-type
orders.

The implication

$\left({Q}_{L}^{\prime \prime }$ $⇒$ $\left({Q}_{L}$follows in a similar manner.  _

The

${Q}^{\prime \prime }$-convergence
orders are norm-independent, being equivalent to the

$Q$-orders.

$R$-orders
and nonlinear residuals

We may complete Theorem 3.17 by the generic relation

 $\left\{R,{R}^{\prime }$

the proof relying on the inequality from Lemma 3.23 .

#### 3.3 Summary of the high (computational) convergence orders

We present the following diagram in order to have a complete
picture on the implications/equivalences of the high orders

$p>1$. As in [20], we use
the symbol ”

$\to$” for the
conjunction of

$⇒$and

$⇍$, while the equivalence of
the

${R}_{\Lambda }$-type orders requires

$1<{q}_{l}$.

### 4 Linear convergence

The

$Q$-linear
convergence is less attractive in numerical applications, but remains an important
theoretical topic (e.g., in the field of Dynamical Systems, as noted in
[20]).

In contrast to the high convergence orders

$p>1$, in
the linear case there are fewer equivalences and more negative results,
illustrated by counterexamples. Beyer, Ebanks and Qualls [20] did most of the
work in providing them, in [20], which we also summarize in the diagram
below:

However, it is worth noting that different relations (e.g.,

$\left\{{C}_{1}\right\},$ ${Q}_{1}^{\prime }$)
were obtained under the strong assumption that

$\left\{{x}_{k}\right\}$is a monotone
sequence from

$ℝ.$Zhang and Wang [62] completed the diagram with two implications. The first one,

${\Lambda }_{1}$was proved
again under the monotony assumption on the sequence of real numbers. The second one,

${\Lambda }_{1}^{\prime }$, was proved
under some even further supplementary assumptions (we mark this by dashed line in the
diagram):

$\mathrm{limsup}{a}_{k}$and

$\mathrm{liminf}{a}_{k}$,
where

${a}_{k+2}$xk+1</msub >xk</msub >,

${a}_{1}$,

${a}_{0}$. The
following completed diagram resulted.

### 5 Conclusions

As the completion

${q}_{u}$for the
definition of the

$Q$-order
of Beyer, Ebanks and Qualls remained unknown, most of the results on the

$Q$-orders
of different iterative methods were given only in terms of the lower

$Q$-order

${q}_{l}$. Ortega and
Rheinboldt used

${q}_{l}$and

${r}_{l}$to measure the speed of convergence of some ”entire” iterative processes
both for fixed point problems (the Ostrowski local attraction theorem [4,
Th. 10.1.3, Th. 10.1.6, Th. 10.1.7]) and for nonlinear systems of equations (the
Newton attraction theorem [4, Th. 10.2.2]). Further results for characterizing the lower

bounds

${q}_{l}$for a single sequence (instead of the whole method) were obtained by us [14], [63]
– in the case of (perturbed) successive approximations – respectively by Dembo,
Eisenstat and Steihaug [7] and us [11], [14], in the case of different Newton-type
iterations in presence of error sources.

It is interesting to note that results on the upper

$Q$-order

${q}_{u}$exist
too. One can find them (where else?) in the book of Ortega and Rheinboldt, both
for the successive approximations [4, Th.10.1.7], and for the Newton method [4,
Th.10.2.2]. It seems interesting to clarify the conditions ensuring the upper

$Q$-order
for individual sequences.

The further consideration only of the lower

$Q$-order

${q}_{l}$may remain
a powerful theoretical tool in measuring the convergence rate of sequences. In the case
when a

$Q$-order

${p}_{0}$exists
(

${p}_{0}$), the use of the
computational quotients

${Q}_{L}^{\prime }$may offer approximations to

${p}_{0}$,
while otherwise (when

${q}_{l}$)
the use of the computational quotients may hopefully offer just an idea about the value
of

${q}_{l}$(and
perhaps

${q}_{u}$too).

Acknowledgments. The author is grateful to an anonymous referee for the
careful reading of the manuscript and for several constructive remarks, that
improved the presentation (specially in Section 3 ).

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

author address: Tiberiu Popoviciu Institute of Numerical Analysis, Romanian

version submitted to Appl. Math. Comput., published with some changes in vol.
343 (2019) 1-20.

### Abstract

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

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. Tight connections
between some asymptotic quantities defined by theoretical and
computational elements are shown to hold.

Keywords. convergent sequences in

${ℝ}^{n}$;
Q-, C-, and R-convergence orders of sequences; computational convergence
orders.

MSC 2010. 65J05.

### 1 Introduction

Consider in the beginning a sequence

$\left\{{x}_{k}$from

$ℝ$, which converges
to a finite limit

${x}^{\ast }$.
Its speed of convergence is characterized by several measures, called convergence
orders, which are fundamental notions in Mathematical and in Numerical
Analysis.

The relation

 $\underset{k\to \infty }{\mathrm{lim}}$ |x∗−xk|p = Cp ∈ (0,+∞) (1)

gives the classical definition of the convergence with order

$p>1$, and,
in a less rigorous but sufficiently intuitive statement, it can be traced back in
1818, to Fourier [1]. Even deeper traces may be found in history: in a letter of
Newton dated in 1675, it appears he was aware of the rough doubling of the
number of correct significant digits in one step (characteristic to the
quadratic convergence) of the iterative method which bears his name
[2].

The existence of the nonzero value

${C}_{p}$is a
very restrictive condition not only in theory, but also in examples from
practice – not to mention that in the multidimensional setting it is a
norm-dependent problem. Moreover, as it requires the knowledge of both

$p$and

${x}^{\ast }$,
relation (1 ) is only of theoretical use.

Still, such definition (along with the

$Q$-superlinear convergence,
which requires

${C}_{1}$for

$p=1$)
is extremely important in studying the local convergence of the main
iterative methods for nonlinear problems, either of Newton type (see,
e.g., [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]), or the more general
successive approximations (see, e.g., [3], [4, ch. 10], [6], [14], [15]) – which,
under some supplementary assumptions (including smoothness) may
be regarded in fact as equally general (see [14]). This definition is also
important in the field of the matrix function theory (polar decomposition,
matrix sign function, computing the matrix inverse by Schulz or different
iterative methods), in a setting where the matrix is an element of a normed
space (see, e.g., [16], [17]); we should also mention the acceleration of
the convergence of sequences (see, e.g., [18]), and perhaps other fields
too.

From a numerical standpoint, the aim is to define the so-called
computational convergence orders, that iteratively approximate the convergence
orders using only the elements of the sequence known at the step

$k$.
To this end (or just for theoretical purposes) new definitions have been
introduced and studied. We attempt to follow the origin of each notion in the
following section; this is a difficult task, as most of the papers treat the
convergence orders as a collateral topic; no survey has been written since two
monographs containing entire chapters devoted to the rigorous analysis of the

$Q$– and

$R$-convergence
orders. The book of Ortega and Rheinboldt [4], though first published in 1970, is
a standard reference in the field of iterative methods and their convergence orders
(see also [6]); it was republished in 2000 in the Classics in Applied Mathematics
series of SIAM. The book of Potra and Pták (1984) [8] contains further,
complementary results.

Two important papers were subsequently published, independently and close
in time, where the equivalence of different definitions of (computational)

$C$– and

$Q$-convergence
orders were given. The complementary results of these two works allow us to
form a complete, simplified and rigorous picture of the topic. The first paper was
written by Potra (1989) [19]; as of November 2017, Scholar Google reported 96
citations of it, which shows that the mathematicians are aware of it. The other
paper however, written by Beyer, Ebanks and Qualls (1990) [20], is almost
unknown to the mathematical community: as of November 2017, we have found
only 6 citations in Google Scholar (out of which two papers in Chinese, two
papers in the field of Bioarchaeology, one paper solving an unsolved problem

from [20], and one paper dealing with the quadratic convergence in period
doubling for trapezoid maps).

Unfortunately, the fact that the paper of Beyer, Ebanks and
Qualls remained unknown, allowed some authors to (honestly)
reinvent such notions, e.g., in [21] and [22]. The very cited paper
[21]1
has the distinctive feature that not only the definition of the
computational convergence order proposed there is not original
(nor proved, nor in fact entirely computational – as it requires

${x}^{\ast }$), but even
the iterative method itself (as it was given in the well known book of J.F. Traub [23,
formula (8-14), p. 164]2 ).

In numerous works, the classical

$Q$
and

$R$-orders
defined by Ortega and Rheiboldt lead to statements expressed as

$\left\{{x}_{k}$has

$Q$-order (at
least)

${p}_{0}$and

$R$-order
(at least)

${p}_{1}$,

${p}_{1}$”. The

$Q$-order
we study here is a completion to the classical one, and no
longer allows such conclusions. Instead, if having at all an order

${p}_{0}$,
the sequences will be only in one of the cases (the

$C$-order
is obtained by taking norms in (1)):

• $\left\{{x}_{k}$has

$R$-order

${p}_{0}$(but neither

$C$
nor

$Q$-order);

• $\left\{{x}_{k}$has the same

$R$
and

$Q$-order

${p}_{0}$(but no

$C$-order);

• $\left\{{x}_{k}$has the same

$R$-,

$Q$
and

$C$-order

${p}_{0}$.

In less words, the relation reads for

${p}_{0}$as

with no converse implication holding in general.

Nonetheless, the comments of Tapia, Dennis Jr. and
Schäfermeyer [13, p. 49] remain true: “The distinction between

$Q$– and

$R$-convergence
is quite meaningful and useful and is essentially sacred to workers in the area of
computational optimization. However, for reasons not well understood,
computational scientists who are not computational optimizers seem to be at
best only tangentially aware of the distinction.”

The structure of this paper is the following: in Section 2 we
review, in chronological order, the definitions of convergence
orders. In Section 3 we analyze the high convergence orders
(

$p>1$), giving full
proofs for

$C$

and

$Q$-orders.
We present first the relation (implication, resp. equivalence) between the introduced
convergence orders; the results are based on the proofs from [19] and [20], which
we can simplify for some cases. Then we review the main properties of the

$R$-orders. We analyze next
the computational

$C$– and

$Q$-convergence orders
based on the corrections

${x}_{k+1}$(the approach was dealt with in [20], but the authors assumed real and monotone
sequences in treating certain cases; a result of Potra and Pták [8] and
the Dennis-Moré lemma [5] allow us to complete the proofs). We also
consider computational convergence orders based on the nonlinear residuals

$F\left({x}_{k}$,
when solving nonlinear systems of equations having the same number
of equations and unknowns. The section ends with a summary of all
the relations obtained, illustrated by a diagram. Finally, in Section 4
we recall, without proofs, the relationships between some convergence
orders in the linear case. Some conclusions are presented in the end of the
paper.

### 2 Definitions – a brief historical review

#### 2.1 C- and Q-type (computational) convergence orders.

##### 2.1.1 Convergence orders based on the errors x*-xk

We turn now our attention to a sequence

$\left\{{x}_{k}$from

${ℝ}^{n}$, which converges
to a finite limit

${x}^{\ast }$;
we prefer the common setting of a normed
space—

${ℝ}^{n}$endowed with
a given norm

$\parallel \cdot \parallel$—though
most of the results from this paper can be presented in a
more general setting, of a metric space. We assume that

${x}_{k}$

$k\ge 0$.
Since this leads to dealing with positive numbers, we can simplify the notations
and consider further only the errors (called such as a short for ”norms of errors”)

 ${e}_{k}$k∥.

As we have mentioned, the oldest definition of convergence with order

${p}_{0}$is
that

 $\underset{k\to \infty }{\mathrm{lim}}$(ek)p0 = Cp0 ∈ (0,+∞), (2)

which we call, as in [20],

$C$-convergence with
order

${p}_{0}$(the linear
convergence requires

${p}_{0}$and

$0<{C}_{1}$)

$Q$comes from quotient. If it exists, it is uniquely defined and it implies that

$\forall >0,\phantom{\rule{0.33em}{0ex}}\exists {k}_{0}$such
that

 $\left({C}_{{p}_{0}}$ ≤ ek+1 ≤ (Cp0 + )ekp0 ,∀ ⁡k ≥ k0, (3)

or, less sharp, the existence of

$A,B>0$such that

 $A{e}_{k}^{{p}_{0}}$ ≤ ek+1 ≤ Bekp0 ,∀ ⁡k ≥ 0. (4)

Remark 2.1
a) For example, if

${p}_{0}$and

${C}_{2}$is not too large, relation (3 ) says that the error is approximately squared at
each step

$k$,
for

$k$sufficiently large.

b) Some authors have used (4 ) to define the exact

$Q$-order
of convergence

${p}_{0}$(see [24], [18], [25], [8], [26], [19]).

c) An inequality of the form (4 ) still holds if the limit in (2 ) does not exist, but

$\mathrm{liminf}$and

$\mathrm{limsup}$are finite and nonzero. Denoting

 $\phantom{\rule{0.33em}{0ex}}{\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{{p}_{0}}$ (5)

relation

 $0<{\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{{p}_{0}}$ (6)

implies (4 ):

$0<{\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{{p}_{0}}$attracts the
first inequality in (4 ), while

${\overline{Q}}_{{p}_{0}}$the second one (see also [4, 9.3.3]).

d) As widely known, the larger the order

${p}_{0}$, the
faster the speed of convergence.

The

$C$-order
is a norm-dependent notion, as the following two examples show for the

$C$-orders

$1$and

$2$.

Example 2.2
[27] Let

${x}^{\ast }$,
the sequence

 ${x}_{k}$(0,2−k+1),k odd. (7)

Consider the norms

 $\parallel \left(u,v\right){\parallel }_{A}$

It follows

${C}_{1}$2for
the

$A$-norm, while

${C}_{1}$does not exist
for the

$B$-norm.

Example 2.3
Let

${x}^{\ast }$and

 ${x}_{k}$(2−2k ,2−2k ),k even. (8)

One obtains

${C}_{2}$when
considering the maximum norm

$\parallel \cdot {\parallel }_{\infty }$,
while

${C}_{2}$does not
exist for the norm

$\parallel \cdot {\parallel }_{1}$.

The following definition for the convergence order seems to be the
second oldest one (we adopt the notation from [20]). The sequence

$\left\{{x}_{k}$has

${Q}_{L}$-convergence
order

${p}_{0}$if:

 ${Q}_{L}$ ln ⁡ ek → p0,as k →∞. (9)

Remark 2.4
a) As noticed by Brezinski in [28] and [26],

${Q}_{L}$is a particular case of a definition used by Bourbaki [29] in order to compare
the convergence orders of two sequences.

The measure

${Q}_{L}$was then independently considered by Wall in 1956 [30] (see also [31], [4,
ch. 9], [32, ch. 2] and [25]), argued – for

$\mathrm{lg}$instead of

$\mathrm{ln}$– as the limit of the quotient of two consecutive numbers of correct decimal
places in

${x}_{k+1}$resp.

${x}_{k}$(see also [8, p. 90]); in this sense, convergence for example with

${Q}_{L}$-order

$2$means intuitively that, from a certain step, the number of correct digits are
doubled the successive elements of

$\left\{{x}_{k}$.
The logarithm base may be taken as any positive real number

$a\ne 1$;
we shall consider further the natural base

$e$for simplicity.

b) In contrast to

${Q}_{{p}_{0}}$,
the expression

${Q}_{L}$does not require the (usually unknown) value

${p}_{0}$of the order, but provides an approximation of it. This may be advantageous
in certain situations when evaluating the order, as it turned out, e.g., in the
books of Wright [33, p. 252 and Ch. 7], Ye [34, p. 210, p. 226] and in two
papers of Potra [35] and [36] (where the authors deal with

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{L}$,
a quantity which we analyze in Proposition 3.3 ).

c) Brent, Winograd and Wolfe noticed in [37] that

$C$-order

${p}_{0}$immediately implies

${Q}_{L}$-order,
of same value.

d) We shall see later, in Remark 3.8 , that this notion (cf. [38]), as well as
the other

$Q$-orders,
is norm-independent (also noted by Potra [19]).

In 1967, Feldstein and Firestone [39] introduced essentially
the following convergence order, which we denote here by

${Q}_{}$.

The sequence

$\left\{{x}_{k}$has

${Q}_{}$-convergence
order

${p}_{0}$if:

$\begin{array}{}\end{array}$lim k</munder >Qp0</msub ></msub > (k) = 0, > 0 with p0</msub > 1, (10)
lim k</munder >Qp0</msub >+</msub > (k) = , > 0.
Condition

${p}_{0}$will be a standing assumption from here on, in all subsequent
formulas, while in Section 3 we will implicitely assume that

${p}_{0}$.

The convergence with

${Q}_{}$-convergence
order

${p}_{0}$can be expressed in a more intuitive form (see also [20]):

$\begin{array}{}\end{array}$lim k</munder >Qp</msub >(k) = 0,1 < p < p0</msub >,
lim k</munder >Qp</msub >(k) = + ,p > p0</msub >.
Two years later, the same authors [40] expressed the above order
in an equivalent form, which in fact will be the same as the

$Q$-order
defined below. Let

$\begin{array}{}\end{array}$λ</msub > = sup {p : lim k</munder >Qp</msub >(k) = 0 }, (11)
λ</msup > = inf {p : lim
k
</munder >Qp</msub >(k) = }.

They noticed that

$0\le {\lambda }_{\ast }$, and
when

${p}_{0}$, the sequence was
said to have the order

${p}_{0}$.

In their classical book from 1970, Ortega and Rheinboldt [4,
Ch. 9] introduced and studied the quantity denoted in (5) by

${\overline{Q}}_{p}$for
all

$p\ge 1$:

 ${}_{}$Q¯p = limsup ⁡ k→∞Qp (k).

They shown the following behavior for

${\overline{Q}}_{p}$as a
”function” of

$p$.

Proposition 2.5
[4, 9.1.2] Exactly one of the following conditions holds:

a)

${\overline{Q}}_{p}$b)

${\overline{Q}}_{p}$c)

$\exists {p}_{0}$s.t.

${\overline{Q}}_{p}$

$\forall p\in \left[1,{p}_{0}$and

${\overline{Q}}_{p}$

$\forall p\in \left({p}_{0}\right).$

The quantity

 inf ⁡ {p ∈ [1,∞) : Q¯p = +∞ } (12)

(called in [4] as

$Q$-convergence
with order at least

${q}_{l}$and denoted by O

${}_{Q}$)
stood there and in many subsequent works for a measure of
the convergence order. We call it, as in [20], the lower

$Q$-order, and
denote it by

${q}_{l}$(instead of

${p}_{l}$used there). We shall see that the lower

$Q$-order needs to be completed
by the upper

$Q$-order for
obtaining the ”full”

$Q$-order;
as a matter of fact,

${q}_{l}$,
the lower order from (11) defined in [40].

Remark 2.6
The sequence

$\left\{{x}_{k}$was said in [4, Def. 9.1.4] that converges faster than

$\left\{{y}_{k}$if for
some

${p}_{0}$we
have

${\overline{Q}}_{{p}_{0}}$(it is interesting to note that if these two upper bounds are finite and nonzero, the
inequality may revert when changing the norm [4, p. 285]). However, in the case
of the even stronger relation

 $0<{C}_{{p}_{0}}$

numerical evidence reveals that

$\left\{{x}_{k}$does not necessarily converge faster than

$\left\{{y}_{k}$,
as shown by Ypma and Igarashi [41].

On the other hand, as noticed by Brezinski [26], the sequences given by

${x}_{k}$kand

${y}_{k}$k2</msup >,

$k\ge 0$,
have the same asymptotical constant,

${Q}_{1}$,
though they converge with quite different speeds.

Some further comments on this topic will be made in Remark 2.11 .

In 1979, Schwetlick [25, B.4.2.3] defined the

$Q$-convergence
order in the following way, which will be used throughout the paper. Let the lower

$Q$-order

${q}_{l}$be
given in (12 ), recall the notation from (5)

 ${}_{}$ Q p = liminf ⁡ k→∞Qp (k)

and define the upper

$Q$-order

${q}_{u}$by

 ${q}_{u}$

Then the

$Q$-convergence
is with order

${p}_{0}$if

${p}_{0}$This definition is equivalent to the one given by Feldstein and Firestone in (11),
as

${q}_{l}$and

${q}_{u}$.

In 1981, Schmidt [42] introduced the following type of convergence order, denoted here,
as in [20], by

${Q}_{\Lambda }$:

$\left\{{x}_{k}$has

${Q}_{\Lambda }$-convergence
order

${p}_{0}$if

 ${Q}_{\Lambda }$ek+1 ln ⁡ ek+1 ek → p0,   as  k →∞. (13)

The author gave no result relating the

$Q$– and

${Q}_{\Lambda }$-orders he introduced,
and assumed that

${Q}_{\Lambda }$is
equivalent to an

$R$-order
(while it is in fact a

$Q$-order,
as we shall see).

In 1982, Beyer and Stein [43] independently introduced
the measure (19 ) below, which is somehow similar to

${Q}_{\Lambda }$, and in fact
equivalent to

${Q}_{\Lambda }^{\prime }$defined in subsubsection 2.1.2 .

In 1985, unaware of [42] and [43], Brezinski [26] considered the

${Q}_{\Lambda }$-convergence order

and, assuming that

$\left\{{x}_{k}$has the exact

$Q$-order of
convergence

${p}_{0}$(as defined
by (4 )), he proved that

$\left\{{x}_{k}$has the

${Q}_{\Lambda }$-order

${p}_{0}$(and

${Q}_{L}$-order

${p}_{0}$as
well).

In 1989, Potra introduced in [19] the following type of convergence order (denoted
here by

${Q}_{I,}$,
with ”

$I$
from ”inequality”), which will be useful in obtaining some simplified
proofs.

The sequence

$\left\{{x}_{k}$has

${Q}_{I,}$-convergence
order

${p}_{0}$if

$\forall >0,$

$\exists a,b>0$such
that

 $a{e}_{k}^{{p}_{0}}$k+1 ≤ bekp0−,∀ ⁡k ≥ 0. (14)

This can be regarded as a generalization of inequalities (4 ).

Potra shown some fundamental results in our study, that

${Q}_{I,}$-convergence
with order

${p}_{0}$is
equivalent to

${Q}_{}$– and
to

${Q}_{L}$-convergence
of same order, which we denote here, as in [20], by symbols between curled
braces:

 $\left\{{Q}_{}\right\}.$

Ortega and Rheinboldt [4, N.R. 9.2.2] have previously noticed a weaker property of

${Q}_{L}$-convergence with order

${p}_{0}$, namely that it implies
(partial)

$R$-convergence
with order

${p}_{0}$(i.e., in the
notations below, that

${p}_{0}$).

In the same paper, Potra has also considered the lower and upper

$Q$-orders, when noting that if
a sequence has exact

$Q$-order

${p}_{0}$then

${q}_{l}$.

In 1990, Beyer, Ebanks and Qualls [20], unaware of the definitions
and the results mentioned above, introduced the definitions of the

$Q$– and

${Q}_{\Lambda }$-orders
and shown some other fundamental results, that convergence with

$Q$-order

${p}_{0}$is equivalent
to

${Q}_{\Lambda }$and to

${Q}_{L}$-convergence
with same order:

 $\left\{Q,{Q}_{L}\right\}.$

They considered

${Q}_{\Lambda }$inspired by the similar measure (19), as we have already mentioned.

We end the historical remarks by noting that the references [44], [45] and [46]
are cited in certain works as containing aspects referring to convergence orders,

but we were not able to consult them.

Lemma 3.1 from Section 3 shows that

${q}_{l}$(which justifies
the terminology lower/upper); when this inequality is strict, the sequence does not have
a

$Q$-order,
as shown below.

Example 2.7
Let

 ${}_{}$xk = { 2−2k,keven, 3−2k ,kodd.

We get

 ${}_{}$Q¯p = { 0, 1 ≤ p < log ⁡ 34, 1, p = log ⁡ 34, +∞,p > log ⁡ 34, and  Q p = { 0, 1 ≤ p < 4log ⁡ 43, 1, p = 4log ⁡ 43, +∞,p > 4log ⁡ 43.

The graphical illustration of

${\overline{Q}}_{p}$and

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}}_{}$ p</msub >for

$p\ge 1$leads to the so called convergence profile of the sequence, as termed in [20]
(see Fig. 1 ).

This sequence does not have a

$Q$-order, as

${q}_{l}$, but it converges
with (exact)

$R$-order

$2$,
as defined later. We note the classical statement in this case,

$\left\{{x}_{k}$converges
with

$Q$-order
(at least)

${\mathrm{log}}_{3}$and
with

$R$-order
(at least)

$2$”,
which no longer holds in the setting of this paper.

An elementary example shows that in case of convergence with

$Q$-order

${p}_{0}$apart
of finite nonzero values of one (or both) of the asymptotical constants

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}}_{}$ p0</msub ></msub > Q¯p0</msub ></msub >, any
of the following relations may hold:

$\begin{array}{}\end{array}$ Q p0</msub ></msub > = Q¯p0</msub ></msub > = +, (15)
Q p0</msub ></msub > = Q¯p0</msub ></msub > = 0, (16)
Q p0</msub ></msub > = 0 andQ¯p0</msub ></msub > = +. (17)

Example 2.8
Let

 ${}_{}$xk = 2−2k∕k,y k = 2−k2k ,zk = { xk,kodd, yk,k even.

It is easy to verify that for

${p}_{0}$the sequences

$\left\{{x}_{k}\right\},\left\{{y}_{k}\right\},\left\{{z}_{k}\right\}$verify respectively (15), (16) and (17) (

$\left\{{x}_{k}$was also considered in [20]).

Schwetlick [25, Ü.4.2.3, p. 93] also considered an example of the type

${x}_{k}$,

${y}_{k}$,
for the errors

${e}_{k}$converging with exact

$Q$-order

$2$.

Potra [19] considered

 ${}_{}$ek = { ek+1 = 2ekek−12,keven, ek+1 = ekek−12, k odd,

which has

$Q$-order

$2$and

${\overline{Q}}_{2}$.

Jay [27] considered the sequence of the type

${x}_{k}$</msup >,

$k$even,

${x}_{k}$</msup >k,

$k$odd, which satisfies (17) for

${p}_{0}$.

One may easily find examples when

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{{p}_{0}}$and

${\overline{Q}}_{{p}_{0}}$is finite, nonzero, or, on the other hand, when

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{{p}_{0}}$is finite, nonzero and

${\overline{Q}}_{{p}_{0}}$.
Jay [27] considered the sequence

${x}_{k}$,

$k$odd,

${x}_{k}$,

$k$even, for

$q\ge 1$,

${x}_{0}$,
for which one obtains

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{q}$and

${\overline{Q}}_{q}$.

Further examples, to be analyzed either elementary or with the aid of the
results from this paper, were given by Potra and Pták [8, p. 93]:

${x}_{k+1}$k </msup >,

$k\ge 1$,

${x}_{1}$,

$0<<1$and also

${x}_{2k}$</msup >,

${x}_{2k+1}$</msup >,

$0<<1.

Remark 2.9
Despite the fact that conditions (15 )–(17 ) may seem rather
abstract (or, at least, suitable for scholar examples), they do occur in relevant
situations from theory or practice. For a nontrivial illustration, we refer
the reader to a computational optimization problem [33, p. 252 and Ch. 7],
where one has

${p}_{0}$and

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{2}$.

Remark 2.10 When

${\overline{Q}}_{1}$i.e.,

 $\underset{k\to \infty }{\mathrm{lim}}$ek = 0,

it is said that the sequence converges (at least)

$Q$-superlinearly.
The convergence is strict

$Q$-superlinearly,
when

${q}_{l}$(i.e.,

${\overline{Q}}_{p}$,

$\forall p>1$);
there are three cases in this situation:

${q}_{l}$,
i.e.,

$Q$-order

$1$(see Example 3.6 ),

$1={q}_{l}$,
resp.

$1={q}_{l}$,

${q}_{u}$(see Example 2.14 b), formula (24)).

It is worth noting that one may also obtain

${q}_{l}$,

${q}_{u}$in lack of

$Q$-superlinear
(and even linear) convergence, shown in Example 2.14 d), formula (25)), as

${\overline{Q}}_{1}$.

Remark 2.11
Returning to the comments from Remark 2.6 ,
we note that the comparison of the convergence of two sequences

$\left\{{x}_{k}$,

$\left\{{y}_{k}$having
errors

$\left\{{e}_{k}^{\prime }$resp.

$\left\{{e}_{k}^{\prime \prime }$is
important in particular in the field of the acceleration of the convergence of sequences,

when sequences with (at most) linear convergence are considered; here it is said that

$\left\{{x}_{k}$converges
faster than

$\left\{{y}_{k}$if (see, e.g., [18]):

 $\underset{k\to \infty }{\mathrm{lim}}$ek′′ = 0(also denoted by ek′ = o(e k′′), as  k →∞). (18)

Now we see that this definition makes sense not only if the

$Q$-orders
of the two sequences are different, but even if they are equal, as the above
condition is norm-independent.

To show this, we consider the terminology proposed by Jay [27] for a sequence converging
with

$Q$-order

${p}_{0}$, who
defined that the convergence is with:

• $Q$-suborder

${p}_{0}$if

${\overline{Q}}_{{p}_{0}}$,

• $Q$-superorder

${p}_{0}$if

${\overline{Q}}_{{p}_{0}}$.

The last terminology is in fact widely used – recall the superquadratic convergence
(for

${\overline{Q}}_{2}$),
supercubic, etc.).

Now we see that for sequences with

$Q$-order

${p}_{0}$we
can briefly say that:

• $Q$-superconvergence
is faster than exact

$Q$-convergence
(in the sense of (4) or (6));

• exact
$Q$-convergence
is faster than

$Q$-subconvergence;

• (consequently)
$Q$-superconvergence
is faster than

$Q$-subconvergence.

This can be illustrated by the following sequences having

$Q$-order

$2$:

$\left\{{2}^{-k{2}^{k}}$</msup >}(

$Q$-superquadratic),

$\left\{{2}^{-{2}^{k}}$</msup >}(exact

$Q$-order

$2$),

$\left\{{2}^{-{2}^{k}}$</msup >}(

$Q$-subquadratic).

##### 2.1.2 Computational convergence orders, based on the corrections xk+1-xk and the nonlinear residuals F(xk)

We consider now some quantities which do not require the limit

${x}^{\ast }$,
for which we adopt the terminology computational convergence
orders
. As some of them require the knowledge of the order

${p}_{0}$itself,
we keep in mind that perhaps a more proper terminology for those would be
semi-computational.

As a notation, for the corresponding convergence orders
based on corrections we shall add a prime mark (e.g.,

${C}^{\prime }$) as in
[20], while for those based on nonlinear residuals, two prime marks (e.g.,

${C}^{\prime \prime }$).

Corrections.

We introduce another notation, for the corrections (including their norms
too):

 ${s}_{k}$

In 1974 Dennis and Moré [5] obtained a result which shows that if a sequence converges

$Q$-superlinearly,
then the errors and corrections converge precisely at the same rate (Lemma 3.14
from subsubsection 3.2.1 ).

In 1984, Potra and Pták [8] shown that a sequence converges

$Q$-superlinearly iff the corrections
converge

$Q$-superlinearly
(Proposition 3.13 ); in 1997 Walker [47] gave a different proof, also presented
later.

In 1982, Beyer and Stein [43] considered, for sequences in

$ℝ$, the
quantity

 $\frac{log\frac{{s}_{k}}{{s}_{k+1}}}{log\frac{{s}_{k-1}}{{s}_{k}}},$ (19)

equivalent to

${Q}_{\Lambda }^{\prime }$defined below.

Three years later, Brezinski [26] proposed the use of the corrections

${x}_{k+1}$instead of the
(unknown) errors

${x}^{\ast }$in the definitions of the convergence orders studied there
(

${Q}_{L}$and exact

$Q$-order).

In 1990, Beyer, Ebanks and Qualls [20], taking the same approach, considered

 ${Q}_{{p}_{0}}^{\prime }$skp0

with the resulted definitions being denoted correspondingly by

${C}^{\prime }$and

${Q}_{\Lambda }^{\prime }$; we use here the same
convention for denoting

${Q}_{}^{\prime }$.
These authors proved the following fundamental results: for

${p}_{0}$, the
errors and corrections converge simultaneously to zero with the same
convergence order, and therefore each convergence order is equivalent to
its corresponding computational one. The following extended relation
resulted:

 $\left\{{C}_{{p}_{0}}\right\}⇒\left\{Q,{Q}_{}$L,QΛ,Q′,Q L′,Q Λ′}.

However, the considered assumptions were very strong when proving

$\left\{{C}_{{p}_{0}}\right\}$and

${Q}^{\prime }$”:

$\left\{{x}_{k}$was assumed as a
monotone sequence from

$ℝ$(see [20, Sect. 4]). We shall see that for the general setting, some results from [8]
and [5] may be used instead to obtain simplified proofs.

In 2010, Grau-Sánchez, Noguera and Gutiérrez [48], while studying connections
between

${Q}_{\Lambda }$– and

${Q}_{\Lambda }^{\prime }$-orders (in the case
of a sequence from

$ℝ$with

$C$-order

$p$)
considered the ”extrapolated convergence order”

 $\frac{ln\frac{|{x}_{k+1}}{|{x}_{k}}}{ln\frac{|{x}_{k}}{|{x}_{k-1}}},$ (20)

where

${\stackrel{~}{\alpha }}_{k}$Δ2</msup >xk2</msub > ,

$k\ge 2,$

$\mathrm{\Delta }{x}_{k}$. This convergence order
was obtained from

${Q}_{\Lambda }$by replacing the unknown limit with an extrapolation

${\stackrel{~}{\alpha }}_{k}$given by
the Aitken

${\mathrm{\Delta }}^{2}$method (in order to use only the terms known at the step

$k$).
However, this convergence order is valid only in the one dimensional case, since the
Aitken

${\mathrm{\Delta }}^{2}$acceleration process does not have a direct extension to

${ℝ}^{n}$.

We do not pursue this definition as, for high convergence orders, it is equivalent
to

${Q}_{\Lambda }^{\prime }$.

In 2012, Grau-Sánchez, Noguera, Grau and Herrero [49] considered another
extrapolated convergence order

 $\frac{ln|{x}_{k}|}{ln|{x}_{k-1}|},$ (21)

for a sequence from

$ℝ$,
having

$C$-order

$p$. Assuming

$C$-order,
they shown some connections to the convergence orders (22) and (23)
below.

In [48] and [49] the authors used the technique of asymptotic
expansions in order to relate different convergence orders of

$\left\{{e}_{k}$,

$\left\{{s}_{k}$and

$\left\{f\left({x}_{k}$for
sequences

$\left\{{x}_{k}$from

$ℝ$.
However, this approach could not be extended to the case of several
dimensions.

Remark 2.12
It is important to keep in mind that the

${C}^{\prime }$-,

${Q}_{}^{\prime }$-,

${Q}_{I,}^{\prime }$-orders
are semi-computational orders, as they require the (supposed) unknown order,
while the logarithm-based

${Q}_{L}^{\prime }$
and

${Q}_{\Lambda }^{\prime }$-orders
are (fully) computational orders, i.e., they require neither the limit

${x}^{\ast }$nor the order

$p$in their expressions.

As certain numerical experiments (see [50] and the references therein)
have not revealed a clear superiority of

${Q}_{\Lambda }^{\prime }$over

${Q}_{L}^{\prime }$,
the later seems to be the most convenient to use in practice (and in theory
too: see [33, p. 252 and Ch. 7], [34, p. 210, p. 226], [35] and [36]). Moreover,
Propositin 3.3 and Theorem 3.15 will show, in terms we will define later,
that

${q}_{l}$and

${q}_{u}$.

Nonlinear residuals.

Other computational convergence orders were considered
in the context of solving nonlinear systems of equations

$F\left(x\right)=0$,

$F:{ℝ}^{n}$, with solution

${x}^{\ast }$. For simplicity,
we shall denote

${F}_{k}$(resp.

${f}_{k}$when

$n=1$).

In 1981, Păvăloiu [51] defined the exact convergence order

${p}_{0}$corresponding to (4) by

 $A\parallel {F}_{k}$ ≤∥Fk+1∥≤ B∥Fk∥p0 ,∀ ⁡k ≥ 0,

while in 1995, he introduced in [52] the expression

 $\frac{ln|{f}_{k+1}}{ln|{f}_{k}}$ (22)

and shown its equivalence to

${Q}_{L}^{\prime }$.
Also, while extending his results from 1999 [53], he introduced in an unpublished
manuscript [54] (posted on his website) the measure

 $\frac{ln|{f}_{k+1}}{ln|{f}_{k}}$ (23)

in connection with the above ones. However, since the manuscript was not
published as of 2011, the credit of first publishing this measure goes to
Petković, who mentioned in passing the above expression in a note published in
SIAM J. Numer. Anal. (2011).

Several computational aspects regarding these quantities were subsequently
analyzed (see, e.g., [50] and the references therein).

#### 2.2 R-convergence orders.

The root convergence order (

$R$-order)
requires weaker conditions than the

$C$
and

$Q$-orders,
but unfortunately at the price of loosing its theoretical importance and its practical
applications. As we have quoted in the Introduction, “The distinction between

$Q$– and

$R$-convergence is […]
essentially sacred to workers in the area of computational optimization” [13, p. 49], meaning
that the

$R$-order
is a much less powerful notion.

In Example 3.11 is presented a sequence for which, depending on the parameters, the
lower

$Q$-order

${q}_{l}$is arbitrary close to

$1$from above, while
the (exact)

$R$-order
is arbitrary high (and even higher is the upper

$Q$-order

${q}_{u}$).

We shall review certain results, but not treat all of them in the same detail as we
will for the

$C$
and

$Q$-orders.

In 1970, Ortega and Rheinboldt [4, 9.2] introduced and
studied the root convergence factors, which we denote here by

${\overline{R}}_{p}$:

 ${}_{}$R¯p {xk} = { limsup ⁡ k→∞(ek)1 k , ifp = 1, limsup ⁡ k→∞(ek) 1 pk , ifp > 1.

Unlike the quotient factors, the root factors do not relate each two
consecutive terms, but consider some averaged asymptotic quantities.

They proved a Proposition similar to Proposition 2.5 , but here

$1$takes
the role of

$+\infty$for

${\overline{Q}}_{p}$.

Proposition 2.13
[4, 9.2.3] Exactly one of the following conditions holds:

a)

${\overline{R}}_{p}$b)

${\overline{R}}_{p}$c)

$\exists {p}_{0}$s.t.

${\overline{R}}_{p}$

$\forall p\in \left[1,{p}_{0}$and

${\overline{R}}_{p}$

$\forall p\in \left({p}_{0}\right).$

The lower

$R$-order,
which we denote here by

${r}_{l}$(instead of

${O}_{R}$in [4, 9.2]), defined by

 ${}_{}$rl = { ∞,   if  R¯p = 0,∀ ⁡p ≥ 1, inf ⁡ {p ∈ [1,∞) : R¯p = 1 },

stood in [4] and in many subsequent works as the definition of the root
convergence order.

Remark 2.14
a) [4, p. 290] If there exists a

${p}_{0}$with

${\overline{R}}_{{p}_{0}}$then for
any

$>0$with

${\overline{R}}_{{p}_{0}}$there
exists a

${k}_{0}$such that either

 ${e}_{k}$0, ifp0 = 1,

or

 ${e}_{k}$,∀ ⁡k ≥ k0, ifp0 > 1.

b) The value

${r}_{l}$arises in the analysis of the speed of convergence of sequences with the errors
satisfying different recurrence inequalities.

The simplest ones appear in usual circumstances for the secant method (see
[3], [9], and also [37]):

 ${e}_{k+1}$

and they yield the well-known

${r}_{l}$-order

$\left(1+\sqrt{5}\right)∕2\approx 1.6$.
It is important to note that the secant method then attains the same value
also for the lower

$Q$-order

${q}_{l}$(see [3], [9]).

For the more general inequalities

 ${e}_{k+1}$i=0m(ek−i)αi,k ≥ m,ck,αk ≥ 0,

Schmidt [42], Burmeister and Schmidt [55], [56] resp. Potra and Pták
[8, p. 107] determined in a series of works the lower

$R$-order

${r}_{l}$.

It is interesting to note that for such inequalities, Herzberger and Metzner
[57] and then Potra [19] gave certain sufficient conditions for the lower bound

${q}_{l}$. Potra
[19] has also noted that the above inequalities do not necessarily attract lower

order

${q}_{l}$greater than one: take

 ekek−1,otherwise, (24)

with

${e}_{0}$2. This sequence
satisfies

${e}_{k+1}$(which
immediately attracts

${\overline{Q}}_{1}$and

${r}_{l}$), but
in fact

${\overline{Q}}_{p}$,

$\forall p>1$(and

${q}_{u}$, as
we shall see).

Certain iterative methods have lead to some different inequalities, of the type
[4, Th. 9.2.9.]

 ${e}_{k+1}$j=0mγ jek−j,∀ ⁡k ≥ m, with γ1,...,γm ≥ 0,

c) As noted in [4, N.R. 9.2.1], the

${\overline{R}}_{1}$factor has been used implicitly in much work concerning iterative processes for
linear systems of equations, see, e.g., Varga [59]. For nonlinear systems, it was
used explicitly by Ortega and Rockoff [60].

d) Some connections between the errors and their majorizing sequences were made
by Dennis Jr. in [61] (see also [13, p. 49]). Potra [19] has noticed that the lower

$R$-order

${r}_{l}$is consistent with the natural ordering of sequences, while the lower

$Q$-order

${q}_{l}$is not:
given

$\left\{{x}_{k}$with
errors

$\left\{{e}_{k}^{\prime }$,
resp.

$\left\{{e}_{k}^{\prime \prime }$, if

${e}_{k}^{\prime }$and

$\left\{{y}_{k}$has

${r}_{l}$order

${p}_{0}$then so
has

$\left\{{x}_{k}$;
however, this statement does not hold for the lower

$Q$-order

${q}_{l}$, as one may take

$\left\{{x}_{k}$from Example
2.7 and

${y}_{k}$</msup >.
A more disturbing example was given by Jay [27], for

${y}_{k}$</msup >and

 ${x}_{k}$2−3k ,kodd. (25)

At the first sight, since

${e}_{k}^{\prime }$,
the speed of

$\left\{{x}_{k}$seems higher
than of the (exact)

$Q$-quadratic

$\left\{{y}_{k}$, but though,

$\left\{{x}_{k}$does not attain even

$Q$-linear convergence, as

${\overline{Q}}_{1}$(we shall see that in this
case the upper

$Q$-order
is

${q}_{u}$).
Another example was given in (24) above.

In 1973, Brent, Winograd and Wolfe [37] considered the following expressions,
which we denote here by

$\begin{array}{}\end{array}$ R L</msub > := liminf k</munder > |ln ek</msub > |1
k
</msup >,
(26)
R¯L</msub > := limsup k</munder > |ln ek</msub > |1
k
</msup >

(in [32] Brent considered in the same year only

${\phantom{\rule{4.1994pt}{0ex}}\phantom{\rule{-4.1994pt}{0ex}}R\phantom{\rule{-4.1994pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.1994pt}{0ex}}}_{L}$). The sequence was said
that converges with

${R}_{L}$-order

${p}_{0}$if

${\phantom{\rule{4.1994pt}{0ex}}\phantom{\rule{-4.1994pt}{0ex}}R\phantom{\rule{-4.1994pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.1994pt}{0ex}}}_{L}$, i.e.,
in a single condition,

$\underset{k\to \infty }{\mathrm{lim}}$</msup >1k =p˙0. (27)

They noted that the

$C$-order

${p}_{0}$easily implies

${R}_{L}$-order

${p}_{0}$.

In 1979, Schwetlick [25] considered the upper

$R$-order

 ${}_{}$ R p {xk} = { liminf ⁡ k→∞(ek)1 k ,  ifp = 1, liminf ⁡ k→∞(ek) 1 pk , ifp > 1,

and (in the notations of this paper)

 ${}_{}$ru = { ∞,   if   R p = 0,∀ ⁡p ≥ 1, sup ⁡ {p ∈ [1,∞) : R p = 0 },

he defined the convergence with

$R$-order

${p}_{0}$if

${p}_{0}$Similarly to the case of the

$Q$-order,
the sequence is said that converges

$R$-superlinearly
if

 $\underset{k\to \infty }{\mathrm{lim}}$k = R¯1 = 0.

Remark 2.15
Analogously to Remark 2.14 , whenever exists a

${p}_{0}$with

${\phantom{\rule{4.1994pt}{0ex}}\phantom{\rule{-4.1994pt}{0ex}}R\phantom{\rule{-4.1994pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.1994pt}{0ex}}}_{{p}_{0}}$then for
any

$>0$with

$0<{\phantom{\rule{4.1994pt}{0ex}}\phantom{\rule{-4.1994pt}{0ex}}R\phantom{\rule{-4.1994pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.1994pt}{0ex}}}_{{p}_{0}}$there
is a

${k}_{0}$such that either

 ${}^{}$ ( R p0 − )k ≤ e k,∀ ⁡k ≥ k0, ifp0 = 1,

or

 ${}^{}$ ( R p0 − )p0k ≤ ek,∀ ⁡k ≥ k0, ifp0 > 1.

In 1989, Potra [19] has introduced and studied some further measures for the

$R$-convergence order,
which we denote here by

${R}_{}$,
resp.

${R}_{I,}$:

• $\left\{{x}_{k}$converges
with

${R}_{}$-order

${p}_{0}$if

$\begin{array}{}\end{array}$lim k</munder >(ek</msub >) 1
(p0</msub >)k</msup >
</msup >
= 0, > 0,
lim k</munder >(ek</msub >) 1
(p0</msub >+)k</msup >
</msup >
= 1, > 0;

• $\left\{{x}_{k}$converges
with

${R}_{I,}$-order

${p}_{0}$if

$\forall >0,$ $\exists a,b>0$and

$0<\eta ,<1$such that

 $a\phantom{\rule{0.17em}{0ex}}{\eta }^{{\left({p}_{0}\right)}^{k}}$ ≤ ek ≤ b (p0−)k ,∀ ⁡k ≥ 0.

The exact

$R$-order
of convergence was also defined here – analogously to the definition of the exact

$Q$-order (4)
– as being

${p}_{0}$if

$\exists A,B>0$,

$0<\eta ,<1$such
that

 $A\cdot {\eta }^{{\left({p}_{0}}^{k}}$ ≤ ek ≤ B ⋅ (p0)k ,∀ ⁡k ≥ 0; (28)

we see that this definition can be immediately connected to the following one,
analogous to (6):

 $0<{\phantom{\rule{4.1994pt}{0ex}}\phantom{\rule{-4.1994pt}{0ex}}R\phantom{\rule{-4.1994pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.1994pt}{0ex}}}_{{p}_{0}}$ (29)

Potra has also considered the lower and upper

$R$-orders, when noting that if
a sequence has exact

$R$-order

${p}_{0}$then

${r}_{l}$.

Comparing the list of

$Q$
and

$R$-orders, the
definition of

${R}_{\Lambda }$was not given before, so we consider it here, by denoting the expressions

 ${}_{}$ R Λ := liminf ⁡ k→∞ |ln ⁡ ek+1 ek |1 k ,R¯Λ := limsup ⁡ k→∞ |ln ⁡ ek+1 ek |1 k

and by defining the

${R}_{\Lambda }$-order

${p}_{0}$when

${\phantom{\rule{4.1994pt}{0ex}}\phantom{\rule{-4.1994pt}{0ex}}R\phantom{\rule{-4.1994pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.1994pt}{0ex}}}_{\Lambda }$, i.e.,
when

 $\underset{k\to \infty }{\mathrm{lim}}$ek |1 k = p0. (30)

### 3 Results relating the high (computational) convergence orders

In this section we consider

$p>1$and present the relationships of different definitions of the
(computational) convergence orders, with full proofs in the case of the

$C$– and

$Q$-orders.

#### 3.1 Relationships between the convergence orders

The equivalences and implications mentioned in the previous section can be
summarized as:

$\begin{array}{}\end{array}$ {Q</msub >,QI,</msub >,QL</msub >} {R</msub >,RI,</msub >,RL</msub >} , [19],
Cp0</msub ></msub > {Q,QL</msub >,QΛ</msub >} , [20].
At this point we can simply merge the equivalences from the two statements
above and write the conclusion.

However, as we intend to provide a self-contained
material, we include the proofs of the equivalence of the

$Q$-orders,
some of them much simplified.

##### 3.1.1 C- and Q-convergence orders

We shall need the three auxiliary results below.

Lemma 3.1
[20, lemma 2.1]

(i)
If

${\overline{Q}}_{p}$then

${Q}_{s}$for all

$s,
and so

$p\le {q}_{l}$

(ii)
if

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{p}$then

${Q}_{s}$for all

$pand so

${q}_{u}$

(iii)
if

$p<{q}_{l}$then

${Q}_{p}$

(iv)
if

${q}_{u}$then

${Q}_{p}$.

Relations (iii) and (iv) show that

${q}_{l}$and justify the terminology ”lower” and ”upper”.

Proof. [20] For (i), if

${\overline{Q}}_{p}$and

$sthen

 ${Q}_{s}$eks = ekp−sQ p(k) → 0,as k →∞.

For (iii), suppose

$p<{q}_{l}$and choose

$s\in \left(p,{q}_{l}\right).$By
definition of

${q}_{l}$then,

${\overline{Q}}_{s}$. Now,
by (i),

${Q}_{p}$The other statements are proved in a similar manner.  _

Remark 3.2
Ortega and Rheinboldt have previously obtained (i) and (ii)
in [4, 9.1], see Proposition 2.5 .

Potra obtained the following general result, interesting in its own, which will
help us relating some quantities to their corresponding computational
versions.

Proposition 3.3
[19, Prop. 1.1] The following relations hold:

 ${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{L}$ ln ⁡ ek = ql (31)

and

 ${}_{}$Q¯L := limsup ⁡ k→∞ln ⁡ ek+1 ln ⁡ ek = qu.

Proof. [19] Let

 $1<{\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{L}$

Then for any

$>0$with

${p}_{0}$, there
exists

${k}_{0}$such that

 $\frac{}{}$ln ⁡ ek+1 ln ⁡ ek > p0 − ,k ≥ k0.

We may assume that

$\mathrm{ln}{e}_{k}$,
and we obtain

 $\mathrm{ln}{e}_{k+1}$

so that

 ${e}_{k+1}$0.

This proves that

${\overline{Q}}_{{p}_{0}}$and, by Lemma 3.1 (i), we get

 ${p}_{0}$

Suppose that

 ${p}_{0}$

It follows by Lemma 3.1 (iii) that

${Q}_{s}$so there
is

$c>0$such
that

${Q}_{s}$

$\forall k\ge 0,$i.e.,

 ${e}_{k+1}$

But then

 $\mathrm{ln}{e}_{k+1}$

 $\underset{k\to \infty }{\mathrm{liminf}}$ln ⁡ ek ≥ s + lim ⁡ k→∞ ln ⁡ c ln ⁡ ek = s > p0,

which is a contradiction. Hence, we have proved that

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}}_{}$ L</msub > = ql</msub >.Analogously,

${\overline{Q}}_{L}$implies

${p}_{1}$ _

Lemma 3.4
[20, lemma 3.3] If

 $\mathrm{lim}\mathrm{inf}{Q}_{\Lambda }$

then

$\left\{{e}_{k}$is monotone decreasing starting from a step

$k\ge {k}_{0}$.

Proof. [20] For some some

$>0$sufficiently small,

$\exists {k}_{0}$such that

${Q}_{\Lambda }$

$\forall k\ge {k}_{0}$Hence, the
numbers

$\mathrm{ln}\frac{{e}_{k+1}}{}$ek</msub >all have
the same sign

$\forall k\ge {k}_{0}$.
If they were positive, then we would have

${e}_{k+1}$, contradictory to the
convergence of

$\left\{{x}_{k}$Therefore,
they are all negative and

${e}_{k+1}$ _

We present now the result relating the

$C$– and

$Q$-orders,
and we choose to incorporate Proposition 3.3 .

Theorem 3.5
(cf. [19], [20]) Consider a given norm

$\parallel \cdot \parallel$in

${ℝ}^{n}$and a convergent
sequence

$\left\{{x}_{k}$to
some

${x}^{\ast }$. Then,
given some

${p}_{0}$,
in the above notations we have:

 ${C}_{{p}_{0}}$ (32)

Moreover,

 (33)

Proof. We give the proof in five steps: A.

${C}_{{p}_{0}}$, B.

$Q⇔{Q}_{}$, C.

${Q}_{}$, D.

$Q⇔{Q}_{L}$and
E.

${Q}_{\Lambda }$A1 (

${C}_{{p}_{0}}$).
The simplest approach is (also mentioned in [37]) to take logarithms in (2). The
proof in [20] used an approach similar to Lemma 3.1 .

A2 (

${Q}_{L}$)
[20] Consider any of the sequences from Example 2.8 .

B. The relation is obvious.

C1 (

${Q}_{}$)
Let

$>0$such
that

${p}_{0}$Since

${Q}_{p-}$, it follows
that exists

$b$such that

${Q}_{{p}_{0}}$

$\forall k\ge 0,$i.e., the second inequality
in the definition of

${Q}_{I,}$The first inequality is obtained in the same way.

C2 (

${Q}_{I,}$) Suppose
for some

${p}_{0}$and

$>0$with

${p}_{0}$we have

 $a{\left({e}_{k}}^{{p}_{0}}$k+1 ≤ b(ek)p0−,∀ ⁡k ≥ 0.

The second inequality implies

${\overline{Q}}_{{p}_{0}}$and by Lemma 3.1 (i),

$\mathrm{lim}{Q}_{q}$s.t.

$1The first inequality (valid for all

$>0$)
implies

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{{p}_{0}}$and again
by Lemma 3.1 (ii),

$\mathrm{lim}{Q}_{s}$with

$p+The assertion follows since

can be taken arbitrarily small.

D. [19, Prop. 1.1] Using the above result, we get

${Q}_{L}$and
vice versa.

E1 (

${Q}_{\Lambda }$)
[20] We notice that, by B and D, we have that

${Q}_{L}$, so it suffices
to show that

${Q}_{\Lambda }$.

Let

${Q}_{\Lambda }$For any

$>0,$we have

${Q}_{\Lambda }$for all

$k$sufficiently large, and using Lemma 3.4 we get

 $\mathrm{ln}\frac{{e}_{k+2}}{}$ek+1 > (p0 + )ln ⁡ ek+1 ek ,

which gives

 $\frac{}{}$ek+2 (ek+1)p0+ > ek+1 (ek)p0+,

i.e.,

${Q}_{{p}_{0}}$

$\forall k\ge {k}_{0}$Considering now

$\frac{}{}$2instead of

$,$we obtain similarly that

 ${Q}_{{p}_{0}}$2 (k + 1) > Qp0+ 2 (k),∀ ⁡k ≥ k1. (34)

We prove that the monotone sequence

$\left\{{Q}_{{p}_{0}}\right\}$is
unbounded:

${Q}_{{p}_{0}}$Otherwise, if

$\exists M>0$s.t.

$\frac{{e}_{k+2}}{}$(ek+1</msub >)p0</msub >+</msup > < M,for
all

$k,$i.e.,

${\overline{Q}}_{{p}_{0}}$then according to Lemma
3.1 (i), this implies

${\overline{Q}}_{{p}_{0}}$2 </msub > = 0in contradiction to the monotony from (34 ).

Similarly, one can prove that for any

$>0$with

${p}_{0}$we have

${Q}_{\Lambda }$for all

$k$sufficiently large
and we get

${Q}_{{p}_{0}}$E2 (

${Q}_{L}$) [20]
Assuming

${Q}_{L}$we have

 $\underset{k\to \infty }{\mathrm{lim}}$QL (k)−1 = p0p0−1 p0−1 = p0.

_

It is interesting to notice that

$Q$-convergence with
order

${p}_{0}$implies

${\overline{Q}}_{1}$However, the reverse is not true
(i.e.,

$Q$-superlinear convergence
does not imply

$Q$-convergence
with order

${p}_{0}$),
as the following examples show.

Example 3.6
a) [4, E 10.1.4] (we follow here [8, p. 94]) Let

$0and
consider the sequence

 ${x}_{k+1}$ln ⁡ xk,k ≥ 0,x0 = c,

which converges superlinearly to

$0.$Suppose there exists

${p}_{0}$such that

$\left\{{x}_{k}\right\}$converges
with

$Q$-order (and
therefore with

${Q}_{I,}$order)

${p}_{0}$Then

$\forall >0$with

${p}_{0}$s.t.

${x}_{k+1}$Hence,

 $-\frac{{x}_{k}}{}$ln ⁡ xk ≤ bxkp0−or ( 1 xk )p0−−1 ln ⁡ 1 xk ≤ b,

for all

$k$sufficiently large, which contradicts the well known limit

$\underset{t\to \infty }{\mathrm{lim}}$ln t = ,
for any

$\alpha >0$.
As

${Q}_{L}$,
we get

${q}_{l}$.

b) Some simpler examples can be found either as explicitly given by

${x}_{k}$[32, p. 22],

${x}_{k}$</msup >[26], [20],

${x}_{k}$k![27], or in the list of exercises, by

${x}_{k}$</msup >,

$c>1$[4, E.9.2.1.j]. To show these assertions we can also use that

${Q}_{L}$or

${Q}_{\Lambda }$.
Expression (24) of Potra gives yet another example.

Remark 3.7
We notice that the orders

${Q}_{}$do not have elements to define the strict

$Q$-superlinear
convergence, when

${\overline{Q}}_{1}$and

${\overline{Q}}_{p}$

$\forall p>1$.

The

$Q$-superlinear
convergence implies

$R$-superlinear
convergence, as in any norm [4, 9.3.1]

 ${}_{}$R¯1 ≤Q¯1.

In fact,

${\overline{R}}_{1}$is a
lower bound for

${\overline{Q}}_{1}$in all norms from

${ℝ}^{n}$.

It is interesting to see that sequences with infinite

$Q$-order exist too,
as one can take

${x}_{k}$</msup >(from the list of exercises in [4, E.9.2.1]).

Remark 3.8
An important aspect we address now is the problem of changing
the norms, as we have seen in Example 2.3 that for a given sequence in

${ℝ}^{n}$the existence of the

$C$-order
is norm-dependent. Ortega and Rheinboldt [4, 9.1.6] proved that the three
relations

${\overline{Q}}_{p}$,

${\overline{Q}}_{p}$finite,

${\overline{Q}}_{p}$are norm-independent. The same hold for

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{p}$,
as one can easily verify (using also Lemma 3.1 ), which shows the norm-independence
of the

$Q$-order.
By the equivalence from Theorem 3.5 , we conclude that the rest of the

$Q$-orders
are norm independent too. This assertion will be further completed by the
computational convergence orders, as we shall see in the following subsection.

It is important to recall that for sequences with

$Q$-order

${p}_{0}$,

while this

$Q$-order
is norm-independent, the values of the quantities

${\overline{Q}}_{{p}_{0}}$and

${\phantom{\rule{4.32828pt}{0ex}}\phantom{\rule{-4.32828pt}{0ex}}Q\phantom{\rule{-4.32828pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{-1.88583pt}{0ex}}\phantom{\rule{4.32828pt}{0ex}}}_{{p}_{0}}$however, if finite, are norm-dependent [4, E.9.1-1].

Remark 3.9
One can easily show that the sequence

$\left\{{x}_{k}$converges with

$Q$-order

${p}_{0}$iff one of the the following sequences converges with the same order:

$\left\{\frac{{e}_{k+1}}{}$ek</msub > },

$\left\{{e}_{k+1}$or

$\left\{{\left({e}_{k}}^{\alpha }$(

$\alpha >0$arbitrary, given).

##### 3.1.2 R-convergence orders

In 1972, Brent [32] stated, while in 1984, Potra and Pták [8] shown
that

 ${r}_{l}$