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

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author address: Tiberiu Popoviciu Institute of Numerical Analysis, Romanian
Academy, P.O. Box 68-1, Cluj-Napoca, Romania. e-mail: ecatinas@ictp.acad.ro,
www.ictp.acad.ro/catinas

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

Keywords. convergent sequences in

;
Q-, C-, and R-convergence orders of sequences; computational convergence
orders.

MSC 2010. 65J05.

1 Introduction

Consider in the beginning a sequence

from

$ℝ$

, which converges
to a finite limit

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

The relation

|x∗</msup >−xk</msub >|p</msup > = Cp</msub > ∈ (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

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

,
relation (1 ) is only of theoretical use.

Still, such definition (along with the

$Q$

-superlinear convergence,
which requires

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

), 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
”

has

$Q$

-order (at
least)

and

$R$

-order
(at least)

,

”. 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

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

$C$

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

• 
  

  has

  $R$

  -order
  

  (but neither

  $C$

  –
  nor

  $Q$

  -order);

• 
  

  has the same

  $R$

  –
  and

  $Q$

  -order
  

  (but no

  $C$

  -order);

• 
  

  has the same

  $R$

  -,
  

  $Q$

  –
  and

  $C$

  -order
  

  .

In less words, the relation reads for

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

(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

,
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

from

, which converges
to a finite limit

;
we prefer the common setting of a normed
space—

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

$k\ge 0$

.
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”)

k</msub >∥.

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

is
that

(ek</msub >)p0</msub ></msup > = Cp0</msub ></msub > ∈ (0,+∞),

(2)

which we call, as in [20],

$C$

-convergence with
order

(the linear
convergence requires

and

)
–

$Q$

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

such
that

</msubsup > ≤ ek+1</msub > ≤ (Cp0</msub ></msub > + )ekp0</msub > </msubsup >,∀ ⁡k ≥ k0</msub >,

(3)

or, less sharp, the existence of

$A,B>0$

such that

</msubsup > ≤ ek+1</msub > ≤ Bekp0</msub > </msubsup >,∀ ⁡k ≥ 0.

(4)

The

$C$

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

$C$

-orders

$1$

and

$2$

.

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

has

-convergence
order

if:

ln ⁡ ek</msub > → p0</msub >,as k →∞.

(9)

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

.

The sequence

has

-convergence
order

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

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

.

The convergence with

-convergence
order

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

, and
when

, the sequence was
said to have the order

.

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

for
all

$p\ge 1$

:

$${}_{}$$ Q¯p</msub > = limsup ⁡ k→∞</munder >Qp</msub > (k).

They shown the following behavior for

as a
”function” of

$p$

.

The quantity

inf ⁡ {p ∈ [1,∞) : Q¯p</msub > = +∞ }

(12)

(called in [4] as

$Q$

-convergence
with order at least

and denoted by O

)
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

(instead of

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,

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

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

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

$${}_{}$$ Q p</msub > = liminf ⁡ k→∞</munder >Qp</msub > (k)

and define the upper

$Q$

-order

by

Then the

$Q$

-convergence
is with order 

if

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

and

.

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

:

has

-convergence
order

if

ek+1</msub > ln ⁡ ek+1</msub > ek</msub > → p0</msub >,   as  k →∞.

(13)

The author gave no result relating the

$Q$

– and

-orders he introduced,
and assumed that

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

, and in fact
equivalent to

defined in subsubsection 2.1.2 .

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

-convergence order

and, assuming that

has the exact

$Q$

-order of
convergence

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

has the

-order

(and

-order

as
well).

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

,
with ”

$I$

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

The sequence

has

-convergence
order

if

$\forall <!--FUNCTION\; APPLICATION-->>0,$

$\exists <!--FUNCTION\; APPLICATION-->a,b>0$

such
that

k+1</msub > ≤ bekp0</msub >−</msubsup >,∀ ⁡k ≥ 0.

(14)

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

Potra shown some fundamental results in our study, that

-convergence
with order

is
equivalent to

– and
to

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

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

-convergence with order

, namely that it implies
(partial)

$R$

-convergence
with order

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

).

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

$Q$

-orders, when noting that if
a sequence has exact

$Q$

-order

then

.

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

$Q$

– and

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

$Q$

-order

is equivalent
to

and to

-convergence
with same order:

They considered

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

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

$Q$

-order,
as shown below.

An elementary example shows that in case of convergence with

$Q$

-order

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

$Q_{}$

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)

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

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

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.,

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

).

Corrections.

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

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

$$<mfrac><mrow>log<!--nolimits-->\; <mfrac><mrow><msub><mrow>s</mrow><mrow>k</mrow></msub\; ></msub></mrow><br>\; <mrow><msub><mrow>s</mrow><mrow>k+1</mrow></msub\; ></msub></mrow></mfrac></mrow><br>\; <mrow>\; log<!--nolimits-->\; <mfrac><mrow><msub><mrow>s</mrow><mrow>k-1</mrow></msub\; ></msub></mrow><br>\; <mrow><msub><mrow>s</mrow><mrow>k</mrow></msub\; ></msub></mrow></mfrac>\; </mrow></mfrac>,$$

(19)

equivalent to

defined below.

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

instead of the
(unknown) errors

in the definitions of the convergence orders studied there
(

and exact

$Q$

-order).

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

skp0</msub ></msubsup >

with the resulted definitions being denoted correspondingly by

and

; we use here the same
convention for denoting

.
These authors proved the following fundamental results: for

, 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:

L</msub >,QΛ</msub >,Q′</msup >,Q L′</msubsup >,Q Λ′</msubsup >}.

However, the considered assumptions were very strong when proving

and
”

”:

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

– and

-orders (in the case
of a sequence from

$ℝ$

with

$C$

-order

$p$

)
considered the ”extrapolated convergence order”

(20)

where

Δ2</msup >xk−2</msub > ,

$k\ge 2,$

. This convergence order
was obtained from

by replacing the unknown limit with an extrapolation

given by
the Aitken

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

acceleration process does not have a direct extension to

.

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

.

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

(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

,

and

for
sequences

from

$ℝ$

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

Nonlinear residuals.

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

$F(x)=0$

,

, with solution

. For simplicity,
we shall denote

(resp.

when

$n=1$

).

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

corresponding to (4) by

</msup > ≤∥Fk+1</msub >∥≤ B∥Fk</msub >∥p0</msub > </msup >,∀ ⁡k ≥ 0,

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

$$<mfrac><mrow>ln<!--nolimits-->|<msub><mrow>f</mrow><mrow>k+1</mrow></msub\; >|</msub></mrow><br>\; <mrow>ln<!--nolimits-->|<msub><mrow>f</mrow><mrow>k</mrow></msub\; >|</msub></mrow></mfrac>$$

(22)

and shown its equivalence to

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

$$<mfrac><mrow>ln<!--nolimits-->|<msub><mrow>f</mrow><mrow>k+1</mrow></msub\; >∕<msub><mrow>f</mrow><mrow>k</mrow></msub\; >|</msub></msub></mrow><br>\; <mrow>ln<!--nolimits-->|<msub><mrow>f</mrow><mrow>k</mrow></msub\; >∕<msub><mrow>f</mrow><mrow>k-1</mrow></msub\; >|</msub></msub></mrow></mfrac>$$

(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

is arbitrary close to

$1$

from above, while
the (exact)

$R$

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

$Q$

-order

).

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

:

$${}_{}$$ R¯p</msub > {xk</msub >} = { limsup ⁡ k→∞</munder >(ek</msub >)1 k </msup >, ifp = 1, limsup ⁡ k→∞</munder >(ek</msub >) 1 pk</msup > </msup >, 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

.

The lower

$R$

-order,
which we denote here by

(instead of

in [4, 9.2]), defined by

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

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

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

). The sequence was said
that converges with

-order

if

, i.e.,
in a single condition,

</msup >1k =p˙0. (27) $$$$ They noted that the $C$ -order easily implies -order . In 1979, Schwetlick [25] considered the upper $R$ -order $${}_{}$$ R p</msub > {xk</msub >} = { liminf ⁡ k→∞</munder >(ek</msub >)1 k </msup >,  ifp = 1, liminf ⁡ k→∞</munder >(ek</msub >) 1 pk</msup > </msup >, ifp > 1, and (in the notations of this paper) $${}_{}$$ ru</msub > = { ∞,   if   R p</msub > = 0,∀ ⁡p ≥ 1, sup ⁡ {p ∈ [1,∞) : R p</msub > = 0 }, he defined the convergence with $R$ -order if Similarly to the case of the $Q$ -order, the sequence is said that converges $R$ -superlinearly if k </msup > = R¯1</msub > = 0. Remark 2.15 Analogously to Remark 2.14 , whenever exists a with then for any $>0$ with there is a such that either $${}^{}$$ ( R p0</msub ></msub > − )k</msup > ≤ e k</msub >,∀ ⁡k ≥ k0</msub >, ifp0</msub > = 1, or $${}^{}$$ ( R p0</msub ></msub > − )p0k</msubsup > </msup > ≤ ek</msub >,∀ ⁡k ≥ k0</msub >, ifp0</msub > > 1. In 1989, Potra [19] has introduced and studied some further measures for the $R$ -convergence order, which we denote here by , resp. : converges with -order if $$\begin{array}{}\end{array}$$ lim ⁡ k→∞</munder >(ek</msub >) 1 (p0</msub >−)k</msup > </msup > = 0,∀ ⁡ > 0, lim ⁡ k→∞</munder >(ek</msub >) 1 (p0</msub >+)k</msup > </msup > = 1,∀ ⁡ > 0; converges with -order if $\forall <!--FUNCTION\; APPLICATION-->>0,$ $\exists <!--FUNCTION\; APPLICATION-->a,b>0$ and $0<\eta ,<1$ such that </msup > ≤ ek</msub > ≤ b (p0</msub >−)k</msup > </msup >,∀ ⁡k ≥ 0. The exact $R$ -order of convergence was also defined here – analogously to the definition of the exact $Q$ -order (4) – as being if $\exists <!--FUNCTION\; APPLICATION-->A,B>0$ , $0<\eta ,<1$ such that </msup > ≤ ek</msub > ≤ B ⋅ (p0</msub >)k</msup > </msup >,∀ ⁡k ≥ 0; (28) we see that this definition can be immediately connected to the following one, analogous to (6): (29) Potra has also considered the lower and upper $R$ -orders, when noting that if a sequence has exact $R$ -order then . Comparing the list of $Q$ – and $R$ -orders, the definition of was not given before, so we consider it here, by denoting the expressions $${}_{}$$ R Λ</msub > := liminf ⁡ k→∞</munder > |ln ⁡ ek+1</msub > ek</msub > |1 k </msup >,R¯Λ</msub > := limsup ⁡ k→∞</munder > |ln ⁡ ek+1</msub > ek</msub > |1 k </msup > and by defining the -order when , i.e., when ek</msub > |1 k </msup > = p0</msub >. (30) 3 Results relating the high (computational) convergence orders In this section we consider $p>1$ and present the relationships of different definitions of the (computational) convergence orders, with full proofs in the case of the $C$ – and $Q$ -orders. 3.1 Relationships between the convergence orders The equivalences and implications mentioned in the previous section can be summarized as: $$\begin{array}{}\end{array}$$ {Q</msub >,QI,</msub >,QL</msub >} ⇒ {R</msub >,RI,</msub >,RL</msub >} , [19], Cp0</msub ></msub > ⇒ {Q,QL</msub >,QΛ</msub >} , [20]. At this point we can simply merge the equivalences from the two statements above and write the conclusion. However, as we intend to provide a self-contained material, we include the proofs of the equivalence of the $Q$ -orders, some of them much simplified. 3.1.1 C- and Q-convergence orders We shall need the three auxiliary results below. Lemma 3.1 [20, lemma 2.1] (i) If then for all $s<p$ , and so (ii) if then for all $p<s$ and so (iii) if then (iv) if then . Relations (iii) and (iv) show that and justify the terminology ”lower” and ”upper”. Proof. [20] For (i), if and $s<p,$ then eks</msubsup > = ekp−s</msubsup >Q p</msub >(k) → 0,as k →∞. For (iii), suppose and choose By definition of then, . Now, by (i), The other statements are proved in a similar manner.  _ Remark 3.2 Ortega and Rheinboldt have previously obtained (i) and (ii) in [4, 9.1], see Proposition 2.5 . Potra obtained the following general result, interesting in its own, which will help us relating some quantities to their corresponding computational versions. Proposition 3.3 [19, Prop. 1.1] The following relations hold: ln ⁡ ek</msub > = ql</msub > (31) and $${}_{}$$ Q¯L</msub > := limsup ⁡ k→∞</munder >ln ⁡ ek+1</msub > ln ⁡ ek</msub > = qu</msub >. Proof. [19] Let Then for any $>0$ with , there exists such that $$\frac{}{}$$ ln ⁡ ek+1</msub > ln ⁡ ek</msub > > p0</msub > − ,k ≥ k0</msub >. We may assume that , and we obtain so that 0</msub >. This proves that and, by Lemma 3.1 (i), we get Suppose that It follows by Lemma 3.1 (iii) that so there is $c>0$ such that $\forall <!--FUNCTION\; APPLICATION-->k\ge 0,$ i.e., But then which leads to ln ⁡ ek</msub > ≥ s + lim ⁡ k→∞</munder > ln ⁡ c ln ⁡ ek</msub > = s > p0</msub >, which is a contradiction. Hence, we have proved that $Q_{}$ L</msub > = ql</msub >.Analogously, implies  _ Lemma 3.4 [20, lemma 3.3] If then is monotone decreasing starting from a step . Proof. [20] For some some $>0$ sufficiently small, such that Hence, the numbers ek</msub >all have the same sign . If they were positive, then we would have , contradictory to the convergence of Therefore, they are all negative and  _ We present now the result relating the $C$ – and $Q$ -orders, and we choose to incorporate Proposition 3.3 . Theorem 3.5 (cf. [19], [20]) Consider a given norm $\parallel \cdot \parallel$ in and a convergent sequence to some . Then, given some , in the above notations we have: (32) Moreover, (33) Proof. We give the proof in five steps: A. , B. , C. , D. and E. A1 ( ). The simplest approach is (also mentioned in [37]) to take logarithms in (2). The proof in [20] used an approach similar to Lemma 3.1 . A2 ( ) [20] Consider any of the sequences from Example 2.8 . B. The relation is obvious. C1 ( ) Let $>0$ such that Since , it follows that exists $b$ such that $\forall <!--FUNCTION\; APPLICATION-->k\ge 0,$ i.e., the second inequality in the definition of The first inequality is obtained in the same way. C2 ( ) Suppose for some and $>0$ with we have k+1</msub > ≤ b(ek</msub >)p0</msub >−</msup >,∀ ⁡k ≥ 0. The second inequality implies and by Lemma 3.1 (i), s.t. The first inequality (valid for all $>0$ ) implies and again by Lemma 3.1 (ii), with $p+<s.$ The assertion follows since $$ can be taken arbitrarily small. D. [19, Prop. 1.1] Using the above result, we get and vice versa. E1 ( ) [20] We notice that, by B and D, we have that , so it suffices to show that . Let For any $>0,$ we have for all $k$ sufficiently large, and using Lemma 3.4 we get ek+1</msub > > (p0</msub > + )ln ⁡ ek+1</msub > ek</msub > , which gives $$\frac{}{}$$ ek+2</msub > (ek+1</msub >)p0</msub >+</msup > > ek+1</msub > (ek</msub >)p0</msub >+</msup >, i.e., Considering now $\frac{}{}$ 2instead of $,$ we obtain similarly that 2 </msub > (k + 1) > Qp0</msub >+ 2 </msub > (k),∀ ⁡k ≥ k1</msub >. (34) We prove that the monotone sequence is unbounded: Otherwise, if $\exists <!--FUNCTION\; APPLICATION-->M>0$ s.t. (ek+1</msub >)p0</msub >+</msup > < M,for all $k,$ i.e., then according to Lemma 3.1 (i), this implies 2 </msub > = 0in contradiction to the monotony from (34 ). Similarly, one can prove that for any $>0$ with we have for all $k$ sufficiently large and we get E2 ( ) [20] Assuming we have QL</msub > (k)−1 = p0</msub >p0</msub >−1 p0</msub >−1 = p0</msub >. _ It is interesting to notice that $Q$ -convergence with order implies However, the reverse is not true (i.e., $Q$ -superlinear convergence does not imply $Q$ -convergence with order ), as the following examples show. Example 3.6 a) [4, E 10.1.4] (we follow here [8, p. 94]) Let $0<c<1∕e$ and consider the sequence ln ⁡ xk</msub >,k ≥ 0,x0</msub > = c, which converges superlinearly to $0.$ Suppose there exists such that converges with $Q$ -order (and therefore with order) Then $\forall <!--FUNCTION\; APPLICATION-->>0$ with s.t. Hence, ln ⁡ xk</msub > ≤ bxkp0</msub >−</msubsup >or ( 1 xk</msub > )p0</msub >−−1</msup > ln ⁡ 1 xk</msub > ≤ b, for all $k$ sufficiently large, which contradicts the well known limit ln ⁡ t = ∞, for any $\alpha >0$ . As , we get . b) Some simpler examples can be found either as explicitly given by [32, p. 22], </msup >[26], [20], k![27], or in the list of exercises, by </msup >, $c>1$ [4, E.9.2.1.j]. To show these assertions we can also use that or . Expression (24) of Potra gives yet another example. Remark 3.7 We notice that the orders do not have elements to define the strict $Q$ -superlinear convergence, when and $\forall <!--FUNCTION\; APPLICATION-->p>1$ . The $Q$ -superlinear convergence implies $R$ -superlinear convergence, as in any norm [4, 9.3.1] $${}_{}$$ R¯1</msub > ≤Q¯1</msub >. In fact, is a lower bound for in all norms from . It is interesting to see that sequences with infinite $Q$ -order exist too, as one can take </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 the existence of the $C$ -order is norm-dependent. Ortega and Rheinboldt [4, 9.1.6] proved that the three relations , finite, are norm-independent. The same hold for , 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 , while this $Q$ -order is norm-independent, the values of the quantities and however, if finite, are norm-dependent [4, E.9.1-1]. Remark 3.9 One can easily show that the sequence converges with $Q$ -order iff one of the the following sequences converges with the same order: ek</msub > }, or ( $\alpha >0$ arbitrary, given). 3.1.2 R-convergence orders In 1972, Brent [32] stated, while in 1984, Potra and Pták [8] shown that k </msup >. (35) In 1989 Potra [19] stated the equivalence , which can be easily proved. In fact we may extend Theorem 3.5 and the above relations. For brevity, we choose to write $\{Q\}$ as a generic form for all the five equivalent $Q$ -type orders, . The same will be written in the sequel for , , $\{R\}$ , and . Theorem 3.10 (cf. [19]) In the assumptions of Theorem 3.5 , we have: (36) (with the equivalence of requiring the additional assumption ). Moreover, (37) Proof. (cf. [19]) We notice first that -order implies -order (and therefore , 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 . The converse ” $Q\Leftarrow R$ ” is false, even if has exact $R$ -convergence order (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 . The equivalence of the -order to the , and therefore all remaining $R$ -orders, is based on the relation ek</msub > = (ln ⁡ ek</msub >) (ln ⁡ ek+1</msub > ln ⁡ ek</msub > − 1 ). We note that there exist sequences (see, e.g., (25)), for which , 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 – can be obtained again from the well known inequalities for positive numbers ak</msub > ≤ liminf ⁡ k→∞</munder > |ak</msub >| 1 k </msup > ≤ limsup ⁡ k→∞</munder > |ak</msub >| 1 k </msup > ≤ limsup ⁡ k→∞</munder >ak+1</msub > ak</msub > taking , and then considering (31 ) and (35 ).  _ The following example shows that one may find sequences with $R$ -order ( ) arbitrary high and arbitrary close to 1 from above. Example 3.11 [19] Given any numbers $1<s<\tau ,$ we construct a sequence which has the exact $R$ -convergence order $\tau$ and for which Let $0<<1$ be given, and take with $q>1$ such that $qs>\tau .$ Define $${}_{}$$ xk</msub > = { τk</msup ></msup >,kodd, ητk</msup > </msup >,keven. The sequence has exact $R$ -convergence order $\tau .$ On the other hand, for $k$ even we obtain xks</msubsup > = (τ</msup >) τk</msup > </msup > (ηs</msup >) τk</msup > </msup > = ( τ</msup > qs</msup > )τk</msup > </msup > →∞, so that In fact, one obtains qwhile ( $>\tau$ , 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 (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 are replaced either by the corrections or by the nonlinear residuals , when solving nonlinear systems having the same number of equations and unknowns. We shall assume that and $\forall <!--FUNCTION\; APPLICATION-->k\ge 0.$ 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 , 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: L</msub >,QΛ</msub >,Q′</msup >,Q L′</msubsup >,Q Λ′</msubsup >}. (38) However, for proving and the considered assumptions were very strong: 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 converges $Q$ -superlinearly to if and only if the sequence converges $Q$ -superlinearly to zero. Proof. Necessity. [8] Suppose converges $Q$ -superlinearly. Then there exists a positive integer such that 2ek</msub >,   for all k ≥ k0</msub >. It follows that for , we have: $$\frac{}{}$$ sk+1</msub > sk</msub > ≤ ek+1</msub >+ek+2</msub > ek</msub >−ek+1</msub > ≤ 3ek+1</msub > ek</msub > , from which we deduce the $Q$ -superlinear convergence of the sequence Sufficiency. [8] If we suppose that the sequence converges $Q$ -superlinearly, then there exists a positive integer such that 3sk−1</msub >,   for all k ≥ k0</msub >. For any we have: $$\begin{array}{}\end{array}$$ ek+1</msub > ≤∑ j=1∞</munderover >sk+j</msub > ≤ sk+2</msub >∑ j=0∞</munderover >3−j</msup > = 3 2sk+1</msub > ≤ 1 2sk</msub >. Finally, writing $$\frac{}{}$$ ek+1</msub > ek</msub > ≤ 3 2sk+1</msub > sk</msub >−ek+1</msub > ≤ 3sk+1</msub > sk</msub > , we infer that 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 converges $Q$ -superlinearly then ek</msub > = 1. Proof. [5] The result follows immediately from the equality ∥x∗</msup >−xk</msub >∥ − x∗</msup >−xk</msub > ∥x∗</msup >−xk</msub >∥ ∥ = ek+1</msub > ek</msub > . _ The assumption of this Lemma also implies that converges $Q$ -superlinearly to zero too. The converse of this result does not hold, as an example given by the same authors shows [5]: k!, , $k\ge 1$ . 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 , $Q$ -superlinearly. Then ek</msub > = ∥−∑ k+1∞</munderover >(xj+1</msub > − xj</msub >)∥ ∥−∑ k∞</munderover >(xj+1</msub > − xj</msub >)∥ ≤ ∑ k+1∞</munderover >sj</msub > sk</msub > −∑ k+1∞</munderover >sj</msub > = ∑ k+1∞</munderover >sj</msub > sk</msub > 1 −∑ k+1∞</munderover >sj</msub > sk</msub >. For each $k$ , define . Note that and for each $i\ge 0$ . Then ek</msub > ≤ ρk</msub > 1−ρk</msub > 1 − ρk</msub > 1−ρk</msub > = ρk</msub > 1 − 2ρk</msub > → 0, and it follows that $Q$ -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: 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¯ ql</msub ></msub > = Q¯ql</msub >′</msubsup >. (40) Proof. The proof will consist of two steps. At step A, we prove , while at step B we prove the equivalence of the $Q$ -type and -type orders. Relations (39) and (40) follow along the way. A1 ( ) Suppose converges with $C$ -order Then converges $Q$ -superlinearly, and by the Potra-Pták and Dennis-Moré Lemmas, we get (ek</msub >)p0</msub ></msup >sk+1</msub > = 1, (41) which shows that converges with -order A2 ( ) A2 is proved similarly to A1. B1 ( $Q$ $\Rightarrow$ ) Suppose converges with some $Q$ -type order Then converges with $Q$ -order and therefore $Q$ -superlinearly; by the Potra-Pták and Dennis-Moré Lemmas, it follows that ln ⁡ sk</msub > = lim ⁡ k→∞</munder >ln ⁡ (sk+1</msub > ek+1</msub >ek+1</msub > ) ln ⁡ (sk</msub > ek</msub >ek</msub > ) = p0</msub >, (42) i.e., converges with -order One can view the previous statement from a different viewpoint: the sequence converges with -order so by the results in Subsection 3.1 , it converges with any $Q$ -type order , whence the conclusion. Relation (42) and Proposition 3.3 attract (39), while (41) implies (40). B2 ( -type $\Rightarrow$ $Q$ -type order) The reverse implication is proved similarly.  _ Remark 3.16 a) Relation (40) shows in particular that if has an exact $Q$ -order of convergence , 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 -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: (with the equivalence of the -orders requiring the additional assumption ). The proof relies on the result [8, Prop. 6.20] which leads by analogy to relation . 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 and have exact $R$ -orders of convergence, then they must have the same exact $R$ -order of convergence. Proposition 3.19 [8, p. 111] If has an exact $R$ -order of convergence this does not imply that 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 $0<c<1<r<s$ and set $${}_{}$$ xk</msub > = { crk</msup ></msup >, k even, crk−1</msup > </msup > + csk−1</msup > </msup >,k odd, yk</msub > = { −crk</msup ></msup >,k even, csk−1</msup > </msup >,k odd. It is easy to see that $r$ is the exact $R$ -order of convergence of the sequences and while the sequences and have no exact $R$ -order of convergence.   Remark 3.21 One can easily show that the sequence converges with $R$ -order iff one of the the following sequences converges with the same order: or ( $\alpha >0$ arbitrary, given). The same holds for ek</msub > }, but under the supplementary assumption that . 3.2.2 Nonlinear residuals – relationships We consider now the solving of a system of nonlinear equations $F(x)=0$ for a nonlinear mapping , having a solution . The quantities – in different formulae – were used not only to control the convergence of a given sequence to , but also to determine at each step $k$ the next element, (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) continuous at C) is invertible at Replacing the errors by the corresponding residuals, we denote ∥Fk</msub >∥p0</msub ></msup > and similarly, the resulted definitions for the – and all the rest of -type orders. $C$ -, $Q$ -orders and nonlinear residuals The following example shows that neither of and implies the other. Example 3.22 a) Consider , with $${}_{}$$ xk</msub > = { (2−2k</msup ></msup >,0), kodd, (2−2k</msup > </msup >,2−2k</msup >−1 </msup >),keven, and the (linear) mapping , $F(u,v)=(u,4v)$ . One can see that conditions A)–C) are satisfied, and further, has $C$ -order $2$ , while does not have $C$ -order (in the max-norm). b) Let , with $${}_{}$$ yk</msub > = { (2−2k</msup ></msup >,0), kodd, (2−2k</msup > </msup >,2−2k</msup >+1 </msup >),keven, and similarly the mapping $G(u,v)=(u,v∕4)$ . Now has no $C$ -order, while has $C$ -order $2$ (in the max-norm). One can also view Example 2.3 in the sense that if one takes as the identity operator, $F=I$ , the sequence from (8) has $C$ -order $2$ , while 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 continuous at the solution , with the Jacobian nonsingular at : . Let 2β,2β }, where . Then there exists $>0$ such that for any with , one has $$\frac{}{}$$ 1 α ∥x − x∗</msup >∥ ≤ ∥F (x)∥ ≤ α ∥x − x∗</msup >∥. 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. $\forall <!--FUNCTION\; APPLICATION-->x$ with we have $$(\frac{1}{}$$ ∥F′</msup >(x∗</msup >)−1</msup >∥ − ) ∥x − x∗</msup >∥≤ ∥F (x)∥ ≤ (∥F′</msup > (x∗</msup >)∥ + )∥x − x∗</msup >∥. 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 converges with some $Q$ -type order, say -order Then, by Lemma 3.23 , we have $$\frac{}{}$$ ln ⁡ ek+1</msub >−ln ⁡ α ln ⁡ ek</msub >+ln ⁡ α ≤ ln ⁡ ∥Fk+1</msub >∥ ln ⁡ ∥Fk</msub >∥ ≤ ln ⁡ ek+1</msub >+ln ⁡ α ln ⁡ ek</msub >−ln ⁡ α , which shows that and further the equivalence of the -type orders. The implication $\Rightarrow$ follows in a similar manner.  _ The -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 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 $\Rightarrow$ and $\nLeftarrow$ , while the equivalence of the -type orders requires the additional assumption . 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., ) were obtained under the strong assumption that is a monotone sequence from $ℝ.$ Zhang and Wang [62] completed the diagram with two implications. The first one, was proved again under the monotony assumption on the sequence of real numbers. The second one, , was proved under some even further supplementary assumptions (we mark this by dashed line in the diagram): and , where xk+1</msub >−xk</msub >, , . The following completed diagram resulted. 5 Conclusions As the completion 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 . Ortega and Rheinboldt used and 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 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 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 may remain a powerful theoretical tool in measuring the convergence rate of sequences. In the case when a $Q$ -order exists ( ), the use of the computational quotients may offer approximations to , while otherwise (when ) the use of the computational quotients may hopefully offer just an idea about the value of (and perhaps 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 ). References [1]    J. 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