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Strong µ‐faster convergence and strong µ‐acceleration
of convergence by regular matrices
A. Aasma a
a Department of Economics , Tallinn University of Technology , Kopli 101, Tallinn, 11712,
Estonia E-mail:
Published online: 14 Oct 2010.
To cite this article: A. Aasma (2008) Strong µ‐faster convergence and strong µ‐acceleration of convergence by regular
matrices, Mathematical Modelling and Analysis, 13:1, 1-6, DOI: 10.3846/1392-6292.2008.13.1-6
To link to this article: http://dx.doi.org/10.3846/1392-6292.2008.13.1-6
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Mathematical Modelling and Analysis
Volume 13 Number 1, 2008, pages 1–6
c
2008 Technika ISSN 1392-6292 print, ISSN 1648-3510 online
STRONG µ-FASTER CONVERGENCE AND
STRONG µ-ACCELERATION OF
CONVERGENCE BY REGULAR MATRICES
A. AASMA
Tallinn University of Technology, Department of Economics
Kopli 101, 11712, Tallinn, Estonia
E-mail: aasma@tv.ttu.ee
Received September 15, 2007; revised October 21, 2007; published online February 15, 2008
Abstract. The present paper continues the study of acceleration of convergence
started in the paper [A. Aasma, Proc. Estonian Acad. Sci. Phys. Math., 2006, 55,
4, 195-209]. The new, non-classical convergence acceleration concept, called strong
µ-acceleration of convergence (µis a positive monotonically increasing sequence), is
introduced. It is shown that this concept allows to compare the speeds of convergence
for a larger set of sequences than the classical convergence acceleration concept.
Regular matrix methods are used to accelerate the convergence of sequences.
Key words: Convergence acceleration, matrix methods, speed of convergence.
1 Introduction
The present paper continues the study of acceleration of convergence of real
or complex sequences started in [1]. Therefore all the notions not defined in
this paper can be found in [1]. Throughout the paper we assume that indices
and summation indices are integers, changing from 0to ∞, if not specified
otherwise.
Classically the convergence acceleration is defined as follows (cf. [5,6]).
Definition 1. Let x= (xk)and y= (yk)be convergent sequences with limits
ςand ξ, respectively. If
lim n
|yn−ξ|
|xn−ς|= 0,(1.1)
then it is said that yconverges faster than x.
Definition 2. The sequence transformation T:x→yis said to accelerate
the convergence of the sequence xif yconverges faster than x.
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2A. Aasma
Some methods, alternative to the classical concept of estimation and com-
parison of speeds of convergence of sequences, are used in [3,4]. In [1] another
alternative method is proposed, where the concept, called µ-faster convergence
(µis a positive monotonically increasing sequence) is introduced. It is shown
that this concept allows the comparison of speeds of convergence for a larger
set of sequences than the classical concept and this comparison is more precise.
Let A= (ank)be a matrix with real or complex entries. A sequence
x= (xk)is said to be A-summable if the sequence Ax= (Anx)is convergent,
where
Anx=X
k
ankxk.
We denote the set of all A-summable sequences by cA. Thus, a matrix A
determines the summability method on cA, which we also denote by A.
A method Ais said to be regular if for each x= (xn)∈c, where cis the
set of all convergent sequences, the equality limnAnx= limnxnholds. The
convergence acceleration and µ-acceleration of convergence by regular matrix
methods were studied correspondingly in [3,4,5,6,7] and [1].
In the present paper the concept of strong µ-faster convergence is defined
and compared with the usual faster convergence concept, determined by Def-
initions 1and 2. It is shown here that the new concept allows the comparison
of speeds of convergence for a larger set of sequences than the classical concept
and this comparison is more precise. It is also proved that if for a sequence
x= (xn)with the limit ςthe sequence of absolute differences (|xn−ς|)is
monotonically decreasing, then the strong µ-faster convergence of a sequence
ywith respect to xcoincides with the usual faster convergence of ywith respect
to x. Also the concept of strong µ-acceleration of convergence by a regular
matrix method is defined and it’s properties are studied.
2 Main Results
Let ϕbe a set of sequences such that
ϕ={x= (xk)|xk= const,if k > k0}
for some k0≥0. For every sequence x∈c\ϕwe denote
µx={µ= (µn)|0< µnր ∞, ln=µn|xn−lim nxn|=O(1), ln6=o(1) }.
Let us remind some notions from [1]. The sequence µis called a speed of
convergence of x. A sequence µ∗= (µ∗
n)∈µxis called the limit speed of
convergence of xif for all µ= (µn)∈µxthe relation µn/µ∗
n=O(1) holds.
The limit speed of convergence µ∗= (µ∗
n)of a sequence yis said to be higher
than the limit speed of convergence λ∗= (λ∗
n)of a sequence xif the ratio
λ∗
n/µ∗
nis upper-bounded, but not lower-bounded. It is said that a sequence y
converges µ-faster than xif the limit speed of convergence of yis higher than
the limit speed of convergence of xor y∈ϕand xdoes not belong to ϕ.
Now we introduce the concept of strong µ-faster convergence.
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Strong µ-Faster Convergence and Strong µ-Acceleration of Convergence 3
Definition 3. Let λ∗= (λ∗
n)and µ∗= (µ∗
n)be correspondingly the limit
speeds of convergence of convergent sequences xand y. We say that yconverges
strongly µ-faster than x, if λ∗
n/µ∗
n−→ 0or y∈ϕand xdoes not belong to ϕ.
Remark 1. It is easy to see that if yconverges strongly µ-faster than x, then
yconverges also µ-faster than x, but not vice versa. If y= (yn)converges
strongly µ-faster than x= (xn), then λ∗
n|yn−ξ|=o(1), where λ∗= (λ∗
n)is
the limit speed of xand ξis the limit of y. But for the case, if yconverges only
µ-faster, but not strongly µ-faster than x, there exists a subsequence (ykn)of
yso that λ∗
n|ykn−ξ| 6=o(1).
It was proved in [1] that if a sequence y= (yn)∈cconverges faster than
x= (xn)∈c\ϕ, then yconverges also µ-faster than x. We prove that the
similar property holds for the concept of strong µ-faster convergence.
Theorem 1. If a sequence y= (yn)∈cconverges faster than x= (xn)∈c\ϕ,
then yconverges also strongly µ-faster than x.
Proof For y∈ϕthe assertion of Theorem 1is clearly true. Thus, suppose
that y∈c\ϕconverges faster than x∈c\ϕ, i.e. relation (1.1) holds, and
show that then yconverges also strongly µ-faster than x. By Corollary 2.1
of [1] there exists the limit speed of convergence λ∗= (λ∗
n)∈λxof x. Using
relation (1.1) we have now
lim n
λ∗
n|yn−ξ|
λ∗
n|xn−ς|= 0.
Consequently, by Proposition 2.1 from [1] there exists ϑ= (ϑn),0< ϑnր ∞,
such that
ϑn
λ∗
n|yn−ξ|
λ∗
n|xn−ς|=O(1).
Denoting ϑnλ∗
n=µn, we get from the last relation that µn|yn−ξ|=O(1)
with 0< µnր ∞ and µn/λ∗
n−→ ∞. Consequently for the limit speed of
convergence µ∗= (µ∗
n)of ywe have µ∗
n/λ∗
n−→ ∞. Thus yconverges strongly
µ-faster than xby Definition 3.
The opposite assertion to Theorem 1, however, is not valid.
Example 1. Let x= (xn)∈c\ϕbe given by the relations
xn=1
(n+ 1)2nif n= 3k,
(n+ 1)38nxn=o(1) if n= 3k+ 1,
2n(n+ 1)2xn6=O(1),2n(n+ 1)xn=o(1) if n= 3k+ 2,
where k= 0,1,.... It was proved in [1] that applying Aitken’s process to the
subsequence (x3k)of xwe get the sequence y= (yn), where
yn=9
8n(1323n3+ 6993n2+ 12024n+ 6736) .
Math. Model. Anal., 13(1):1–6, 2008.
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4A. Aasma
It is easy to see now that yconverges not faster than xand xconverges
not faster than y, but yconverges strongly µ-faster than x. Indeed, we can
determine the limit speeds of convergence of xand yrespectively by λ∗= (λ∗
n)
and µ∗= (µ∗
n), where
λ∗
n= 2n(n+ 1), µ∗
n= 23n(n+ 1)3.
As µ∗
n/λ∗
n−→ ∞, then yconverges strongly µ-faster than xby Definition 3.
Suppose now that x= (xn)∈c\ϕwith the limit ςbe a sequence for which
the sequence of absolute differences (|xn−ς|)is monotonically decreasing.
We show that in this case the strong µ-faster convergence coincides with the
classical faster convergence.
Theorem 2. Let x= (xn)∈c\ϕbe a sequence (with the limit ς), for which
the sequence of absolute differences (|xn−ς|)is monotonically decreasing. If
a sequence y= (yn)(with limit ξ) converges strongly µ-faster than x, then y
converges also faster than x.
Proof It is not difficult to see that the limit speed λ∗= (λ∗
n)of a sequence x
can be defined by the equality λ∗
n= 1/|xn−ς|. If µ∗= (µ∗
n)is the limit speed
of y, then we get
µ∗
n|yn−ξ|
λ∗
n|xn−ς|=µ∗
n|yn−ξ|=O(1).
Last relation implies equality (1.1), since µ∗
n/λ∗
n−→ ∞.
It is said (see [1]) that a regular method Aµ-accelerates the convergence
of a sequence x∈cif the sequence Axconverges µ-faster than x.
Definition 4. We say that a matrix method Astrongly µ-accelerates the
convergence of a sequence x∈cif the sequence Axconverges strongly µ-faster
than x.
Theorem 3. For every x∈c\ϕthere exists a regular matrix A, which strongly
µ-accelerates the convergence of x.
Proof By Corollary 2.1 of [1] every x∈c\ϕhas the limit speed λ∗= (λ∗
n).
We show that there exists a regular matrix Aso that the limit speed of the
sequence (Anx)is higher than λ∗. As every x= (xn)∈c(with limit ς) can
be presented in the form
x=x0+ςe, where x0=x0
n∈c0and e= (1,1, ...),(2.1)
where c0is the set of sequences, converging to zero, then we get
λ∗
n|xn−ς|=λ∗
nx0
n=O(1) or x0
n=O1
λ∗
nand λ∗
nx0
n6=o(1).
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Strong µ-Faster Convergence and Strong µ-Acceleration of Convergence 5
As the limit speed λ∗is a monotonically increasing unbounded sequence, then
there exists a subsequence λ∗
knof λ∗such that λ∗
kn/λ∗
n−→ ∞. We define a
matrix A= (ank)by the equalities
ank =(1k=kn,
0k6=kn.
With the help of Theorem 2.3.7 from [2] it is not difficult to check that the
matrix Ais regular. Now we have
Anx0=X
k
ankx0
k=x0
kn=O1
λ∗
kn
or, equivalently,
λ∗
knAnx0=O(1).
Denoting µ= (µn) = λ∗
kn, we get
µnAnx0=O(1),where µn/λ∗
n−→ ∞.
Therefore Astrongly µ-accelerates the convergence of x0. As Ane= 1, then
with the help of (2.1) we conclude
µn|Anx−ς|=µnAnx0+ςAne−ς=µnAnx0.
Consequently Astrongly µ-accelerates also the convergence of x.
We note that the assertion of Theorem 3does not hold for the concept
of classical faster convergence. Indeed, it is not possible to accelerate the
convergence of x= (xn)∈c\ϕby any regular matrix method if, for example,
xis defined by the relation
xn=
1
n+ 1 n= 2k,
0n= 2k+ 1.
It follows from the proof of Theorem 3.2 of [1] that for every triangular
regular matrix Athere exists a convergent sequence x, which converges µ-
faster than its A-transform Ax. For strong µ-acceleration of convergence we
can extract from the proof of Theorem 3.2 of [1] the following result.
Proposition 1. If a triangular regular matrix Ahas a column with infinite
number of non-zero elements, then there exists a sequence x, converging strongly
µ-faster than its A-transform Ax.
As we see from Proposition 1, for some triangular regular methods Ait
is possible to choose a sequence x, converging strongly µ-faster than its A-
transform Ax, but it is not so for all triangular regular matrices.
Math. Model. Anal., 13(1):1–6, 2008.
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6A. Aasma
Example 2. Let A= (ank )be defined by the relation
ank =
δnk n= 2j,
1
2n= 2j+ 1, k =n−1, n,
0k < n −1,
where j= 0,1,.... Then for every convergent sequence x= (xk)we get
Anx=
xnn= 2j,
1
2(xn−1+xn)n= 2j+ 1,
where j= 0,1,.... Now a sequence xcan converge µ-faster than its A-
transform Ax only in the case, if xn−1/xn6=O(1).But never xcan converge
strongly µ-faster than its A-transform Ax.
References
[1] A. Aasma. µ-faster convergence and µ-acceleration of convergence by regular
matrices. Proc. Estonian Acad. Sci. Phys. Math.,55(4):195–209, 2006.
[2] J. Boos. Classical and Modern Methods in Summability. Oxford University Press,
Oxford, 2000.
[3] C. Brezinski. Limiting relationships and comparison theorems for sequences.
Rend. Circ. Mat. Palermo,28(2):273–280, 1979.
[4] C. Brezinski. Contraction properties of sequence transformations. Numer. Math.,
54(3):565–574, 1989.
[5] C. Brezinski. Convergence acceleration during the 20th century. J. Comput.
Appl. Math.,122(1):1–21, 1999.
[6] I. Kornfeld. Nonexistence of universally accelerating linear summability methods.
J. Comput. Appl. Math.,53(2):309–321, 1994.
[7] I. Tammeraid. Several remarks on acceleration of convergence using generalized
linear methods of summability. J. Comput. Appl. Math.,159(2):365–373, 2003.
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