NEW METHOD FOR GENERATING
CONVERGENCE ACCELERATION
ALGORITHMS
EDOUN M.2; OUAMBO R.1; KUITCHE A.3 Department of Automatic, Energetic and Electrical EngineeringENSAI, P.O BOX 455 University of Ngaoundere – Cameroon
[email protected] ; 2- [email protected], 3- [email protected]
ABSTRACT: In this work, we show a new method of generating convergence acceleration algorithms, by performing an adequate approximation technique. The Aitken’s
2 algorithm can be derived by this way, as well as the more general E-algorithm. Moreover, a family of algorithms for logarithmic sequences have been derived and tested with success on a set of logarithmic sequences.Keywords: Method; Generation; Convergence, Algorithms; Acceleration
INTRODUCTION
Let
x
n be a sequence of real numbers converging top
. This convergence will be said to be linear if there exist a number
0
,
1
verifying
x
p
p
x
n n n
1
lim
(1).If
1
then
x
n is said to be logarithmic. This type of sequence is generally difficult to accelerate. In this paper, we shall present a method of generating acceleration algorithms, which will allow us to build very strong algorithms for logarithmic sequences.Firstly we will explain the idea sustaining this method, as well as some theoretical results. After it, we will see two practical implementations of the method, and some examples of algorithms will be derived. This will directly enable the development of a family of algorithms for logarithmic sequences. The last part will be the test of these algorithms on some logarithmic sequences, followed by a concluding comment.
1- THE IDEA OF THE METHOD
Let consider the serie
1
1
0
(
)
k
k k
x
x
x
A
(2). For a givenn
, we can writen n
k
k
k
x
x
x
x
1
1
0
(
)
(3). Thus, as the sequence(
x
n)
converges top
, then the serieA
converges to the same value. It is easy to see that this serie can also be written
1
1
)
(
k
k n k n
n
x
x
x
A
(4). Considering this form, the idea of this method is to approximate theserie by another converging serie
1
0
(
)
(
)
ˆ
k k
A
(5) where:
0,
1,
2,
...
,
m
is a set ofm
1
parameters, withm
1
; The sequence
k(
)
converges to zero.The parameters are chosen to give the best approximation. For a given
n
, there can be many approximation criteria. We will choose two criteria, defined by the following equations:
x
k
m
x
x
k n k n k
n
...
1
,
)
(
)
(
1 0
m
k
x
x
x
k n k n k
n
n n
k k
...
1
,
)
(
)
(
)
(
1 1 0
(7)
Knowing the parameters, the limit of the serie
A
ˆ
is also known. We will call itx
ˆ
n. By this way we can then construct the sequence
x
ˆ
n , which is the accelerated sequence. Now, let us establish the following theoretical results.Theorem 1 :
If the sequence
x
n is monotone and converges top
, and if the sequence
k(
)
is also monotone, then the sequence
x
ˆ
n also converges top
.Proof
We will make this proof only for the approximation criterion given by the equation (6). For the equation (7), the proof can be done by a similar way.
Let us consider the family of sequences
u
k(n) defined by
(
)
,
1
)
(
) (1 ) (
0 ) ( 0
k
u
u
u
k n k n k
n
(8)
It is clear that:
(P1): Due to the equation (6),
u
x
n kk
m
nk
,
0
...
)
(
;
(P2): For a given
n
, as
k(
)
converges to zero, then
) (n k
u
is also convergent; (P3): For a given
n
, as
k(
)
is monotone, then
) (n k
u
is also monotone.The approximating serie can then be written
1
) (
1 ) ( )
(
0
(
)
ˆ
k
n k n k n
u
u
u
A
(9).Due to the fact that
ik
n k n k n
n
i
u
u
u
u
1
) (
1 ) ( )
( 0 ) (
)
(
and thatA
ˆ
converges tox
ˆ
n, then we have (P4):n n k
k
u
x
ˆ
lim
( )
.
(P1) implies that
u
p
k
m
n k
n
,
0
...
lim
( )
(10).
As
u
k(n) is monotone for a givenn
, then) (
1 ) ( ) ( ) (
1
n m n m n m n
m
u
u
u
u
.Due to (10), it is easy to see that we will then have
u
mnp
n
) (
1
lim
.We can then do the same for
m
2
,
m
3
,
...
and finally writeu
knp
k
n
,
lim
( ) (11).Then as
n
, the family of sequences
u
k(n) tends to a constant sequence. Thus considering (P4), it is easyto conclude that
x
ˆ
n converges top
.Now we know that the accelerated sequence will converge to the same value as the original one. But how fast will it converge? The answer is given by the following proposition.
Proposition 1
The convergence speed of
x
ˆ
n can be as higher as we want, depending on the quality of the approximation.Proof
As with the preceding theorem, this proof will consider only the approximating equation (6), as the other can be done by a similar way. Lets defined the quality of the approximation at each step by the following
sequence : kn n k k
n
Max
u
x
d
( )..
It is clear that
x
ˆ
n
p
d
n andlim
0
n
n
d
. Thus,
x
nˆ
will converge at least as fast as
d
n .2- GENERATION OF ACCELERATING ALGORITHMS
We will see here two ways of generating algorithms, in other words two ways of finding approximating series.
2-1 Using Taylor developments
Let consider a parameterized function
f
(
t
)
verifying:
0,
1,
2,
...
,
m
is a set ofm
1
parameters, withm
1
;
f
(
t
)
has a Taylor development int
0
, written k kk
t
t
f
0
)
(
)
(
and that also
converges for
t
1
; The sequence
k(
)
is monotone and converges to zero.Then
f
(
1
)
can be a valid approximating serie.Let take for example the function
t
t
t
t
t
t
f
2 0 2 1 2 0 11
1
1
)
(
.Knowing that
01
1
k k
for
1
, then we can write
1 2 1 1 2 0 1 0 1 2 0 0 21
(
)
)
(
k k k k k k k k k kt
t
t
t
f
, assuming that the hypothesis of convergence and monotony are fulfilled.Thus, the approximation equations given by (6) yield:
1 2 2 1 0 2 2 2 1 2 0 1 2 1 0 1)
(
n nn n n
x
x
x
x
x
implying
n n n n n n n n n nx
x
x
x
x
x
x
x
x
x
1 2 2 1 1 0 1 1 2 2 1
This shows that if
(
x
n)
is strictly monotone, then0
2
1
ensuring thus the convergence and monotony hypothesis on the approximating serie.This approximating serie converges to
2 1 0
1
ˆ
)
1
(
x
nf
. Finally, the recurrence formula isn n n n n n n
x
x
x
x
x
x
x
1 2 2 1 22
ˆ
(13), which is the formula of the Aitken’s
2 method. We would have obtained thesame result with the approximating equation (7).
This result is not a surprise, because we have proved that taking
f
(
t
)
as a rational function and using Padé approximants, this method lead to the well known E-algorithm whose the Aitken’s
2 is a particular case.Let now have a look on the second way of generating algorithms. 2-2 Using a linear combination of known series
Let consider the following family of series
S
S
q
m
k q k q...
1
,
1 ) ( ) (
, where for a given
q
theLet take the approximating serie
1 ) ( ) 1 ( 1 0 ) 1 ( 10
...
(
...
)
ˆ
k m k m k mm
S
S
S
S
A
Using the equations given by (7), we can write :
) ( ) 1 ( 1 0 1 ) ( ) 1 ( 1 1 ) ( ) 1 ( 1 0...
ˆ
...
1
,
...
)
...
(
m m n k n k n m k n m k n n n k m k m kS
S
x
m
k
x
x
S
S
x
S
S
(14)By using the following notation
n k q k qS
n
S
1 ) ( ) ()
(
, we have:
)]
(
[
...
)]
(
[
ˆ
...
1
,
...
) ( ) ( ) 1 ( ) 1 ( 1 1 ) ( ) 1 ( 1n
S
S
n
S
S
x
x
m
k
x
x
S
S
m m m n n k n k n m k n m k n
(15)As we can see, at each step of the algorithm, a linear system of order
m
has to be solved to get the parameters)
(
i .Now we will see how we can chose
S
k(q) to accelerate logarithmic sequences. Let establish first the following theorem.Theorem 2:
If
(
x
n)
is a logarithmic sequence, then(
x
n
x
n1)
is also a logarithmic sequence.Proof
As
(
x
n
x
n1)
converges to zero, it is logarithmic iflim
1
1 1
n n n n nx
x
x
x
(16).
Let
(
x
n)
be a linear sequence, that islim
1
,
1
x
p
p
x
n n n If we prove that
lim
(
)
1 1
f
x
x
x
x
n n n nn
(17),
then (16) will be true if
lim
(
)
1
f
(18).Let establish (17) first.
1
(
)
1
1
1
1
lim
lim
1 1 1 1f
p
x
p
x
p
x
p
x
x
x
x
x
n n n n n n n n n nIt is thus clear that
lim
(
)
1
f
, hence the result.Let take for example
!
1
)
)...(
1
(
1
1 ) ( )
( ) (
q
q
S
S
q
k
k
k
S
k q k q
q k
(19). For a given
q
,
S
k(q) is monotone and converges tozero.
In addition,
1
1
lim
lim
( )) (
1
k
q
k
S
S
k q kq k
k thus
) (q k
S
is logarithmic and can be a good choice forlogarithmic sequences.
3- EXAMPLES FOR LOGARITHMIC SEQUENCES
Using the family of sequences given by (19), we have written a Matlab function (see the appendixes 1 and 2) which accelerates logarithmic sequences. The following sample sequences are the same used in [2] to test their method. This will allow us to have a first appreciation of the power of our algorithm.
Example 1:
0
2
)
1
(
1
k n
k
x
,p
1
.
6449340668
n
x
nn
x
ˆ
m=1
n
x
ˆ
m=2
n
x
ˆ
m=3
n
x
ˆ
m=4
n
x
ˆ
m=5
n
x
ˆ
m=6
n
x
ˆ
Example 2:
2
1
1
0n n
n
x
x
x
x
,
p
0
To have an idea of the strength of this algorithm, we will say that: if we consider only those two examples, then on average with the method used in [2], to get the same precision as our with
m
6
, you need to compute nearly 10 times more terms of the original sequence(
x
n)
.CONCLUSION
In this paper, we have established a new method of generating acceleration algorithms. Theoretically an infinite number of algorithms, comprising the E-algorithm can be derived by this way. The one generates here for logarithmic sequences was just an example, although its test was unbelievable successful.
We have no doubt that there is still a lot of work to be done in order to better master this method, and hence it will surely open new perspectives in the acceleration of the convergence.
REFERENCES
[1] DION J-G., GAUDET R.; Méthodes d’analyse numérique, De la théorie à l’application, Modulo Editeur ; 1996.
[2] BREZINSKI C.: Convergence acceleration by extraction of linear subsequences; SIAM journal Vol. 20 No. 6; December 1983. [3] BREZINSKI C.; Algorithmes d’accélération de la convergence, Etude numérique ; Editions Technip ; 1978.
n
x
nn
x
ˆ
m=1
n
x
ˆ
m=2
n
x
ˆ
m=3
n
x
ˆ
m=4
n
x
ˆ
m=5
n
x
ˆ
m=6
n
