A family of the Levin-type algorithms for accelerating convergence of sequence

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A family of the Levin-type algorithms for

accelerating convergence of sequence

R. Thukral

Padé Research Centre 39 Deanswood Hill, Leeds West Yorkshire, LS17 5JS, England

Received 16 Jan 04, Revised 28 Mar and 3 Apr 04, Accepted 11 Apr 04

Abstract

In this paper we demonstrate further development of the Levin-type algorithms. We consider the use of the Levin-type algorithms to accelerate the convergence of scalar sequence. The effectiveness of the Levin-type algorithms for approximating the partial sum of a given series is illustrated. In process we shall demonstrate the similarity between the super enhanced Levin algorithm, the super modified Levin algorithm and the Lubkin’s transformation. The approximate solutions of the super enhanced Levin algorithms are found to be substantially more accurate than the other Levin-type algorithms and the Lubkin’s transformation.

Key words and phrases: Super enhanced Levin algorithm, super improved Levin algorithm, improved Levin algorithm, super modified Levin algorithm, modified Levin algorithm, Lubkin transformation.

2000 Mathematics Subject Classification: 41A99.

1. Introduction

In this paper, we introduce another version of the Levin-type algorithms, namely the super

enhanced Levin algorithm and further development of the improved Levin algorithm and the

modified Levin algorithm. The Levin-type algorithms are essentially for accelerating the

convergence of scalar sequence. The super enhanced Levin algorithm, the super modified

Levin algorithm and the super improved Levin algorithm are of similar form as the Levin

transformation and the other Levin-type algorithms given in [1-13, 15]. The prime motive for

the development of these new algorithms was to improve the improved Levin algorithm and

the modified Levin algorithm, [11]. We have found in the previous study [11] that the modified

Levin algorithm and the improved Levin algorithm has low precision of the estimates when we

compared them with the Brezinski’s theta algorithm. Hence, the new Levin type algorithms are

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Lubkin’s transformation and the Brezinski’s theta algorithm are very similar; therefore we do

not consider the Brezinski’s theta algorithm in this paper. This paper is actually a continuation

of the previous study [11].

We examine the effectiveness of the Levin-type algorithms with the Lubkin

transformation by showing the accuracy of the approximate solution. In process we shall

demonstrate similarity between the super enhanced Levin algorithm, the super modified Levin

algorithm and the Lubkin’s transformation. Furthermore, we demonstrate the performance of

the super enhanced Levin algorithm, the super improved Levin algorithm, the super modified

Levin algorithm, the improved Levin algorithm, the modified Levin algorithm and the Lubkin

transformation. We describe the essentials of the Levin-type algorithm, and finally the

successor the super enhanced Levin algorithm. We have found that the super enhanced

algorithm is consistent, stable and much more accurate than the other similar algorithms

considered.

Proposition. Let us assume that

,

∑

∞

= =

0 i

i

c

s (1)

is slowly convergent or divergent sequence, whose elements are the partial sum of an

infinite series, given as

n s

∑

=

= n

i i

n c

s

0

. (2)

The basic assumption of all sequence transformation is that a sequence element can be for

all indices to be partitioned into a limit s and a truncation error according to n s

0

≥

n e_n

n

n s e

s = + . (3)

The conventional approach of evaluating an infinite series consists of adding up so

many terms that the error ultimately becomes zero. Unfortunately, this is not always

possible because of obvious practical limitations. Moreover, adding up further terms does not

work in the case of a divergent series. Therefore the development of the new algorithms plays

an important role. In this paper, we see how the new algorithms can be used to accelerate the

convergence of a series having the form (1). We shall review the recently introduced

Levin-type algorithm and then we shall observe modifications made to the improved Levin algorithm

(3)

For each of the following algorithms we use the initial estimate given by

1.1 The improved Levin algorithm (ILA)

The formula of the improved Levin algorithm is expressed as

( )

(

)

[

(

)

]

[

(

)

]

sequel, on the variable n, for example

(

n−1,k

)

= (n,k)− (n−1,k)

Δε ε ε . (6)

The most important differences between the improved Levin algorithm and the original Levin

transformation was to replace sn with ε(n,k−1). The purpose of this is to use the latest approximation ε(n,k) produced by the improved Levin algorithm, whereas the original Levin transformation uses . Hence we found that the accuracy of the improved Levin algorithm

has increased substantially. n s

1.2 The super improved Levin algorithm (SILA)

The formula of the super improved Levin algorithm is expressed as

( )

(

)

[

(

)

]

[

(

)

]

differences between the super improved Levin algorithm and the improved Levin algorithm;

firstly we use (k+1) terms of the summation in the numerator and denominator and the

second is that replace the binomial coefficient _⎟⎟ with . We calculate the new

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estimates of the super improved Levin algorithm in the same manner as with the improved

Levin algorithm.

1.3 The modified Levin algorithm (MLA)

We express the modified Levin algorithm as

( )

[

(

)

]

[

(

)

]

the improved Levin algorithm and the modified Levin algorithm is that we discard the factor

of in the modified Levin algorithm. We know that the modified Levin algorithm

has fewer calculations to perform in order to approximate the solution of the given power

series. Actually the modified Levin algorithm is the modified version of the improved Levin

algorithm. 2

) (n+i+k k−

1.4 The super modified Levin algorithm (SMLA)

The formula of the super modified Levin algorithm is expressed as

( )

[

(

)

]

[

(

)

]

between the super modified Levin algorithm and the modified Levin algorithm is that replace

the binomial coefficient with . We found that the precision of the estimates had

increased substantially.

⎟⎟

1.5 The super enhanced Levin algorithm (SELA)

The most efficient of the Levin-type algorithms, namely the super enhanced Levin algorithm,

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( )

(

)

[

(

)

]

[

(

)

]

super improved Levin algorithm. The essential difference between the super enhanced Levin

algorithm and the super improved Levin algorithm is that we replace with

in the super improved Levin algorithm to obtain the super enhanced Levin

algorithm. Simply by replacing this factor we have found that the accuracy, stability and

consistency of the super enhanced Levin algorithm is much better than the other Levin-type

algorithm.

Each estimate of the Levin-type algorithms can be represented in following table:

O

The structure of this paper is as follows. In section 2 we briefly review a well-known

algorithm, the Lubkin’s transformation. In section 3 we demonstrate the similarity between the

super enhanced Levin algorithm, the super modified Levin algorithm and the Lubkin’s

transformation. Moreover, in section 4 we examine the effectiveness of the Levin-type

algorithms for determining the approximate solution of a given power series. We make three

distinct comparisons of the estimates derived from the Levin-type algorithms. First, we

compare estimates formed using the row sequence the super enhanced Levin algorithm of type

(n, 1) with corresponding estimates derived from the modified Levin algorithm of type (n+1,1),

the super improved Levin algorithm of type (n, 1), the super modified Levin algorithm of type

(n, 1), the improved Levin algorithm of type (n+1, 1) and the Lubkin’s transformation of type

(n, 1). Then we compare estimates based on another row sequence of the super enhanced

Levin algorithm of type (n, 2) with corresponding estimates derived from the super modified

Levin algorithm of type (n, 2), the super improved Levin algorithm of type (n, 2), the improved

Levin algorithm of type (n+2, 2), the modified Levin algorithm of type (n+2, 2) and the

Lubkin’s transformation of type (n+1, 2). The third comparison of estimates is based on

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estimates derived from the super modified Levin algorithm of type (n, 3), the super improved

Levin algorithm of type (n, 3), the improved Levin algorithm of type (n+3, 3), the modified

Levin algorithm of type (n+3, 3) and the Lubkin’s transformation of type (n+3, 3). The

effectiveness of these new methods for accelerating the convergence of a scalar sequence was

investigated. The super enhanced Levin algorithm is proved to be the most effective of the

algorithms considered.

2. The Lubkin’s transformation (LT)

The Lubkin’s transformation is well-established [8, 11, 14, 15]; therefore we shall state the

essential expression used in order to evaluate the approximate solution of a given series, we

use the following formula

)

3. Equivalence of the estimates

Here we shall demonstrate the similarity between the super enhanced Levin algorithm, the

super modified Levin algorithm and the Lubkin’s transformation. We shall observe how these

three algorithms produce similar formula of the approximate solution of the given power

series. For a particular case we shall justify that the super enhanced Levin algorithm reduces to

the super modified Levin algorithm, hence both algorithms are identical.

First we begin by expanding (9), the formula of the super modified Levin algorithm of

type (n,1)

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( )

We simplify the equation (14) by multiplying the numerator and denominator by

)

algorithm

( ) ( ) (

₍

) (

_{) (}

)

(

₎

) ( ) (

_{( ) (}

₎

) (

_{( ) (}

) ( ) (

₎

)

Similarly, we know the formula of the the Lubkin’s transformation of type (n,1) is

given as

We replace the second order difference operator with first order,

( )

,0 ( 1,0) ( ,0)

Thus (16) becomes

[

]

Expanding the denominator of (19), we get

[

]

We find that the denominator of (20), the expression of the Lubkin’s transformation, is

identical to the denominator of the super modified Levin algorithm given by (15).

The next stage is to demonstrate the numerator of the Lubkin’s transformation is

identical to the numerator of the super modified Levin algorithm. We begin by expressing (20)

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(

) (

)

( ) (

)

( ) (

)

{

}

[

( 1,0) 1,0 2,0 2 ,0 2,0 ,0 1,0

) 1 ,

(n = ε n+ Δε n+ Δε n+ − Δε n Δε n+ +Δε n Δε n+

ε

{

Δ + −Δ +

}

]

÷

Δ + Δ

− ( 1,0) ( ,0) ( 2,0) ( 1,0)

ε n ε n ε n ε n

(

) (

)

( ) (

)

( ) (

)

[

1,0 2,0 2 ,0 2,0 ,0 1,0

]

Δε n+ Δε n+ − Δε n Δε n+ +Δε n Δε n+ . (21)

Expanding (21), we obtain

(

) (

)

( ) (

)

[

( 1,0) 1,0 2,0 2 ( 1,0) ,0 2,0

) 1 ,

(n = ε n+ Δε n+ Δε n+ − ε n+ Δε n Δε n+

ε

( ) (

Δ +

)

−Δ + Δ Δ + +Δ

{

Δ +

}

]

÷

Δ +

+ 2

) 0 , 1 ( ) 0 , ( ) 0 , 2 ( ) 0 , ( ) 0 , 1 ( 0 , 1 0

, ) 0 , 1

(n ε n ε n ε n ε n ε n ε n ε n

ε

(

) (

)

( ) (

)

( ) (

)

[

Δε n+1,0Δε n+2,0 −2Δε n,0Δε n+2,0 +Δε n,0Δε n+1,0

]

. (22)

We observe the expression (22) and find that we have one of the terms of the numerator of

the super modified Levin algorithm, which is

( ) (

,0 2,0

)

) 0 , 1 (

2 + Δ Δ +

− ε n ε n ε n . (23)

We shall now pair the remaining four terms in the numerator of (22). The first pair is

( ) (

,0 1,0

)

) 0 , 1

(n+ Δε n Δε n+

ε (24)

and

{

}

2

) 0 , 1 ( ) 0 ,

( Δ +

Δε n ε n . (25)

Replacing the component Δε(n+1,0) of (25) with ε(n+2,0)−ε(n+1,0) and then adding (24) and (25), we find that the term (24) cancels and obtain the following term

( ) (

,0 1,0

)

) 0 , 2

(n+ Δε n Δε n+

ε . (26)

The required term (26) is the last term of the numerator of the super modified Levin

algorithm, given in (15). We progress to the next pair,

(

1,0

) (

2,0

)

) 0 , 1

(n+ Δε n+ Δε n+

ε (27)

and

( ) (

,0 2,0

)

) 0 , 1

( + Δ Δ +

Δ

− ε n ε n ε n . (28)

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(

1,0

) (

2,0

)

) 0 ,

(n Δε n+ Δε n+

ε . (29)

This is the remaining identical term of the numerator of the super modified Levin algorithm.

Collecting (23), (26) and (29), the terms of the numerator of the Lubkin’s transformation, we

have obtained an identical expression to that of the super modified Levin algorithm. Hence

we have shown that the rational form of the super modified Levin algorithm is identical to the

Lubkin’s transformation.

In order to demonstrate that the super enhanced Levin algorithm of type (n,1) is

identical to the above two algorithms we shall verify that the formula of the super enhanced

Levin algorithm of type (n,1) is equivalent to the formula of the super modified Levin

algorithm of type (n,1).

We know by the fundamental law of indices, the following term simplifies to unity,

1 ) (_n+_i+_k k−1 =

, (30)

for n∈ℵ, i∈

[

0,k

]

and given k=1.

Therefore the formula of the super enhanced Levin algorithm of type (n,1) given by

(10 ) reduces to the formula of the super modified Levin algorithm of type (n,1). Hence the

three algorithms, namely the super enhanced Levin algorithm of type (n,1), the super modified

Levin algorithm of type (n,1) and the Lubkin’s transformation of type (n,1) are identical. The

similarity between these three algorithms is also evident in the Tables 1 and 4.

4. Application of the new Levin type algorithms

We shall demonstrate the further development of the Levin-type algorithms by two familiar

power series. We determine the consistency and stability of results by examining the

convergence of the new algorithms for three particular types of row sequence. The findings are

generalised by illustrating the effectiveness of the Levin-type algorithms for determining the

approximate solution of two power series. Consequently, we shall list the errors of the

estimates of the approximate solution produced by the algorithms described.

The following procedure is common to both of the illustrative numerical examples.

We illustrate the convergence of the algorithms described by making three distinct

comparisons of the estimates based on three particular types of row sequence. The first of the

comparisons is based on the estimates obtained by the row sequence of the super enhanced

Levin algorithm of type (n, 1) with corresponding estimates derived from the super improved

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Levin algorithm of type (n+1, 1), the modified Levin algorithm of type (n+1, 1) and the

Lubkin’s transformation of type (n, 1). The second comparison is based on another row

sequence of the super enhanced Levin algorithm of type (n,2) with corresponding estimates

derived from the super improved Levin algorithm of type (n, 2), the super modified Levin

algorithm of type (n, 2), the improved Levin algorithm of type (n+2, 2), the modified Levin

algorithm of type (n+2, 2) and the Lubkin’s transformation of type (n+1, 2). The next

comparison is formed using the row sequence of the super enhanced Levin algorithm of type

(n, 3) with corresponding estimates derived from the super improved Levin algorithm of type

(n, 3), the super modified Levin algorithm of type (n, 3), the improved Levin algorithm of type

(n+3, 3), the modified Levin algorithm of type (n+3, 3) and the Lubkin’s transformation of

type (n+3, 3). In each case, the comparisons with other algorithms were made using a similar

amount of data, which is, using a similar number of terms of (1). The numerical computations

listed in the tables were performed on an algebra system called Maple. Also the errors

displayed are absolute value.

4.1 Numerical example 1

We investigate the convergence of the Levin-type algorithms for the power series given as

( )

1 ₁

1

1 (2 1) 0.78539816.... 4

i

s i π

∞ ₊

− =

=

∑

− − = = , (31)

and the partial sum is given as

( )

1 ₁

1

1 (2 1) n

i n

i

s + i −

=

∑

− − . (32)

The following tables contain the numerical results of the six iterative algorithms for

three particular types of row sequence. In Tables 1 to 3 are the errors obtained by each of the

algorithms described. In Table 1 we observe the similarity between the super enhanced Levin

algorithm, the super modified Levin algorithm and the Lubkin’s transformation and find that

these algorithms produce better estimates than the super improved Levin algorithm, the

improved Levin algorithm and the modified Levin algorithm. In Table 2 and 3 we see that the

precision of the super enhanced Levin algorithm is substantially more accurate than the

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Table 1. Errors occurring in the estimates of the approximate solution by the six algorithms

described, when k=1.

n ILA(n+1,1) SILA(n,1) MLA(n+1,1) SELA=SMLA=LT(n,1)

1 2 3 4 5

0.810(-2) 0.525(-2) 0.365(-2) 0.268(-2) 0.204(-2)

0.228(-2) 0.130(-2) 0.748(-3) 0.462(-3) 0.303(-3)

0.206(-2) 0.911(-3) 0.478(-3) 0.280(-3) 0.178(-3)

0.127(-2) 0.364(-3) 0.140(-3) 0.643(-4) 0.335(-4)

n ILA(n+2,2) SILA(n,2) MLA(n+2,2) SMLA(n,2) SELA(n,2) LT(n+1,2)

1 2 3 4 5

0.471(-5) 0.252(-5) 0.141(-5) 0.824(-6) 0.506(-6)

0.256(-5) 0.118(-6) 0.279(-7) 0.245(-7) 0.144(-7)

0.129(-5) 0.508(-6) 0.226(-6) 0.110(-6) 0.578(-7)

0.194(-6) 0.544(-7) 0.174(-7) 0.627(-8) 0.251(-8)

0.801(-7) 0.149(-7) 0.343(-8) 0.940(-9) 0.296(-9)

0.473(-6) 0.153(-6) 0.571(-7) 0.239(-7) 0.110(-7)

1 2 3 4 5

0.157(-9) 0.426(-10) 0.138(-10) 0.511(-11) 0.208(-11)

0.102(-7) 0.151(-8) 0.470(-10) 0.303(-11) 0.417(-12)

0.197(-10) 0.795(-11) 0.338(-11) 0.152(-11) 0.715(-12)

0.176(-11) 0.178(-12) 0.170(-13) 0.629(-15) 0.153(-14)

0.583(-13) 0.885(-14) 0.150(-14) 0.283(-15) 0.585(-16)

0.230(-10) 0.897(-11) 0.374(-11) 0.166(-11) 0.774(-12)

4.2 Numerical example 2

In this subsection we take another power series. We shall demonstrate the convergence of the

enhanced algorithm for the series given as

,

.... 612375348 .

2 ) 5 . 1 (

0 5 .

1 = =

=

∑

∞ =

− _ς

i

s (33)

and the partial sum is given as

,

∑

= −

= n

i

n i

s

0 5 . 1

(34)

We repeat the procedure of the comparison in the previous example; that is, we

compare the partial sum produced by each of the algorithm described. As before, in the

following tables are the numerical results of the six iterative algorithms for three particular

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described. As expected, we observe in Table 4 the similarity between the super enhanced

Levin algorithm, the super modified Levin algorithm and the Lubkin’s transformation, and

find that these algorithms produce better estimates than the super improved Levin algorithm,

the improved Levin algorithm and the modified Levin algorithm. In Table 5 and 6 we see that

the precision of the super enhanced Levin algorithm is substantially more accurate than the

Lubkin’s transformation and other similar Levin-type algorithms.

n ILA(n+1,1) SILA(n,1) MLA(n+1,1) SELA=SMLA=LT(n,1)

1 2 3 4 5

0.133 0.094 0.071 0.056 0.046

0.622 0.424 0.312 0.243 0.196

0.710 0.627 0.568 0.522 0.486

0.216(-1) 0.105(-1) 0.598(-2) 0.379(-2) 0.257(-2)

n ILA(n+2,2) SILA(n,2) MLA(n+2,2) SMLA(n,2) SELA(n,2) LT(n+1,2)

1 2 3 4 5

0.649(-4) 0.404(-4) 0.264(-4) 0.181(-4) 0.128(-4)

0.729(-2) 0.387(-2) 0.228(-2) 0.145(-2) 0.967(-3)

0.619(-2) 0.415(-2) 0.295(-2) 0.218(-2) 0.167(-2)

0.396(-4) 0.179(-4) 0.908(-5) 0.501(-5) 0.296(-5)

0.233(-5) 0.782(-6) 0.316(-6) 0.145(-6) 0.735(-7)

0.132(-4) 0.632(-5) 0.334(-5) 0.191(-5) 0.116(-5)

1 2 3 4 5

0.559(-7) 0.217(-7) 0.952(-8) 0.460(-8) 0.240(-8)

0.123(-4) 0.273(-5) 0.769(-6) 0.249(-6) 0.876(-7)

0.175(-4) 0.101(-4) 0.622(-5) 0.403(-5) 0.271(-5)

0.257(-6) 0.610(-7) 0.199(-7) 0.795(-8) 0.364(-8)

0.923(-12) 0.150(-11) 0.762(-12) 0.362(-12) 0176.(-12)

0.364(-8) 0.176(-8) 0.922(-9) 0.513(-9) 0.301(-9)

5. Remarks and conclusion

In this paper we have demonstrated further development of the Levin-type algorithms and

their effectiveness has been investigated in many examples. These new Levin-type algorithms

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are actually based on the original Levin transformation [1-13, 15]. The main purpose of this

paper is to show the improvement of the recently introduced algorithms, namely the modified

Levin algorithm and the improved Levin algorithm. The prime motive of the development of

the Levin-type algorithm was to increase the precision of the improved Levin algorithm and

the modified Levin algorithm. We have shown in the two numerical examples that the super

enhanced Levin algorithm is substantially more accurate in approximating the solution of a

given power series than the super improved Levin algorithm, the improved Levin algorithm,

the super modified Levin algorithm, the modified Levin algorithm and the Lubkin’s

transformation.

The purpose of demonstrating these algorithms for three particular types of row

sequences was to illustrate the accuracy of the estimates, the stability of the convergence and

the consistency of the Levin-type algorithms. In process we have demonstrated the similarity

between the super enhanced Levin algorithm, the super modified Levin algorithm and the

Lubkin’s transformation for a particular type of row sequence. Also we have found that the

super improved Levin algorithm is inconsistent in approximating the solution of a given power

series. In some cases the super improved Levin algorithm produce very satisfactory estimates.

In all the test examples we have found that the super modified Levin algorithm is much more

reliable than the super improved Levin algorithm and the super enhanced improved Levin

algorithm is much more efficient than the other similar algorithms. The numerical results of

the super enhanced Levin algorithm may be consider a very good alternative to the Lubkin’s

transformation and Brezinski’s theta algorithm. Finally, an analytical investigation of the

Levin-type algorithms is a subject of further research.

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