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(âˆ’z)m

n (nâˆ’2<n=2=) (nâˆ’2<n=2=) (nâˆ’3<n=3=

z n+1 <n=2= z n+1 â€™2<n=2= z n+1 <n=3=

n (z) (z) )(z)

m=0 m+1

Eq. (2.19) Eq. (3.14) Eq. (4.18)

0:1000000000 Â· 101

0 0 0 0

âˆ’0:1500000000 Â· 101

1 0 0 0

0:6833333333 Â· 101 âˆ’0:6410256410 Â· 101 âˆ’0:6410256410 Â· 101

2 0

âˆ’0:2441666667 Â· 102 0:2467105263 Â· 102 0:2467105263 Â· 102 0:2480158730 Â· 102

3

0:1005833333 Â· 103 âˆ’0:1002174398 Â· 103 âˆ’0:1002155172 Â· 103 âˆ’0:1002604167 Â· 103

4

âˆ’0:4202500000 Â· 103 0:4205996885 Â· 103 0:4205974843 Â· 103 0:4206730769 Â· 103

5

0:1811892857 Â· 104 âˆ’0:1811533788 Â· 104 âˆ’0:1811532973 Â· 104 âˆ’0:1811533744 Â· 104

6

âˆ’0:7953732143 Â· 104 0:7954089807 Â· 104 0:7954089068 Â· 104 0:7954089765 Â· 104

7

0:3544904563 Â· 105 âˆ’0:3544868723 Â· 105 âˆ’0:3544868703 Â· 105 âˆ’0:3544868636 Â· 105

8

âˆ’0:1598634544 Â· 106 0:1598638127 Â· 106 0:1598638125 Â· 106 0:1598638127 Â· 106

9

0:7279206365 Â· 106 âˆ’0:7279202782 Â· 106 âˆ’0:7279202781 Â· 106 âˆ’0:7279202782 Â· 106

10

The results in Table 2 show that a sequence transformation accomplishes a summation of a

divergent series by constructing approximations to the actual remainders. Both the partial sums

as well as the actual remainders diverge individually if their indices become large, but the linear

combination of the partial sum and the remainder has a constant and Ã¿nite value for every index.

The fact, that the transformation terms in (2.19), (3.14), and (4.18) approach the negative of the

corresponding partial sums of course also implies that one should not try to sum a divergent series

in this way. The subtraction of two nearly equal terms would inevitably lead to a serious loss of

signiÃ¿cant digits.

In the next example, the transformation terms in (2.19), (3.14), and (4.18) will be used to make

predictions for unknown series coe cients. For that purpose, it is recommendable to use a computer

algebra system like Maple, and do all calculations symbolically. If the coe cients of the series to be

transformed are exact rational numbers, the resulting rational expressions are then computed exactly.

We use the symbolic expressions for the partial sums n (âˆ’z)m =(m + 1) with 06n612 of

m=0

the inÃ¿nite series (5.2) for ln(1 + z)=z as input data in the recursive schemes (2:20); (3:15), and

(4:19). The resulting rational expressions z 13 6 (z), z 13 â€™(0) (z), and z 13 4 with unspeciÃ¿ed z are

(0) (4)

12

then expanded, yielding predictions for the next series coe cients that are exact rational numbers.

Only in the Ã¿nal step, the predictions for the next series coe cients are converted to oating point

numbers in order to improve readability:

12

(âˆ’z)m

A(0) âˆ’ 0:07142857137 z 13 + 0:06666666629 z 14

=

6

m+1

m=0

âˆ’ 0:06249999856 z 15 + 0:05882352524 z 16 + O(z 17 ); (5.7a)

12

(âˆ’z)m

(0)

âˆ’ 0:07142854717 z 13 + 0:06666649774 z 14

=

12

m+1

m=0

âˆ’ 0:06249934843 z 15 + 0:05882168762 z 16 + O(z 17 ); (5.7b)

350 E.J. Weniger / Journal of Computational and Applied Mathematics 122 (2000) 329â€“356

12

(âˆ’z)m

(0)

âˆ’ 0:07142857148 z 13 + 0:06666666684 z 14

J4 =

m+1

m=0

âˆ’ 0:06249999986 z 15 + 0:05882352708 z 16 + O(z 17 ); (5.7c)

12

(âˆ’z)m

ln(1 + z)

âˆ’ 0:07142857143 z 13 + 0:06666666667 z 14

=

z m+1

m=0

âˆ’ 0:06250000000 z 15 + 0:05882352941 z 16 + O(z 17 ): (5.7d)

The accuracy of the prediction results in (5.7) is quite remarkable. The coe cients m =(âˆ’1)m =(m+1)

with 06m612 are the only information that was used for the construction of the transformation

terms z 13 6 (z), z 13 â€™(0) (z), and z 13 4 , which were expanded to yield the results in (5.7). The

(0) (0)

12

accuracy of the approximations to the next four coe cients should su ce for many practical appli-

cations.

As in all other application, Wynnâ€™s epsilon algorithm is in (5.7) slightly but signiÃ¿cantly less

e ective than Aitkenâ€™s iterated 2 process and Brezinskiâ€™s iterated theta algorithm.

Instead of computing the transformation terms z 13 6 (z), z 13 â€™(0) (z), and z 13 4 , it is of course

(0) (0)

12

also possible to compute A(0) , 12 , and J4 directly via their deÃ¿ning recursive schemes, and to

(0) (0)

6

expand the resulting rational expressions with a symbolic system like Maple. This would lead to

the same results. However, in order to extract the partial sum 12 (âˆ’z)m =(m + 1) from the rational

m=0

(0) (0) (0)

approximants A6 , 12 , and J4 , one would have to compute their 12th-order derivatives, and only

the next derivatives would produce predictions to unknown series coe cients. Thus, this approach

can easily become very expensive. In contrast, the use of the transformation terms requires only

low-order derivatives of rational expressions.

If only the prediction of a single unknown term is to be done, then it is of course much more

e cient to use the recursive schemes (2:23); (3:17), and (4:21). The input data of these recursive

schemes are the coe cients of the series to be transformed, and no di erentiations have to be done.

6. Summary and conclusions

As already mentioned in Section 1, it has become customary in certain branches of theoreti-

cal physics to use PadÃƒ approximants to make predictions for the leading unknown coe cients of

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strongly divergent perturbation expansions. This can be done by constructing symbolic expressions

for PadÃƒ approximants from the known coe cients of the perturbation series. A Taylor expansion

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of su ciently high order of such a PadÃƒ approximants then produces the predictions for the series

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coe cients which were not used for the construction of the PadÃƒ approximant. The Taylor expansion

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of the symbolic expression can be done comparatively easily with the help of powerful computer

algebra systems like Maple or Mathematica, which are now commercially available for a wide range

of computers.

It is the purpose of this article to overcome two principal shortcomings of the approach sketched

above: Firstly, it is not necessary to rely entirely on the symbolic capabilities of computers. Instead,

it is possible to construct recursive schemes, which either facilitate considerably the symbolic tasks

computers have to perform, or which permit a straightforward computation of the prediction for

E.J. Weniger / Journal of Computational and Applied Mathematics 122 (2000) 329â€“356 351

the leading unknown coe cient. Secondly, it is possible to use instead of PadÃƒ approximants other

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sequence transformations, as proposed by Sidi and Levin [85] and Brezinski [18]. It was shown in

[105] that this may lead to more accurate predictions.

In this article, the prediction properties of Aitkenâ€™s iterated 2 process, Wynnâ€™s epsilon algorithm,

and Brezinskiâ€™s iterated theta algorithm are studied.

As is well known [4,8], a PadÃƒ approximant can be considered to be the solution of a system

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of linear equations for the coe cients of its numerator and denominator polynomials. If this system

of linear equations has a solution, then it is automatically guaranteed that the PadÃƒ approximant

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satisÃ¿es the accuracy-through-order relationship (1.6). In the case of other sequence transformations,

the situation is usually much more di cult. They are usually not deÃ¿ned as solutions of systems of

linear equations, but via (complicated) nonlinear recursive schemes.

Since accuracy-through-order relationships of the type of (1.6) play a very important role for the

understanding of the prediction properties of sequence transformations, it was necessary to derive

accuracy-through-order relationships for Aitkenâ€™s iterated 2 process, Wynnâ€™s epsilon algorithm, and

Brezinskiâ€™s iterated theta algorithm on the basis of their deÃ¿ning recursive schemes.

Unfortunately, the deÃ¿ning recursive schemes (2:4), (3:1), and (4:3) are not suited for a construc-

tion of accuracy-through-order relationships. They Ã¿rst had to be modiÃ¿ed appropriately, yielding

the mathematically equivalent recursive schemes (2:11); (3:8), and (4:11).

These alternative recursive schemes were the starting point for the derivation of the accuracy-

through-order relationships (2.13), (3.9), and (4.13) and the corresponding recursive schemes (2:14),

(3.10), and (4:14) for the transformation error terms. These relationships describe how the rational

(n) (n) (n)

approximants Ak , 2k , and Jk di er from the function f(z) which is to be approximated.

With the help of these accuracy-through-order relationships, a second group of results could be

derived â€” (2.19), (3.14), and (4.18) and the corresponding recursive schemes (2:20); (3:15), and

(n) (n) (n)

(4:19) â€” which describe how the rational approximants Ak , 2k , and Jk di er from the par-

tial sums which were used for their construction. These di erences are expressed by the terms

(n) (n) (n)

z n+2k+1 k (z), z n+2k+1 â€™2k (z), and z n+3k+1 k (z) which can be computed via the recursive schemes

(2:20); (3:15), and (4:19).

The predictions for the leading unknown series coe cients can be obtained by expanding symbolic

expressions for these transformation terms. The advantage of this approach is that the partial sums,

(n) (n) (n)

which are used for the construction of the rational approximants Ak , 2k , and Jk as well as of the

(n) (n) (n)

transformation terms z n+2k+1 k (z), z n+2k+1 â€™2k (z), and z n+3k+1 k (z), are already explicitly separated.

Consequently, only derivatives of low order have to be computed. Moreover, the predictions for

the leading unknown series coe cient can be computed conveniently via the recursive schemes

(2:23); (3:17), and (4:21). In this way, it is neither necessary to construct symbolic expressions nor

to di erentiate them.

Finally, in Section 5 some applications of the new results were presented. In all applications of

this article, Wynnâ€™s epsilon algorithm was found to be less e ective than Aitkenâ€™s iterated 2 process

or Brezinskiâ€™s iterated theta algorithm. Of course, it remains to be seen whether this observation is

speciÃ¿c for the inÃ¿nite series (5.2) for ln(1 + z)=z, which was used as the test system, or whether

it is actually more generally valid. Nevertheless, the results presented in Section 5 provide further

evidence that suitably chosen sequence transformations may indeed be more e ective than PadÃƒ e

approximants. Consequently, one should not assume that PadÃƒ approximants produce by default the

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best results in convergence acceleration and summation processes, and it may well be worth while to

352 E.J. Weniger / Journal of Computational and Applied Mathematics 122 (2000) 329â€“356

investigate whether sequence transformations can be found which are better adapted to the problem

under consideration.

Acknowledgements

My interest in PadÃƒ approximants, sequence transformation, convergence acceleration, and the

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summation of divergent series â€” which ultimately led to this article â€” was aroused during a stay

as a Postdoctoral Fellow at the Faculty of Mathematics of the University of Waterloo, Ontario,

Ã„Ã„

Canada. Special thanks to Prof. J. CÃƒzek for his invitation to work with him, for numerous later

invitations to Waterloo, for his friendship, and the inspiring atmosphere which he has been able to

provide. Many thanks also to PD Dr. H. Homeier for stimulating and fruitful discussions. Financial

support by the Fonds der Chemischen Industrie is gratefully acknowledged.

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ee e

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ee e

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