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m=0

(’z)m
n (n’2<n=2=) (n’2<n=2=) (n’3<n=3=
z n+1 <n=2= z n+1 ™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 · 101
0 0 0 0
’0:1500000000 · 101
1 0 0 0
0:6833333333 · 101 ’0:6410256410 · 101 ’0:6410256410 · 101
2 0
’0:2441666667 · 102 0:2467105263 · 102 0:2467105263 · 102 0:2480158730 · 102
3
0:1005833333 · 103 ’0:1002174398 · 103 ’0:1002155172 · 103 ’0:1002604167 · 103
4
’0:4202500000 · 103 0:4205996885 · 103 0:4205974843 · 103 0:4206730769 · 103
5
0:1811892857 · 104 ’0:1811533788 · 104 ’0:1811532973 · 104 ’0:1811533744 · 104
6
’0:7953732143 · 104 0:7954089807 · 104 0:7954089068 · 104 0:7954089765 · 104
7
0:3544904563 · 105 ’0:3544868723 · 105 ’0:3544868703 · 105 ’0:3544868636 · 105
8
’0:1598634544 · 106 0:1598638127 · 106 0:1598638125 · 106 0:1598638127 · 106
9
0:7279206365 · 106 ’0:7279202782 · 106 ’0:7279202781 · 106 ’0:7279202782 · 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 ÿnite 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
signiÿcant 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 (’z)m =(m + 1) with 06n612 of
m=0
the inÿnite 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 ™(0) (z), and z 13 4 with unspeciÿed z are
(0) (4)
12
then expanded, yielding predictions for the next series coe cients that are exact rational numbers.
Only in the ÿnal step, the predictions for the next series coe cients are converted to oating point
numbers in order to improve readability:
12
(’z)m
A(0) ’ 0:07142857137 z 13 + 0:06666666629 z 14
=
6
m+1
m=0

’ 0:06249999856 z 15 + 0:05882352524 z 16 + O(z 17 ); (5.7a)
12
(’z)m
(0)
’ 0:07142854717 z 13 + 0:06666649774 z 14
=
12
m+1
m=0

’ 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) 329“356

12
(’z)m
(0)
’ 0:07142857148 z 13 + 0:06666666684 z 14
J4 =
m+1
m=0

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

12
(’z)m
ln(1 + z)
’ 0:07142857143 z 13 + 0:06666666667 z 14
=
z m+1
m=0

’ 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 =(’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 ™(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, Wynn™s epsilon algorithm is in (5.7) slightly but signiÿcantly less
e ective than Aitken™s iterated 2 process and Brezinski™s iterated theta algorithm.
Instead of computing the transformation terms z 13 6 (z), z 13 ™(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 deÿning 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 (’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 Padà approximants to make predictions for the leading unknown coe cients of
e
strongly divergent perturbation expansions. This can be done by constructing symbolic expressions
for Padà approximants from the known coe cients of the perturbation series. A Taylor expansion
e
of su ciently high order of such a Padà approximants then produces the predictions for the series
e
coe cients which were not used for the construction of the Padà approximant. The Taylor expansion
e
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) 329“356 351


the leading unknown coe cient. Secondly, it is possible to use instead of Padà approximants other
e
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 Aitken™s iterated 2 process, Wynn™s epsilon algorithm,
and Brezinski™s iterated theta algorithm are studied.
As is well known [4,8], a Padà approximant can be considered to be the solution of a system
e
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 Padà approximant
e
satisÿes 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 deÿned 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 Aitken™s iterated 2 process, Wynn™s epsilon algorithm, and
Brezinski™s iterated theta algorithm on the basis of their deÿning recursive schemes.
Unfortunately, the deÿning recursive schemes (2:4), (3:1), and (4:3) are not suited for a construc-
tion of accuracy-through-order relationships. They ÿrst had to be modiÿed 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 ” (2.19), (3.14), and (4.18) and the corresponding recursive schemes (2:20); (3:15), and
(n) (n) (n)
(4:19) ” 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 ™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 ™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, Wynn™s epsilon algorithm was found to be less e ective than Aitken™s iterated 2 process
or Brezinski™s iterated theta algorithm. Of course, it remains to be seen whether this observation is
speciÿc for the inÿnite 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 Padà e
approximants. Consequently, one should not assume that Padà approximants produce by default the
e
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) 329“356


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


Acknowledgements

My interest in Padà approximants, sequence transformation, convergence acceleration, and the
e
summation of divergent series ” which ultimately led to this article ” was aroused during a stay
as a Postdoctoral Fellow at the Faculty of Mathematics of the University of Waterloo, Ontario,
„„
Canada. Special thanks to Prof. J. CÃzek 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.


References

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ee e
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