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)
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
âˆ’0:2441666667 Â· 102 0:2467105263 Â· 102 0:2467105263 Â· 102 0:2480158730 Â· 102
0:1005833333 Â· 103 âˆ’0:1002174398 Â· 103 âˆ’0:1002155172 Â· 103 âˆ’0:1002604167 Â· 103
âˆ’0:4202500000 Â· 103 0:4205996885 Â· 103 0:4205974843 Â· 103 0:4206730769 Â· 103
0:1811892857 Â· 104 âˆ’0:1811533788 Â· 104 âˆ’0:1811532973 Â· 104 âˆ’0:1811533744 Â· 104
âˆ’0:7953732143 Â· 104 0:7954089807 Â· 104 0:7954089068 Â· 104 0:7954089765 Â· 104
0:3544904563 Â· 105 âˆ’0:3544868723 Â· 105 âˆ’0:3544868703 Â· 105 âˆ’0:3544868636 Â· 105
âˆ’0:1598634544 Â· 106 0:1598638127 Â· 106 0:1598638125 Â· 106 0:1598638127 Â· 106
0:7279206365 Â· 106 âˆ’0:7279202782 Â· 106 âˆ’0:7279202781 Â· 106 âˆ’0:7279202782 Â· 106
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
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
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
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:
A(0) âˆ’ 0:07142857137 z 13 + 0:06666666629 z 14
âˆ’ 0:06249999856 z 15 + 0:05882352524 z 16 + O(z 17 ); (5.7a)
âˆ’ 0:07142854717 z 13 + 0:06666649774 z 14
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
âˆ’ 0:07142857148 z 13 + 0:06666666684 z 14
âˆ’ 0:06249999986 z 15 + 0:05882352708 z 16 + O(z 17 ); (5.7c)
ln(1 + z)
âˆ’ 0:07142857143 z 13 + 0:06666666667 z 14
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
accuracy of the approximations to the next four coe cients should su ce for many practical appli-
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
also possible to compute A(0) , 12 , and J4 directly via their deÃ¿ning recursive schemes, and to
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
(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
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
of su ciently high order of such a PadÃƒ approximants then produces the predictions for the series
coe cients which were not used for the construction of the PadÃƒ approximant. The Taylor expansion
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
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
sequence transformations, as proposed by Sidi and Levin  and Brezinski . It was shown in
 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
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
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
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
My interest in PadÃƒ approximants, sequence transformation, convergence acceleration, and the
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,
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.
 A.C. Aitken, On Bernoulliâ€™s numerical solution of algebraic equations, Proc. Roy. Soc. Edinburgh 46 (1926)
 M. Arai, K. Okamoto, Y. Kametaka, Aitken acceleration and Fibonacci numbers, Japan J. Appl. Math. 5 (1988)
 G.A. Baker Jr., The theory and application of the PadÃƒ approximant method, Adv. Theoret. Phys. 1 (1965) 1â€“58.
 G.A. Baker Jr., Essentials of PadÃƒ Approximants, Academic Press, New York, 1975.
 G.A. Baker Jr., Quantitative Theory of Critical Phenomena, Academic Press, San Diego, 1990, pp. 211â€“346.
 G.A. Baker Jr., The PadÃƒ approximant and related material, in: D. Bessis (Ed.), CargÃ‚ se Lectures in Physics,
Vol. 5, Gordon and Breach, New York, 1972, pp. 349 â€“383.
 G.A. Baker Jr., J.L. Gammel (Eds.), The PadÃƒ Approximant in Theoretical Physics, Academic Press, New York,
 G.A. Baker Jr., P. Graves-Morris, PadÃƒ Approximants, 2nd Edition, Cambridge University Press, Cambridge, 1996.
 J.L. Basdevant, The PadÃƒ approximation and its physical applications, Fortschr. Phys. 20 (1972) 283â€“331.
 G.E. Bell, G.M. Phillips, Aitken acceleration of some alternating series, BIT 24 (1984) 70â€“77.
 S. Bhowmick, R. Bhattacharya, D. Roy, Iterations of convergence accelerating nonlinear transforms, Comput. Phys.
Comm. 54 (1989) 31â€“36.
 P. BjHrstad, G. Dahlquist, E. Grosse, Extrapolations of asymptotic expansions by a modiÃ¿ed Aitken 2 -formula,
BIT 21 (1981) 56â€“65.
 C. Brezinski, AccÃƒ lÃƒ ration de suites a convergence logarithmique, C. R. Acad. Sci. Paris 273 (1971) 727â€“730.
 C. Brezinski, A bibliography on PadÃƒ approximation and related matters, in: H. Cabannes (Ed.), PadÃƒ Approximation
Method and Its Application to Mechanics, Springer, Berlin, 1976, pp. 245â€“267.
 C. Brezinski, AccÃƒ lÃƒ ration de la Convergence en Analyse NumÃƒ rique, Springer, Berlin, 1977.
 C. Brezinski, Algorithmes dâ€™AccÃƒ lÃƒ ration de la Convergence â€“ Etude NumÃƒ rique, Editions Technip, Paris, 1978.
 C. Brezinski, PadÃƒ -type Approximation and General Orthogonal Polynomials, Birkhauser, Basel, 1980.
 C. Brezinski, Prediction properties of some extrapolation methods, Appl. Numer. Math. 1 (1985) 457â€“462.
 C. Brezinski, History of Continued Fractions and PadÃƒ Approximants, Springer, Berlin, 1991.
 C. Brezinski, A Bibliography on Continued Fractions, PadÃƒ Approximation, Extrapolation and Related Subjects,
Prensas Universitarias de Zaragoza, Zaragoza, 1991.
 C. Brezinski (Ed.), Continued Fractions and PadÃƒ Approximants, North-Holland, Amsterdam, 1991.
 C. Brezinski, Extrapolation algorithms and PadÃƒ approximations: a historical survey, Appl. Numer. Math. 20 (1996)
 C. Brezinski, Projection Methods for Systems of Equations, Elsevier, Amsterdam, 1997.
E.J. Weniger / Journal of Computational and Applied Mathematics 122 (2000) 329â€“356 353