Many numerical techniques produce results which are actually sequences. Examples are iterative methods, discretization methods, perturbation methods, and series expansions.
Numerical techniques of that kind are practically useful only if the resulting sequences converge sufficiently fast. As is well known, this is not always the case: It frequently happens that the resulting sequence either converges too slowly to be practically useful, or not at all.
Sequence transformations are tools to overcome convergence
problems of that kind. In this approach, a slowly convergent or
divergent sequence 
-- whose elements may for instance
be the partial sums
of an infinite series -- is converted into a new sequence 
with hopefully better numerical properties.
Problems with slow convergence or divergence were of course already
encountered in the early days of calculus. Accordingly, numerical
techniques for the acceleration of convergence or the summation of
divergent series are almost as old as calculus itself. According to
Knopp (see p. 249 of [1]), the first series transformation was
published by Stirling [2] already in 1730, and in 1755 Euler
[3] published the series transformation which now bears his
name. In rudimentary form, convergence acceleration methods are even
older. On pp. 90 - 91 of Brezinski's book [4] it is mentioned
that convergence acceleration methods were already used in 1654 by
Huygens and in 1674 by Seki Kowa, the probably most famous Japanese
mathematician. Both tried to obtain better approximations to 
.
Huygens used a linear extrapolation scheme which is a special case of
what we now call Richardson extrapolation [5], and Seki Kowa
used the so-called 
process, which is usually attributed to
Aitken [6].
Those who would like to know more about the history of sequence
transformations should consult a preprint by Brezinski [7],
where the historical development of extrapolation methods and Padé
approximants starting from the 17th century until today is
described
.
Before the invention of electronic computers, mainly linear
sequence transformations were used, which compute the elements of the
transformed sequence 
as weighted
averages of the input sequence 
according to
The theoretical properties of linear sequence transformations are now
very well understood [1,8,9,10,11,12]. The main
appeal of these matrix transformations lies in the fact that for the
weights 
in eq. (1.2) some necessary and
sufficient conditions could be formulated which guarantee that the
application of such a matrix transformation to a convergent
sequence 
yields a transformed sequence 
converging to the same limit.
From a theoretical point of view, this regularity is an extremely desirable property. However, from a practical point of view, regular matrix transformations are not particularly useful since the requirement that they should work in all cases prevents them from producing spectacular results in special cases. Hence, regular linear transformations are in general at most moderately powerful. Accordingly, the popularity of most linear transformations has declined considerably in recent years.
Modern nonlinear sequence transformations as for instance Wynn's
[13] and 
[14] algorithm or Brezinski's
algorithm [15] have largely complementary properties.
They are nonregular, which means that is not guaranteed that the
transformed sequence converges, let alone to the correct limit, and
their theoretical properties are far from being completely understood.
Nevertheless, they are often able to accomplish spectacular convergence
acceleration and summation results. Consequently, the more powerful but
also theoretically more complicated nonlinear transformations now
dominate both in mathematical research as well as in practical
applications, as documented by the large number of recent books
[12,16,17,18,19,20,21] and review articles
[22,23] on this topic.
However, there is considerable evidence that the culprit for the frequently unsatisfactory performance of regular matrix transformation is not their linearity, but their regularity. In [24] it was shown that suitably chosen linear but nonregular transformations can at least for special problems be as efficient as the most powerful nonlinear transformations.
Padé approximants [25,26,27] are able to accelerate the convergence of many power series or to sum them outside their radii of convergence. They accomplish this by converting the partial sums
of a power series 
into rational functions. The Padé
approximant
to 
is the ratio of two polynomials
and
of degrees 
and m, respectively. The coefficients 
, 
,
, 
, and 
, 
, 
, 
are chosen in
such a way that the Taylor expansion of 
agrees with
the power series for 
as far as possible:
It is relatively easy to understand why the transformation of the
partial sums (1.3) into rational functions 
can improve convergence or sum the power series in the case of
divergence. Because of the asymptotic condition (1.7),
the Taylor expansion of 
at z = 0 reproduces
the power series for 
up to terms of order 
:
The truncation error 
of this Taylor expansion,
which is generated automatically in the process of transforming the
partial sums (1.3) into the Padé approximant, is in
general different from zero.
Any difference in the numerical properties of the Padé approximants
and the partial sums, which are used for their
construction, must be attributed to the truncation errors 
. In the case of an arbitrary power series 
, it
is not a priori clear whether these truncation errors will
improve the situation or not. In the case of divergent Stieltjes series,
which will be discussed in sections 6 and 7 of this report, it can be
shown rigorously that certain sequences of Padé approximants are
able to sum them.
In any way, convergence is obviously improved or a divergent series is
summed if the truncation errors 
of the Padé
approximants provide sufficiently accurate approximations to the
truncation errors of the formal power series.
Padé approximants are special sequence transformations. In many branches of applied mathematics, in theoretical physics and in theoretical chemistry, Padé approximants are now used almost routinely and with considerable success. However, Padé approximants cannot help in all convergence acceleration and summation problems since they are in principle limited to power series, and even in the case of power series, they are not always the best choice.
In this report, the emphasis will be on some alternative sequence transformations (see sections 7 - 9 of [23] or section 2.7 of [19]). Like Padé approximants, they transform the partial sums (1.3) of a power series into rational functions. However, their numerator and denominator polynomials differ from those of Padé approximants. Unlike Padé approximants, they are not restricted to power series. Moreover, they incorporate into the approximation scheme explicit estimates for the remainders of the slowly convergent or divergent sequence or series which is to be transformed. Because of the explicit incorporation of additional information, these transformations are potentially very powerful. In particular, they are apparently very well suited for the summation of strongly divergent series as they for instance occur in special function theory or in quantum mechanical perturbation theory [23,28,29,30,31,32,33,34].
