The mappings from the original to the transformed series may be
linear (as defined in the article
sequence transformations), or non-linear. In general, the non-linear sequence transformations tend to be more powerful.
For
alternating series, several powerful techniques, offering convergence rates from all the way to for a summation of terms, are described by Cohen et al.[3]
Euler's transform
A basic example of a
linear sequence transformation, offering improved convergence, is Euler's transform. It is intended to be applied to an alternating series; it is given by
If the original series, on the left hand side, is only slowly converging, the forward differences will tend to become small quite rapidly; the additional power of two further improves the rate at which the right hand side converges.
can be written as , where the
functionf is defined as
The function can have
singularities in the
complex plane (
branch point singularities,
poles or
essential singularities), which limit the
radius of convergence of the series. If the point is close to or on the boundary of the disk of convergence, the series for will converge very slowly. One can then improve the convergence of the series by means of a
conformal mapping that moves the singularities such that the point that is mapped to ends up deeper in the new disk of convergence.
The conformal transform needs to be chosen such that , and one usually chooses a function that has a finite
derivative at w = 0. One can assume that without loss of generality, as one can always rescale w to redefine . We then consider the function
Since , we have .We can obtain the series expansion of by putting in the series expansion of because ; the first terms of the series expansion for will yield the first terms of the series expansion for if . Putting in that series expansion will thus yield a series such that if it converges, it will converge to the same value as the original series.
A simple nonlinear sequence transformation is the Aitken extrapolation or delta-squared method,
defined by
This transformation is commonly used to improve the
rate of convergence of a slowly converging sequence; heuristically, it eliminates the largest part of the
absolute error.
Herbert H. H. Homeier: Scalar Levin-Type Sequence Transformations, Journal of Computational and Applied Mathematics, vol. 122, no. 1–2, p 81 (2000). Homeier, H. H. H. (2000). "Scalar Levin-type sequence transformations". Journal of Computational and Applied Mathematics. 122 (1–2): 81–147.
arXiv:math/0005209.
Bibcode:
2000JCoAM.122...81H.
doi:
10.1016/S0377-0427(00)00359-9.,
arXiv:
math/0005209.
Brezinski Claude and Redivo-Zaglia Michela : "The genesis and early developments of Aitken's process, Shanks transformation, the -algorithm, and related fixed point methods", Numerical Algorithms, Vol.80, No.1, (2019), pp.11-133.
Delahaye J. P. : "Sequence Transformations", Springer-Verlag, Berlin, ISBN 978-3540152835 (1988).
Sidi Avram : "Vector Extrapolation Methods with Applications", SIAM, ISBN 978-1-61197-495-9 (2017).
Brezinski Claude, Redivo-Zaglia Michela and Saad Yousef : "Shanks Sequence Transformations and Anderson Acceleration", SIAM Review, Vol.60, No.3 (2018), pp.646–669. doi:10.1137/17M1120725 .
Brezinski Claude : "Reminiscences of
Peter Wynn", Numerical Algorithms, Vol.80(2019), pp.5-10.
Brezinski Claude and Redivo-Zaglia Michela : "Extrapolation and Rational Approximation", Springer, ISBN 978-3-030-58417-7 (2020).