Scalar Levin-type Sequence Transformations

Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. The basic idea is to construct from a given sequence { sn} a new sequence { s ′ n} = T ({ sn} ) where each s ′ n depends on a finite number of elements sn1 , . . . , snm . Often, the sn are the partial sums of an infinite series. The aim is to find transformations such that { sn} converges faster than (or sums) { sn} . Transformations T ({ sn} , {ωn} ) that depend not only on the sequence elements or partial sums sn but also on an auxiliary sequence of so-called remainder estimates ωn are of Levin-type if they are linear in the sn, and nonlinear in the ωn. Such remainder estimates provide an easy-to-use possibility to use asymptotic information on the problem sequence for the construction of highly efficient sequence transformations. As shown first by Levin, it is possible to obtain such asymptotic information easily for large classes of sequences in such a way that the ωn are simple functions of a few sequence elements sn. Then, nonlinear sequence transformations are obtained. Special cases of such Levin-type transformations belong to the most powerful currently known extrapolation methods for scalar sequences and series. Here, we review known Levin-type sequence transformations and put them in a common theoretical framework. It is discussed how such transformations may be constructed by either a model sequence approach or by iteration of simple transformations. As illustration, two new sequence transformations are derived. Common properties and results on convergence acceleration and stability are given. For important special cases, extensions of the general results are presented. Also, guidelines for the application of Levin-type sequence transformations are discussed, and a few numerical examples are given.