Prediction Properties of Aitken's Iterated Delta^2 Process, of Wynn's Epsilon Algorithm, and of Brezinski's Iterated Theta Algorithm

The prediction properties of Aitken's iterated Delta^2 process, Wynn's epsilon algorithm, and Brezinski's iterated theta algorithm for (formal) power series are analyzed. As a first step, the defining recursive schemes of these transformations are suitably rearranged in order to permit the derivation of accuracy-through-order relationships. On the basis of these relationships, the rational approximants can be rewritten as a partial sum plus an appropriate transformation term. A Taylor expansion of such a transformation term, which is a rational function and which can be computed recursively, produces the predictions for those coefficients of the (formal) power series which were not used for the computation of the corresponding rational approximant.Comment: 34 pages, LaTe

Matrix Shanks Transformations

Shanks' transformation is a well know sequence transformation for accelerating the convergence of scalar sequences. It has been extended to the case of sequences of vectors and sequences of square matrices satisfying a linear difference equation with scalar coefficients. In this paper, a more general extension to the matrix case where the matrices can be rectangular and satisfy a difference equation with matrix coefficients is proposed and studied. In the particular case of square matrices, the new transformation can be recursively implemented by the matrix $arepsilon$-algorithm of Wynn. Then, the transformation is related to matrix Pad\ue9-type and Pad\ue9 approximants. Numerical experiments showing the interest of this transformation end the paper

Abstract Fixpoint Computations with Numerical Acceleration Methods

Static analysis by abstract interpretation aims at automatically proving properties of computer programs. To do this, an over-approximation of program semantics, defined as the least fixpoint of a system of semantic equations, must be computed. To enforce the convergence of this computation, widening operator is used but it may lead to coarse results. We propose a new method to accelerate the computation of this fixpoint by using standard techniques of numerical analysis. Our goal is to automatically and dynamically adapt the widening operator in order to maintain precision

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. Transformations that depend not only on the sequence elements or partial sums $s_n$ but also on an auxiliary sequence of so-called remainder estimates $\omega_n$ are of Levin-type if they are linear in the $s_n$, and nonlinear in the $\omega_n$. Known Levin-type sequence transformations are reviewed and put into 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.Comment: 59 pages, LaTeX, invited review for J. Comput. Applied Math., abstract shortene