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

Shanks function transformations in a vector space

In this paper, we show how to construct various extensions of Shanks transformation for functions in a vector space. They are aimed at transforming a function tending slowly to its limit when the argument tends to infinity into another function with better convergence properties. Their expressions as ratio of determinants and recursive algorithms for their implementation are given. A simplified form of one of them is derived. It allows us to obtain a convergence result for an important class of functions. An application to integrable systems is discussed

Generalizations of Aitken's process for accelerating the convergence of sequences

When a sequence or an iterative process is slowly converging, a convergence acceleration process has to be used. It consists in transforming the slowly converging sequence into a new one which, under some assumptions, converges faster to the same limit. In this paper, new scalar sequence transformations having a kernel (the set of sequences transformed into a constant sequence) generalizing the kernel of the Aitken's delta2 process are constructed. Then, these transformations are extended to vector sequences. They also lead to new fixed point methods which are studied