Matrix Shanks Transformations

Abstract

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 $\varepsilon$-algorithm of Wynn. Then, the transformation is related to matrix Pad\'{e}-type and Pad\'{e} approximants. Numerical experiments showing the interest of this transformation end the paper.