On condition numbers of the Shanks transformation

Quasi-linear functions generate sequence transformation methods whose conditioning depends upon the nature of the sequence to be accelerated. These methods are often well conditioned when they are applied to alternating sequences; however, they are relatively ill-conditioned in case of monotonic convergence. The condition numbers of the Shanks transformation ek(sn) are given in order to prove that the closely related ε-algorithm to a such transformation is ill-conditioned when performed on the set of totally monotonic sequences. In the same way, we show that this algorithm is well conditioned on the set of totally oscillating sequences.