The Set of Logarithmically Convergent Sequences Cannot be Accelerated

Abstract

Some theorems (Pennacchi, Germain-Bonne, Smith and Ford) state that methods of a certain form which are exact on geometric sequences accelerate linear convergence. But no corresponding theorem is known for logarithmic convergence. Our study shows the reason why: There is no algorithm which can accelerate all logarithmically convergent sequences. We obtain this result with a generalization of “remanence”, which is a sufficient property for a set of sequences to be unaccelerable.