General Overview

Linear and nonlinear extrapolation methods extrapolation method are used for accelerating the convergence of sequences and series. Iterative procedures are widely employed in applied sciences. If the convergence of a sequence (of numbers, vectors, matrices, tensors,…) is slow, or even when divergence occurs, these procedures have only a restricted practical interest. In some cases, it is possible to modify the iterative scheme to produce a sequence with better convergence properties. In other cases, one has no access to the algorithm building the sequence, which comes from a “black box.” The idea is then to transform the sequence into another one converging faster, under certain assumptions, to the same limit by a sequence transformation. When they are linear, sequence transformations are known and have been studied for a long time under the name of summation processes summation method . They are attractive, since under quite mild conditions (Toeplitz’s theorem) Toeplitz theorem , they transform any convergent sequence into a sequence converging to the same limit. However, their main drawback is the fact that each of them is only able to accelerate the convergence of quite particular classes of sequences.