Some unusual results on extrapolation methods
The paper discusses two fundamental questions in connection with extrapolation methods. Assuming a convergent sequence \((S_n)\) is given, the essence of such methods is that one may attempt to extrapolate the sequence, i.e., to construct, by the well known Shanks transformation, a second sequence \((T_n)\), say, that has the same limit but that hopefully converges faster than the original sequence. The first main result of the paper is a set of necessary conditions that the original sequence \((S_n)\) must satisfy in order for the sequence \((T_n)\) to converge faster than \((S_n)\). Based on these conditions, the authors then construct stopping criteria allowing the user to terminate the acceleration procedure when a best possible precision is reached. The second main result then deals with an ``inverse problem'' in the sense that properties of the transformed sequence \((T_n)\) are prescribed and it is then demonstrated which properties the original sequence \((S_n)\) must have in order for these properties to be present.
scientific article; zbMATH DE number 7221098
scientific article