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Summary.In this paper a general formalism for linear and rational extrapolation processes is developped. This formalism includes most of the sequence transformations actually used for convergence acceleration. A general recursive algorithm for implementing the method is given. Convergence results and convergence acceleration results are proved. The vector case and some other extensions are also studied. Extrapolation processes are very useful in numerical analysis since many methods produce sequences of approximations {S.} (often depending on a sequence of parameters {x.}) of the exact result S. If the error has the form:where the g/are known sequences, then more accurate approximations to S can be obtained by using an extrapolation method. A typical example of this situation is given by the well known trapezoidal rule for quadratures whose results can be extrapolated by Romberg method.The aim of this paper is to provide a general formalism to study such extrapolation methods in the general case. This formalism includes most of the sequence transformations actually used for convergence acceleration. A recursive algorithm for implementing this general extrapolation method is given and convergence theorems are proved. The vector case is also studied and some extensions are developed.

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