What should be a rate of convergence ?

— An aiiempt to set up a method for estimating convergence of itérative processes using inequalities of the type where (Ù is a "small" fonction as deflned by the author. This method is compared with the classical notion of a rate of convergence ; for convex fonctions (ù inequalities using the distance of xn from the solution and those using the distance of two consécutive steps are shown to be equivalent. Some advantages of this new approach are pointed out. 1. HEURISTICS The classical notion of the order of convergence or rate of convergence which reputedly goes back to the last century is defined as follows. Given an itérative process which yields a séquence xn of éléments of a complete metric space (E, d) converging to an element x e E we say that the convergence is of order p if there exists a constant a such that Clearly it is immaterial whether we require this for all n or only asymptotically. Let us point out two difficultés which seem to arise if this point of view is adopted. 1° If p > 1 then the above inequality contains a certain amount of information about the process ; the information, however, is more of a qualitative nature since it relates quantities which we are not able to measure at any finite stage of the process. The obvious meaning of the above inequality seems to consist rather in the fact that, at each stage of the process, the following step of the itération yields a significant improvement of the estimate. () Text of a lecture (unpublished) given at the Dublin Conference on Numexicat Analysis. August 1976. Manuscrit reçu le 10 novembre 1976. () Vlastimil Ptâk, Ceskoslovenska Akademie Ved, Matematicky Ustav, Praha, Tchécoslovaquie. R.A.I.R.O. Analyse Numérique/Numerical Analysis. vol 11. n° 3, 1977