Numer. Math. 73: 521–531 (1996) Numerische Mathematik c  Springer-Verlag 1996 An acceleration theorem for the ρ-algorithm Naoki Osada Nagasaki Institute of Applied Science, 536 Abamachi, Nagasaki City, 851-01, Japan; e-mail: osada@comm.nias.ac.jp Received October 20, 1994 / Revised version received July 2, 1995 Summary. The ρ-algorithm of Wynn is an excellent device for accelerating the convergence of some logarithmically convergent sequences. Until now a convergence theorem and an acceleration theorem for the ρ-algorithm have not been obtained. The purpose of this paper is to give an acceleration theorem for the ρ-algorithm. Moreover, it is proved that the ρ-algorithm cannot accelerate linear convergence. Numerical examples are given. Mathematics Subject Classification (1991): 65B05 1. Introduction In 1956, P. Wynn [5] proposed the ρ-algorithm defined by ρ (n ) -1 =0, ρ (n ) 0 = s n , n =1, 2,..., ρ (n ) k = ρ (n +1) k -2 + k ρ (n +1) k -1 − ρ (n ) k -1 , n =1, 2,... ; k =1, 2,..., (1.1) where (s n ) is a real or complex sequence. This algorithm can efficiently accel- erate the convergence of certain logarithmic sequences but it is not very often used. Because the ρ-algorithm has been considered to be unreliable for the three following reasons: 1. The ρ-algorithm lacks a convergence theorem and an acceleration theorem. 2. The ρ-algorithm sometimes cannot accelerate the convergence of a sequence satisfying (1.2) s n ∼ s + n θ  c 0 + c 1 n + c 2 n 2 + ...  , as n →∞,