A bound on the multiplication efficiency of iteration

Abstract

For a convergent sequence {xi} generated by xi+1 = @@@@(xi,x1,...,xi-d+1), define the multiplication efficiency measure E to be p1/M, where p is the order of convergence, and M is the number of multiplications or divisions (except by 2) needed to compute @@@@. Then, if @@@@ is any multivariate rational function, E ≤ 2. Since E = 2 for the sequence {xi} generated by xi+1 = 1/2(xi +a/x i)with the limit @@@@a, the bound on E is sharp.