A bound on the multiplication efficiency of iteration

For a convergent sequence {x<subscrpt>i</subscrpt>} generated by x<subscrpt>i+1</subscrpt> &equil; @@@@(x<subscrpt>i</subscrpt>,x<subscrpt>1</subscrpt>,...,x<subscrpt>i−d+1</subscrpt>), define the multiplication efficiency measure E to be p<supscrpt>&frac1M;</supscrpt>, 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 &equil; 2 for the sequence {x<subscrpt>i</subscrpt>} generated by x<subscrpt>i+1</subscrpt> &equil; ½(x<subscrpt>i</subscrpt> +&fracax; <subscrpt>i</subscrpt>)with the limit @@@@a, the bound on E is sharp.