A Bound on the Multiplicative Efficiency of Iteration

For a convergent sequence { x i } generated by x i+1 =( x i , x i−1 ,…, x i−d+1 ), define the multiplicative efficiency measure E to be (log 2 p )/ M , where p is the order of convergence and M is the number of multiplications or divisions needed to compute . Then, ifis any multivariate rational function, E ≤1. Since E =1 for the sequence { x i } generated by x i+1 = x i 2 + x i −1/4 with the limit −1/2, the bound on E is sharp. Let P M denote the maximal order for a sequence generated by an iteration with M multiplications. Then P M ≤2 M for all positive integers M . Moreover this bound is sharp.