Linear Algorithms with Finite Deviation

We discuss the computation of If], where I is a positive functional on C a; b]. Any algorithm starts with some input, in typical situations this input consists of function values (f (x 1) ; : : : ; f (x n)) and the more global information f 2 K, where K C a; b] is a given set. The algorithm which makes the fullest use of the input is called the strongly optimal algorithm. We shall show that in many cases of practical interest these algorithms are not linear in (f (x 1) ; : : : ; f (x n)), in sharp contrast to the better known optimal algorithms. A weakening of the notion of strong optimality leads to algorithms with nite deviation, for these we obtain almost the same results. Furthermore some examples of linear algorithms with small deviation are given. 2