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By J.F. Traub

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When f: R there. R ) , since the mean value theorem can be used The problem becomes nontrivial for nonscalar cases. 1, which works for Banach spaces. 7)) of the theorem could be weakened by complicating the algorithm used in the proof. 7) used here, a more com­ plicated assumption involving β, k, was used. 1. 15). 15) is equiva­ Since lent to from which N(hQ,6) can be calculated. 17) it appears that we should use small δ in the algorithm. 19». 2. | of f^ can be found by Newton's method. This is clearly not the case in practice.

In this section we turn our attention to the com- plexity index. Recall that z = c/lg p. We begin our analysis of z by considering the cost per step, c. We distinguish between two kinds of problems. We say a problem is explicit if the formula for f is given explicitly. For example, the calculation of al/~ by 2 solving f = x _a is an explicit problem. The complexity of explicit problems has been studied by Paterson [72] and Kung [7~], [73h]. ) (Paterson and Kung take the efficiency index as We do not treat explicit problems here.

K. x O • 2 and ·comp s: c 19(1+t). To derive a lower bound on complexity one must make an assumption about the closest machine-representable number to 2 1/2 • We do not pursue that here. 5. • BOUNDS ON THE COMPLEXITY INDEX We have shown that provided w is not too close to unity, then for fixed c, complexity depends only on the complexity index z. In this section we turn our attention to the com- plexity index. Recall that z = c/lg p. We begin our analysis of z by considering the cost per step, c.

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