For which error criteria can we solve nonlinear equations?

Abstract

For which error criteria can we solve a nonlinear scalar equation f ( x) = 0, where f is a real function on the interval [ a, b]? The information on f consists of adaptive evaluations of arbitrary linear functionals and an algorithm is any mapping based on these evaluations. The error of an algorithm is defined by its worst performance. For the root criterion we prove there does not exist an algorithm to find a point x such that ∥ x− α∥≤ ϵ, where α is a zero of f and ϵ < (b−a) 2 . This holds for rbitrary information and for the class of infinitely many times differentiable functions with all simple zeros. We do not assume that f( a) f( b) ≤ 0. For the residual criterion we exhibit almost optimal information and algorithm. More precisely, we prove that if x is the value computed by our algorithm using n function values then f( x) = O( n − r ), where r measures the smoothness of the class of functions f. Finally a general error criterion is introduced and some of our results are generalized.