The problem of locating and computing with certainty all the simple roots of a twice continuously differentiable function f : [a, b] ⊂ R → R is studied when some additional information on the distribution of the roots in the interval is available. The framework is the one proposed by [SIAM J. Sci. Comput., 17 (1996), pp. 1232-1248, where only the uniform case was examined. This paper settles some of the problems posed there and generalizes some of its results by considering an arbitrary distribution of the roots in [a, b]. The theoretical results are accompanied by simulations in a number of problems of various size.Key words. zeros isolation, Kronecker-Picard theory, topological degree, locating simple roots, computing simple roots, zeros identifications, bisection method, distribution of the roots, expected complexity of algorithms AMS subject classifications. 65H05, 65Y20PII. S1064827598333806 1. Introduction. Many problems in different areas of science, such as mechanics, physical sciences, statistics, operations research, etc., are reduced to the problem of finding all the roots or the extrema of a function in a given interval. The importance of the problem has attracted the attention of many research efforts and as a result many different approaches to the problem exist. We briefly mention here the deflation techniques used for the calculation of further solutions [3] and more recently interval analysis based methods (see, e.g., [6,7,14,15,16,17]) and the method described in [10].Evaluating the performance of an algorithm usually involves experimental observations in known problem instances drawn from the literature and/or randomly constructed ones. Few analytical estimations of widely accepted performance measures exist and these mostly concern the worst-case behavior of an algorithm [2,20]. The worst-case behavior, however, can sometimes be very conservative for real-life applications, while the expected behavior seems to be a more natural and informative measure. Contrary to the apparent importance of the expected behavior of an algorithm, however, even fewer analytical results exist [4]. One reason for this might be the difficulty inherent in such a study and the lack of a suitable framework.In [10] a framework for the study of the expected complexity of the problem of finding with certainty all the simple roots of a function was presented and some results were shown for the case when the roots are uniformly distributed in the interval. While this models the situation where no information of the distribution of the roots is available, it is clearly a severe restriction of the general problem, as sometimes additional information about concentrations of roots is available. Such information can in turn be modeled as a mathematical probability distribution and thus the clustering behavior can be taken into account.