Locating and Computing All the Simple Roots and Extrema of a Function

This paper describes and analyzes two algorithms for locating and computing with certainty all the simple roots of a twice continuously differentiable function f : (a, b) ⊂ R→ R and all the extrema of a three times continuously differentiable function in (a, b). The first algorithm locates and computes all the simple roots or all the extrema, while the second one is more efficient in the case where both simple roots and extrema are required. This paper also gives analytical estimation of the expected complexity of the algorithms based on the distribution of the roots in (a, b). Here only the case of uniform distribution is examined, which is also the approach to be followed when no statistical data are available for the function at hand. The algorithms have been implemented and tested. Performance information for a well-known Bessel function is reported.