LOCATING AND COMPUTING ALL THE SIMPLE ROOTS AND EXTREMA OF A FUNCTION

This paper describes and analyzes two algorithms for locating and comp uting with certainly all the simple roots of a twice continuously diff erentiable function f: (a, b) subset of R --> R and all the extrema of a three times continuously differentiable function in (a, b). The fir st algorithm locates and computes all the simple roots or all the extr ema, while the second one is more efficient in the case where both sim ple roots and extrema are required. This paper also gives analytical e stimation of the expected complexity of the algorithms based on the di stribution of the roots in (a, b). Here only the case of uniform distr ibution is examined. which is also the approach to be followed when no statistical data are available for the function at hand. The algorith ms have been implemented and tested. Performance information for a wel l-known BesseI function is reported.