Numerical analysis of a bisection-exclusion method to find zeros of univariate analytic functions

We state precise results on the complexity of a classical bisection-exclusion method to locate zeros of univariate analytic functions contained in a square. The output of this algorithm is a list of squares containing all the zeros. It is also a robust method to locate clusters of zeros. We show that the global complexity depends on the following quantities: the size of the square, the desired precision, the number of clusters of zeros in the square, the distance between the clusters and the global behavior of the analytic function and its derivatives. We also prove that, closed to a cluster of zeros, the complexity depends only on the number of zeros inside the cluster. In particular, for a polynomial which has d simple roots separated by a distance greater than sep, we will prove the bisection-exclusion algorithm needs O(d3log(d/sep)) tests to isolate the d roots and the number of squares suspected to contain a zero is bounded by 4d. Moreover, always in the polynomial case, we will see the arithmetic complexity can be reduced to O(d2(logd)2log(d/sep)) using ⌈logd⌉ steps of the Graeffe iteration.