A derivative-free algorithm for computing zeros of analytic functions

Let W be a simply connected region in C, f : W --> C analytic in W and gamm a a positively oriented Jordan curve in W that does not pass through any ze ro of f. We present an algorithm for computing all the zeros of f that lie in the interior of gamma. It proceeds by evaluating certain integrals along gamma numerically and is based on the theory of formal orthogonal polynomi als. The algorithm requires only f and not its first derivative f'. We have found that it gives accurate approximations for the zeros. Moreover, it is self-starting in the sense that it does not require initial approximations . The algorithm works for simple zeros as well as multiple zeros, although it is unable to compute the multiplicity of a zero explicitly. Numerical ex amples illustrate the effectiveness of our approach. AMS Subject Classification: 65H05.