A Derivative-Free Algorithm for Computing Zeros of Analytic Functions

Abstract.

Let W be a simply connected region in $\Bbb C$, f:W\rightarrow\Bbb C analytic in W and γ a positively oriented Jordan curve in W that does not pass through any zero of f. We present an algorithm for computing all the zeros of f that lie in the interior of γ. It proceeds by evaluating certain integrals along γ numerically and is based on the theory of formal orthogonal polynomials. 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 examples illustrate the effectiveness of our approach.