Computing the Zeros of a Holomorphic Function Using Quadrature-Based Subdivision and Rational Approximation of the Logarithmic Derivative

We introduce a new method that uses AAA approximation to reliably compute all the zeros of a holomorphic function in a specified search region in the complex plane. Specifically, the method is based on rational approximation of the logarithmic derivative in combination with subdivision of the search region based on Cauchy's argument principle. This is motivated by the fact that, while it is straightforward to compute the zeros of a AAA rational approximation, there is no guarantee that all of the zeros of the function being approximated will be found. Many of the ideas presented are also applicable to computing both the zeros and the poles of a meromorphic function. A implementation of the method is provided by the Python package skzeros.