In this book we consider the problem of computing zeros of analytic functions and several related problems in computational complex analysis.
We start by studying the problem of computing all the zeros of an analytic function / that lie inside a positively oriented Jordan curve 7. Our principal means of obtaining information about the location of the zeros is a certain symmetric bilinear form that can be evaluated via numerical integration along 7. This form involves the logarithmic derivative /'// of /. Our approach could therefore be called a logarithmic residue based quadrature method. It can be seen as a continuation of the pioneering work done by Delves and Lyness. We shed new light on their approach by considering a different set of unknowns and by using the theory of formal orthogonal polynomials. Our algorithm computes not only approximations for the zeros but also their respective multiplicities. It does not require initial approximations for the zeros and we have found that it gives accurate results. The algorithm proceeds by solving generalized eigenvalue problems and a Vandermonde system. A Fortran 90 implementation is available (the package ZEAL). We also present an approach that uses only / and not its first derivative /'. These results are presented in Chapter 1.
In Chapter 2 we focus on the problem of locating clusters of zeros of analytic functions. We show how the approach presented in Chapter 1 can be used to compute approximations for the centre of a cluster and the total number of zeros in this cluster. We also attack the problem of computing all the zeros of / that lie inside 7 in an entirely different way, based on rational interpolation at roots of unity. We show how the new approach complements the previous one and how it can be used effectively in case 7 is the unit circle.
In Chapter 3 we show how our logarithmic residue based approach can be used to compute all the zeros and poles of a meromorphic function that lie in the interior of a Jordan curve.
In Chapter 4 we consider systems of analytic equations. A multidimensional logarithmic residue formula is available in the literature. This formula involves the integral of a differential form. We transform it into a sum of Rie-mann integrals and show how the zeros and their respective multiplicities can be computed from these integrals by solving a generalized eigenvalue problem that has Hankel structure and by solving several Vandermonde systems.
