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.