ZEAL: A mathematical software package for computing zeros of analytic functions

Abstract

We present a reliable and portable software package for computing zeros of analytic functions. The package is named ZEAL (ZEros of AnaLytic functions). Given a rectangular region W in the complex plane and a function f:W→C that is analytic in W and does not have zeros on the boundary of W, ZEAL localizes and computes all the zeros of f that lie inside W, together with their respective multiplicities. ZEAL is based on the theory of formal orthogonal polynomials. It proceeds by evaluating numerically certain integrals along the boundary of W involving the logarithmic derivative f′/f and by solving generalized eigenvalue problems. The multiplicities are computed by solving a linear system of equations that has Vandermonde structure. ZEAL is written in Fortran 90.