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We present a scheme for finding all roots of an analytic function in a squaredomain in the complex plane. The scheme can be viewed as a generalization ofthe classical approach to finding roots of a function on the real line, byfirst approximating it by a polynomial in the Chebyshev basis, followed bydiagonalizing the so-called ''colleague matrices''. Our extension of theclassical approach is based on several observations that enable theconstruction of polynomial bases in compact domains that satisfy three-termrecurrences and are reasonably well-conditioned. This class of polynomial basesgives rise to ''generalized colleague matrices'', whose eigenvalues are rootsof functions expressed in these bases. In this paper, we also introduce aspecial-purpose QR algorithm for finding the eigenvalues of generalizedcolleague matrices, which is a straightforward extension of the recentlyintroduced componentwise stable QR algorithm for the classical cases (See[Serkh]). The performance of the schemes is illustrated with several numericalexamples.

Finding roots of complex analytic functions via generalized colleague matrices