Finding roots of complex analytic functions via generalized colleague matrices

TL;DR

This paper introduces a novel method for locating all roots of complex analytic functions within a square domain using generalized colleague matrices and a specialized QR algorithm.

Contribution

It extends classical root-finding techniques by constructing polynomial bases in compact domains and developing a stable eigenvalue algorithm for generalized matrices.

Findings

The method accurately finds roots in complex domains.

The generalized colleague matrices have well-conditioned eigenvalues.

Numerical examples demonstrate the scheme's effectiveness.

Abstract

We present a scheme for finding all roots of an analytic function in a square domain in the complex plane. The scheme can be viewed as a generalization of the classical approach to finding roots of a function on the real line, by first approximating it by a polynomial in the Chebyshev basis, followed by diagonalizing the so-called ''colleague matrices''. Our extension of the classical approach is based on several observations that enable the construction of polynomial bases in compact domains that satisfy three-term recurrences and are reasonably well-conditioned. This class of polynomial bases gives rise to ''generalized colleague matrices'', whose eigenvalues are roots of functions expressed in these bases. In this paper, we also introduce a special-purpose QR algorithm for finding the eigenvalues of generalized colleague matrices, which is a straightforward extension of the recently introduced componentwise stable QR algorithm for the classical cases (See [Serkh]). The performance of the schemes is illustrated with several numerical examples.