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 structured stable QR algorithm for the classical cases (see Serkh and Rokhlin 2021). The performance of the schemes is illustrated with several numerical examples.

我们提出了一种在复平面的平方域中查找解析函数的所有根的方案。该方案可以被视为在实线上求函数根的经典方法的推广，首先通过切比雪夫基中的多项式对其进行近似，然后对所谓的“同事矩阵”进行对角化。我们对经典方法的扩展基于一些观察，这些观察使得能够在满足三项递归并且条件相当良好的紧凑域中构造多项式基。这类多项式基产生了“广义同事矩阵”，其特征值是用这些基表示的函数的根。在本文中，我们还介绍了一种用于查找广义同事矩阵特征值的专用 QR 算法，这是最近针对经典情况引入的结构化稳定 QR 算法的直接扩展（参见 Serkh 和 Rokhlin 2021）。通过几个数值示例说明了该方案的性能。