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.

翻译：我们提出了一种在复平面正方形区域内寻找解析函数所有根的方案。该方案可视为经典实轴上函数求根方法的推广：首先用切比雪夫基下的多项式逼近函数，然后对角化所谓的“伴随矩阵”。我们对经典方法的推广基于若干关键观察，这些观察使得在紧致域上构造满足三项递推关系且条件数合理的多项式基成为可能。此类多项式基催生了“广义伴随矩阵”，其特征值为以这些基表示的函数之根。本文还引入了一种专用的QR算法，用于求广义伴随矩阵的特征值，该算法是近期提出的经典情形下分量稳定QR算法（参见Serkh文献）的直接推广。最后通过多个数值算例验证了所提方案的有效性。