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解析函数全部根的求解方案。该方案可视为经典实轴函数求根方法（先通过切比雪夫基多项式逼近原函数，再对角化所谓的"伴随矩阵"）的推广。我们对经典方法的扩展基于多项观察结果，这些观察使得在紧致域中构造满足三项递推关系且条件数合理稳健的多项式基成为可能。这类多项式基衍生出"广义伴随矩阵"，其特征值即为用这些基表示的函数之根。本文还引入了专用于求解广义伴随矩阵特征值的QR算法，该算法是近期提出的经典情形分量稳定QR算法（参见[Serkh]）的直接推广。通过若干数值算例验证了该方案的有效性。