Multi-delayed systems, especially the neutral ones, have infinitely many and complex distributed characteristic roots that are crucial for system dynamics. The definite integral method, which determines the system stability by using only a definite integral, is extended in this paper for calculating all the characteristic roots in an arbitrarily given area on the complex plane of both retarded and neutral multi-delayed systems with constant discrete delays. Two simple algorithms are proposed for implementing the proposed method, by first calculating the distribution of the real parts of all the characteristic roots, then the imaginary parts by using an iteration method. The real part distribution can be used for the quick estimation of key characteristic roots such as the rightmost ones or the corresponding accumulation point(s), thus allowing adjusting the upper limit of the integral to further simplify the calculation procedure. Examples are given to show the feasibility and the efficiency of the proposed method through numerical analyses.

多时滞系统，尤其是中性系统，具有无限多且复杂的分布式特征根，这对系统动力学至关重要。本文扩展了仅使用定积分来确定系统稳定性的定积分法，用于计算具有恒定离散延迟的延迟和中性多延迟系统复平面上任意给定区域内的所有特征根. 提出了两种简单的算法来实现所提出的方法，首先计算所有特征根的实部分布，然后使用迭代方法计算虚部的分布。实部分布可用于快速估计关键特征根，例如最右边的根或相应的累积点，从而允许调整积分的上限以进一步简化计算过程。举例说明了通过数值分析所提出方法的可行性和有效性。