The Durand-Kerner algorithm is a widely used iterative technique for simultaneously finding all the roots of a polynomial. However, its convergence heavily depends on the choice of initial approximations. This paper introduces two novel approaches for determining the initial values: New bound 1 and the lambda maximal bound, aimed at improving the stability and convergence speed of the algorithm. Theoretical analysis and numerical experiments were conducted to evaluate the effectiveness of these bounds. The lambda maximal bound consistently ensures that all the roots lie within the complex circle, leading to faster and more stable convergence. Comparative results demonstrate that while New bound 1 guarantees convergence, but it yields excessively large radii.

翻译：Durand-Kerner算法是一种广泛使用的迭代技术，用于同时求解多项式的所有根。然而，其收敛性在很大程度上依赖于初始近似值的选择。本文介绍了两种确定初始值的新方法：新界1和λ最大界，旨在提高算法的稳定性和收敛速度。通过理论分析和数值实验评估了这些界的有效性。λ最大界始终确保所有根位于复平面圆内，从而实现更快且更稳定的收敛。对比结果表明，虽然新界1能保证收敛，但其产生的半径过大。