This article develops a new predictor-corrector algorithm for numerical path tracking in the context of polynomial homotopy continuation. In the corrector step, it uses a newly developed Newton corrector algorithm which rejects an initial guess if it is not an approximate zero. The algorithm also uses an adaptive step size control that builds on a local understanding of the region of convergence of Newton’s method and the distance to the closest singularity following Telen, Van Barel, and Verschelde. To handle numerically challenging situations, the algorithm uses mixed precision arithmetic. The efficiency and robustness are demonstrated in several numerical examples.

本文开发了一种新的预测器-校正器算法，用于在多项式同伦连续的上下文中进行数值路径跟踪。在校正器步骤中，它使用新开发的牛顿校正器算法，如果初始猜测不是近似零，则拒绝初始猜测。该算法还使用自适应步长控制，该控制建立在对牛顿方法收敛区域的局部理解以及与 Telen、Van Barel 和 Verschelde 之后的最近奇点的距离的基础上。为了处理具有挑战性的数值情况，该算法使用混合精度算法。几个数值例子证明了效率和鲁棒性。