发表机构
Rose-Hulman Institute of Technology; University of Notre Dame（罗斯-赫尔曼理工学院; 圣母大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出螺旋终局算法，通过对数螺旋采样插值Puiseux级数，平衡数值条件与更新效率，用于计算同伦路径的奇异端点，实验证明优于Cauchy终局。
AI 中文摘要
我们开发了一种新的终局算法，用于通过同伦路径数值计算复解析系统的奇异解，该路径的端点即为待计算的解。当解为奇异时，预测-校正路径跟踪可能在端点处失败。终局算法不是将路径一直跟踪到终点，而是在进入路径上采样非奇异点，并通过截断的Puiseux级数对样本进行插值来近似端点。通过沿复平面中的对数螺旋收集样本点，所提出的螺旋终局算法在圆形采样的优越数值条件（称为Cauchy终局）和线性采样的更新效率（称为幂级数终局）之间取得平衡，这两种方法作为新方法的两个极端被联系在一起。对方法的数值条件进行了分析和比较，计算实例说明了新的螺旋采样方法与Cauchy终局的圆形采样方法相比的有效性。
英文摘要
We develop a new endgame for numerically computing singular solutions to complex analytic systems by means of a homotopy path whose endpoint is the solution to be computed. When the solution is singular, prediction-correction path tracking may fail at the endpoint. Instead of tracking the path all the way to its end, endgames sample nonsingular points on the incoming path and approximate the endpoint by interpolating the samples with a truncated Puiseux series. By collecting sample points along a logarithmic spiral in the complex plane, the proposed spiral endgame balances between the superior numerical conditioning of a circular sample (known as the Cauchy endgame) and the update efficiency of a linear sample (known as the power series endgame), these being tied together as two extremes of the new approach. The numerical conditioning of the methods are analyzed and compared, and computational examples illustrate the effectiveness of the new spiral sampling method compared with the circular sampling method of the Cauchy endgame.
