奇异解的代数曲线扫描
Sweeping Algebraic Curves for Singular Solutions
AI总结：
针对多项式系统参数变化时的奇异解追踪问题，提出通过监测雅可比矩阵行列式来检测路径上的奇异点，并分析紧缩法的操作范围及其与缠绕数的关系。
摘要：
许多问题产生多项式系统。这些系统通常有多个参数，我们感兴趣的是当参数值改变时解如何变化。使用预测-校正方法追踪解路径。当雅可比矩阵秩亏损时，解路径上的点是临界点。二次转折点的最简单情况已被充分理解，但这些方法不再适用于一般类型的奇异性。为了不遗漏路径上的任何奇异解，我们建议监测雅可比矩阵的行列式。我们检验了紧缩法的操作范围，并将紧缩法的有效性与缠绕数联系起来。给出了来自不同应用领域的系统的计算实验。
英文摘要：
Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the solution paths. A point along a solution path is critical when the Jacobian matrix is rank deficient. The simplest case of quadratic turning points is well understood, but these methods no longer work for general types of singularities. In order not to miss any singular solutions along a path we propose to monitor the determinant of the Jacobian matrix. We examine the operation range of deflation and relate the effectiveness of deflation to the winding number. Computational experiments on systems coming from different application fields are presented.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
