Sweeping algebraic curves for singular solutions
Sweeping algebraic curves for singular solutions
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DOI：
10.1016/j.cam.2009.07.056
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发表时间：
2008-09
期刊：
影响因子：
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通讯作者：
Kathy Piret;J. Verschelde
中科院分区：
文献类型：
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作者：
Kathy Piret;J. Verschelde
许多问题都涉及多项式系统。这些系统通常有几个参数，我们有兴趣研究如何改变的解决方案时，我们改变的参数值。使用预测-校正方法，我们跟踪解决方案的路径。当雅可比矩阵秩亏时，沿解路径的沿着点是关键的。最简单的二次转折点的情况是很好理解的，但这些方法不再适用于一般类型的奇点。为了不错过任何奇异的解决方案沿着的路径，我们建议监控的雅可比矩阵的行列式。我们研究了放气的操作范围，并将放气的有效性与绕组数联系起来。对来自不同应用领域的系统进行了计算实验。
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.