论文标题
在多项式同喻中定位最接近的奇异性
Locating the Closest Singularity in a Polynomial Homotopy
论文作者
论文摘要
多项式同喻是多项式系统的家族，其中家族中的系统取决于一个参数。如果对于参数的一个值，我们知道一个常规解决方案，那么多项式同型中的解决方案是奇异的参数的最接近值？对于此问题，我们应用Fabry的比率定理。理查森外推有效地加速了由同型定义的溶液路径系列膨胀系数的比率的收敛性。为了获得数值稳定性，我们将重新基于同义。为了计算该系列的系数，我们提出了四个傅立叶变换。我们将最接近的奇点计算定位在常规解决方案中，避免了奇异性附近的数值困难。
A polynomial homotopy is a family of polynomial systems, where the systems in the family depend on one parameter. If for one value of the parameter we know a regular solution, then what is the nearest value of the parameter for which the solution in the polynomial homotopy is singular? For this problem we apply the ratio theorem of Fabry. Richardson extrapolation is effective to accelerate the convergence of the ratios of the coefficients of the series expansions of the solution paths defined by the homotopy. For numerical stability, we recondition the homotopy. To compute the coefficients of the series we propose the quaternion Fourier transform. We locate the closest singularity computing at a regular solution, avoiding numerical difficulties near a singularity.
