Locating the Closest Singularity in a Polynomial Homotopy
Locating the Closest Singularity in a Polynomial Homotopy
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DOI：
10.48550/arxiv.2205.07380
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发表时间：
2022-05
期刊：
影响因子：
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通讯作者：
J. Verschelde;Kylash Viswanathan
中科院分区：
文献类型：
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作者：
J. Verschelde;Kylash Viswanathan
多项式同伦是多项式系统族，其中族中的系统取决于一个参数。如果对于参数的一个值，我们知道一个正则解，那么多项式同伦中的解是奇异的，最接近的参数值是多少？对于这个问题，我们应用法布里比率定理。理查森外推法对于加速由同伦定义的解路径的级数展开式的系数比的收敛是有效的。为了数值稳定性，我们重新调整同伦。为了计算级数的系数，我们提出了四元数傅立叶变换。我们将最接近的奇点计算定位在正则解处，避免了奇点附近的数值困难。
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.