The Method of Gauss-Newton to Compute Power Series Solutions of Polynomial Homotopies
The Method of Gauss-Newton to Compute Power Series Solutions of Polynomial Homotopies
复制标题
计算多项式同伦幂级数解的高斯-牛顿法
DOI：
10.1016/j.laa.2017.10.022
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
J. Verschelde
中科院分区：
文献类型：
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作者：
N. Bliss;J. Verschelde
本文将Gauss-Newton方法从复浮点运算推广到复浮点系数截断幂级数领域。通过线性化，我们建立了一个线性系统，其中系数矩阵是一个矩阵系数的级数，并根据增广系统的代数簇，给出了矩阵级数正则时的一个特征。线性系统的结构导致块三角系统。在常规情况下，求解线性系统等价于求解Hermite插值问题。我们表明，这种解决方案的成本立方的问题的大小。在一般情况下，在奇点，我们依靠热带代数几何的方法来计算Puixix级数。通过几个示例，我们展示了多项式同伦延拓的应用。
We consider the extension of the method of Gauss–Newton from complex floating-point arithmetic to the field of truncated power series with complex floating-point coefficients. With linearization we formulate a linear system where the coefficient matrix is a series with matrix coefficients, and provide a characterization for when the matrix series is regular based on the algebraic variety of an augmented system. The structure of the linear system leads to a block triangular system. In the regular case, solving the linear system is equivalent to solving a Hermite interpolation problem. We show that this solution has cost cubic in the problem size. In general, at singular points, we rely on methods of tropical algebraic geometry to compute Puiseux series. With a few illustrative examples, we demonstrate the application to polynomial homotopy continuation.