具有孤立奇异解的多项式系统的高阶消去法
Higher-Order Deflation for Polynomial Systems with Isolated Singular Solutions
AI总结：
针对多项式系统的孤立多重解，提出一种利用高阶偏导数的高阶消去算法，以加速牛顿法的二次收敛并降低解的重数。
摘要：
给定多项式方程组的一个多重孤立解的近似，我们提供了一种符号-数值消去算法来恢复牛顿法的二次收敛。利用系统中多项式的一阶导数，我们的方法创建了一个增广方程组，该方程组将原始系统的多重孤立解作为正则根。在本文中，我们考虑了两种计算奇异孤立解的“重数结构”的方法。其中一种方法的思想催生了我们新的高阶消去法。利用原始多项式的高阶偏导数，新算法比我们的第一种方法更快地降低了重数，适用于需要多个一阶消去步骤的系统。我们还提出了一种预测消去阶数的算法。
英文摘要：
Given an approximation to a multiple isolated solution of a polynomial system of equations, we have provided a symbolic-numeric deflation algorithm to restore the quadratic convergence of Newton's method. Using first-order derivatives of the polynomials in the system, our method creates an augmented system of equations which has the multiple isolated solution of the original system as a regular root. In this paper we consider two approaches to computing the ``multiplicity structure'' at a singular isolated solution. An idea coming from one of them gives rise to our new higher-order deflation method. Using higher-order partial derivatives of the original polynomials, the new algorithm reduces the multiplicity faster than our first method for systems which require several first-order deflation steps. We also present an algorithm to predict the order of the deflation.
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。
