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Higher-Order Deflation for Polynomial Systems with Isolated Singular Solutions

Anton Leykin·Jan Verschelde·Ailing Zhao
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Abstract

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.

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Anton Leykin, Jan Verschelde, Ailing Zhao. 2006-02-01. Higher-Order Deflation for Polynomial Systems with Isolated Singular Solutions. https://arxiv.org/abs/math/0602031

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