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Two-step Newton's method for deflation-one singular zeros of analytic systems

Kisun Lee, Nan Li, Lihong Zhi

TL;DR

The paper addresses refining isolated singular zeros of square analytic systems, specifically deflation-one singularities where deflation terminates after one step. It introduces a two-step Newton refinement that leverages a singular value decomposition to identify the corank and project onto the kernel, followed by a second step on a reduced system derived from $D^2f(\xi)$; this yields quadratic convergence under a proximity assumption. A key contribution is the invertibility-based characterization of deflation-one zeros via the smaller matrix $\\mathcal{B}(\xi)$, enabling a efficient, matrix-size-reduced Newton update. Experimental results on benchmark systems demonstrate faster convergence and greater robustness than existing deflation-based methods, highlighting practical benefits for singular-zero isolation and clustering tasks. The approach offers a scalable, numerically stable pathway to refine deflation-one singular zeros with potential extensions to higher-order deflation and cluster isolation scenarios.

Abstract

We propose a two-step Newton's method for refining an approximation of a singular zero whose deflation process terminates after one step, also known as a deflation-one singularity. Given an isolated singular zero of a square analytic system, our algorithm exploits an invertible linear operator obtained by combining the Jacobian and a projection of the Hessian in the direction of the kernel of the Jacobian. We prove the quadratic convergence of the two-step Newton method when it is applied to an approximation of a deflation-one singular zero. Also, the algorithm requires a smaller size of matrices than the existing methods, making it more efficient. We demonstrate examples and experiments to show the efficiency of the method.

Two-step Newton's method for deflation-one singular zeros of analytic systems

TL;DR

The paper addresses refining isolated singular zeros of square analytic systems, specifically deflation-one singularities where deflation terminates after one step. It introduces a two-step Newton refinement that leverages a singular value decomposition to identify the corank and project onto the kernel, followed by a second step on a reduced system derived from ; this yields quadratic convergence under a proximity assumption. A key contribution is the invertibility-based characterization of deflation-one zeros via the smaller matrix , enabling a efficient, matrix-size-reduced Newton update. Experimental results on benchmark systems demonstrate faster convergence and greater robustness than existing deflation-based methods, highlighting practical benefits for singular-zero isolation and clustering tasks. The approach offers a scalable, numerically stable pathway to refine deflation-one singular zeros with potential extensions to higher-order deflation and cluster isolation scenarios.

Abstract

We propose a two-step Newton's method for refining an approximation of a singular zero whose deflation process terminates after one step, also known as a deflation-one singularity. Given an isolated singular zero of a square analytic system, our algorithm exploits an invertible linear operator obtained by combining the Jacobian and a projection of the Hessian in the direction of the kernel of the Jacobian. We prove the quadratic convergence of the two-step Newton method when it is applied to an approximation of a deflation-one singular zero. Also, the algorithm requires a smaller size of matrices than the existing methods, making it more efficient. We demonstrate examples and experiments to show the efficiency of the method.
Paper Structure (17 sections, 7 theorems, 52 equations, 1 figure, 4 tables, 1 algorithm)

This paper contains 17 sections, 7 theorems, 52 equations, 1 figure, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local dual space
  4. Newton's method with deflation
  5. Characterizing deflation-one singularity
  6. Two-step Newton's method
  7. Perturbations
  8. Quadratic convergence
  9. Examples
  10. Experiments
  11. Performance
  12. Effectiveness
  13. Stability
  14. Comparison
  15. Efficiency
  16. ...and 2 more sections

Key Result

Theorem 4

Let $\xi\in\mathbb{C}^n$ be a deflation-one singular zero of a polynomial system $f\in\mathbb{C}[X_1,\dots, X_n]^n$, then $Dg(\xi,0)$ is of full column rank for almost all choices of $\boldsymbol{\lambda}_2\in\mathbb{C}^{\kappa}$ if and only if the linear operator is invertible for almost all choices of $v\in \ker Df(\xi)$, where $\Pi_{\ker Df(\xi)}$ is the Hermitian projection to $\ker Df(\xi)$.

Figures (1)

  • Figure 1: The description of Example \ref{['running_example_1.1']}. The point $x$ is depicted in red, while the blue point $x'$ and the black point $x"$ are obtained through the refinement steps. The orange plane represents the kernel of the Jacobian of $f$. The origin of the coordinate axes has been shifted to a suitable point due to scaling issues. The second figure shows the projection onto $y=x$ plane. The solid blue and red lines are contours of $\|f\|=10^{-3.5}$ and $10^{-3}$ respectively and dotted circles describe the distance from $\xi$ to $x$ and $x'$. The distance from $\xi$ to $x"$ is negligible compared to the other distances and thus omitted.

Theorems & Definitions (23)

  • Definition 1
  • Remark 2
  • Example 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Example 6: Example \ref{['running_example_1']} continued
  • Remark 7
  • Corollary 8
  • ...and 13 more