Approximation Num\'erique de Racines Isol\'ees Multiples de Syst\`emes Analytiques

TL;DR

This paper introduces a certified symbolic-numeric method for approximating multiple isolated roots of systems, using deflation sequences and Bergman kernel-based analysis to ensure numerical stability and regularization.

Contribution

It develops a novel deflation sequence construction from approximate points and provides a rigorous numerical analysis using Bergman kernel and  theory, without relying on thresholds.

Findings

Constructed deflation sequence from approximate roots.

Provided a numerical analysis framework using Bergman kernel.

Developed an -theory-based algorithm free of  thresholds.

Abstract

The approximation of a multiple isolated root is a difficult problem. In fact the root can even be a repulsive root for a fixed point method like the Newton method. However there exists a huge literature on this topic but the answers given are not satisfactory. Numerical methods allowing a local convergence analysis work often under specific hypotheses. This viewpoint favouring numerical analysis forgets the geometry and the structure of the local algebra. Thus appeared so-called symbolic-numeric methods, yet full of lessons, but their precise numerical analysis is still missing. We propose in this paper a method of symbolic-numeric kind, whose numerical treatment is certified. The general idea is to construct a finite sequence of systems, admitting the same root, and called the deflation sequence, so that the multiplicity of the root drops strictly between two successive systems. So the root becomes regular. Then we can extract a regular square system we call deflated system. We described already the construction of this deflated sequence when the singular root is known. The originality of this paper consists on one hand to construct a deflation sequence from a point close to the root and on the other hand to give a numerical analysis of this method. Analytic square integrable functions build the functional frame. Using the Bergman kernel, reproducing kernel of this functional frame, we are able to give a $\alpha$-theory {\`a} la Smale. Furthermore we present new results on the determination of the numerical rank of a matrix and the closeness to zero of the evaluation map. As an important consequence we give an algorithm computing a deflation sequence free of $\epsilon$, threshold quantity measuring the numerical approximation, meaning that the entry of this algorithm does not involve the variable $\epsilon$.