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

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