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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 higherorder 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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