Given an approximation to a multiple isolated solution of a system of polynomial equations, we 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 first-order deflation method creates an augmented system that has the multiple isolated solution of the original system as a regular solution.
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. In particular: the number of higher-order deflation steps is bounded by the number of variables.
2000 Mathematics Subject Classification. Primary 65H10. Secondary 14Q99, 68W30.
Key words and phrases. Deflation, isolated singular solutions, Newton's method, multiplicity, polynomial systems, reconditioning, symbolic-numeric computations.