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The aim of this paper is to study how efficiently we evaluate a system of multivariate polynomials and their partial derivatives in homotopy continuation methods. Our major tool is an extension of the Horner scheme, which is popular in evaluating a univariate polynomial, to a multivariate polynomial. But the extension is not unique, and there are many Horner factorizations of a given multivariate polynomial which require different numbers of multiplications. We present exact method for computing a minimum Horner factorization, which can process smaller size polynomials, as well as heuristic methods for a smaller number of multiplications, which can process larger size polynomials. Based on these Horner factorization methods, we then present methods to evaluate a system of multivariate polynomials and their partial derivatives. Numerical results are shown to verify the effectiveness and the efficiency of the proposed methods.

EFFICIENT EVALUATION OF POLYNOMIALS AND THEIR PARTIAL DERIVATIVES IN HOMOTOPY CONTINUATION METHODS