Efficient Evaluation of Polynomials and Their Partial Derivatives in Homotopy Continuation Methods

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.