Fast and Accurate Multi-Argument Interval Evaluation of Polynomials

The verification of the existence of certain spherical t- designs involves the evaluation of a degree-t polynomial Jt at a very large number of (interval) arguments. To make the overall verification process feasible computationally, this evaluation must be fast, and the enclosures for the function values must be affected with only modest over-estimation. We discuss several approaches for multi-argument interval evaluation of the polynomial Jt and show how they can be adapted to other polynomials p. One particularly effective new method is based on expanding the polynomial p around several points xi_j and truncating each resulting expansion Pepsiv_J to a lower-degree polynomial.