Long Integers and Polynomial Evaluation with Estrin's Scheme

Abstract

In this paper the problem of univariate polynomial evaluation is considered. When both polynomial coefficients and the evaluation "point" are integers, unbalanced multiplications (one factor having many more digits than the other one) in classical Ruffini-Horner rule do not let computations completely benefit of sub quadratic methods, like Karatsuba, Toom-Cook and Schonhage-Strassen's. We face this problem by applying an approach originally proposed by Estrin to augment parallelism exploitation in computation. We show that it is also effective in the sequential case, whenever data dimensions grow, e.g. in the long integer case. We add some adjustments to Estrin's proposal obtaining a smoother behavior around corner cases, and to avoid performance degradation when most of the coefficients are zero. This way, a new general algorithm is obtained, improving both theoretical complexity and actual performance. The algorithm itself is very simple, and its use can be usefully extended to evaluation of polynomials on rationals or on polynomials (polynomial composition). Some tests, results and comparisons obtained with PARI/GP are also presented, for both dense and "sparse" polynomials.