Abstract

Abstract. We present algorithms which use only O(x / nonscalar multiplications (i.e. multiplications involving "x " on both sides) to evaluate polynomials of degree n, and proofs that at least x/ are required. These results have practical application in the evaluation of matrix polynomials with scalar coefficients, since the "matrix matrix " multiplications are relatively expensive, and also in determining how many multiplications are needed for polynomials with rational coefficients, since multiplications by integers can in principle be replaced by several additions. Key words. Polynomial evaluation, nonscalar multiplications, rational coefficients, matrix polynomial. 1. Introduction. A well-known result given by Motzkin [23 and Winograd [6 is that, even with preliminary adaptation of the coefficients, at least n/2 multiplications are required to evaluate a polynomial of degree n if the coefficients of the polynomial are algebraically independent. However we frequently wish to evaluate polynomials with rational or integer coefficients for which this result does not

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