A unified approach to evaluation algorithms for multivariate polynomials

We present a unified framework for most of the known and a few new evaluation algorithms for multivariate polynomials expressed in a wide variety of bases including the Bernstein-Bézier, multinomial (or Taylor), Lagrange and Newton bases. This unification is achieved by considering evaluation algorithms for multivariate polynomials expressed in terms of L-bases, a class of bases that include the Bernstein-Bézier, multinomial, and a rich subclass of Lagrange and Newton bases. All of the known evaluation algorithms can be generated either by considering up recursive evaluation algorithms for L-bases or by examining change of basis algorithms for L-bases. For polynomials of degree $n$ in $s$ variables, the class of up recursive evaluation algorithms includes a parallel up recurrence algorithm with computational complexity $O(n^{s+1})$, a nested multiplication algorithm with computational complexity $O(n^s \log n)$ and a ladder recurrence algorithm with computational complexity $O(n^s)$. These algorithms also generate a new generalization of the Aitken-Neville algorithm for evaluation of multivariate polynomials expressed in terms of Lagrange L-bases. The second class of algorithms, based on certain change of basis algorithms between L-bases, include a nested multiplication algorithm with computational complexity $O(n^s)$, a divided difference algorithm, a forward difference algorithm, and a Lagrange evaluation algorithm with computational complexities $O(n^s)$, $O(n^s)$ and $O(n)$ per point respectively for the evaluation of multivariate polynomials at several points.