Fast polynomial evaluation and composition

TL;DR

The paper introduces the fast_polynomial library for Sage, enabling efficient multivariate polynomial evaluation with various schemes, including a novel method for non-power-of-two term counts, optimized for performance and extensibility.

Contribution

It presents a new polynomial evaluation scheme that improves upon classical methods for non-power-of-two term counts and offers a flexible, parallelized implementation in Sage.

Findings

New evaluation scheme improves performance for non-power-of-two terms

Library supports multiple evaluation schemes and types

Parallelization and memory optimization enhance efficiency

Abstract

The library \emph{fast\_polynomial} for Sage compiles multivariate polynomials for subsequent fast evaluation. Several evaluation schemes are handled, such as H\"orner, divide and conquer and new ones can be added easily. Notably, a new scheme is introduced that improves the classical divide and conquer scheme when the number of terms is not a pure power of two. Natively, the library handles polynomials over gmp big integers, boost intervals, python numeric types. And any type that supports addition and multiplication can extend the library thanks to the template design. Finally, the code is parallelized for the divide and conquer schemes, and memory allocation is localized and optimized for the different evaluation schemes. This extended abstract presents the concepts behind the \emph{fast\_polynomial} library. The sage package can be downloaded at \url{http://trac.sagemath.org/sage_trac/ticket/13358}.