Computing Clipped Products

TL;DR

This paper introduces algorithms for efficiently computing specific segments of a product's coefficients or digits, which is useful in various numerical and polynomial computations, improving performance in practical scenarios.

Contribution

It generalizes the 'middle product' concept to compute arbitrary coefficient ranges within products, enhancing efficiency in classical and Karatsuba multiplication methods.

Findings

Algorithms for computing specific coefficient ranges

Improved performance in polynomial and integer multiplication

Effective combination of classical and Karatsuba methods

Abstract

Sometimes only some digits of a numerical product or some terms of a polynomial or series product are required. Frequently these constitute the most significant or least significant part of the value, for example when computing initial values or refinement steps in iterative approximation schemes. Other situations require the middle portion. In this paper we provide algorithms for the general problem of computing a given span of coefficients within a product, that is the terms within a range of degrees for univariate polynomials or range digits of an integer. This generalizes the "middle product" concept of Hanrot, Quercia and Zimmerman. We are primarily interested in problems of modest size where constant speed up factors can improve overall system performance, and therefore focus the discussion on classical and Karatsuba multiplication and how methods may be combined.