In-place accumulation of fast multiplication formulae

TL;DR

This paper introduces a novel automatic method to design in-place, fast algorithms for bilinear formulae like polynomial and matrix multiplication, overcoming traditional space and in-place computation challenges.

Contribution

It presents a new automatic approach to create in-place, fast algorithms for bilinear and linear accumulation problems, including polynomial and matrix multiplication.

Findings

Developed algorithms for in-place polynomial multiplication

Extended methods to in-place Strassen-like matrix multiplication

Achieved algorithms that restore inputs after modification

Abstract

This paper deals with simultaneously fast and in-place algorithms for formulae where the result has to be linearly accumulated: some of the output variables are also input variables, linked by a linear dependency. Fundamental examples include the in-place accumulated multiplication of polynomials or matrices, C+=AB. The difficulty is to combine in-place computations with fast algorithms: those usually come at the expense of (potentially large) extra temporary space, but with accumulation the output variables are not even available to store intermediate values. We first propose a novel automatic design of fast and in-place accumulating algorithms for any bilinear formulae (and thus for polynomial and matrix multiplication) and then extend it to any linear accumulation of a collection of functions. For this, we relax the in-place model to any algorithm allowed to modify its inputs, provided that those are restored to their initial state afterwards. This allows us, in fine, to derive unprecedented in-place accumulating algorithms for fast polynomial multiplications and for Strassen-like matrix multiplications.