On parallelizing the Clifford algebra product for CLIFFORD

TL;DR

This paper introduces a parallelized method for computing Clifford products in the CL_{p,q} algebra using Maple's multithreading capabilities, aiming to improve computational efficiency.

Contribution

A new parallel procedure `cmulWpar' for Clifford product computation in Maple, leveraging multithreading and Walsh functions, with benchmarking against existing methods.

Findings

`cmulWpar' outperforms existing procedures in benchmarks.

Parallelization significantly reduces computation time.

Potential for further improvements with multi-core processors.

Abstract

We present, as a proof of concept, a way to parallelize the Clifford product in CL_{p,q} for a diagonalized quadratic form as a new procedure `cmulWpar' in the \Clifford package for \Maple(R). The procedure uses a new `Threads' module available under Maple 15 (and later) and a new \Clifford procedure `cmulW' which computes the Clifford product of any two Grassmann monomials in \CL_{p,q} with a help of Walsh functions. We benchmark `cmulWpar' and compare it to two other procedures `cmulNUM' and `cmulRS' from \Clifford. We comment on how to improve `cmulWpar' by taking advantage of multi-core processors and multithreading available in modern processors.