A Low-Memory Time-Efficient Implementation of Outermorphisms for Higher-Dimensional Geometric Algebras

TL;DR

This paper presents a memory-efficient and time-competitive method for implementing outermorphisms in high-dimensional geometric algebras, enhancing practical software applications.

Contribution

It introduces a novel low-memory implementation of outermorphisms that maintains performance, addressing scalability issues in geometric algebra software.

Findings

Requires significantly less memory than traditional methods

Maintains comparable computational speed for high-dimensional algebras

Facilitates practical prototyping in geometric algebra applications

Abstract

From the beginning of David Hestenes rediscovery of geometric algebra in the 1960s, outermorphisms have been a cornerstone in the mathematical development of GA. Many important mathematical formulations in GA can be expressed as outermorphisms such as versor products, linear projection operators, and mapping between related coordinate frames. Over the last two decades, GA-based mathematical models and software implementations have been developed in many fields of science and engineering. As such, efficient implementations of outermorphisms are of significant importance within this context. This work attempts to shed some light on the problem of optimizing software implementations of outermorphisms for practical prototyping applications using geometric algebra. The approach we propose here for implementing outermorphisms requires orders of magnitude less memory compared to other common approaches, while being comparable in time performance, especially for high-dimensional geometric algebras.