Geometric algebra: an overview about different implementation representations and memory requirements

Several implementations of geometric algebra are available, and many have been approved in the past and can be found in various software codes. For low-dimensional algebras, representations as a sum of basis blades are typically the most suitable. However, isomorphic or irreducible matrix representations are also options. Techniques such as compression on a per-coordinate, per-grade or per-group basis, as well as the use of sparse matrices, can help reduce memory consumption. In the case of high-dimensional algebras, factorized representations can significantly decrease the memory needed for blades and versors, reducing it to O(n2), as opposed to the O(2n) required to represent a complete basis of blades. This is particularly relevant in the context of quantum computing, where high dimensionality and memory usage are increasingly important considerations. This paper reviews various implementation strategies, classifies the corresponding software tools and libraries and highlights recent advancements aimed at reducing memory consumption within the tool Gaalop. This article is part of the theme issue 'Modern applications of geometric algebra'.