Computational Aspects of Geometric Algebra Products of Two Homogeneous Multivectors

TL;DR

This paper provides a comprehensive computational analysis of geometric algebra products for full homogeneous multivectors, establishing tight bounds on arithmetic operations and demonstrating optimal algorithms for these cases.

Contribution

It introduces a complete study of geometric algebra products for full homogeneous multivectors, including tight bounds and optimal algorithms for practical applications.

Findings

Tight bounds on arithmetic operations for products of full homogeneous multivectors.

Existence of algorithms achieving these bounds.

Focus on practical geometric algebra applications.

Abstract

Studies on time and memory costs of products in geometric algebra have been limited to cases where multivectors with multiple grades have only non-zero elements. This allows to design efficient algorithms for a generic purpose; however, it does not reflect the practical usage of geometric algebra. Indeed, in applications related to geometry, multivectors are likely to be full homogeneous, having their non-zero elements over a single grade. In this paper, we provide a complete computational study on geometric algebra products of two full homogeneous multivectors, that is, the outer, inner, and geometric products of two full homogeneous multivectors. We show tight bounds on the number of the arithmetic operations required for these products. We also show that algorithms exist that achieve this number of arithmetic operations.