论文标题
两种均匀多生物的几何代数产物的计算方面
Computational Aspects of Geometric Algebra Products of Two Homogeneous Multivectors
论文作者
论文摘要
关于几何代数中产品的时间和记忆成本的研究仅限于多个等级的多生只有非零元素的情况。这允许为通用目的设计有效的算法；但是，它不能反映几何代数的实际用法。实际上，在与几何形状有关的应用中，多生效可能是完全均匀的，其非零元素在单个等级上。在本文中，我们提供了两种完全均匀多生物的几何代数产物的完整计算研究，即两个完整的均质多生动物的外部，内部和几何产物。我们对这些产品所需的算术操作的数量显示了紧密的界限。我们还表明，存在实现这一数量算术操作的算法。
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.
