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.

中文翻译：

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两个同质多矢量的几何代数积的计算方面

几何代数中乘积的时间和存储成本的研究仅限于具有多个等级的多向量仅具有非零元素的情况。这样可以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，在与几何相关的应用中，多矢量可能是完全同质的，它们的非零元素在单个等级上。在本文中，我们对两个完全齐次的多重矢量的几何代数乘积，即两个完全齐次的多重向量的外，内和几何乘积提供了完整的计算研究。我们对这些产品所需的算术运算数量显示出严格的界限。我们还表明存在可以实现这种数量的算术运算的算法。