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**1 - 2**of**2**### GraVisMa 2009 Geometric Algebra Computing on the CUDA Platform

"... Geometric Algebra (GA) is a mathematical framework that allows a compact, geometrically intuitive description of geometric relationships and algorithms. These algorithms require significant computational power because of the intrinsically high dimensionality of geometric algebras. Algorithms in an n ..."

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Geometric Algebra (GA) is a mathematical framework that allows a compact, geometrically intuitive description of geometric relationships and algorithms. These algorithms require significant computational power because of the intrinsically high dimensionality of geometric algebras. Algorithms in an n-dimensional GA require2 n elements to be computed for each multivector. GA is not restricted to a maximum of dimensions, so arbitrary geometric algebras can be constructed over a vector space Vn. Since computations in GA can be highly parallelized, the benefits of a parallel computing architecture can lead to a significant speed-up compared to standard CPU implementations, where elements of the algebra have to be calculated sequentially. An upcoming approach of coping with parallel computing is to use general-purpose computation on graphics processing units (GPGPU). In this paper, we focus on the Compute Unified Device Architecture (CUDA) from NVIDIA [9]. We present a code generator that takes as input the description of an arbitrary geometric algebra and produces an implementation of geometric products for the underlying algebra on the CUDA platform. Keywords: Geometric Algebra, Geometric Computing, GPU, CUDA.

### ∗ Embedded Systems and Applications Group

"... Abstract—Geometric Algebra (GA), a generalization of quaternions, is a very powerful form for intuitively expressing and manipulating complex geometric relationships common to engineering problems. The actual evaluation of GA expressions, though, is extremely compute intensive due to the high-dimens ..."

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Abstract—Geometric Algebra (GA), a generalization of quaternions, is a very powerful form for intuitively expressing and manipulating complex geometric relationships common to engineering problems. The actual evaluation of GA expressions, though, is extremely compute intensive due to the high-dimensionality of data being processed. On standard desktop CPUs, GA evaluations take considerably longer than conventional mathematical formulations. GPUs do offer sufficient throughput to make the use of concise GA formulations practical, but require power far exceeding the budgets for most embedded applications. While the suitability of low-power reconfigurable accelerators for evaluating specific GA computations has already been demonstrated, these often required a significant manual design effort. We present a proof-of-concept compile flow combining symbolic and hardware optimization techniques to automatically generate accelerators from the abstract GA descriptions without user intervention that are suitable for high-performance embedded computing. I.