Adv. Appl. Clifford Algebras 27 (2017), 599–620 c  2016 Springer International Publishing 0188-7009/010599-22 published online June 10, 2016 DOI 10.1007/s00006-016-0694-6 Advances in Applied Clifford Algebras Geometric Algebra Computing for Heterogeneous Systems D. Hildenbrand ∗ , J. Albert, P. Charrier and Chr. Steinmetz Abstract. Starting from the situation 15 years ago with a great gap between the low symbolic complexity on the one hand and the high numeric complexity of coding in Geometric Algebra on the other hand, this paper reviews some applications showing, that, in the meantime, this gap could be closed, especially for CPUs. Today, the use of Geo- metric Algebra in engineering applications relies heavily on the avail- ability of software solutions for the new heterogeneous computing archi- tectures. While most of the Geometric Algebra tools are restricted to CPU focused programming languages, in this paper, we introduce the new Gaalop (Geometric Algebra algorithms optimizer) Precompiler for heterogeneous systems (CPUs, GPUs, FPGAs, DSPs ...) based on the programming language C++ AMP (Accelerated Massive Parallelism) of the HSA (Heterogeneous System Architecture) Foundation. As a proof- of-concept we present a raytracing application together with some com- puting details and first performance results. Mathematics Subject Classification. Primary 99Z99; Secondary 00A00. Keywords. Geometric Algebra Computing, Ray tracer, C++ AMP, Gaalop. 1. Introduction Especially since the introduction of CGA (Conformal Geometric Algebra) by David Hestenes et al. [11, 16] there has been an increasing interest in using Geometric Algebra (GA) in engineering. The use of Geometric Algebra in engineering applications relies heavily on the availability of an appropri- ate computing technology. The main problem of Geometric Algebra Comput- ing is the exponential growth of data and computations compared to linear algebra, since the multivector of an n-dimensional Geometric Algebra is 2 n - dimensional. For the 5-dimensional Conformal Geometric Algebra, as used in this article, the multivector is already 32-dimensional. In 2000, Gerald Sommer stated in the preface of his book [21]: Today we have to accept a great gap between the low symbolic complexity on the one hand and the high * Corresponding author.