Efficient Development of Competitive Mathematica Solutions Based on Geometric Algebra with GAALOPWeb
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Advances in Applied Clifford Algebras
Efficient Development of Competitive Mathematica Solutions Based on Geometric Algebra with GAALOPWeb R. Alves∗ , D. Hildenbrand, C. Steinmetz and P. Uftring Abstract. In this paper we present a new tool for Mathematica users, based on the new web-based geometric algebra algorithm optimizer (GAALOP). GAALOPWeb for Mathematica now supports the Mathematica user with an intuitive interface for the development, testing and visualization of geometric algebra algorithms, combining the geometric intuitiveness of geometric algebra with an efficient development of algorithms for Wolfram mathematica. We particularly illustrate this integration using an implementation of a distance geometry Problem, which consists of finding three-dimensional embeddings of graphs, that is very suitable for geometric algebra. Keywords. GAALOP, GAALOPWeb, Mathematica, Geometric algebra (GA), Conformal geometric algebra (CGA).
1. Introduction In this paper we present our approach for the integration of Geometric AlR gebra to Wolfram Mathematica . It is based on a new web interface of GAALOP [11], the geometric algebra algorithm optimizer that produces robust and fast implementations in geometric algebra [8]. Mathematica is a software developed by Stephen Wolfram around 1980, and is a widely used and very powerful tool. It has many implemented functions, a great documentation, and it is suitable for symbolic computation [25]. There are already Geometric Algebra implementations for mathematica, the GrassmannAlgebra package [2], the CGA (Conformal Geometric Algebra)Package of Kondo et al [13], which is a package for 5D CGA used to solve an This article is part of the ENGAGE 2019 Topical Collection on Geometric Algebra for Computing, Graphics and Engineering edited by Linwang Yuan (EiC), Werner Benger, Dietmar Hildenbrand, and Eckhard Hitzer. ∗ Corresponding
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origami problem; the geometric algebra library of Terje Vold, a package for three-dimensional space and four-dimensional spacetime computations [23]. Mathematica is also used to perform symbolic computations for GMac, a tool for generating optimized geometric algebra code for C# among other languages, developed by Ahmad Eid [7]. Our approach for the implementation of geometric algebra in mathematica is based on geometric algebra algorithm optimizer (GAALOP) as described in Sect. 4. GAALOP and GMac are not just another libraries but mainly differ from the other tools by focusing on the optimization of geometric algebra algorithms in terms of runtime and robustness. While GMac uses the commercial mathematica as computer algebra system for the symbolic optimization, GAALOP uses the free available Maxima as computer algebra system. We briefly describe the conformal geometric algebra (CGA) in Sect. 2, because it is required for the distance geometry application (see Sect. 3), which we chose for illustration purposes of this paper. After the description of GAALOP in Sect. 4 we
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