An Extended Implementation Framework for Geometric Algebra Operations on Systems of Coordinate Frames of Arbitrary Signature

There is a steadily increasing interest in applying Geometric Algebra (GA) in diverse fields of science and engineering. Consequently, we need better software implementations to accommodate such increasing demands that widely vary in their possible uses and goals. For large-scale complex applications having many integrating parts, such as Big Data and Geographical Information Systems, we should expect the need for integrating several GAs to solve a given problem. Even within the context of a single GA space, we often need several interdependent systems of coordinates to efficiently model and solve the problem at hand. Future GA software implementations must take such important issues into account in order to scale, extend, and integrate with existing software systems, in addition to developing new ones, based on the powerful language of GA. This work attempts to provide GA software developers with a self-contained description of an extended framework for performing linear operations on GA multivectors within systems of interdependent coordinate frames of arbitrary metric. The work explains the mathematics and algorithms behind this extended framework and discusses some of its implementation schemes and use cases. If properly implemented, the extended framework can significantly reduce the memory requirements for implementing Geometric Algebras with larger dimensions, especially for systems based on the symbolic processing of multivector scalar coefficients.