21 - Fundamental algorithms for nonlinear products

Abstract

The previous chapter looked at how to implement the linear products in geometric algebra. The linearity of these products allows implementation of them using linear algebra or through a simple double loop. However, there are other operations in geometric algebra that are nonlinear (such as inverse, meet, join, and factorization). These cannot be implemented in the same way. This chapter discusses the implementation of such nonlinear geometric algebra operations. The nonlinearity results in more complex algorithms, still reasonably efficient but typically an order of magnitude more time-consuming than linear operations. One needs to compute the inverse of the elements that are constructed in geometric algebra. Those are almost exclusively blades or versors (the only exceptions were the bi-vectors in an exponent, and there is no need to invert those). Invertible blades are always versors, since a k-blade (the outer product of k vectors) can always be written as the geometric product of k vectors (i.e., a k-versor). So if a versor V is inverted, it can also invert blades and thus most of the elements constructed. This chapter gives algorithms for the inverse, for exponentiation, for testing whether multi-vector is a blade or a versor, for blade factorization, and for the efficient computation of meet and join.