Numerics

So far, it has been shown how geometric algebra can be used to describe geometry elegantly and succinctly. The goal of this chapter is to show how numerical evaluation methods can be applied to geometric-algebra expressions. It will be seen that such expressions have a straightforward representation as tensor contractions, to which standard linear-algebra evaluation methods may be applied. This chapter therefore starts with a discussion of the mapping from geometric-algebra expressions to tensor operations in Sect. 5.1. Next it is shown, in Sect. 5.2, how linear multivector equations can be solved using the tensor representation. The representation of uncertain multivectors is then introduced in Sect. 5.3, and its limits are investigated in Sect. 5.4, where an expression for the bias term in the error propagation of general bilinear functions is derived. Uncertain multivectors are particularly useful in the representation of uncertain geometric entities and uncertain transformations. Geometric algebra is ideally suited for the latter in particular. Sections 5.5, 5.6, and 5.7 give details of the representation of uncertain geometric entities in projective, conformal, and conic space, respectively.

Two linear least-squares estimation methods, the Gauss{Markov and the Gauss{Helmert method, are discussed in some detail in Sects. 5.8 and 5.9. The Gauss{Helmert method is particularly interesting, since it incorporates the uncertainty of the data into the optimization process. While the Gauss{ Markov and Gauss{Helmert estimation methods are well known, it is a long way from the de nition of a stochastic model to the corresponding estimation algorithm, which is why these estimation methods, which are used later on, are described here in some detail. In this way, the notation necessary for later chapters is introduced. The chapter concludes with a discussion, in Sect. 5.10, of how the Gauss{Helmert and Gauss{Markov methods may be applied to practical problems.

The foundations for this approach to solving general multivector equations were laid by the author and Sommer in [146], and were extended to uncertain algebraic entities in collaboration with W. Forstner [136].