Multidimensional Computational Methods

Abstract

Scientists and engineers wishing to compute with the mathematical objects and operations of dimensioned linear algebra require software tools somewhat different from the tools currently available. The study of computational methods for dimensioned quantities can be called multidimensional methods, analogous to traditional numerical methods. This chapter presents compact data structures for representing dimensioned quantities and efficient algorithms for performing the standard operations of linear algebra. Manipulations such as products, inverses, eigenstructure decomposition, singular value decomposition can be carried out with dimensioned scalars, vectors, and matrices, but we need efficient algorithms to check constraints and compute dimensions. Although the dimensional structures may be more intricate, it is shown that the space and time complexities required for dimensioned operations are no higher than the complexities of the corresponding standard algorithms for dimensionless quantities. Thus, there is no significant computational burden added due to the dimensioned nature of the matrices, for large matrices at least.

The purpose of computing is insight, not numbers. —Richard Hamming