Exact Linear Algebra Algorithmic: Theory and Practice

Exact linear algebra is a core component of many symbolic and algebraic computations, as it often delivers competitive theoretical complexities and also better harnesses the efficiency of modern computing infrastructures. In this tutorial we will present an overview on the recent advances in exact linear algebra algorithmic and implementation techniques, and highlight the few key ideas that have proven successful in their design. As an illustration, we will study in more details the computation of some matrix normal forms over a finite field or the ring of polynomials, specific to computer algebra. In particular, we will give a special care to the design and implementation of parallel exact linear algebra routines, trying to emphasize the similarities and distinctness with parallel numerical linear algebra. We aim to provide the working computer algebraist with a set of best practices for the use or the design of exact linear algebra software, together with an overview on a few still unresolved algorithmic problems in the field.