Beyond and behind linear algebra

Over the last century, linear algebra theory and matrix computations became irreplaceable, not only for high-tech industries, but also in every corner of our computerised society. Most of the time, any given problem (linear or not) is reduced to finding the solution of a linear system. Thus, the possibility of solving large linear systems in a reasonable amount of time using parallel algorithms has cast a shadow on any other kind of computing approach. However, physical problems are not linear in general: they often represent strongly coupled phenomena which inherently resist software parallelisation. Devising ways to overcome or circumvent computational difficulties has been the research core of the Qualitative Computing (QC) group at Cerfacs ever since it was established in 1987, [Chaitin-Chatelin and Frayssé, 1996, Chatelin, 2012a, Chatelin, 2012b, Chatelin, 2016]. In this summary we review some of the reasons why number tools other than matrices may be better fitted for extreme computing and how they offer a radically new perspective on the current field of linear algebra.