Abstract only

Abstract

Most of the methods of linear algebra, even the iterative ones, are exact in principle but are applied approximately. Over the past half century numerical analysis has made great strides in the understanding of approximation in matrix computation. But also, roughly over the past three decades, a solid algorithmic basis for exact sparse and dense matrix computation has emerged. We argue that the computer algebra community has the interests, techniques, motivations, and applications to develop high performance computational capabilities for exact linear algebra It is a tale of two equivalence relations, similarity and matrix equivalence. These were characterized in the 19th century via the Jordan canonical form and Smith normal form respectively. Associated with these are the computational concerns of numerical linear algebra, eigenvalues and linear system solving respectively. But applications of exact linear algebra tend to need some or all of the invariants themselves associated with these equivalences. From rank to characteristic polynomial, computation of these invariants resists pure numerical solution and provides an important role for computer algebra. We discuss the state of the art and current issues in algorithms and software system design for exact linear algebra over the integers, the rational numbers, and over finite fields. We show some of the applications in which large scale exact linear algebra has been used.