Exact Linear Algebra Algorithmic

Abstract

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.

Simultaneous Conversions with the Residue Number System Using Linear Algebra

“…Furthermore, using Strassen's subcubic algorithm [Strassen 1969] and a stronger bound on m i one can reach even better performance [Dumas et al 2008;FFLAS-FFPACK-Team 2016]. This approach provides the best performance per bit for modular matrix multiplication [Pernet 2015] and thus we rely on it to build our RNS conversion algorithms from Section 3.…”

Computing floating-point logarithms with fixed-point operations

“…Indeed, the floating-point units and memory subsytems of workstations processors are designed to achieve maximum floating-point performance on essential numerical kernels, such as the vector product around which all linear algebra can be built. At the time of writing this article, it remains true that a high-end processor can achieve more floating-point operations than integer operations per seconds [2]. This is mainly due to the wide vector units (such as Intel's AVX extensions) not fully supporting 64-bit integer multiplication.…”