Dense Linear Algebra over Finite Fields: the FFLAS and FFPACK packages
Résumé
In the last past two decades, several efforts have been made to reduce exact linear algebra problems to matrix multiplication in order to provide algorithms with optimal asymptotic complexity. To provide efficient implementations of such algorithms one need to be careful with the underlying arithmetic. It is well know that modular technique such as Chinese remainder algorithm or p-adic lifting allow in practice to achieve better performances especially when word size arithmetic are used. Therefore, finite field arithmetics becomes an important core for efficient exact linear algebra libraries. In this paper we study different implementations of finite fields in order to achieve efficiency for basic linear algebra routines such as dot product or matrix multiplication; our goal being to provide an exact alternate to numerical BLAS library. Following matrix multiplication reductions, our kernel has many symbolic linear algebra applications: symbolic triangularization, system solving, exact determinant computation and matrix inversion are then studied and we demonstrate the efficiency of these reductions in practice.