Section: New Results
Algorithms and Software for High Performance Linear Algebra
Our study of asymptotically fast algorithms for the most basic operations on polynomial matrices has appeared in  . There, the target matrices have entries in K[x] for K an arbitrary commutative field and the emphasis is on reductions to the multiplication problem. In particular, we highlight the role played by two problems when designing asymptotically fast algorithms for any of the operations above: computing minimal bases of some matrix Padé approximants, and expanding/reconstructing polynomial matrix fractions.
For linear algebra over a field , we have studied fast algorithms both for dense matrices and some structured matrices. We propose in  some algorithms for LSP/LQUP matrix decomposition that rely on matrix multiplication and whose complexity is shown to be rank-sensitive. In  we present some asymptotically fast probabilistic algorithms for structured linear system solving (and their applications). There, the target matrices are Toeplitz-, Hankel- or Vandermonde-like matrices with large displacement rank.
We have found a new algorithm for exact sparse linear system solving  . It is the first algorithm whose worst-case complexity estimate may be less than the one of matrix multiplication (sparse enough matrices). The first published version is based on a conjecture for structured projections in the block-Krylov approach. Since then a full proof has been obtained.