## Section: New Results

Keywords : polynomial matrix, structured matrix, matrix factorization, algebraic complexity, reduction to matrix multiplication, asymptotically fast algorithm.

### Algorithms and Software for High Performance Linear Algebra

Participants : C.-P. Jeannerod, G. Villard.

Our study of asymptotically fast algorithms for the most basic
operations on *polynomial matrices* has appeared in [23] .
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 [54] some algorithms for LSP/LQUP matrix decomposition
that rely on matrix multiplication and
whose complexity is shown to be rank-sensitive.
In [58] 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 [39] . 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.