## Section: New Results

Keywords : polynomial matrix, matrix fraction, matrix multiplication, asymptotic complexity, rank, nullspace, determinant, reduced form, inversion, Padé approximants.

### Algorithms and Software for High Performance Linear Algebra

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

We have pursued our study of asymptotically fast algorithms for the most basic operations on polynomial matrices, with an emphasis on reductions to the multiplication problem. The target matrices are typically
n×
n of degree
dwith univariate entries in
for
an arbitrary commutative field. Among the studied operations are computing the
*rank* , a
*nullspace* basis, the
*determinant* or a
*reduced form* ; there is also the task of computing the matrix of fractions which is equal to the
*inverse* of a generic polynomial matrix
[6] . In
[51] 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* . We show that reducing these two problems to polynomial matrix multiplication implies the same kind of reductions for all the other operations, hence cost estimates in
operations in
where
is the exponent of scalar matrix multiplication, and with
and
two real constants. The latter type of reduction has been developed in
[46] for establishing that a polynomial nullspace basis (almost minimal) can be computed using about the same number of
operations as for multiplying two polynomial matrices. These studies, and an algorithm we propose for computing the Kalman form
[53] , may be useful for the treatment of multivariable linear systems in control.

Part of these theoretical developments yield software components in the LinBox library (see § 5.6 ).