Computing the Rank and a Small Nullspace Basis of a Polynomial Matrix

We reduce the problem of computing the rank and a nullspace basis of a univariate polynomial matrix to polynomial matrix multiplication. For an input n x n matrix of degree d over a field K we give a rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension n and degree d. If the latter multiplication is done in MM(n,d)=softO(n^omega d) operations, with omega the exponent of matrix multiplication over K, then the algorithm uses softO(MM(n,d)) operations in K. The softO notation indicates some missing logarithmic factors. The method is randomized with Las Vegas certification. We achieve our results in part through a combination of matrix Hensel high-order lifting and matrix minimal fraction reconstruction, and through the computation of minimal or small degree vectors in the nullspace seen as a K[x]-moduleComment: Research Report LIP RR2005-03, January 200

Asymptotically fast polynomial matrix algorithms for multivariable systems

We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer algebra techniques, minimal basis computations and matrix fraction expansion/reconstruction, and to polynomial matrix multiplication. Such reductions eventually imply that all these problems can be solved in about the same amount of time as polynomial matrix multiplication

Algorithms for zero-dimensional ideals using linear recurrent sequences

Inspired by Faug\`ere and Mou's sparse FGLM algorithm, we show how using linear recurrent multi-dimensional sequences can allow one to perform operations such as the primary decomposition of an ideal, by computing the annihilator of one or several such sequences.Comment: LNCS, Computer Algebra in Scientific Computing CASC 201

Computational linear algebra over finite fields

We present here algorithms for efficient computation of linear algebra problems over finite fields

General computational approach for optimal fault detection

We propose a new computational approach to solve the optimal fault detection problem in the most general setting. The proposed procedure is free of any technical assumptions and is applicable to both proper and non-proper systems. This procedure forms the basis of an integrated numerically reliable state-space algorithm, which relies on powerful descriptor systems techniques to solve the underlying computational subproblems. The new algorithm has been implemented into a Fault Detection Toolbox for Matlab