Asymptotically fast polynomial matrix algorithms for multivariable systems
Claude-Pierre Jeannerod, Gilles Villard · International Journal of Control · 2006
We present the asymtotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank; nullspace; determinant; generic inverse reduced form (Giorgi et al. 2003, Storjohann 2003 Storjohann, A. 2003. “High-order lifting and integrality certification”. In J. Symb. Comp. Edited by: Giusti, M and Pardo, LM. Vol. 36, 613–648. Nice, France, 3, , USA Special issue International Symposium on Symbolic and Algebraic Computation (ISSAC’2002). Guest editors:[Crossref] , [Google Scholar], Jeannerod and Villard 2005 Jeannerod, C-P and Villard, G. 2005. Essentially optimal computation of the inverse of generic polynomial matrices. J. Comp., 21: 72–86. [Crossref] , [Google Scholar], Storjohann and Villard 2005 Storjohann, A and Villard, G. July 2005. “Computing the rank and a small nullspace basis of a polynomial matrix”. In Proc. International Symposium on Symbolic and Algebraic Computation, 309–316. Beijing, China: ACM Press. [Crossref] , [Google Scholar]). 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. The algorithms are deterministic, or randomized with certified output in a Las Vegas fashion.