Abstract:
here has been a revolution in our computational capabilities for integer matrix problems, particularly for sparse and structured matrices. The complexity has dropped dramatically forproblems such as Diophantine and rational system solution, rank, determinant, characteristic polynomial, and Smith normal form (invariant factors). High performance implementations are starting to appear. I'll present the key methods that have driven this development and discuss a few applications which illustrate the state of the art regarding the size of problem that can be handled by a system such as LinBox. Generally speaking, approximation rules in scientific computation, but fails utterly on some questions. I'll offer some speculations about the role exact computation may come to play.