Numerical techniques for computing the inertia of products of matrices of rational numbers

Abstract

Consider a rational matrix, particularly one whose entries have large numerators and denominators, but which is presented as a product of very sparse matrices with relatively small entries. We report on a numerical algorithm which computes the inertia of such a matrix in the nonsingular case and effectively exploits the product structure. We offer a symbolic/numeric hybrid algorithm for the singular case. We compare these methods with previous purely symbolic ones. By "purely symbolic" we refer to methods which restrict themselves to exact arithmetic and can assure that errors of approximation do not affect the results. Using an application in the study of Lie Groups as a plentiful source of examples of problems of this nature we explore the relative speeds of the numeric and hybrid methods as well as the range of applicability without error.