On the generation of very ill-conditioned integer matrices

Abstract

In this paper we study the generation of an ill-conditioned integer matrix A=[aij] with |aij|≤µ for some given constant µ. Let n be the order of A. We first give some upper bounds of the condition number of A in terms of n and µ. We next propose new methods to generate extremely ill-conditioned integer matrices. These methods are superior to the well-known method by Rump in some respects, namely, the former has a simple algorithm to generate a larger variety of ill-conditioned matrices. In particular we propose a method to generate ill-conditioned matrices with a choice of desirable singular value distributions as benchmark matrices.