Accurate Computations and Applications of Some Classes of Matrices

Abstract

Performing an algorithm with high relative accuracy is a very desirable goal. High relative accuracy means that the relative errors of the computations are of the order of machine precision, independently of the size of the condition number. This goal is difficult to assure although in recent years there have been some advances, in particular in the field of Numerical Linear Algebra. Up to now, computations with high relative accuracy are guaranteed only for a few classes of matrices, mainly for some subclasses of M-matrices and for some subclasses of totally positive matrices. Previously, a reparametrization of the matrices is needed. We review this procedure related with the high relative accuracy computations of these matrices. We also present some recent applications of the two classes of matrices mentioned previously. On the one hand, applications of M-matrices to the linear complementarity problem. On the other hand, applications of totally positive matrices to Computer Aided Geometric Design.