On a perturbation analysis of Higham squared maximum Gaussian elimination growth matrices

Alan Edelman, John Urschel, Bowen Zhu

Abstract: Gaussian elimination is the most popular technique for solving a dense linear system. Large errors in this procedure can occur in floating point arithmetic when the matrix's growth factor is large. In the study of numerical linear algebra, it is often valuable to study and characterize the worst case examples. To this end, in their 1989 paper, Higham and Higham characterized the complete set of real n by n matrices that achieves the maximum growth factor under partial pivoting. Left undone is a sensitivity analysis for these matrices under perturbations. The growth factor of these and nearby matrices is the subject of this work. Through theoretical insights and empirical results, we illustrate the high sensitivity of the growth factor of these matrices to perturbations and show how subtle changes can be strategically applied to matrix entries to significantly reduce the growth.