Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices

Abstract

Let \orig{A} be any matrix and let $A$ be a slight random perturbation of \orig{A}. We prove that it is unlikely that $A$ has large condition number. Using this result, we prove it is unlikely that $A$ has large growth factor under Gaussian elimination without pivoting. By combining these results, we bound the smoothed precision needed by Gaussian elimination without pivoting. Our results improve the average-case analysis of Gaussian elimination without pivoting performed by Yeung and Chan (SIAM J. Matrix Anal. Appl., 1997).Comment: corrected some minor mistake