  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).
