Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices

Let A be any matrix and let A be a slight random perturbation of 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). Partially supported by NSF grant CCR-0112487 Partially supported by an Alfred P. Sloan Foundation Fellowship, and NSF grant CCR-0112487 Partially supported by an Alfred P. Sloan Foundation Fellowship, and NSF grant CCR-9972532.