Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices
Arvind Sankar, Daniel A. Spielman, Shang-Hua Teng
TL;DR
This work develops a smoothed-analysis framework for Gaussian perturbations of arbitrary matrices to bound the likelihood of large condition numbers and large growth factors in Gaussian elimination without pivoting. The authors derive explicit tail bounds for the condition number $\kappa(A)$ and for the growth factors $\rho_U(A)$ and $\rho_L(A)$, and show how these bounds translate into a logarithmic-smoothed-precision bound for solving linear systems via Gaussian elimination without pivoting. They extend the analysis to zero-preserving perturbations of symmetric matrices with diagonals, obtaining analogous control of inverse norms and growth factors. The results unify and extend prior average-case analyses (e.g., Demmel, Edelman, Yeung-Chan) by providing bounds that hold in every neighborhood of inputs, and they outline open problems, including extensions to pivoting strategies and other perturbation models, as well as recent progress improving the bounds.
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).