How Accurate is Gaussian Elimination?

Abstract

J.H. Wilkinson put Gaussian elimination (GE) on a sound numerical footing in the 1960's when he showed that with partial pivoting the method is stable in the sense of yielding a small backward error. He also derived bounds proportional to the condition number $\kappa(A)$ for the forward error $\| x - \hat{x} \|$, where $\hat{x}$ is the computed solution to $Ax = b$. More recent work has furthered our understanding of GE, largely through the use of componentwise rather than normwise analysis. We survey what is known about the accuracy of GE in both the forward and backward error senses. Particular topics include: classes of matrix for which it is advantageous not to pivot; how to estimate or compute the backward error; iterative refinement in single precision; and how to compute efficiently a bound on the forward error. Key Words: Gaussian elimination, partial pivoting, rounding error analysis, backward error, forward error, condition number, iterative refinement in single precision, growth factor, componentwise bounds, condition estimator.