# Error Analysis of Triangular Linear Systems

## Abstract

The vast majority of the occurrences of condition numbers in the study of linear systems of equations involve the normwise condition number *κ*(*A*). Almost invariably, the use of *κ*(*A*) is enough to provide a satisfying explanation of the phenomena observed in practice.

The case of triangular systems of linear equations provides, in contrast, an example in which *κ*(*A*) turns out to be inadequate. Practitioners long observed that triangular systems of equations are generally solved to high accuracy in spite of being, in general, ill-conditioned. Thus, for instance, J.H. Wilkinson wrote: “In practice one almost invariably finds that if *L* is ill-conditioned, so that ∥*L*∥∥*L* ^{−1}∥≫1, then the computed solution of *Lx*=*b* (or the computed inverse) is far more accurate than [what forward stability analysis] would suggest.”

*L*,

*b*) instead of the normwise condition number. The second goal of this chapter is to prove that when

*L*and \(b\in \mathbb {R}^{n}\) are drawn from standard Gaussian distributions, then we have \(\mathbb {E}(\log \mathsf {Cw}(L,b))=\mathcal {O}(\log n)\). This bound, together with some backward error analysis, yields bounds for \(\mathbb {E}(\mathsf {LoP}(L^{-1}b))\) that are much smaller than the one above, as well as closer to the loss of precision observed in practice.