Componentwise condition numbers of random sparse matrices

TL;DR

This paper establishes an O(log n) bound on the expected logarithm of the componentwise condition number for random sparse matrices, implying improved bounds on numerical stability for solving triangular systems.

Contribution

It provides the first probabilistic bound on the componentwise condition number of random sparse matrices, linking it to numerical stability in linear algebra.

Findings

Expected log condition number is O(log n) for random sparse matrices.

Small average loss of accuracy in triangular system solutions.

Implications for numerical stability analysis.

Abstract

We prove an O(log n) bound for the expected value of the logarithm of the componentwise (and, a fortiori, the mixed) condition number of a random sparse n x n matrix. As a consequence, small bounds on the average loss of accuracy for triangular linear systems follow.