Smoothed analysis of componentwise condition numbers for sparse matrices

TL;DR

This paper conducts a smoothed analysis of componentwise condition numbers for sparse matrices, revealing that their expected logarithmic bounds grow logarithmically with matrix size, which implies improved stability in computations.

Contribution

It provides the first smoothed analysis bounds for componentwise condition numbers of sparse matrices in determinant, inversion, and linear system solving.

Findings

Expected log condition numbers are of order O(log n)

Small bounds on smoothed loss of accuracy for triangular systems

Improved understanding of numerical stability for sparse matrices

Abstract

We perform a smoothed analysis of the componentwise condition numbers for determinant computation, matrix inversion, and linear equations solving for sparse n times n matrices. The bounds we obtain for the ex- pectations of the logarithm of these condition numbers are, in all three cases, of the order O(log n). As a consequence, small bounds on the smoothed loss of accuracy for triangular linear systems follow.