Large Growth Happens: Gaussian Elimination with Partial Pivoting on Random Matrices

2026-10-06

AI Summary

This study addresses a longstanding open problem in smoothed analysis: the absence of non-trivial lower bounds on the probability of large growth factors during Gaussian elimination on random matrices. By integrating techniques from probability theory and linear algebra, this work establishes the first non-trivial probability lower bound for large growth phenomena in Gaussian random matrices, proving it to be at least inverse quasi-polynomial, specifically Ω(exp(−c log²(ρ) log(n))). This result departs from traditional average-case analysis assumptions by demonstrating that large growth occurs with substantially higher probability than previously anticipated, thereby refuting standard conjectures in the field. Ultimately, these findings provide a critical theoretical foundation for understanding the numerical stability of Gaussian elimination and related algorithms in practical computation.

Abstract

We prove that the probability that Gaussian elimination with partial pivoting on $n \times n$ random matrices has growth $ρ$ is at least inverse quasi-polynomial: $Ω(\exp(-c \log^2 (ρ)\log(n)))$ for some constant $c > 0$. This lower bound breaks standard conjectures in the smoothed and average-case analysis of Gaussian elimination. To the best of our knowledge, it is the first non-trivial lower bound on the probability of large growth for Gaussian random matrices.

Problem

• Gaussian elimination
• partial pivoting
• growth factor
• random matrices
• smoothed analysis

Innovation

• Gaussian Elimination
• Partial Pivoting
• Random Matrices
• Growth Factor
• Smoothed Analysis