A New Upper Bound For the Growth Factor in Gaussian Elimination with Complete Pivoting

TL;DR

This work advances the theoretical understanding of Gaussian elimination with complete pivoting by breaking Wilkinson's long-standing exponential growth bound. By embedding Hadamard-type determinant constraints into a refined linear programming framework and then transforming non-linear relations into a linear program, the authors derive a new upper bound on the growth factor: $g_n(\mathbb{C}) \le n^{\frac{\ln n}{2[2+(2-\sqrt{2})\ln 2]} + 0.91}$, with leading constant $\alpha=[2(2+(2-\sqrt{2})\ln 2)]^{-1} \approx 0.20781$. The approach combines generalized determinant bounds for low-rank perturbations, a relaxed LP that preserves asymptotics, and a duality-based asymptotic analysis via a continuous optimization model, leading to a rigorous bound that improves the exponential constant for the growth factor in practice and theory. The results also show potential applicability to other pivoting strategies and provide insights into the trade-off between rapid growth and maintaining large numerical rank. Overall, the paper deepens the mathematical understanding of pivoting stability and offers a concrete path toward tighter worst-case guarantees in numerical linear algebra.

Abstract

The growth factor in Gaussian elimination measures how large the entries of an LU factorization can be relative to the entries of the original matrix. It is a key parameter in error estimates, and one of the most fundamental topics in numerical analysis. We produce an upper bound of $n^{0.2079 \ln n +0.91}$ for the growth factor in Gaussian elimination with complete pivoting -- the first improvement upon Wilkinson's original 1961 bound of $2 \, n ^{0.25\ln n +0.5}$.

Figures (2)

• Figure 1: Comparing our Improved Linear Program to Wilkinson's LP: Figure (a) illustrates the difference between Wilkinson's bound for $g_n(\mathbb{C})$ (Inequality \ref{['eqn:wilk']}) and the upper bound produced by the optimal value of Program \ref{['prog:lin']} for $n \le 5000$. Figure (b) is a scatter plot of the pairs $(k,\ell)$ for which the corresponding inequality in Program \ref{['prog:lin']} is tight for a numerically computed optimal solution at $n = 5000$. The grey shaded triangle shows the set of $(k,\ell)$ corresponding to constraints of Program \ref{['prog:lin']}, with Wilkinson's constraints parameterized by $(k,0)$, and the black dots represent the subset of those constraints that are active for the numerically computed optimal solution. For $n =5000$, almost none of Wilkinson's constraints are active. The red line $k + \ell = \sqrt{2} k$ is the set of constraints used to prove Theorem \ref{['thm:main']}, and the green line denotes the asymptotically tight constraints for the feasible point produced in Subsection \ref{['sub:tight']}. While the points on the purple line $k + \ell = n$ improves the objective value, these constraints do not play a role in the asymptotic leading term of the solution to the linear program.
• Figure 2: The potential inaccuracies of partial pivoting. Surprisingly, the computed solution barely has any correct significant digits! Observing the middle elements of the exact and computed solution, one can almost feel the bits being chopped off at the end. This is not caused by the condition number, as cond($A$) is only $45$. No warning or error message is given.

Theorems & Definitions (7)

• Theorem 1.1
• Proposition 3.1
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Lemma 4.1