Some New Results on the Maximum Growth Factor in Gaussian Elimination

TL;DR

This work investigates the maximum growth factor $g$ in Gaussian elimination with complete pivoting, addressing a long-standing question by blending numerical optimization with rigorous analysis. The authors prove a stability lemma linking near-CP computations to exact CP cases, analyze growth under constrained-entry sets, and establish a tight connection between floating-point and exact arithmetic for large mantissas. They provide computer-assisted lower bounds that yield $g[CP_n(\mathbb{R})] \ge 1.0045\,n$ for $n>10$ and $\limsup_n g[CP_n(\mathbb{R})]/n \ge 2.525$, plus a super-linear bound for rook pivoting $g[RP_n(\mathbb{R})] > \tfrac{1}{641}n^{1.669}$, and demonstrate that the maximum growth can exceed $n$ for fairly small $n$ (e.g., $n\ge 11$). The results imply that the old conjecture of growth bounded by $n$ is unlikely to hold asymptotically, and they illuminate the landscape of growth across pivoting strategies, including both Hadamard-related cases and advanced numerical constructions. The combination of optimization-based search, careful stability analysis, and extrapolation yields both concrete lower bounds and qualitative insight into the growth phenomenon, with practical implications for numerical linear algebra and stability analyses.

Abstract

This paper combines modern numerical computation with theoretical results to improve our understanding of the growth factor problem for Gaussian elimination. On the computational side we obtain lower bounds for the maximum growth for complete pivoting for $n=1:75$ and $n=100$ using the Julia JuMP optimization package. At $n=100$ we obtain a growth factor bigger than $3n$. The numerical evidence suggests that the maximum growth factor is bigger than $n$ if and only if $n \ge 11$. We also present a number of theoretical results. We show that the maximum growth factor over matrices with entries restricted to a subset of the reals is nearly equal to the maximum growth factor over all real matrices. We also show that the growth factors under floating point arithmetic and exact arithmetic are nearly identical. Finally, through numerical search, and stability and extrapolation results, we provide improved lower bounds for the maximum growth factor. Specifically, we find that the largest growth factor is bigger than $1.0045n$ for $n>10$, and the lim sup of the ratio with $n$ is greater than or equal to $3.317$. In contrast to the old conjecture that growth might never be bigger than $n$, it seems likely that the maximum growth divided by $n$ goes to infinity as $n \rightarrow \infty$.

Figures (4)

• Figure 1: We compare the (modulus of the) determinant and pivots of $A^{(k)}$, $k = 1,...,n$, under GECP for three examples of size $n=100$: Red: Wilkinson's bound; Yellow: a particular $n=100$ Hadamard matrix; Blue: our observed maximum matrix. (a) reveals that at least on an admittedly muted log scale, the observed determinant curve qualitatively is bending in a manner resembling Wilkinson's bound, while the Hadamard data feels qualitatively different, and thus, less relevant. (b) suggests the same conclusions as those of (a) and also suggests that "slow and steady wins the race" rather than "greedy."
• Figure 2: The ratio between numerically observed growth factors and matrix size for $n$ equals $1$ to $75$ and $100$. Only the values for sizes $n = 1,2,3,4$ are known mathematically to be the exact maximal growth factor though we suspect at least for the smaller values of $n$ we are achieving the maximum with our JuMP software. This data leads us to make Conjecture \ref{['conj:superlinear']}.
• Figure 3: Our run_model function performs growth optimization from a random start. We encourage the reader to examine the constraints: they correspond to the mathematical constraints a completely pivoted matrix satisfies and are easy to read. The variable $x$ is a three-dimensional array that stores the Gaussian elimination "pyramid,” e.g., $x[(i,j,k)]$ is the $(i,j)^{th}$ entry of the $k^{th}$ step of Gaussian elimination and the $(k,k,k)$ entry is the $k^{th}$ pivot.
• Figure 4: The above figure shows that we can go from matrices that are completely pivoted in floating point to matrices that are completely pivoted in exact arithmetic. Lemma \ref{['lm:cp_eps_stable']} proves that this is possible and Algorithm \ref{['alg:eps_stable']} provides a pseudocode implementation (a Julia implementation may be found in the https://github.com/alanedelman/CompletePivotingGrowthourrepo). For instance, Algorithm \ref{['alg:eps_stable']} has fully automated Edelman's exact arithmetic extension of Gould's finite precision counterexample to Conjecture \ref{['conj:cryer']}, and provides some answers to Edelman's perturbation question for growth factor edelman1992complete.

Theorems & Definitions (31)

• Conjecture 1.1: Folklore?
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Simplified Version of Theorem \ref{['thm:restrict']}
• Theorem 1.5: Simplified Version of Theorem \ref{['thm:float']}
• Conjecture 1.6
• Conjecture 1.7
• Lemma 2.1
• proof
• Corollary 2.2