Complete pivoting growth of butterfly matrices and butterfly Hadamard matrices

Abstract

The growth problem in Gaussian elimination (GE) remains a foundational question in numerical analysis and numerical linear algebra. Wilkinson resolved the growth problem in GE with partial pivoting (GEPP) in his initial analysis from the 1960s, while he was only able to establish an upper bound for the GE with complete pivoting (GECP) growth problem. The GECP growth problem has seen a spike in recent interest, culminating in improved lower and upper bounds established by Bisain, Edelman, and Urschel in 2023, but still remains far from being fully resolved. Due to the complex dynamics governing the location of GECP pivots, analysis of GECP growth for particular input matrices often estimates the actual growth rather than computes the growth exactly. We present a class of dense random butterfly matrices on which we can present the exact GECP growth. We extend previous results that established exact growth computations for butterfly matrices when using GEPP and GE with rook pivoting (GERP) to now also include GECP for particular input matrices. Moreover, we present a new method to construct random Hadamard matrices using butterfly matrices.

Figures (4)

• Figure 1: Sparsity patterns for (computed, within the given tolerance $\operatorname{tol} = 10^4 \cdot \epsilon_{\operatorname{machine}}$) matrix factors (a) $P$, (b) $L+U$, and (c) $Q$ from then GECP factorization $PBQ = LU$ for $B = B(\tilde{\boldsymbol{\theta}}) \in \operatorname{B}_s(N)$ for $N = 2^{10}$ and $\boldsymbol{\theta}\sim \operatorname{Uniform}([0,2\pi)^{10})$.
• Figure 2: Sparsity patterns for the computed GECP factors of $B$ from \ref{['fig: lex CP']}, now using GECP without an added tolerance parameter to identify potential candidates during each intermediate pivot search.
• Figure 3: Sparsity patterns for the computed GECP factors of $B \in \operatorname{B}_s(2^{10})$ not satisfying \ref{['thm: new']}, with (a)-(c) corresponding to GECP with an added tolerance parameter during the pivot candidate compilation, and (d)-(f) without the added tolerance setting.
• Figure 4: Overlaid histograms of computed growth factors (a) $\rho(B)$ and (b) $\rho_\infty(B)$ where $B \sim \operatorname{B}_s(N,\Sigma_S)$, using GEPP and both GECP with and without an added tolerance parameter to enforce the column-major lexicographic tie-breaking strategy, using $10^4$ trials with $N = 2^8$ and a logarithmic scaling.

Theorems & Definitions (26)

• Proposition 2.1: Tr19
• Proposition 2.2
• Theorem 2.3: PT23
• Lemma 2.4: PT23
• Corollary 2.5
• Theorem 2.6
• Remark 2.7
• Proposition 2.8
• Proof 1
• Lemma 2.9