A note on the growth factor in Gaussian elimination for Higham matrices

TL;DR

The paper addresses the growth factor in Gaussian elimination for generalized Higham matrices $A=B+iC$, with $B$ and $C$ Hermitian positive definite. It derives a κ-dependent bound, $\frac{4\kappa}{(1+\kappa)^2} \le \rho_n(A) \le \frac{2(1+\kappa^2)}{(1+\kappa)^2} \le 2$, where $\kappa=\max\{\kappa(B),\kappa(C)\}$, and shows this bound tight as $\kappa\to\infty$, thereby confirming Higham's conjecture that $\rho_n(A) < 2$ for Higham matrices. The results are obtained through Schur-complement-based analysis and eigenvalue/condition-number inequalities, complemented by numerical experiments on PDE-inspired discretizations that corroborate the theoretical bounds. Overall, the work resolves a long-standing open problem by tying the growth factor to the conditioning of the Hermitian parts and establishing a universal upper bound of 2 for this class of matrices.

Abstract

The Higham matrix is a complex symmetric matrix A=B+iC, where both B and C are real, symmetric and positive definite and $\mathrm{i}=\sqrt{-1}$ is the imaginary unit. For any Higham matrix A, Ikramov et al. showed that the growth factor in Gaussian elimination is less than 3. In this paper, based on the previous results, a new bound of the growth factor is obtained by using the maximum of the condition numbers of matrixes B and C for the generalized Higham matrix A, which strengthens this bound to 2 and proves the Higham's conjecture.

Figures (1)

• Figure 1: Left: the variation curve of $f(\kappa)=\frac{4\kappa}{(1+\kappa)^2}$ with $\kappa$ decreasing. Right: the variation curve of $g(\kappa)=\frac{2(1+\kappa ^2)}{(1+\kappa )^2}$ with $\kappa$ increasing.

Theorems & Definitions (16)

• Lemma 2.1: I.K
• Lemma 2.2: I.K
• Lemma 2.3: H.J
• Lemma 2.4: FZ
• Lemma 2.5: FZ
• Theorem 2.6: H.J
• Corollary 2.1: Wang
• Lemma 2.7: ML
• Corollary 2.2: ML
• Theorem 2.8