Sharp condition-number bounds for growth factors of Higham matrices in Gaussian elimination

TL;DR

This paper refines bounds on the growth factors of Higham matrices in Gaussian elimination, providing sharp, condition-number-dependent estimates that improve upon previous universal bounds.

Contribution

It establishes sharp, condition-number-dependent bounds for Higham matrices' growth factors, refining Drury's universal bound of 2 with a quantitative approach.

Findings

Derived sharp scalar Schur-complement inequality

Provided condition-number-dependent bounds for Higham matrices

Improved entrywise growth bounds for accretive-dissipative matrices

Abstract

Higham's conjecture on the growth factor of complex symmetric positive definite matrices is a longstanding problem in the stability theory of Gaussian elimination without pivoting. It asserts that every complex matrix $A=B+iC$ with $B$ and $C$ real symmetric positive definite, is called Higham matrix and has growth factor $\rho_n(A)<2$. In 2013, Drury [Linear Algebra Appl. \textbf{439} (2013), no.~10, 3129--3133] proved that $\rho_n(A)\le 2$. In fact, we will see his sectorial determinant method can be refined to give the strict bound $\rho_n(A)<2$ for each fixed Higham matrix; however, the resulting constant $1+\delta_A^2$ depends on the matrix $A$. In this paper, we establish sharp condition-number-dependent lower and upper bounds for the growth factors of Higham matrices, thereby providing a quantitative refinement of Drury's result. The main ingredient is a sharp scalar Schur-complement inequality, proved via a two-dimensional domination.We also obtain corresponding sharp scalar and diagonal estimates for accretive-dissipative matrices, and an improved entrywise growth bound for that broader class.