A note on the growth factor in Gaussian elimination for accretive-dissipative matrices (Q2017962)

The \(n\times n\) complex matrix \textbf{A} is accretive-dissipative if it can be written as \(\mathbf{A}= \mathbf{B}+i \mathbf{C}\), where \(\mathbf{B} = \frac{1}{2}(\mathbf{A}+\mathbf{A}^\ast)\) and \(\mathbf{C} = \frac{1}{2i}(\mathbf{A}-\mathbf{A}^\ast)\) are both (Hermitian) positive definite. If \textbf{B}, \textbf{C} are real symmetric positive definite, then \textbf{A} is called a Higham matrix. The growth factor (in Gaussian elimination) of \textbf{A} is defined as a ratio of the maximum element (in absolute value) of any intermediate result of the Gaussian elimination process to the maximum element of the original matrix \textbf{A}. The presented short note proves that if \textbf{A} is accretive-dissipative, then its growth factor in Gaussian elimination is less than 4. If \textbf{A} is a Higham matrix then the growth factor is less than \(2\sqrt{2}\). This result improves those obtained recently and is one step closer to the final solution of Higham's conjecture.