A note on a paper by P. Amodio and F. Mazzia (Q5952353)

Let \(A\in\mathbb{R}^{n\times n}\) be a nonsingular matrix, let \(A^{(t)}\), \(t = 2,\dots,n\), be the matrices of the Gaussian elimination directly after the \(t\)-th elimination before pivoting is applied. With \(A^{(1)}:= A\), \(U := A^{(n)}\) the particular growth factor \(\rho_n(A)\) of \textit{P. Amodio} and \textit{F. Mazzia} [BIT 39, No.~3, 385-402 (1999; Zbl 0944.65026)] is defined by \[ \rho_n(A):=\frac{\max_t\|A^{(t)}\|_\infty}{\|A\|_\infty} , \] where \(\|\cdot \|_\infty\) denotes the usual row sum norm.~With the condition number \(\kappa(A):=\|A\|_\infty\|A^{-1}\|_\infty\) the author shows by an example that \(\kappa(U)>\rho_n(A)\kappa(A)\) can hold for an \(M\)-matrix \(A\) and some pivoting strategy (in contrast to a result of Amodio and Mazzia). In addition, he presents a pivoting strategy for \(M\)-matrices which guarantees \(\rho_n(A)=1\) and \(\kappa(U)\leq \kappa(A)\).