A note on Jacobi Being More Accurate Than $QR

In [SIAM J. Matrix Anal. Appl., 13 (1992) pp. 1204-1245], Demmel and Veselic present a theoretical and experimental analysis to show that the Jacobi method is more accurate than the $QR$ method when computing the eigenvalues of positive definite matrices. They show that the error caused by the Jacobi method depends on the size of a factor $\rho $, which is related to the singular values of certain matrices associated with the Jacobi iterates. Their experiments suggest that $\rho = O( 1 )$. However, in this note a family of matrices and orderings is presented for which $\rho = O( N )$, where $N$ is the dimension of the matrix.