In a series of papers the author and others have studied an asymptotic expansion of the errors of the eigenvalue approximation, using the spectral symbol, in connection with Toeplitz (and Toeplitz-like) matrices, that is, $E_{ j,n}$ in $\lambda_j(A_n)=f(\theta_{ j,n})+E_{ j,n}$, $A_n=T_n(f)$, $f$ real-valued cosine polynomial. In this paper we instead study an asymptotic expansion of the errors of the equispaced sampling grids $\theta_{ j,n}$, compared to the exact grids $\xi_{ j,n}$ (where $\lambda_j(A_n)=f(\xi_{ j,n})$), that is, $E_{ j,n}$ in $\xi_{ j,n}=\theta_{ j,n}+E_{ j,n}$. We present an algorithm to approximate the expansion. Finally we show numerically that this type of expansion works for various kind of Toeplitz-like matrices (Toeplitz, preconditioned Toeplitz, low-rank corrections of them). We critically discuss several specific examples and we demonstrate the superior numerical behavior of the present approach with respect to the previous ones.
Categories: math.NA cs.NA