Approximating the Perfect Sampling Grids for Computing the Eigenvalues of Toeplitz-like Matrices Using the Spectral Symbol

TL;DR

This work addresses the problem of accurately computing eigenvalues of Toeplitz-like matrices by exploiting the spectral symbol $f$ and a perfect sampling grid $\xi_{j,n}$ with $\lambda_j(A_n)=f(\xi_{j,n})$. It develops an $α$-order expansion $ξ_{j,n}=θ_{j,n}+\sum_{k=1}^α d_k(θ_{j,n}) h^k$ and introduces two matrix-less algorithms to (i) estimate the expansion coefficients $d_k$ on a small grid and (ii) interpolate/extrapolate these coefficients to arbitrary $n$ to obtain eigenvalue estimates $λ_j(A_n) \approx f(ξ_{j,n}^{(α)})$, all within the generalized locally Toeplitz (GLT) framework. Numerical experiments on Laplacian, bilaplacian, and preconditioned Toeplitz-like matrices demonstrate that the proposed method yields accurate eigenvalue predictions and outperforms prior grid-based approaches, particularly for large $n$ and near $θ=0$. The results show promise for fast, matrix-free spectrum computation across a broad class of Toeplitz-like operators and motivate future work on non-monotone symbols and block/multilevel extensions.

Abstract

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 $λ_j(A_n)=f(θ_{j,n})+E_{j,n}$, $A_n=T_n(f)$, $f$ real-valued cosine polynomial. In this paper we instead study an asymptotic expansion of the errors of the equispaced sampling grids $θ_{j,n}$, compared to the exact grids $ξ_{j,n}$ (where $λ_j(A_n)=f(ξ_{j,n})$), that is, $E_{j,n}$ in $ξ_{j,n}=θ_{j,n}+E_{j,n}$. 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.

Figures (10)

• Figure 1: Three grids $\theta_{j,n_k}$ with $k=1,2,4$, $n_k=2^{k-1}(n_1+1)-1$, and $n_1=3$. The subgrids $\theta_{j_k,n_k}$, which is the same for any $k$, is indicated by blue circles ($j_k=2^{k-1}j_1$, $j_1=\{1,\ldots, n_1\}$).
• Figure 2: Example \ref{['exmp:main:errorlambdavsxi']}: Symbol $f(\theta)=(2-2\cos(\theta))^2$ (dashed pink line) and the two expansions approximate the errors $E_{j,n}^{\lambda,\theta}=\lambda_j(T_n(f))-f(\theta_{j,n})=f(\xi_{j,n})-f(\theta_{j,n})$ and $E_{j,n}^\xi=\xi_{j,n}-\theta_{j,n}$.
• Figure 3: Example \ref{['exmp:main:errorxi']}: Errors $(E_{j,n}^\xi=\xi_{j,n}-\theta_{j,n})$ and scaled errors $(E_{j,n}^\xi/h)$. Left: Errors $E_{j,n_k}^\xi$, for $n_k=2^{k-1}(n_1+1)-1$, $n_1=100$, and $k=1,\ldots,4$. Right: Scaled errors $E_{j,n_k}^\xi/h_k$ where $h_k=1/(n_k+1)$.
• Figure 4: Example \ref{['exmp:numerical:laplace']}: Expansion of the grid $\xi_{j,n_1}$ associated with the matrix $A_{n_1}=T_{n_1}(f)+R_{n_1}$, where $f(\theta)=2-2\cos(\theta)$ and $R_n$ is a small rank perturbation (coming from a Neumann boundary condition). The functions $d_k(\theta)$ are well approximated by $\tilde{d}_k(\theta)$.
• Figure 5: Example \ref{['exmp:numerical:bilaplace']}: Computed expansion function approximations $\tilde{d}_k(\theta_{j,n_1})$, $k=1,\ldots,\alpha$ for $\alpha=2$ (left) and $\alpha=3$ (right), and $n_1=100$. Note the erratic behavior for $\tilde{d}_2(\theta_{j,n_1})$ and $\tilde{d}_3(\theta_{j,n_1})$ close to $\theta=0$.

Theorems & Definitions (6)

• Example 1
• Example 2
• Remark 1
• Example 3
• Example 4
• Example 5