David Meadon

Let { Λ n = { λ 1 , n , … , λ d n , n } } n ${\left\{{{\Lambda}}_{n}=\left\{{\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right\}\right\}}_{n}$ be a sequence of finite multisets of real numbers such that d _n → ∞ as n → ∞, and let f : Ω ⊂ R d → R $f:{\Omega}\subset {\mathbb{R}}^{d}\to \mathbb{R}$ be a Lebesgue measurable function defined on a domain Ω with 0 < μ _d (Ω) < ∞, where μ _d is the Lebesgue measure in R d ${\mathbb{R}}^{d}$ . We say that { Λ n } n ${\left\{{{\Lambda}}_{n}\right\}}_{n}$ has an asymptotic distribution described by f , and we write { Λ n } n ∼ f ${\left\{{{\Lambda}}_{n}\right\}}_{n}\sim f$ , if (*) lim n → ∞ 1 d n ∑ i = 1 d n F ( λ i , n ) = 1 μ d ( Ω ) ∫ Ω F ( f ( x ) ) d x ${\mathrm{lim}}_{n\to \infty }\frac{1}{{d}_{n}}{\sum }_{i=1}^{{d}_{n}}F\left({\lambda }_{i,n}\right)=\frac{1}{{\mu }_{d}\left({\Omega}\right)}{\int }_{{\Omega}}F\left(f\left(\boldsymbol{x}\right)\right)\mathrm{d}\boldsymbol{x}$ for every continuous function F with bounded support. If Λ _n is the spectrum of a matrix A _n , we say that { A n } n ${\left\{{A}_{n}\right\}}_{n}$ has an asymptotic spectral distribution described by f and we write { A n } n ∼ λ f ${\left\{{A}_{n}\right\}}_{n}{ \sim }_{\lambda } f$ . In the case where d = 1, Ω is a bounded interval, Λ _n ⊆ f (Ω) for all n , and f satisfies suitable conditions, Bogoya, Böttcher, Grudsky, and Maximenko proved that the asymptotic distribution (*) implies the uniform convergence to 0 of the difference between the properly sorted vector [ λ 1 , n , … , λ d n , n ] $\left[{\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right]$ and the vector of samples [ f ( x 1 , n ) , … , f ( x d n , n ) ] $\left[f\left({x}_{1,n}\right),\dots ,f\left({x}_{{d}_{n},n}\right)\right]$ , i.e., (**) lim n → ∞ max i = 1 , … , d n | f ( x i , n ) − λ τ n ( i ) , n | = 0 , ${\mathrm{lim}}_{n\to \infty } {\mathrm{max}}_{i=1,\dots ,{d}_{n}}\vert f\left({x}_{i,n}\right)-{\lambda }_{{\tau }_{n}\left(i\right),n}\vert =0,$ where x 1 , n , … , x d n , n ${x}_{1,n},\dots ,{x}_{{d}_{n},n}$ is a uniform grid in Ω and τ _n is the sorting permutation. We extend this result to the case where d ⩾ 1 and Ω is a Peano–Jordan measurable set (i.e., a bounded set with μ _d (∂Ω) = 0). We also formulate and prove a uniform convergence result analogous to (**) in the more general case where the function f takes values in the space of k × k matrices. Our derivations are based on the concept of monotone rearrangement (quantile function) as well as on matrix analysis arguments stemming from the theory of generalized locally Toeplitz sequences and the observation that any finite multiset of numbers Λ n = { λ 1 , n , … , λ d n , n } ${{\Lambda}}_{n}=\left\{{\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right\}$ can always be interpreted as the spectrum of a matrix A n = diag ( λ 1 , n , … , λ d n , n ) ${A}_{n}=\mathrm{diag}\left({\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right)$ . The theoretical results are illustrated through numerical experiments, and a reinterpretation of them in terms of vague convergence of probability measures is hinted.

Sequences of structured matrices of increasing size, such as generalized locally Toeplitz sequences, arise in many scientific applications and especially in the numerical discretization of linear differential problems. We assume that the eigenvalues of a matrix $$X_n$$ X n , belonging to a sequence of such kind, are given by a regular expansion. Under this working hypothesis, we propose a method for computing approximations of the eigenvalues of $$X_n$$ X n for large n and we provide a theoretical analysis of its convergence. The method is called matrix-less because it does not operate on the matrix $$X_n$$ X n but on a few similar matrices of smaller size combined with an interpolation-extrapolation strategy. The working hypothesis and the performance of the proposed eigenvalue approximation method are benchmarked on several numerical examples, with a special attention to matrices arising from the discretization of variable-coefficient differential problems.

In the current work, we study the eigenvalue distribution results of a class of non-normal matrix-sequences which may be viewed as a low rank perturbation, depending on a parameter $β>1$, of the basic Toeplitz matrix-sequence $\{T_n(e^{\mathbf{i}θ})\}_{n\in\mathbb{N}}$, $\mathbf{i}^2=-1$. The latter of which has obviously all eigenvalues equal to zero for any matrix order $n$, while for the matrix-sequence under consideration we will show a strong clustering on the complex unit circle. A detailed discussion on the outliers is also provided. The problem appears mathematically innocent, but it is indeed quite challenging since all the classical machinery for deducing the eigenvalue clustering does not cover the considered case. In the derivations, we resort to a trick used for the spectral analysis of the Google matrix plus several tools from complex analysis. We only mention that the problem is not an academic curiosity and in fact stems from problems in dynamical systems and number theory. Additionally, we also provide numerical experiments in high precision, a distribution analysis in the Weyl sense concerning both eigenvalues and singular values is given, and more results are sketched for the limit case of $β=1$

We study the spectral properties of flipped Toeplitz matrices of the form $H_n(f)=Y_nT_n(f)$, where $T_n(f)$ is the $n\times n$ Toeplitz matrix generated by the function $f$ and $Y_n$ is the $n\times n$ exchange (or flip) matrix having $1$ on the main anti-diagonal and $0$ elsewhere. In particular, under suitable assumptions on $f$, we establish an alternating sign relationship between the eigenvalues of $H_n(f)$, the eigenvalues of $T_n(f)$, and the quasi-uniform samples of $f$. Moreover, after fine-tuning a few known theorems on Toeplitz matrices, we use them to provide localization results for the eigenvalues of $H_n(f)$. Our study is motivated by the convergence analysis of the minimal residual (MINRES) method for the solution of real non-symmetric Toeplitz linear systems of the form $T_n(f)\mathbf x=\mathbf b$ after pre-multiplication of both sides by $Y_n$, as suggested by Pestana and Wathen.