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The present work is devoted to the eigenvalue asymptotic expansion of the Toeplitz matrix $T_{n}(a)$ whose generating function $a$ is complex valued and has a power singularity at one point. As a consequence, $T_{n}(a)$ is non-Hermitian and we know that the eigenvalue computation is a non-trivial task in the non-Hermitian setting for large sizes. We follow the work of Bogoya, B\"ottcher, Grudsky, and Maximenko and deduce a complete asymptotic expansion for the eigenvalues. After that, we apply matrix-less algorithms, in the spirit of the work by Ekstr\"om, Furci, Garoni, Serra-Capizzano et al, for computing those eigenvalues. Since the inner and extreme eigenvalues have different asymptotic behaviors, we worked on them independently, and combined the results to produce a high precision global numerical and matrix-less algorithm. The numerical results are very precise and the computational cost of the proposed algorithms is independent of the size of the considered matrices for each eigenvalue, which implies a linear cost when all the spectrum is computed. From the viewpoint of real world applications, we emphasize that the matrix class under consideration includes the matrices stemming from the numerical approximation of fractional diffusion equations. In the final conclusion section a concise discussion on the matter and few open problems are presented., Comment: 21 pages

We consider the numerical solution of time-dependent space tempered fractional diffusion equations. The use of Crank-Nicolson in time and of second-order accurate tempered weighted and shifted Gr\"unwald difference in space leads to dense (multilevel) Toeplitz-like linear systems. By exploiting the related structure, we design an ad-hoc multigrid solver and multigrid-based preconditioners, all with weighted Jacobi as smoother. A new smoothing analysis is provided, which refines state-of-the-art results expanding the set of the suitable Jacobi weights. Furthermore, we prove that if a multigrid method is effective in the non-tempered case, then the same multigrid method is effective also in the tempered one. The numerical results confirm the theoretical analysis, showing that the resulting multigrid-based solvers are computationally effective for tempered fractional diffusion equations.

Under appropriate technical assumptions, the simple-loop theory allows to deduce various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function $f$. Independently and under the milder hypothesis that $f$ is even and monotonic over $[0,\pi]$, matrix-less algorithms have been developed for the fast eigenvalue computation of large Toeplitz matrices, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions predicted by the simple-loop theory, combined with the extrapolation idea. Here we focus our attention on a change of variable, followed by the asymptotic expansion of the new variable, and we adapt the matrix-less algorithm to the considered new setting. Numerical experiments show a higher precision (till machine precision) and the same linear computation cost, when compared with the matrix-less procedures already presented in the relevant literature. Among the advantages, we concisely mention the following: a) when the coefficients of the simple-loop function are analytically known, the algorithm computes them perfectly; b) while the proposed algorithm is better or at worst comparable to the previous ones for computing the inner eigenvalues, it is extremely better for the computation of the extreme eigenvalues.

The eigenvalues of Toeplitz matrices $T_{n}(f)$ with a real-valued symbol $f$, satisfying some conditions and tracing out a simple loop over the interval $[-\pi,\pi]$, are known to admit an asymptotic expansion with the form \[ \lambda_{j}(T_{n}(f))=f(d_{j,n})+c_{1}(d_{j,n})h+c_{2}(d_{j,n})h^{2}+O(h^{3}), \] where $h=\frac{1}{n+1}$, $d_{j,n}=\pi j h$, and $c_k$ are some bounded coefficients depending only on $f$. The numerical results presented in the literature suggests that the effective conditions for the expansion to hold are weaker and reduce to an even character of $f$, to a fixed smoothness, and to its monotonicity over $[0,\pi]$. \\ In this note we investigate the superposition caused over this expansion, when considering a linear combination of symbols that is \[ \lambda_{j}\big(T_{n}(f_0)+\beta_{n}^{(1)} T_{n}(f_{1}) + \beta_{n}^{(2)} T_{n}(f_{2}) +\cdots\big), \] where $ \beta_{n}^{(t)}=o\big(\beta_{n}^{(s)}\big)$ if $t>s$ and the symbols $f_{j}$ are either simple loop or satisfy the weaker conditions mentioned before. We prove that the asymptotic expansion holds also in this setting under mild assumptions and we show numerically that there is much more to investigate, opening the door to linear in time algorithms for the computation of eigenvalues of large matrices of this type. The problem is of concrete interest in particular in the case where the coefficients of the linear combination are functions of $h$, considering spectral features of matrices stemming from the numerical approximation of standard differential operators and distributed order fractional differential equations, via local methods such as Finite Differences, Finite Elements, Isogeometric Analysis etc.

The analysis of the spectral features of a Toeplitz matrix-sequence $\left\{T_{n}(f)\right\}_{n\in\mathbb N}$, generated by a symbol $f\in L^1([-\pi,\pi])$, real-valued almost everywhere (a.e.), has been provided in great detail in the last century, as well as the study of the conditioning, when $f$ is nonnegative a.e. Here we consider a novel type of problem arising in the numerical approximation of distributed-order fractional differential equations (FDEs), where the matrices under consideration take the form \[ \mathcal{T}_{n}=c_0T_{n}(f_0)+c_{1} h^h T_{n}(f_{1})+c_{2} h^{2h} T_{n}(f_{2})+\cdots+c_{n-1} h^{(n-1)h}T_{n}(f_{n-1}), \] $c_0,c_{1},\ldots, c_{n-1} \in [c_*,c^*]$, $c^*\ge c_*>0$, independent of $n$, $h=\frac{1}{n}$, $f_j\sim g_j$, $g_j=|\theta|^{2-jh}$, $j=0,\ldots,n-1$. Since the resulting generating function depends on $n$, the standard theory cannot be applied and the analysis has to be performed using new ideas. Few selected numerical experiments are presented, also in connection with matrices that come from distributed-order FDE problems, and the adherence with the theoretical analysis is discussed together with open questions and future investigations.

In the present note we consider a type of matrices stemming in the context of the numerical approximation of distributed order fractional differential equations (FDEs): from one side they could look standard, since they are, real, symmetric and positive definite. On the other hand they present specific difficulties which prevent the successful use of classical tools. In particular the associated matrix-sequence, with respect to the matrix-size, is ill-conditioned and it is such that a generating function does not exists, but we face the problem of dealing with a sequence of generating functions with an intricate expression. Nevertheless, we obtain a real interval where the smallest eigenvalue belongs, showing also its asymptotic behavior. We observe that the new bounds improve those already present in the literature and give a more accurate spectral information, which are in fact used in the design of fast numerical algorithms for the associated large linear systems, approximating the given distributed order FDEs. Very satisfactory numerical results are presented and critically discussed, while a section with conclusions and open problems ends the current note.

The goal of this paper is to create a fruitful bridge between the numerical methods for approximating partial differential equations (PDEs) in fluid dynamics and the (iterative) numerical methods for dealing with the resulting large linear systems. Among the main objectives are the design of new efficient iterative solvers and a rigorous analysis of their convergence speed. The link we have in mind is either the structure or the hidden structure that the involved coefficient matrices inherit, both from the continuous PDE and from the approximation scheme: in turn, the resulting structure is used for deducing spectral information, crucial for the conditioning and convergence analysis, and for the design of more efficient solvers. As specific problem we consider the incompressible Navier-Stokes equations, as numerical technique we consider a novel family of high order accurate Discontinuous Galerkin methods on staggered meshes, and as tools we use the theory of Toeplitz matrices generated by a function (in the most general block, multi-level form) and the more recent theory of Generalized Locally Toeplitz matrix-sequences. We arrive at a quite complete picture of the spectral features of the underlying matrices and this information is employed for giving a forecast of the convergence history of the conjugate gradient method, together with a discussion on new more advanced techniques (involving preconditioning, multigrid, multi-iterative solvers). Several numerical tests are provided and critically illustrated in order to show the validity and the potential of our analysis.

In this note, we study the fast solution of Toeplitz linear systems with coefficient matrix $T_n(f)$, where the generating function $f$ is nonnegative and has a unique zero at zero of any real positive order $\theta$. As preconditioner we choose a matrix ${\tau}_n(f)$ belonging to the so-called $\tau$ algebra, which is diagonalized by the sine transform associated to the discrete Laplacian. In previous works, the spectral equivalence of the matrix sequences $\{{\tau}_n(f)\}_n $ and $\{T_n(f) \}_n$ was proven under the assumption that the order of the zero is equal to $2$: in other words the preconditioned matrix sequence $\{{\tau}^{-1}_n(f)T_n(f) \}_n $ has eigenvalues, which are uniformly away from zero and from infinity. Here we prove a generalization of the above result when $\theta<2$. Furthermore, by making use of multiple step preconditioning, we show that the matrix sequences $\{{\tau}_n(f)\}_n $ and $\{T_n(f) \}_n$ are essentially spectrally equivalent for every $\theta>2$, i.e., for every $\theta>2$, there exist $m_\theta$ and a positive interval $[\alpha_\theta,\beta_\theta]$ such that all the eigenvalues of $\{{\tau}^{-1}_n(f)T_n(f) \}_n $ belong to this interval, except at most $m_\theta$ outliers larger than $\beta_\theta$. Such a nice property, already known only when $\theta$ is an even positive integer greater than 2, is coupled with the fact that the preconditioned sequence has an eigenvalue cluster at one, so that the convergence rate of the associated preconditioned conjugate gradient method is optimal. As a conclusion we discuss possible generalizations and we present selected numerical experiments.

Let A(x) be a holomorphic family of bounded self-adjoint operators on a separable Hilbert space H and let A(x)_n be the orthogonal compressions of A(x) to the span of first n elements of an orthonormal basis of H. The problem considered here is to approximate the spectrum of A(x) using the sequence of eigenvalues of A(x)_n. We show that the bounds of the essential spectrum and the discrete spectral values outside the bounds of essential spectrum of A(x) can be approximated uniformly on all compact subsets by the sequence of eigenvalue functions of A(x)_n. The known results for a bounded selfadjoint operator, are translated into the case of a holomorphic family of operators. Also an attempt is made to predict the existence of spectral gaps that may occur between the bounds of essential spectrum of A(0) = A and study the effect of holomorphic perturbation of operators in the prediction of spectral gaps. As an example, gap issues of some block Toeplitz-Laurent operators are discussed. The pure linear algebraic approach is the main advantage of the results here., Comment: 23 Pages

In recent contributions, algebraic multigrid methods have been designed and studied from the viewpoint of the spectral complementarity. In this note we focus our efforts on specific applications and, more precisely, on large linear systems arising from the approximation of weighted Laplacian with various boundary conditions. We adapt the multigrid idea to this specific setting and we present and critically discuss a wide numerical experimentation showing the potentiality of the considered approach., Comment: arXiv admin note: substantial text overlap with arXiv:0804.3016