In previous work by several authors, the behavior of the condition numbers of banded Toeplitz matrices was studied as the matrix size tends to infinity. In the present contribution, two main directions are pursued. As a first step, we extend this study to block Toeplitz matrices with blocks of fixed size N. As in the scalar case, we show that even when the symbol generates a Fredholm infinite Toeplitz operator, the condition numbers of the finite matrices may grow at least exponentially. Upper and lower bounds for the condition numbers are obtained, and examples showing that they may grow arbitrarily fast are presented. Then, as a second step, we apply the developed theory to the stability analysis of space-time Galerkin methods, where in time an Isogeometric approach is used with regularity r, 1≤ r≤ p-1, p being the employed polynomial degree. These stability issues are related exactly to the conditioning of block Toeplitz-like matrices with blocks of size N=p-r. Specific examples are treated in detail and related numerical experiments are presented and critically discussed. We finally present a short list of relevant open problems.

In 2021, B & ouml;ttcher, Gasca, Grudsky, and Kozak showed that the limit set of the spectra of tetradiagonal Toeplitz matrices consists of one, two, or three analytic arcs in the complex plane. Building on this result, in two recent papers, the authors constructed and rigorously justified a uniform asymptotic expansion for all eigenvalues in the case where the limit set is a single arc. In the present work, we take the next natural step in addressing the asymptotic approximation of the corresponding eigenvectors in this single-arc setting. The resulting formulas explicitly reveal the structure of the eigenvectors. We also provide numerical examples that illustrate the high accuracy and linear-time computability of the proposed approximations.

We study a family of non-Hermitian tetradiagonal Toeplitz matrices having a limiting set consisting of one analytic arc only. We derive individual asymptotic expansions for all eigenvalues as the matrix size grows to infinity. Additionally, we provide specific expansions for the extreme eigenvalues, which are those approaching the endpoints of the limiting set. Although this family does not belong to the simple-loop class, we managed to extend the existing theory to this case. Our results reveal the intricate details of the eigenvalue structure and allow a high accuracy direct calculation.

We consider Hermitian Toeplitz matrices emerging from finite linear combinations with non-negative coefficients of the differential operators (-1)kd2k/dx2k over the interval (0, 1) after discretizing them on a uniform grid of step size 1/(n + 1). The collective distribution in the Szeg & odblac;-Weyl sense of the eigenvalues of these matrices as n goes to infinity can be described by GUT theory. However, we focus on the asymptotic behavior of the individual eigenvalues, on both the inner eigenvalues in the bulk and on the extreme eigenvalues. The difficulty of the problem is that not only the order of the matrices depends on n but also their so-called symbols. Our main results are third order asymptotic formulas for the eigenvalues in the case k 2. These results reveal some basic phenomena one should expect when considering the problem in full generality. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org /licenses /by-nc-nd /4 .0/).

The present work is devoted to the eigenvalue asymptotic expansion of the Toeplitz matrix Tn(a), whose generating function a is complex -valued and has a power singularity at one point. As a consequence, Tn(a) is non -Hermitian and we know that in this setting, the eigenvalue computation is a nontrivial task for large sizes. First we follow the work of Bogoya, Bo"\ttcher, Grudsky, and Maximenko and deduce a complete asymptotic expansion for the eigenvalues. In a second step, we apply matrixless algorithms, in the spirit of the work by Ekstro"\m, 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 matrixless 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 the entire spectrum is computed. From the viewpoint of real -world applications, we emphasize that the class under consideration includes the matrices stemming from the numerical approximation of fractional diffusion equations. In the final section a concise discussion on the matter and a few open problems are presented.

The eigenvalues of Toeplitz matrices Tn(f) with a real-valued generating function f, satisfying some conditions and tracing out a simple loop over the interval [−π,π], are known to admit an asymptotic expansion with the formλj(Tn(f))=f(σj,n)+c1(σj,n)h+c2(σj,n)h2+O(h3), where h=1/(n+1), σj,n=πjh, and ck are some bounded coefficients depending only on f. The numerical results presented in the literature suggest that the effective conditions for the expansion to hold are weaker and reduce to a fixed smoothness and to having only two intervals of monotonicity over [−π,π].In this article we investigate the superposition caused over this expansion, when considering the following linear combinationλj(Tn(f0)+βn,1Tn(f1)+βn,2Tn(f2)), where βn,1,βn,2 are certain constants depending on n and the generating functions f0,f1,f2 are either simple loop or satisfy the weaker conditions mentioned before.We formally obtain an asymptotic expansion in this setting under simple-loop related 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 including a multilevel setting.The problem is of concrete interest, 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, and Isogeometric Analysis.

In this paper we consider a family of tetradiagonal (= four non-zero diagonals) Toeplitz matrices with a limiting set consisting in one analytic arc only and obtain individual asymptotic expansions for all the eigenvalues, as the matrix size goes to infinity. Additionally, we provide specific expansions for the extreme eigenvalues which are the eigenvalues approaching the extreme points of the limiting set. In contrast to previous related works, we study non-Hermitian Toeplitz matrices having non-canonical distribution and a real limiting set. The considered family does not belong to the so-called simple-loop class, nevertheless we manage to extend the theory to this case. The achieved formulas reveal the fine details of the eigenvalue structure and allow us to directly calculate high accuracy eigenvalues, even for matrices of relatively small size.

It is known that the generating function of a sequence of Toeplitz matrices may not describe the asymptotic distribution of the eigenvalues of the considered matrix sequence in the non-Hermitian setting. In a recent work, under the assumption that the eigenvalues are real, admitting an asymptotic expansion whose first term is the distribution function, fast algorithms computing all the spectra were proposed in different settings. In the current work, we extend this idea to non-Hermitian Toeplitz matrices with complex eigenvalues, in the case where the range of the generating function does not disconnect the complex field or the limiting set of the spectra, as the matrix-size tends to infinity, has one nonclosed analytic arc. For a generating function having a power singularity, we prove the existence of an asymptotic expansion, that can be used as a theoretical base for the respective numerical algorithm. Different generating functions are explored, highlighting different numerical and theoretical aspects; for example, non-Hermitian and complex symmetric matrix sequences, the reconstruction of the generating function, a consistent eigenvalue ordering, the requirements of high-precision data types. Several numerical experiments are reported and critically discussed, and avenues of possible future research are presented.

The present work is devoted to the construction of an asymptotic expansion for the eigenvalues of a Toeplitz matrix T-n(a) as n goes to infinity, with a continuous and real-valued symbol a having a power singularity of degree gamma with 1 < gamma < 2, at one point. The resulting matrix is dense and its entries decrease slowly to zero when moving away from the main diagonal, we apply the so called simple-loop (SL) method for constructing and justifying a uniform asymptotic expansion for all the eigenvalues. Note however, that the considered symbol does not fully satisfy the conditions imposed in previous works, but only in a small neighborhood of the singularity point. In the present work: (i) We construct and justify the asymptotic formulas of the SL method for the eigenvalues lambda(j)(T-n(a)) with j ? epsilon n, where the eigenvalues are arranged in nondecreasing order and epsilon is a sufficiently small fixed number. (ii) We show, with the help of numerical calculations, that the obtained formulas give good approximations in the case j < epsilon n. (iii) We numerically show that the main term of the asymptotics for eigenvalues with j < epsilon n, formally obtained from the formulas of the SL method, coincides with the main term of the asymptotics constructed and justified in the classical works of Widom and Parter.

The present paper is a survey of some of the authors’ results on the asymptotic behavior of individual eigenvalues and eigenvectors of sequences of Toeplitz matrices when their size tends to infinity. The symbols of the matrices are supposed to have power singularities and are special cases of so-called Fisher–Hartwig symbols.

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 previous works, Bohemian matrices have attracted the attention of several researchers for their rich combinatorial structure, and they have been studied intensively from several points of view, including height, determinants, characteristic polynomials, normality, and stability. Here we consider a selected number of examples of upper Hessenberg and Toeplitz Bohemian matrix sequences whose entries belong to the population P = {0, +/- 1}, and we propose a connection with the spectral theory of Toeplitz matrix sequences and Generalized Locally Toeplitz (GLT) matrix sequences in order to give results on the localization and asymptotical distribution of their spectra and singular values. Numerical experiments that support the mathematical study are reported. A conclusion section ends the note in order to illustrate the applicability of the proposed tools to more general cases.

Under appropriate technical assumptions, the simple-loop theory allows to derive 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 monotone over [0, π ], 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 vastly better for the computation of the extreme eigenvalues; a mild deterioration in the quality of the numerical experiments is observed when dense Toeplitz matrices are considered, having generating function of low smoothness and not satisfying the simple-loop assumptions.

In the present article 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 cause 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 to, showing also its asymptotic behavior. We observe that the new bounds improve those already present in the literature and give more accurate pieces of 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 work.

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.

Low-density polyethylene (LDPE) sheets (3.0 ± 0.1 cm) received sequential treatment, first by the action of direct-current low-pressure plasma (DC-LPP) with a 100% oxygen partial pressure, 3.0 × 10−2 mbar pressure, 600 V DC tension, 5.6 cm distance, 6-min treatment. Then, sheets were submitted to TiO2 photocatalysis at UV radiation at 254 nm (TiO2/UV) with a pH value of 4.5 ± 0.2 and a TiO2 concentration of 1 gL−1. We achieved a complementary effect on the transformation of LDPE films. With the first treatment, ablation was generated, which increased hydrophilicity. With the second treatment, the cavities appeared. The changes in the LDPE sheets’ hydrophobicity were measured using the static contact angle (SCA) technique. The photocatalytic degradation curve at 400 h revealed that the DC-LPP photocatalysis sequential process decreased SCA by 82°. This was achieved by the incorporation of polar groups, which increased hydrophilicity, roughness, and rigidity by 12 and 38%, respectively. These sequential processes could be employed for LDPE and other material biodegradation pretreatment.

Different types of structures and substrates are used for urban extensive green roofs. However, there is not enough information about the performance of these structures and substrates for growing edible plants in tropical climate conditions. This study evaluates the best combination of three different modular extensive green roof structures and two types of local substrates (compost and soil + biochar) for Lactuca sativa L. var. crispa (lettuce) growth. Physicochemical and biological properties of substrates and growth variables of plants were measured for the different green roof structures, at the beginning and at the end of the experiment (seven weeks after), and under rainy and dry climatic conditions. According to the model obtained, the monolithic multilayer structure, high concentrations of Mg, elevated amounts of 600 ?m aggregates and less substrate bulk density were linked with a higher L. sativa yield. The substrate used, however, was not associated with this result. Additionally, it was found that some of the physicochemical and microbiological properties of substrates changed at the end of the experiment. However, these properties were dependent on climatic conditions. It is recommended to use the monolithic multilayer green roof structure and a substrate with added biochar for higher L. sativa yield. Further evaluations of this structure and substrate for other edible plants may be useful for supporting vulnerable people and enhancing agroecological functions of green roofs in Latin-America.

The implementation of state tests at the end of the different academic cycles, to pursue high standard education, is used by many countries in the world. For the Colombian case there are two classic exams: one of them is for all the students that are about to finish high school, called Saber 11, and the other one is for people that are finishing the university cycle, called Saber Pro. Both are measured by quantitative scores. This case study was considered because it was possible to know the Saber 11 and Saber Pro scores for the same individual. The machine learning k-means technique was used for defining different groups of Saber Pro scores. Then a multinomial logistic model was applied to estimate the probability of a student to fall in every one of those groups, given his own characteristics. In other words, the distribution probability of the Saber Pro results was produced through the different defined groups using k-means. This splitting technique is far better than the standard even subdivisions (like quartiles or quintiles), because it produces optimal distance-wise clusters that allows more natural statistical results. This work gives numerical insights regarding the performance for different student profiles, and shows that the educational achievement at the end of an academic program, is influenced by the initial academic level, the socioeconomic condition, and the university status, but also that more input variables must be analyzed and included. In particular, for the considered samples, the research concluded that the individuals with the least favorable socioeconomic condition have a tendency to underperform when compared with the rest of the population, and the students of the non-accredited institutions tend to obtain lower educational achievements than the accredited ones. Of course, in all the investigated groups, students with a high result in Saber 11 tend to achieve an outstanding performance in Saber Pro, regardless of their socioeconomic status or the accreditation status of the institution. On the other hand, the gender shows no general behavior or whatsoever.

Multi-objective optimization problems (MOPs) naturally arise in many applications. Since for such problems one can expect an entire set of optimal solutions, a common task in set based multi-objective optimization is to compute N solutions along the Pareto set/front of a given MOP. In this work, we propose and discuss the set based Newton methods for the performance indicators Generational Distance (GD), Inverted Generational Distance (IGD), and the averaged Hausdorff distance Δp for reference set problems for unconstrained MOPs. The methods hence directly utilize the set based scalarization problems that are induced by these indicators and manipulate all N candidate solutions in each iteration. We demonstrate the applicability of the methods on several benchmark problems, and also show how the reference set approach can be used in a bootstrap manner to compute Pareto front approximations in certain cases.

A brief but comprehensive review of the averaged Hausdorff distances that have recently been introduced as quality indicators in multi-objective optimization problems (MOPs) is presented. First, we introduce all the necessary preliminaries, definitions, and known properties of these distances in order to provide a stat-of-the-art overview of their behavior from a theoretical point of view. The presentation treats separately the definitions of the ( p , q ) -distances GD p , q , IGD p , q , and Δ p , q for finite sets and their generalization for arbitrary measurable sets that covers as an important example the case of continuous sets. Among the presented results, we highlight the rigorous consideration of metric properties of these definitions, including a proof of the triangle inequality for distances between disjoint subsets when p , q ⩾ 1 , and the study of the behavior of associated indicators with respect to the notion of compliance to Pareto optimality. Illustration of these results in particular situations are also provided. Finally, we discuss a collection of examples and numerical results obtained for the discrete and continuous incarnations of these distances that allow for an evaluation of their usefulness in concrete situations and for some interesting conclusions at the end, justifying their use and further study.