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The eigenvalues of Toeplitz matrices T n (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 form λ j (T n (f)) = f (σ j , n) + c 1 (σ j , n) h + c 2 (σ j , n) h 2 + O (h 3) , where h = 1 / (n + 1) , σ j , n = π j h , and c k 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 combination λ j (T n (f 0) + β n , 1 T n (f 1) + β n , 2 T n (f 2)) , where β n , 1 , β n , 2 are certain constants depending on n and the generating functions f 0 , f 1 , f 2 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. [ABSTRACT FROM AUTHOR]

We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order n , where one edge has weight α ∈ C , with Re (α) < 0 , and all the others have weights 1. This paper is a sequel of a previous one where we considered Re (α) ∈ [ 0 , 1 ] (Grudsky et al., 2022 [12]). We prove that for Re (α) < 0 and n > Re (α − 1) / Re (α) , one eigenvalue is negative while the others belong to [ 0 , 4 ] and are distributed as the function x ↦ 4 sin 2 ⁡ (x / 2). Additionally, we prove that as n tends to ∞, the outlier eigenvalue converges exponentially to 4 Re (α) 2 / (2 Re (α) − 1). We give exact formulas for half of the inner eigenvalues, while for the others we justify the convergence of Newton's method and fixed-point iteration method. We find asymptotic expansions, as n tends to ∞, both for the eigenvalues belonging to [ 0 , 4 ] and the outlier. We also compute the eigenvectors and their norms. [ABSTRACT FROM AUTHOR]

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 i θ) } n ∈ N , 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. [ABSTRACT FROM AUTHOR]

Let { A n } n be a sequence of square matrices such that size (A n) = d n → ∞ as n → ∞. We say that { A n } n has an asymptotic spectral distribution described by a Lebesgue measurable function f : D ⊂ R k → C if, for every continuous function F : C → C with bounded support, lim n → ∞ ⁡ 1 d n ∑ i = 1 d n F (λ i (A n)) = 1 μ k (D) ∫ D F (f (x)) d x , where μ k is the Lebesgue measure in R k and λ 1 (A n) , ... , λ d n (A n) are the eigenvalues of A n. In the last decades, the asymptotic spectral distribution of increasing size matrices has become a subject of investigation by several authors. Special attention has been devoted to matrices A n arising from the discretization of differential equations, whose size d n diverges to ∞ along with the mesh-fineness parameter n. However, despite the popularity that the topic has reached nowadays, the role of the so-called test functions F appearing in the above limit relation has been inexplicably neglected so far. In particular, a natural question such as "Which is the largest set of test functions F for which the above limit relation is satisfied?" is still unanswered. In the present paper, we provide a definitive answer to this question by identifying the largest set of test functions F for which the above limit relation is satisfied. We also present some applications of this result to the analysis of spectral clustering, including new asymptotic estimates on the number of outliers. A special attention is devoted to the so-called "inner outliers" of Hermitian block Toeplitz matrices. We conclude the paper with an interpretation of the main result in the context of the vague convergence of probability measures. [ABSTRACT FROM AUTHOR]

The main aim of the present paper is to establish inversion formulas of Gohberg-Semencul type for Toeplitz-plus-Hankel matrices. In particular, it is shown how the inverse of such a structured matrix A n of order n is computed by means of their first two and last two columns or rows under the additional assumption that a certain 2 × 2 matrix is nonsingular. Moreover, a formula for the inverse of the Toeplitz-plus-Hankel matrix A n − 2 of order n − 2 in the center of A n is established and sufficient invertibility criteria for both A n and A n − 2 are obtained. Hereby a main tool is to use known inversion formulas involving not only rows or columns of the inverse but also solutions of equations the right hand sides of which depend on entries of the Toeplitz-plus-Hankel matrix itself. [ABSTRACT FROM AUTHOR]

A new efficient nonparametric estimator for Toeplitz covariance matrices is proposed. This estimator is based on a data transformation that translates the problem of Toeplitz covariance matrix estimation to the problem of mean estimation in an approximate Gaussian regression. The resulting Toeplitz covariance matrix estimator is positive definite by construction, fully data driven and computationally very fast. Moreover, this estimator is shown to be minimax optimal under the spectral norm for a large class of Toeplitz matrices. These results are readily extended to estimation of inverses of Toeplitz covariance matrices. Also, an alternative version of the Whittle likelihood for the spectral density based on the discrete cosine transform is proposed. [ABSTRACT FROM AUTHOR]

In Lin et al. (2021) [21] and Zhao et al. (2023) [37] , two-sided preconditioning techniques are proposed for non-local evolutionary equations, which possesses (i) mesh-size independent theoretical bound of condition number of the two-sided preconditioned matrix; (ii) small and stable iteration numbers in numerical tests. In this paper, we modify the two-sided preconditioning by multiplying the left-sided and the right-sided preconditioners together as a single-sided preconditioner. Such a single-sided preconditioner essentially derives from approximating the spatial matrix with a fast diagonalizable matrix and keeping the temporal matrix unchanged. Clearly, the matrix-vector multiplication of the single-sided preconditioning is faster to compute than that of the two-sided one, since the single-sided preconditioned matrix has a simpler structure. More importantly, we show theoretically that the single-sided preconditioned generalized minimal residual (GMRES) method has a convergence rate no worse than the two-sided preconditioned one. As a result, the one-sided preconditioned GMRES solver requires less computational time than the two-sided preconditioned GMRES solver in total. Numerical results are reported to show the efficiency of the proposed single-sided preconditioning technique. • The operation cost for implementing the single-sided preconditioner is linearithmic, which is nearly optimal. • The single-sided preconditioner is shown to be better than the two-sided one by theoretical and numerical evidences. • The iteration number of the proposed preconditioning technique is not sensitive to the parameters of the problem. • The complexity of the proposed solver is linearithmic, which is nearly optimal. [ABSTRACT FROM AUTHOR]

By considering the tiling of an N-board (a linear array of N square cells of unit width) with new types of tile that we refer to as combs, we give a combinatorial interpretation of the product of two consecutive generalized Fibonacci numbers $ s_n $ s n (where $ s_{n}=\sum _{i=1}^q v_i s_{n-m_i} $ s n = ∑ i = 1 q v i s n − m i , $ s_0=1 $ s 0 = 1 , $ s_{n \lt 0}=0 $ s n < 0 = 0 , where $ v_i $ v i and $ m_i $ m i are positive integers and $ m_1 \lt \cdots \lt m_q $ m 1 < ⋯ < m q ) each raised to an arbitrary non-negative integer power. A $ (w,g;m) $ (w , g ; m) -comb is a tile composed of m rectangular sub-tiles of dimensions $ w\times 1 $ w × 1 separated by gaps of width g. The interpretation is used to give combinatorial proof of new convolution-type identities relating $ s_n^2 $ s n 2 for the cases q = 2, $ v_i=1 $ v i = 1 , $ m_1=M $ m 1 = M , $ m_2=m+1 $ m 2 = m + 1 for M = 0, m to the permanent of a (0,1) Toeplitz matrix with 3 nonzero diagonals which are $ -2 $ − 2 , M−1, and m above the leading diagonal. When m = 1, these identities reduce to ones connecting the Padovan and Narayana's cows numbers. [ABSTRACT FROM AUTHOR]

Matrix theory is essential for addressing practical problems and executing computational tasks. Matrices related to Hermite polynomials are essential due to their applications in quantum mechanics, numerical analysis, probability, and signal processing. Their orthogonality, recurrence relations, and spectral properties make them a valuable tool for both theoretical research and practical applications. From a different perspective, we introduced a variant of the Hermite matrix that incorporates triple factorials and demonstrated that this matrix satisfies various properties. By utilizing effective matrix algebra techniques, various algebraic properties of this matrix have been determined, including the product formula, inverse matrix and eigenvalues. Additionally, we extended this matrix to a more generalized form and derived several identities. [ABSTRACT FROM AUTHOR]

The BC-method is the approach to inverse problems based on their connection with control theory and theory of systems. The main fragment of its numerical implementation is the matrix inversion of the so-called connecting operator. In multidimensional problems, its matrix is ill-conditioned and large in size, which yields the rapid growth of the volume of computations. The paper reveals block-Toeplitz structure of this matrix, using which one can significantly reduce the amount of computations. [ABSTRACT FROM AUTHOR]

We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian (− Δ) α 2 . Numerical analysis and experiments are provided to study its performance. Our method has the same symbol | ξ | α as the fractional Laplacian (− Δ) α 2 at the discrete level, and thus it can be viewed as the exact discrete analogue of the fractional Laplacian. This unique feature distinguishes our method from other existing methods for the fractional Laplacian. Note that our method is different from the Fourier pseudospectral methods in the literature which are usually limited to periodic boundary conditions (see Remark 1.1). Numerical analysis shows that our method can achieve a spectral accuracy. The stability and convergence of our method in solving the fractional Poisson equations were analyzed. Our scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms can be developed for efficient matrix-vector multiplications. The computational complexity is O (2 N log ⁡ (2 N)) , and the memory storage is O (N) with N the total number of points. Extensive numerical experiments verify our analytical results and demonstrate the effectiveness of our method in solving various problems. [ABSTRACT FROM AUTHOR]

Among the directional angle calculation models of Bluetooth 5.1, the rectangular hollow array offers the advantage of shorter sampling time due to its fewer elements compared to traditional planar arrays. However, the antimultipath algorithms suitable for traditional planar arrays cannot be applied to rectangular hollow arrays. Therefore, this study proposes a virtual array filling algorithm, wherein four virtual matrices are inserted into the hollow matrix to transform the array into a uniform rectangular array. This algorithm ensures translation invariance of the rectangular array, enabling the application of the antimultipath coherent source algorithm to a rectangular hollow array. An algorithm for reconstructing Toeplitz matrices in two‐dimensional uniform planar arrays is also proposed. Through the analysis of the spatial spectrum and angle estimation results of various algorithms, the effectiveness of the signal angle of arrival estimation theory is verified. [ABSTRACT FROM AUTHOR]

The primary objective of this paper is to create a novel infinite Toeplitz matrix by leveraging Tetranacci numbers. This matrix serves as the foundation for defining new sequence spaces denoted as c 0 ⁢ ( G ) {c_{0}(G)} , c ⁢ ( G ) {c(G)} , ℓ ∞ ⁢ ( G ) {\ell_{\infty}(G)} , and ℓ p ⁢ ( G ) {\ell_{p}(G)} , where 1 ≤ p < ∞ {1\leq p<\infty} . By utilizing this newly constructed matrix, the paper also explores and examines various algebraic and topological properties inherent to the sequence spaces c 0 ⁢ ( G ) {c_{0}(G)} , c ⁢ ( G ) {c(G)} , ℓ ∞ ⁢ ( G ) {\ell_{\infty}(G)} , and ℓ p ⁢ ( G ) {\ell_{p}(G)} for values of p within the range of 1 ≤ p < ∞ {1\leq p<\infty} . At last, we also prove existence theorem with example for infinite systems of differential equations in ℓ p ⁢ ( G ) {\ell_{p}(G)} . [ABSTRACT FROM AUTHOR]

In [V. M. Abramov, Bull. Austral. Math. Soc., 104, 108 (2021)], the fixed-point equation was studied for an infinite nonnegative particular Toeplitz matrix. In the present work, we provide an alternative proof of the existence of a positive solution in the general case. This proof is based on the application of a version of the M. A. Krasnosel'skii fixed-point theorem. The results are then extended to the equations with infinite matrices of the general type. [ABSTRACT FROM AUTHOR]

In the present paper, we define and study a new family of holomorphic functions which involve the Bazilevič functions and the λ -pseudo-starlike functions. We establish coefficient estimates for the first four determinants of the symmetric Toeplitz matrices T 2 (2) , T 2 (3) , T 3 (1) and T 3 (2) for the functions in this family. Further, we investigate several special cases and consequences of our results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We present a procedure which enables the computation and the description of structures of isotropy subgroups of the group of complex orthogonal matrices with respect to the action of *congruence on Hermitian matrices. A key ingredient in our proof is an algorithm giving solutions of a certain rectangular block (complex-alternating) upper triangular Toeplitz matrix equation. [ABSTRACT FROM AUTHOR]

The Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) are known to exist in all dimensions ⩽ 151 and many larger dimensions as high as 39 604 . All known solutions with the exception of the Hoggar solutions are covariant with respect to the Weyl-Heisenberg group and in the case of dimension 3, it has been proven that all SIC-POVMs are Weyl-Heisenberg group covariant. In this work, we explore this by SIC-POVMs in dimensions 4–7 without the assumption of group covariance. We introduce two functions with which SIC-POVM Gram matrices can be generated without the group covariance constraint, and analytically show that the SIC-POVM Gram matrices exist on critical points of the surfaces defined by the two functions on a subspace of Hermitian matrices. In dimensions 4–7, all known SIC-POVM Gram matrices lie in disjoint continuous sets of solution. Thus, we define an equivalent class that relates Gram matrices based on the trivial symmetries of the two functions. In dimensions 4–7, we generated { 1.7 × 10 6 , 1.1 × 10 5 , 169 , 50 } Gram matrices, respectively. For each of the Gram matrices, we generate the symmetry group of their respective disjoint sets. In all cases, the symmetry group was isomorphic to a subgroup of the Clifford group containing the Weyl-Heisenberg group matrices and the order-3 unitaries. In dimensions 4–6, All constructed solutions belonged to a single equivalence class of Gram matrices whereas in dimension 7, we find two distinct families of SIC-POVMs. In dimensions 4 and 5, the absence of new classes of SIC-POVM strongly suggests that the functions do not have a non-trivial symmetry. Furthermore, in all the dimensions tested, the only equivalence classes found correspond to the distinct stabilizers of fiducial vectors of each dimension. [ABSTRACT FROM AUTHOR]

This paper shows both analytically and numerically that the skin effect in systems of non-Hermitian subwavelength resonators is robust with respect to random imperfections in the system. The subwavelength resonators are highly contrasting material inclusions that resonate in a low-frequency regime. The non-Hermiticity is due to the introduction of a directional damping term (motivated by an imaginary gauge potential), which leads to a skin effect that is manifested by the system's eigenmodes accumulating at one edge of the structure. We elucidate the topological protection of the associated (real) eigenfrequencies and illustrate numerically the competition between the two different localization effects present when the system is randomly perturbed: the non-Hermitian skin effect and the disorder-induced Anderson localization. We show numerically that, as the strength of the disorder increases, more and more eigenmodes become localized in the bulk. Our results are based on an asymptotic matrix model for subwavelength physics and can be generalized also to tight-binding models in condensed matter theory. [ABSTRACT FROM AUTHOR]

In an earlier publication of these authors, a unified approach was proposed to the construction of matrix pairs that solve the -commutation problem for Toeplitz and Hankel matrices. Here, this approach is applied for deriving new classes of solutions. [ABSTRACT FROM AUTHOR]

Aiming at solving the problem that the capability of direction of arrival method for estimation of nested arrays decreases sharply under unknown mutual coupling, an iterative weighted low‐rank matrix reconstruction algorithm is proposed here. First, the received data covariance matrix is extended, and a weighted low‐rank matrix recovery problem is formulated for joint estimations of the Toeplitz matrix and mutual coupling coefficients. Next, the direction of arrival from recovered Toeplitz matrix is retrieved by using the root multiple signal classification algorithm. Simulation results show that iterative weighted low‐rank matrix reconstruction algorithm can effectively avoid the influence of mutual coupling while solving the grid‐mismatch problem so that the accuracy and the resolution of estimation are improved. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the various properties of the Toeplitz operators acting on the Dirichlet spaces. First, we consider the matrix representation of Toeplitz operators with harmonic and monomial symbols. Second, we establish the expansivity and contractivity of the Toeplitz operators T φ with monomial symbols φ . Third, we give a necessary and sufficient conditions for the normality and hyponormality of the Toeplitz operators T φ with such symbols on the Dirichlet spaces. [ABSTRACT FROM AUTHOR]

In this article, we give an explicit expression for calculating the arbitrary positive integer powers of the Kronecker sum of two tridiagonal Toeplitz matrices. [ABSTRACT FROM AUTHOR]

In this paper we use the superpotential W q for the flag variety G L n / B and particular coordinate systems that we call 'ideal coordinates for i ', to construct polytopes P λ i inside R R + , associated to highest weight representations V λ of G L n (C). Here i is a reduced expression of the longest element of the Weyl group and R + is the set of positive roots of G L n. The lattice points of P λ i can be used to encode a basis of the representation V λ , with the weights encoded in a projection onto the weight polytope. For a specific choice of i , the polytope P λ i is unimodularly equivalent to a Gelfand-Tsetlin polytope. The construction of the polytopes involves tropicalisation of the superpotential, namely P λ i is { Trop (W t λ i) ≥ 0 } written in a tropical version of the ideal coordinates for i. Using work of Judd ([13]) we have that there is a unique 'positive' critical point of W t λ over the field of Puiseux series, which projects onto the centre of mass of the weight polytope. Its coordinates, in terms of the ideal coordinates for i , are positive in the sense of having positive leading term. The remarkable property of our new polytopes relates to the tropical version of this critical point, which, for every choice of i , gives a point in R R + that lies in the interior of the polytope P λ i. We prove that this tropical critical point is independent of the reduced expression i , and that it is given by a pattern called the 'ideal filling' for λ that was introduced by Judd. Finally, combining these results with work of Rietsch ([22]) relating critical points of the superpotential with Toeplitz matrices, we show that for a totally positive lower-triangular Toeplitz matrix over the field of Puiseux series factorised into simple root subgroups, the valuations of the factors give an ideal filling. [ABSTRACT FROM AUTHOR]

Coprime arrays (CPA) are extensively applied to estimate the direction of arrival (DOA). Yet, the effectiveness of existing DOA estimation algorithms for coprime arrays is predicated on the precise calibration of the array. Gain-phase errors in the sensors, when present, are magnified by the co-array, greatly impacting the performance of DOA estimation algorithms. To address the issue, this paper proposes a joint DOA estimation algorithm combining auxiliary arrays with atomic norm minimization. Exploiting the sub-array decomposition feature of coprime array, gain-phase error estimation is accomplished with the addition of precisely calibrated auxiliary arrays. Besides, the estimated gain-phase errors are used to correct the covariance of the signals received by the array. By combining array interpolation, all virtual sensors are fully utilized, and the interpolated virtual received signals are reconstructed into a Toeplitz matrix. The covariance matrix of these signals is then refined using atomic norm minimization, preparing the data for the final DOA estimation performed by subspace algorithms. In addition, for comparative analysis, the Cramér–Rao bound has also been derived under the condition of gain-phase errors, providing a benchmark for the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Asymptotic formulas are derived that admit a uniform estimate of the remainder for Toeplitz matrices of size as in the case when their symbol has the form . This result is a generalization of the result of Stukopin et al. (2021), who obtained similar asymptotic formulas for a seven-diagonal Toeplitz matrix with a similar symbol in the case . The resulting formulas are of high computational efficiency and generalize the classical results of Parter and Widom on asymptotics of extreme eigenvalues. [ABSTRACT FROM AUTHOR]

Understanding the eigenvalue distribution of sequence Toeplitz matrices has advanced significantly in recent years. Notable contributors include Bogoya, Grudsky, Böttcher, and Maximenko, who have derived precise asymptotic expansions for these eigenvalues under certain conditions related to the generating function as the matrix size increases. Building on this foundation, the Stefano Serra-Capizzano conjectured that, under certain assumptions about Ω and Φ, a similar expansion may hold for the eigenvalues of a sequence of preconditioned Toeplitz matrices T-1n (Φ)Tn(Ω), given a monotonic ratio r=Ω/Φ. In contrast to current eigenvalue solvers, this work presents a novel method for efficiently calculating the eigenvalues of a sequence of large preconditioned banded symmetric Toeplitz matrices (PBST). Our algorithm uses a higher-order spline fitting extrapolation technique to gather spectral data from a smaller sequence of PBST matrices in order to forecast the spectrum of bigger matrices. [ABSTRACT FROM AUTHOR]

We estimated convex-structured covariance/correlation matrices by minimizing the entropy loss corresponding to the given matrix. We first considered the estimation of the Weighted sum of known Rank-one matrices with unknownWeights (W-Rank1-W) structural covariance matrices, which appeared commonly in array signal processing tasks, e.g., direction-of-arrival (DOA) estimation. The associated minimization problem is convex and can be solved using the primal-dual interior-point algorithm. However, the objective functions (the entropy loss function) can be bounded above by a sequence of separable functions--we proposed a novel estimation algorithm based on this property under the Majorization-Minimization (MM) algorithmic framework. The proposed MM algorithm exhibited very low computational complexity in each iteration, and its convergence was demonstrated theoretically. Subsequently, we focused on the estimation of Toeplitz autocorrelation matrices, which appeared frequently in time-series analysis. In particular, we considered cases in which the autocorrelation coefficient decreased as the time lag increased. We transformed the Toeplitz structure into aW-Rank1-W structure via special variable substitution, and proposed anMMalgorithm similar to that for the W-Rank1-W covariance estimation. However, each MM iteration involved a second-order cone programming SOCP problem that must be resolved. Our numerical experiments demonstrated the high computational efficiency and satisfactory estimation accuracy of the proposedMMalgorithms in DOA and autocorrelation matrix estimation. [ABSTRACT FROM AUTHOR]

In the present paper, we mainly consider the direct solution of periodic pentadiagonal Toeplitz linear systems. By exploiting the low-rank and Toeplitz structure of the coefficient matrix, we derive a new matrix decomposition of periodic pentadiagonal Toeplitz matrices. Based on this matrix decomposition form and combined with the Sherman-Morrison-Woodbury formula, we propose an efficient algorithm for numerically solving periodic pentadiagonal Toeplitz linear systems. Furthermore, we present a fast and reliable algorithm for evaluating the determinants of periodic pentadiagonal Toeplitz matrices by a certain type of matrix reordering and partitioning, and linear transformation. Numerical examples are given to demonstrate the performance and effectiveness of our algorithms. All of the experiments are performed on a computer with the aid of programs written in Matlab. [ABSTRACT FROM AUTHOR]

When uncorrelated signals are incident on a uniform linear array, the array covariance matrix is of the Toeplitz form. An adaptive beamforming method is proposed based on the signal-plus-interference (SI) subspace via the Toeplitz rectification of the sample matrix. The rectified matrix is shown to be more accurate in a norm sense than the modified matrix according to the centro-Hermitian property. Since the former also is centro-Hermitian we can efficiently obtain its eigen-decomposition from a real matrix and then the weight vector in the estimated SI subspace. The proposed method, showing robustness to pointing errors, is not only computationally efficient but also very quickly converges to the optimum performance as demonstrated in the simulation. [ABSTRACT FROM AUTHOR]

It is established that the σ-commutation problem for a Toeplitz and a Hankel matrices has no solutions in a particular case. [ABSTRACT FROM AUTHOR]

The paper is devoted to the problem of determining the number of subalgebras in the matrix algebra that are generated by nonderogatory matrices and are distinct up to similarity. A similarity criterion in terms of isomorphism of quotient algebras of the algebra of polynomials for matrix algebras of the type indicated is established. The existence of infinite fields for which the number of distinct algebras is infinite for all values of the matrix order is proved. For algebraically closed fields, the field of reals, and finite fields of sufficiently large cardinalities, the number of distinct algebras generated by nonderogatory matrices is determined as a function of the matrix order. [ABSTRACT FROM AUTHOR]

When dealing with the discretization of differential equations on non‐rectangular domains, a careful treatment of the boundary is mandatory and may result in implementation difficulties and in coefficient matrices without a prescribed structure. Here we examine the numerical solution of a two‐dimensional constant coefficient distributed‐order space‐fractional diffusion equation with a nonlinear term on a convex domain. To avoid the aforementioned inconvenience, we resort to the volume‐penalization method, which consists of embedding the domain into a rectangle and in adding a reaction penalization term to the original equation that dominates in the region outside the original domain and annihilates the solution correspondingly. Thanks to the volume‐penalization, methods designed for problems in rectangular domains are available for those in convex domains and by applying an implicit finite difference scheme we obtain coefficient matrices with a 2‐level Toeplitz structure plus a diagonal matrix which arises from the penalty term. As a consequence of the latter, we can describe the asymptotic eigenvalue distribution as the matrix size diverges as well as estimate the intrinsic asymptotic ill‐conditioning of the involved matrices. On these bases, we discuss the performances of the conjugate gradient with circulant and τ$$ \tau $$‐preconditioners and of the generalized minimal residual with split circulant and τ$$ \tau $$‐preconditioners and conduct related numerical experiments. [ABSTRACT FROM AUTHOR]

We consider cyclic algebras, Milnor's symbol algebras, and certain graded algebra structures on them. We classify these gradings with respect to both isomorphism and equivalence relations. Some of them induce gradings on matrix algebras, which we also classify. As an application, we obtain the classification of all group gradings on the algebra M p (F) , where p is a prime number and F is an arbitrary field. [ABSTRACT FROM AUTHOR]

Experimental and theoretical evidence has shown that adaptive step size methods such as the backward differentiation formula (BDF) are more robust over a wider range of step sizes compared to those used in fixed step methods. Acknowledging the computational efficiency and accuracy obtained with such strategies, an adaptive step size version of the block backward differentiation formula (BBDF) in a diagonally implicit structure is proposed for solving stiff ordinary differential equations (ODEs), particularly in addressing the challenges posed by the chemical reaction problem within the domains of applied and industrial mathematics. The diagonally implicit structure with a lower triangular matrix and constant diagonal inputs offers significant advantages in evaluating the Jacobian and the lower-upper decomposition. The stability properties that were investigated show that the new class is zero−stable, A0−stable and almost A−stable. Comparative evaluations reveal the superior performance of the proposed method compared to the existing fully implicit BBDF and ode15s conducted in MATLAB software. [ABSTRACT FROM AUTHOR]

In this paper, keeping in view the idea of difference sequence space E(Δ) of Kızmaz [20], we availed an opportunity to introduce new kind of summing squence spaces E(∇), E ∊ {l∞,c,c0} by exploring the sum of two consecutive terms. In addition to this we computed the continuous as well Köthe-Toeplitz duals of these spaces. Like E(Δ) (the difference sequence spaces of Kızmaz) new sequence spaces E(∇) turned out to be much wider than E. [ABSTRACT FROM AUTHOR]

Minor perturbations to a linear operator may drastically change its spectrum, and hence the difficulty of deciding whether or not a numerically computed quantity is zero causes problems in spectral theory. The purpose of this expository paper is to illustrate such instability phenomena by some examples with Toeplitz-like operators and matrices. [ABSTRACT FROM AUTHOR]

In this paper, we study several properties of an orthonormal basis { N n (z) } for the Newton space N 2 (P) . In particular, we investigate the product of N m and N m and the orthogonal projection P of N n ‾ N m that maps from L 2 (P) onto N 2 (P) . Moreover, we find the matrix representation of Toeplitz operators with respect to such an orthonormal basis on the Newton space N 2 (P) . [ABSTRACT FROM AUTHOR]

In this paper, a Toeplitz construction method based on eigenvalues and eigenvectors is proposed to combine with traditional denoising algorithms, including fractional low-order moment (FLOM), phased fractional low-order moment (PFLOM), and correntropy-based correlation (CRCO) methods. It can improve the direction of arrival (DOA) estimation of signals in impulsive noise. Firstly, the algorithm performs eigenvalue decomposition on the received covariance matrix to obtain eigenvectors and eigenvalues, and then the Toeplitz matrix is created according to the eigenvectors corresponding to its eigenvalues. Secondly, the spatial averaging method is used to obtain an unbiased estimate of the Toeplitz matrix, which is then weighted and added based on the corresponding eigenvalues. Next, the noise subspace of the Toeplitz matrix is reconstructed to obtain the one that has less angle information. Finally, the DOA of the coherent signal is estimated using the Multiple Signal Classification (MUSIC) algorithm. The improved method based on the Toeplitz matrix can not only suppress the effect of impulsive noise but can also solve the problem of aperture loss due to its decoherence. A series of simulations have shown that they have better performances than other algorithms. [ABSTRACT FROM AUTHOR]

Artificially determining the model order is necessary for time domain modal identification, which affects the identification efficiency and accuracy. An automatic identification method is proposed to determine the best estimation of the model order. Based on the covariance-driven stochastic subspace identification method (SSI-COV), the singular value decomposition of the Toeplitz matrix is carried out to determine the range of the lower triangular matrix. The similarity coefficient and distance function are introduced to cluster the modes, and the poles of the false modes are removed to obtain the clustering stabilization diagram. The model order is taken as the number of clustering centers of the fuzzy c-means (FCM) algorithm, and all the poles of the stabilization diagram are clustered to get the modal parameters of each order. The effectiveness of the proposed method is verified by adopting a thin-walled cylindrical column, a triangular-prism truss in simulation, and a flexible beam in experiment. The robustness of the method is illustrated by adding white noise to the simulated response data. The advantage of the clustering stabilization diagram is demonstrated by comparing with the traditional stabilization diagram. [ABSTRACT FROM AUTHOR]

The paper considers algorithms for solving inverse scattering problems based on the discretization of the Gelfand–Levitan–Marchenko integral equations, associated with the system of nonlinear Schrödinger equations of the Manakov model. The numerical algorithm of the first order approximation for solving the scattering problem is reduced to the inversion of a series of nested block Toeplitz matrices using the Levinson-type bordering method. Increasing the approximation accuracy violates the Toeplitz structure of block matrices. Two algorithms are described that solve this problem for second order accuracy. One algorithm uses a block version of the Levinson bordering algorithm, which recovers the Toeplitz structure of the matrix by moving some terms of the systems of equations to the right-hand side. Another algorithm is based on the Toeplitz decomposition of an almost block-Toeplitz matrix and the Tyrtyshnikov bordering algorithm. The speed and accuracy of calculations using the presented algorithms are compared on an exact solution (the Manakov vector soliton). [ABSTRACT FROM AUTHOR]

This paper focuses on the reversibility of multidimensional linear cellular automata with an intermediate boundary condition. We begin by addressing the matrix representation of these automata, and the question of reversibility boils down to the invertibility of this matrix representation. We introduce a decomposition method that factorizes the matrix representation into a Kronecker sum of significantly smaller matrices. The invertibility of the matrix hinges on determining whether zero can be expressed as the sum of eigenvalues of these smaller matrices, which happen to be tridiagonal Toeplitz matrices. Notably, each of these smaller matrices represents a one-dimensional cellular automaton. Leveraging the rich body of research on the eigenvalue problem of Toeplitz matrices, our result provides an efficient algorithm for addressing the reversibility problem. As an application, we show that there is no reversible nontrivial linear cellular automaton over Z 2 . This paper focuses on the reversibility of multidimensional linear cellular automata with an intermediate boundary condition. We begin by addressing the matrix representation of these automata, and the question of reversibility boils down to the invertibility of this matrix representation. We introduce a decomposition method that factorizes the matrix representation into a Kronecker sum of significantly smaller matrices. The invertibility of the matrix hinges on determining whether zero can be expressed as the sum of eigenvalues of these smaller matrices, which happen to be tridiagonal Toeplitz matrices. Notably, each of these smaller matrices represents a one-dimensional cellular automaton. Leveraging the rich body of research on the eigenvalue problem of Toeplitz matrices, our result provides an efficient algorithm for addressing the reversibility problem. As an application, we show that there is no reversible nontrivial linear cellular automaton over . [ABSTRACT FROM AUTHOR]

We consider weighted sums of independent random variables regulated by an increment sequence and provide operative conditions that ensure a strong law of large numbers for such sums in both the centred and non-centred case. The existing criteria for the strong law are either implicit or based on restrictions on the increment sequence. In our setup we allow for an arbitrary sequence of increments, possibly random, provided the random variables regulated by such increments satisfy some mild concentration conditions. In the non-centred case, convergence can be translated into the behaviour of a deterministic sequence and it becomes a game of mass when the expectation of the random variables is a function of the increment sizes. We identify various classes of increments and illustrate them with a variety of concrete examples. [ABSTRACT FROM AUTHOR]

This paper presents an efficient two-dimensional (2D) direction of arrival (DOA) estimation method, termed as decoupled projected atomic norm minimization (D-PANM), to solve the angle-ambiguity problem. It first introduces a novel atomic metric via projecting the original atom set onto a smoothing space, based on which we formulate an equivalent semi-definite programming (SDP) problem. Then, two relatively low-complexity decoupled Toeplitz matrices can be obtained to estimate the DOAs. We further exploit the structural information hidden in the newly constructed data to avoid pair matching for the azimuth and elevation angles when the number of sensors is odd, and then propose a fast and feasible decoupled alternating projections (D-AP) algorithm, reducing computational complexity to a great extent. Numerical simulations are performed to demonstrate that the proposed algorithm is no longer restricted by angle ambiguity scenarios, but instead provides a more stable estimation performance, even when multiple signals share the same angles in both azimuth and elevation dimensions. Additionally, it greatly improves the resolution, with control of the computation load compared with the existing atomic norm minimization (ANM) algorithm. [ABSTRACT FROM AUTHOR]

We consider linear isotopes of commutative groups, i.e., central quasigroups and study the invertibility and orthogonality conditions. It turns out that it is sufficient to study these conditions solely for the unitary isotopes, i.e., for isotopes, which have an idempotent. We establish criteria for the possession of each invertibility property (inverse property, crossed inverse property, and mirroring) for unitary central and matrix quasigroups. In particular, for matrices of the second order, we describe the corresponding matrix quasigroups over the fields of characteristics 2 and 3. The orthogonality criteria are obtained for matrix quasigroups with the indicated invertibility properties. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to exploit the multiple-Toeplitz matrices reconstruction method combined with quadratic spatial smoothing processing to improve the direction-of-arrival (DOA) estimation performance of coherent signals at low signal-to-noise ratio (SNRs). Design/methodology/approach: An improved multiple-Toeplitz matrices reconstruction method is proposed via quadratic spatial smoothing processing. Our proposed method takes advantage of the available information contained in the auto-covariance matrices of individual Toeplitz matrices and the cross-covariance matrices of different Toeplitz matrices, which results in a higher noise suppression ability. Findings: Theoretical analysis and simulation results show that, compared with the existing Toeplitz matrix processing methods, the proposed method improves the DOA estimation performance in cases with a low SNR. Especially for the cases with a low SNR and small snapshot number as well as with closely spaced sources, the proposed method can achieve much better performance on estimation accuracy and resolution probability. Research limitations/implications: The study investigates the possibility of reusing pre-existing designs for the DOA estimation of the coherent signals. The proposed technique enables achieve good estimation performance at low SNRs. Practical implications: The paper includes implications for the DOA problem at low SNRs in communication systems. Originality/value: The proposed method proved to be useful for the DOA estimation at low SNR. [ABSTRACT FROM AUTHOR]

In this article, we consider the rank‐structured approximation of one important type of Cauchy matrix. This approximation plays a key role in some structured matrix methods such as stable and efficient direct solvers and other algorithms for Toeplitz matrices and certain kernel matrices. Previous rank‐structured approximations (specifically hierarchically semiseparable, or HSS, approximations) for such a matrix of size n$$ n $$ cost at least O(n)$$ O(n) $$ complexity. Here, we show how to construct an HSS approximation with sublinear (specifically, O(log3n)$$ O\left({\log}^3n\right) $$) complexity. The main ideas include extensive computation reuse and an analytical far‐field compression strategy. Low‐rank compression at each hierarchical level is restricted to just a single off‐diagonal block row, and a resulting basis matrix is then reused for other off‐diagonal block rows as well as off‐diagonal block columns. The relationships among the off‐diagonal blocks are rigorously analyzed. The far‐field compression uses an analytical proxy point method where we optimize the choice of some parameters so as to obtain accurate low‐rank approximations. Both the basis reuse ideas and the resulting analytical hierarchical compression scheme can be generalized to some other kernel matrices and are useful for accelerating relevant rank‐structured approximations (though not subsequent operations like matrix‐vector multiplications). [ABSTRACT FROM AUTHOR]

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, Böttcher, 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 Ekström, 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. [ABSTRACT FROM AUTHOR]

Annotation: In their preceding publication, the authors proposed a unified approach to the construction of matrix pairs that solve the -commutation problem for Toeplitz and Hankel matrices. Here, this approach is applied to the derivation of two classes of solutions that were earlier found by V.N. Chugunov from entirely different considerations. [ABSTRACT FROM AUTHOR]

In this paper we try to approximate any properties of quasi-block Toeplitz matrix (QBT), by means of a finite number of parameters. A quasi-block Toeplitz (QBT) matrix is a semi-infinite block matrix of the kind F = T(F) + E where... These matrices are a Banach algebra with the standard arithmetic operations. We try to analysis some structures and computational properties for arithmetic operations of QBT matrices with some MATLAB commands. [ABSTRACT FROM AUTHOR]