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Let G be a connected graph of order n , A ( G ) the adjacency matrix of G and D ( G ) the diagonal matrix of the row-sums of A ( G ) . For every real α ∈ [ 0 , 1 ] , Nikiforov defined the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . In this paper, we obtain a factorization of the A α -characteristic polynomial of the some families of graphs and determine in these families the conditions about α so that A α ( G ) is positive semidefinite.

By employing off-diagonal matrix Mittag-Leffler functions and stability theory for line fractional functional differential equations, a new technique is proposed to investigate the existence, uniqueness and global asymptotical stability of S -asymptotically periodic solution for a class of semilinear Caputo fractional functional differential equations. Some better results are derived, which improve and extend the existing research findings in recent years. As an application of the general theory, some decision theorems are established for the asymptotically dynamical behaviors for fractional four-neuron BAM neural networks. The methods used in this paper could be applied to the study of other fractional differential systems in the areas of science and engineering.

In this paper, the robust filtering problem for uncertain complex networks with time-varying state delay and stochastic nonlinear coupling based on H ∞ performance criterion is studied. The random connections of coupling nodes are represented by utilizing independent random variables and the multiple fading measurements phenomenon is characterized by introducing diagonal matrices with independent stochastic elements. Moreover, the probabilistic time-varying delays in the measurement outputs are described by white sequences with the Bernoulli distributions. Furthermore, All system’s matrices are supposed to have uncertainty and a quadratic bound is assumed for nonlinear part of the network. This bound can be obtained by solving a sum of squares (SOS) optimization problem. By applying the Lyapunov theory, we design a robust filter for each node of the network so that the filtering error system is asymptomatically stable and the H ∞ performances are met. Then, the parameters of the filters are achieved by solving a linear matrix inequality (LMI) feasibility problem. Finally, the applicability and performance of the proposed H ∞ filtering approach are demonstrated via a practical example.

For a graph G , the signless Laplacian matrix Q ( G ) is defined as Q ( G ) = D ( G ) + A ( G ) , where A ( G ) is the adjacency matrix of G and D ( G ) the diagonal matrix whose main entries are the degrees of the vertices in G . The Q -spectrum of G is that of Q ( G ) . In the present paper, we are interested in the minimum values of the second largest signless Laplacian eigenvalue q 2 ( G ) of a connected graph G . We find the five smallest values of q 2 ( G ) over the set of connected graphs G with given order n . We also characterize the corresponding extremal graphs.

In a previous paper based on the interpolation finite difference method, a calculation system was shown for calculating 1D (one-dimensional) Laplace’s equation and Poisson’s equation using high-order difference schemes. Finite difference schemes, from the usual second-order to tenth-order differences, including odd number order differences, were systematically and instantaneously derived over equally/unequally spaced grid points based on the Lagrange interpolation function. Using the direct method with the band diagonal matrix algorithm, 1D Poisson equations were numerically calculated under double precision floating arithmetic, but it became clear that high accurate calculations could not be secured in high-order differences because “digit-loss errors” caused by the finite precision of computations occurred in the calculations when using the high-order differences. The double precision calculation corresponds to 15 (significant) digit calculation. In this paper, we systematically investigate how the calculation accuracy changes by high precision calculations (30-digit, and 45-digit calculations). Under 45-digit calculation, where the digit-loss error can be almost ignored, the high-order differences enable extremely high-accurate calculations.

In the radial basis function-based state-dependent autoregressive (RBF-AR) models with regression weights, the local linear models are included between the hidden layers and the output layers of the networks. The parameter estimation for the RBF-AR models with regression weights is studied in this brief. Considering the separable feature of the models, two criterion functions based on the increasing data length are defined to fit the observation data of the whole dynamical process. Two sub-algorithms are proposed by minimizing the criterion functions. Aiming to overcome the existence of the singular matrix during the Newton search and to make the algorithm more stable, a positive definite diagonal matrix is introduced to the algorithm. Based on the hierarchical principle, a hierarchical Newton recursive algorithm is proposed, which can realize the on-line parameter estimation. Simulation results verify the validity.

In this paper, experimental work to investigate the failure sequence in cross-ply CFRP beams under quasi-static and low-velocity impact (LVI) loadings is presented. [05/903]s CFRP beams with thick clustered plies are investigated for clear observation of damage mechanisms. A non-standard LVI test setup is designed and built in order to conduct line impact experiments and to allow in-situ observation of damage evolution using a high-speed camera. Full-field strain measurement on the visible edge prior to initial failure is performed with digital image correlation (DIC) analysis. In both static and dynamic loading cases, failure sequence is demonstrated to initiate in the middle 90° plies as a diagonal matrix crack leading to delaminations at the upper and lower 0/90 interfaces. Major diagonal matrix crack formation is shown to occur consistently at a local transverse shear strain peak in conjunction with a tensile transverse normal strain. Two additional distinct types of matrix damage, namely hairline cracks and micro-matrix cracks, are discovered in post-mortem examination. Delamination propagation in a composite beam subjected to line loading is monitored using high-speed photography at half-a-million frames-per-second for the first time. It is observed that, in both static and impact cases, delaminations are not only dynamic, but they also reach the same speed plateau of 850 – 900 m/s. We suggest that the experimental crack tip speed data can be used as a benchmark to fine-tune interlaminar damage models.

We develop a preconditioned fast divided-and-conquer finite element approximation for the initial-boundary value problem of variable-order time-fractional diffusion equations. Due to the impact of the time-dependent variable order, the coefficient matrix of the resulting all-at-once system does not have a Toeplitz-like structure. In this paper we derive a fast approximation of the coefficient matrix by the means of a sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires O ( M N log 3 ⁡ N ) computational complexity and O ( M N log 2 ⁡ N ) memory with M and N being the numbers of degrees of freedom in space and time, respectively. Furthermore, a preconditioner is introduced to reduce the number of iterations caused by the bad condition number of the coefficient matrix. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method.

Summary This paper introduces a new data-driven methodology for estimating sparse covariance matrices of the random coefficients in logit mixture models. Researchers typically specify covariance matrices in logit mixture models under one of two extreme assumptions: either an unrestricted full covariance matrix (allowing correlations between all random coefficients), or a restricted diagonal matrix (allowing no correlations at all). Our objective is to find optimal subsets of correlated coefficients for which we estimate covariances. We propose a new estimator, called MISC (mixed integer sparse covariance), that uses a mixed-integer optimization (MIO) program to find an optimal block diagonal structure specification for the covariance matrix, corresponding to subsets of correlated coefficients, for any desired sparsity level using Markov Chain Monte Carlo (MCMC) posterior draws from the unrestricted full covariance matrix. The optimal sparsity level of the covariance matrix is determined using out-of-sample validation. We demonstrate the ability of MISC to correctly recover the true covariance structure from synthetic data. In an empirical illustration using a stated preference survey on modes of transportation, we use MISC to obtain a sparse covariance matrix indicating how preferences for attributes are related to one another.

The covariance matrix, which should be estimated from the data, plays an important role in many multivariate procedures, and its positive definiteness (PDness) is essential for the validity of the procedures. Recently, many regularized estimators have been proposed and shown to be consistent in estimating the true matrix and its support under various structural assumptions on the true covariance matrix. However, they are often not PD. In this paper, we propose a simple modification to make a regularized covariance matrix be PD while preserving its support and the convergence rate. We focus on the matrix $$\ell_{\infty }$$ norm error in covariance matrix estimation because it could allow us to bound the error in the downstream multivariate procedure relying on it. Our proposal in this paper is an extension of the fixed support positive-definite (FSPD) modification by Choi et al. (2019) from spectral and Frobenius norms to the matrix $$\ell_{\infty }$$ norm. Like the original FSPD, we consider a convex combination between the initial estimator (the regularized covariance matrix without PDness) and a given form of the diagonal matrix minimize the $$\ell_{\infty }$$ distance between the initial estimator and the convex combination, and find a closed-form expression for the modification. We apply the procedure to the minimum variance portfolio (MVP) optimization problem and show that the vector $$\ell_{\infty }$$ error in the estimation of the optimal portfolio weight is bounded by the matrix $$\ell _{\infty }$$ error of the plug-in covariance matrix estimator. We illustrate the MVP results with S&P 500 daily returns data from January 1978 to December 2014.

The error variance-covariance matrices of the monthly GRACE gravity field models are usually well-structured (e.g., order-leading) to contain the error information of the monthly gravity field models, and they are important information to further improve the accuracy of the estimated mass transportation by a post-processing scheme. Intensive studies have been performed to understand the impact of different approximations of the full error variance-covariance matrix on the mass estimates obtained with the conventional GRACE-post processing methods (e.g., the de-striping method). In this study, based on the variants of the mascon approach which treat monthly GRACE solutions as input, we consider four different structures to the error variance-covariance matrix, (i) full matrix, (ii) block diagonal matrix, (iii) diagonal matrix, and (iv) identity matrix, and examine their impact on the mass transport estimates in Greenland. Using both synthetic and real data, we analyze the results at four temporal scales: (i) the long-term decadal scale, (ii) the inter-annual scale, (iii) the seasonal scale, and (iv) the monthly scale. Based on the synthetic study, we find that for the recovery of the long-term trend, the application of the diagonal structure obtains the best estimates. This is caused by the amplification of the model error in the mascon approach when considering the full and block diagonal structures of error variance-covariance matrix, not due to any imperfection of them. Therefore, we emphasize that one should be aware of mascon model deficiency in the mascon approach. Furthermore, the best inter-annual, seasonal and monthly mass estimates are derived by considering the block diagonal and full structures. This is caused by the behavior of the bias and the unique parameterization error in the case of different structures. A similar finding is also presented in the real data case. Finally, our analysis denotes the necessity of releasing the block diagonal structure together with the official monthly gravity field model for the GRACE Follow-On mission, instead of releasing only the diagonal structure as done for the GRACE mission.

Let G be a graph, whose adjacency matrix and degree diagonal matrix are denoted by A ( G ) and D ( G ) , respectively. In 2017 Nikiforov (2017) merged the A - and Q -spectral theories to proposed the A α matrix: A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) , α ∈ [ 0 , 1 ] . Its largest eigenvalue is called the A α -index of G . In this paper, we focus on specifying the property of the A α matrix. On the one hand, we show that among the set of connected bipartite graphs of fixed order and size, the extremal graphs with the maximum A α -index are the double nested graphs, i.e., bipartite chain graphs. On the other hand, we establish a series of inequalities respecting the entries of the Perron vector of A α ( G ) of double nested graphs. Then we obtain some lower and upper bounds on the A α -index of double nested graphs using the eigenvector and matrix techniques. Finally, we provide some computational results to compare these bounds.

This article considers a two-way multiple-input multiple-output (MIMO) Rayleigh block fading wiretap channel with two full-duplex nodes (Alice and Bob) exchanging messages simultaneously and wiretapped by a $ {N}_{ {e}}$ -antennas eavesdropper (Eve). The channel state remains unchanged over a coherence interval T and is unknown to all terminals, including Alice, Bob, and Eve, at the beginning of the coherence interval. We first show the expression of the optimal signal waveform for Alice and Bob, which maximizes the sum secrecy rate, should be the products of independent random diagonal matrices and isotropically distributed unitary matrices. Then, conditioned on large T and small $ {N}_{ {e}}$ , the upper-bound for the secure degree of freedom (s.d.o.f.) of the two-way wiretap channel is derived, and a constant-norm-signaling (CNS) scheme is presented to achieve this theoretical bound. Finally, we formulate an optimization problem of the s.d.o.f. achieved by the CNS scheme for general cases with arbitrary T and $ {N}_{ {e}}$ , and solve this problem by exploring the monotonicity of the achievable s.d.o.f.. Our result shows that the optimal s.d.o.f. can be independent of coherence time T and becomes the product of the numbers of legitimate nodes’s transmitter antennas (NLNTA) for large $ {N}_{ {e}}$ and small NLNTA.

In this paper, we propose two methods for solving unconstrained multiobjective optimization problems. First, we present a diagonal steepest descent method, in which, at each iteration, a common diagonal matrix is used to approximate the Hessian of every objective function. This method works directly with the objective functions, without using any kind of a priori chosen parameters. It is proved that accumulation points of the sequence generated by the method are Pareto-critical points under standard assumptions. Based on this approach and on the Nesterov step strategy, an improved version of the method is proposed and its convergence rate is analyzed. Finally, computational experiments are presented in order to analyze the performance of the proposed methods.

The net Laplacian matrix of a signed graph G is defined to be N G = D G ± − A G , where D G ± and A G are the diagonal matrix of net-degrees and the adjacency matrix of G , respectively. For a binary vector b , the pair ( N G , b ) is controllable if N G has no eigenvector orthogonal to b ; we also say that G is net Laplacian controllable for b . In this study we consider the net Laplacian controllability of joins of signed graphs. In particular, we establish all controllable pairs ( N G , b ) , where G is a signed threshold graph determined by a ( 0 , 1 , − 1 ) -generating sequence. This result contains all controllable pairs ( L G , b ) , where L G is the Laplacian matrix of a graph G .

We consider hypothesis testing for high-dimensional covariance structures in which the covariance matrix is a (i) scaled identity matrix, (ii) diagonal matrix, or (iii) intraclass covariance matrix. Our purpose is to systematically establish a nonparametric approach for testing the high-dimensional covariance structures (i)–(iii). We produce a new common test statistic for each covariance structure and show that the test statistic is an unbiased estimator of its corresponding test parameter. We prove that the test statistic establishes the asymptotic normality. We propose a new test procedure for (i)–(iii) and evaluate its asymptotic size and power theoretically when both the dimension and sample size increase. We investigate the performance of the proposed test procedure in simulations. As an application of testing the covariance structures, we give a test procedure to identify an eigenvector. Finally, we demonstrate the proposed test procedure by using a microarray data set.

We prove ultradifferentiable Chevelley restriction theorems for a wide range of ultradifferentiable classes. As a special case we find that isotropic functions, i.e., functions defined on the vector space of real symmetric matrices invariant under the action of the special orthogonal group by conjugation, possess some ultradifferentiable regularity if and only if their restriction to diagonal matrices has the same regularity.

In this paper, we consider a large scale portfolio selection problem with and without a sparsity constraint. Neutral constraints on industries are included as well as investment styles. To develop fast algorithms for the use in the real financial market, we shall expose the special structure of the problem, whose Hessian is the summation of a diagonal matrix and a low rank modification. Specifically, an interior point algorithm taking use of the Sherman–Morrison–Woodbury formula is designed to solve the problem without any sparsity constraint. The complexity in each iteration of the proposed algorithm is shown to be linear with the problem dimension. In the occurrence of a sparsity constraint, we propose an efficient three-block alternating direction method of multipliers, whose subproblems are easy to solve. Extensive numerical experiments are conducted, which demonstrate the efficiency of the proposed algorithms compared with some state-of-the-art solvers.

This paper investigates the general decay stability on systems represented by stochastic functional differential equations with semi-Markovian switching and Levy noise (SFDEs-SMS-LN). Based on functional Ito’s formula, multiple degenerate Lyapunov functionals and nonnegative semi-martingale convergence theorem, new pth moment and almost surely stability criteria with general decay rate for SFDEs-SMS-LN are established. Meanwhile, the diffusion operators are allowed to be controlled by multiple auxiliary functions with time-varying coefficients, which can be more adaptable to the non-autonomous SFDEs-SMS-LN with high-order nonlinear coefficients. Furthermore, as applications of the presented stability criteria, new delay-dependent sufficient conditions for general decay stability of the stochastic delayed neural network with semi-Markovian switching and Levy noise (SDNN-SMS-LN) and the scalar non-autonomous SFDE-SMS-LN with non-global Lipschitz condition are respectively obtained in terms of binary diagonal matrices (BDMs) and linear matrix inequalities (LMIs). Finally, two numerical examples are given to demonstrate the effectiveness of the proposed results.

This paper presents a class of low memory quasi-Newton methods with standard backtracking line search for large-scale unconstrained minimization. The methods are derived by means of least change updating technique analogous to that for the DFP method except that the full quasi-Newton matrix has been replaced by some diagonal matrix. We establish convergence properties for some particular members of the class under line search with Armijo condition. Sufficient conditions for the methods to be superlinearly convergent are also given. Numerical results are then presented to illustrate the usefulness of these methods in large-scale minimization.

We extend the positive-definite and skew-Hermitian splitting (PSS) iterative method to EPSS (for extended PSS) for solving non-Hermitian positive-definite systems of linear equations. In this kind of extension, the shift matrix is replaced by a Hermitian positive-definite matrix Σ . It is proved that the method is unconditionally convergent. Then, the EPSS method with a 2 × 2-block diagonal matrix Σ as the shift matrix is applied to generalized saddle point problems and the convergence of the method is verified. Naturally, the induced preconditioner can be applied to generalized saddle point problems. In some special cases, this preconditioner coincides with some existing preconditioners. A special case of the EPSS preconditioner, say SEPSS, is used to accelerate the convergence of GMRES. Effectiveness of this preconditioner is investigated by some numerical experiments.

We develop a fast divide-and-conquer indirect collocation method for the homogeneous Dirichlet boundary value problem of variable-order space-fractional diffusion equations. Due to the impact of the space-dependent variable order, the resulting stiffness matrix of the numerical scheme does not have a Toeplitz structure. In this paper, we derive a fast approximation of the coefficient matrix by the means of a finite sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires $O(N\log ^{2} N)$ memory and $O(N\log ^{3} N)$ computational complexity with N being the numbers of unknowns. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method.

An accelerated diagonal quasi-Newton updating algorithm for unconstrained optimization is presented. The elements of the diagonal matrix approximating the Hessian are determined as scaling of the f...

This letter proposes a method to obtain integer reversible discrete cosine transforms for generic transform-based coding schemes. The novelty of the proposed method, which is based on decomposition of the DCT-II matrix into two triangular and one diagonal matrices, is twofold: (i) the new matrices can be of arbitrary size, i.e., any square $N\times N$ dimension, thus suitable for applications where non power-of-2 dimensions are required; (ii) they can be designed with adjustable precision in a trade-off with the number of representation bits. Furthermore, improvements are also proposed over the base scheme to avoid numerical issues when working with large matrices and to obtain more reliable approximations. The performance evaluation demonstrate the effectiveness of the proposed transforms to approximate the coding gain capabilities of the original DCT-II.

The structure of the Hessian matrix obtained from the locational cost used in coverage control is investigated to provide conditions on the optimality of coverage control solutions. It is shown that in arbitrary dimensions, the Hessian matrix is composed of the direct sum of three well-structured matrices: 1) a diagonal matrix; 2) a block-diagonal matrix; and 3) block-Laplacian matrix. This structure is exploited in the one-dimensional case, where an alternative proof of a sufficient condition for optimality is given. A relationship is shown between centroidal Voronoi tessellation (CVT) configurations and the sufficient condition for optimality via the spatial derivative of the density provided in the cost. A decomposition is used to provide insight into the terms which most affect optimality. Several classes of density functions are analyzed under the proposed condition. Experiments on a multi-robot team are shown to verify theoretical results.

Memoryless quasi–Newton updating formulas of BFGS (Broyden–Fletcher–Goldfarb–Shanno) and DFP (Davidon–Fletcher–Powell) are scaled using well-structured diagonal matrices. In the scaling approach, diagonal elements as well as eigenvalues of the scaled memoryless quasi–Newton updating formulas play significant roles. Convergence analysis of the given diagonally scaled quasi–Newton methods is discussed. At last, performance of the methods is numerically tested on a set of CUTEr problems as well as the compressed sensing problem.

The random walk model describing the super-diffusion competition phenomenon of particles can derive the spatial fractional diffusion equation. For irregular diffusion and super-diffusion phenomena, the use of such equation can obtain more accurate and realistic results, so it has a wide application background in practice. The implicit finite-difference method derived from the shifted Grunwald scheme is used to discretize the spatial fractional diffusion equation. The coefficient matrix of discrete system is in the form of the sum of a diagonal matrix and a Toeplitz matrix. In this paper, a preconditioner is constructed, which transforms the coefficient matrix into the form of an identity matrix plus a diagonal matrix multiplied by Toeplitz matrix. On this basis, a new quasi-Toeplitz trigonometric transform splitting iteration format (abbreviated as QTTTS method) is proposed. We theoretically verify the unconditional convergence of the new method, and obtain the effective optimal form of the iteration parameter. Finally, numerical simulation experiments also demonstrate the accurateness and efficiency of the new method.

In this paper, we construct random two-faced families of matrices with non-Gaussian entries to approximate a bi-free central limit distribution with a positive definite covariance matrix. We prove that, under modest conditions weaker than independence, a family of random two-faced families of matrices with non-Gaussian entries is asymptotically bi-free from a two-faced family of constant diagonal matrices.

This paper studies the trade-off performance between tracking performance and control input energy of the multi-input multi-output(MIMO), linear and time-invariant(LTI) system over an additive coloured Gaussian noise(ACGN) channel and the encoder-decoder strategies. The restriction that filter in the encoder-decoder strategy must be diagonal matrix is not necessary. And some new results are derived according to the inner-outer factorization. The results show that the trade-off performance is correlated to the unstable pole, non-minimum phase zero of the system. Also new poles and zeros generated by the non-diagonal encoder-decoder strategies may affect the trade-off performance. At last, two examples with different filters and different encoder-decoder strategies are discussed to validate the conclusions. The various encoder-decoder strategies revealed by the simulations may enhance or deteriorate the trade-off performance proposed in this paper.

While the literature on numerical methods (e.g. finite elements and, to a certain extent, virtual elements) concentrates on convex elements, there is a need to probe beyond this limiting constraint so that the field can be further explored and developed. Thus, in this paper, we employ the virtual element method for non-convex discretizations of elastodynamic problems in 2D and 3D using an explicit time integration scheme. In the formulation, a diagonal matrix-based stabilization scheme is proposed to improve performance and accuracy. To address efficiency, a critical time step is approximated and verified using the maximum eigenvalue of the local (rather than global) system. The computational results demonstrate that the virtual element method is able to consistently handle general nonconvex elements and even non-simply connected elements, which can lead to convenient modeling of arbitrarily-shaped inclusions in composites.

An algorithm for solving large-scale systems of nonlinear equations based on the transformation of the Newton method with the line search into a derivative-free descent method is introduced. Main idea used in the algorithm construction is to approximate the Jacobian by an appropriate diagonal matrix. Furthermore, the step length is calculated using inexact line search procedure. Under appropriate conditions, the proposed method is proved to be globally convergent under mild conditions. The numerical results presented show the efficiency of the proposed method.

This paper investigates the general decay stability on systems represented by stochastic functional differential equations with semi-Markovian switching and Levy noise (SFDEs-sMS-LN). Based on generalized multidimensional Ito’s formula and multiple Lyapunov functions, a new pth moment stability criterion with general decay rate is established. Meanwhile, as an applications of the presented stability criterion, we consider the stabilization problem of stochastic delayed neural networks with semi-Markovian switching and Levy noise (SDNN-sMS-LN). A vertex approach is proposed to design the controller in terms of binary diagonal matrices (BDMs) and linear matrix inequalities (LMIs). Finally, a numerical example is presented to demonstrate the effectiveness of the proposed results.

We introduce and study the combinatorial optimization problem with interaction costs (COPIC). COPIC is the problem of finding two combinatorial structures, one from each of two given families, such that the sum of their independent linear costs and the interaction costs between elements of the two selected structures is minimized. COPIC generalizes the quadratic assignment problem and many other well studied combinatorial optimization problems, and hence covers many real world applications. We show how various topics from different areas in the literature can be formulated as special cases of COPIC. The main contributions of this paper are results on the computational complexity and approximability of COPIC for different families of combinatorial structures (e.g. spanning trees, paths, matroids), and special structures of the interaction costs. More specifically, we analyze the complexity if the interaction cost matrix is parameterized by its rank and if it is a diagonal matrix. Also, we determine the structure of the intersection cost matrix, such that COPIC is equivalent to independently solving linear optimization problems for the two given families of combinatorial structures.

The paper studies two numerical characteristics of matrix incidence algebras over finite fields associated with generating sets of such algebras: the minimal cardinality of a generating set and the length of an algebra. Generating sets are understood in the usual sense, the identity of the algebra being considered a word of length 0 in generators, and also in the strict sense, where this assumption is not used. A criterion for a subset to generate an incidence algebra in the strict sense is obtained. For all matrix incidence algebras, the minimum cardinality of a generating set and a generating set in the strict sense are determined as functions of the field cardinality and the order of the matrices. Some new results on the lengths of such algebras are obtained. In particular, the length of the algebra of “almost” diagonal matrices is determined, and a new upper bound for the length of an arbitrary matrix incidence algebra is obtained.

Chebyshev polynomial spectral methods are very accurate, but are plagued by the cost and ill-conditioning of dense discretization matrices. Modified schemes, collectively known as “integration sparsification”, have mollified these problems by discretizing the highest derivative as a diagonal matrix. Here, we examine five case studies where the highest derivative diagonalization fails. Nevertheless, we show that Galerkin discretizations do yield banded matrices that retain most of the advantages of “integration sparsification”. Symbolic computer algebra greatly extends the reach of spectral methods. When spectral methods are implemented using exact rational arithmetic, as is possible for small truncation N in Maple, Mathematica and their ilk, roundoff error is irrelevant, and sparsification failure is not worrisome. When the discretization contains a parameter L , symbolic algebra spectral methods return, as answer to an eigenproblem, not discrete numbers but rather a plane algebraic curve defined as the zero set of a bivariate polynomial P ( λ , L ) ; the optimal approximations to the eigenvalues λ j are in the middle of the straight portions of the zero contours of P ( λ ; L ) where the isolines are parallel to the L axis.

A new diagonal quasi-Newton updating algorithm for unconstrained optimization is presented. The elements of the diagonal matrix approximating the Hessian are determined as scaled forward finite dif...

We investigate a kind of generalized equations involving absolute values of variables as $|A|x-|B||x|=b$ , where $A \in R^{n\times n}$ is a symmetric matrix, $B \in R^{n\times n}$ is a diagonal matrix, and $b\in R^{n}$ . A sufficient condition for unique solvability of the proposed generalized absolute value equations is also given. By utilizing an equivalence relation to the unconstrained optimization problem, we propose a modified HS conjugate gradient method to solve the transformed unconstrained optimization problem. Only under mild conditions, the global convergence of the given method is also established. Finally, the numerical results show the efficiency of the proposed method.

Let G be a graph of order n with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For any real α ∈ [0, 1], write Aα(G) for the matrix A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . This paper shows some extremal results on the spectral radius ρα(G) of Aα(G). We determine the upper bound on ρα(G) if α ∈ (0, 1) and G is a graph with no K2, t (t ≥ 3) minor. We also show that the unique outerplanar graph of order n with maximum ρα(G) is the join of a vertex and a path P n − 1 . Moreover, we prove that the unique planar graph of order n with maximum signless Laplacian spectral radius is the join of an edge and a path P n − 2 .

In this paper, we show a novel algorithm for strong line-of-sight (LoS) multiple-input-multiple-output (MIMO) channel equalization. With optimally spaced antennas under specific arrangements, the LoS channels can be made spatially orthogonal. In practice, antenna displacements are expected. Conventional algorithms like zero-forcing (ZF) do not consider the special properties of the LoS MIMO channel and result in high complexity. We show that the LoS MIMO channel can be factorized into a product of three matrices. Thereby, the two diagonal matrices at the outer product positions are the most varying terms and should be compensated dynamically. Being a good tradeoff between complexity and robustness, the proposed sequential channel equalization is applied in a reverse order of the factorization. The algorithm can be applied to LoS MIMO systems with uniform linear or rectangular arrays, which are making use of digital or analog equalization. By considering the usage of the Winograd butterfly and Butler matrices, the number of multiplications in both digital and analog implementations of the proposed solution is increasing approximately linearly with respect to the number of antennas, while the complexity of the state-of-the-art designs grows quadratically. As found numerically and verified experimentally, the proposed method performs nearly as well as the ZF-based algorithms with near-optimal arrangements, while having significantly lower complexity.

In this paper, we solve the nonlinear space fractional complex Ginzburg-Landau equations. The motivation of this work is to give a fast numerical method to solve the linear system, which is obtained from the nonlinear space fractional complex Ginzburg-Landau equations. The coefficient matrix of the linear system is the sum of a complex diagonal matrix and a real Toeplitz matrix. The significance of this work is that the new method has a superiority in computation because we can use the circulant preconditioner and the fast Fourier transform to solve the linear system. Numerical examples are tested to illustrate the advantage of the numerical method.

Given a convex and differentiable objective $Q(\M)$ for a real symmetric matrix $\M$ in the positive definite (PD) cone -- used to compute Mahalanobis distances -- we propose a fast general metric learning framework that is entirely projection-free. We first assume that $\M$ resides in a space $\cS$ of generalized graph Laplacian matrices corresponding to balanced signed graphs. $\M \in \cS$ that is also PD is called a graph metric matrix. Unlike low-rank metric matrices common in the literature, $\cS$ includes the important diagonal-only matrices as a special case. The key theorem to circumvent full eigen-decomposition and enable fast metric matrix optimization is Gershgorin disc perfect alignment (GDPA): given $\M \in \cS$ and diagonal matrix $\S$, where $S_{ii} = 1/v_i$ and $\v$ is $\M$'s first eigenvector, we prove that Gershgorin disc left-ends of similarity transform $\B = \S \M \S^{-1}$ are perfectly aligned at the smallest eigenvalue $\lambda_{\min}$. Using this theorem, we replace the PD cone constraint in the metric learning problem with tightest possible linear constraints per iteration, so that the alternating optimization of the diagonal / off-diagonal terms in $\M$ can be solved efficiently as linear programs via the Frank-Wolfe method. We update $\v$ using Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) with warm start as entries in $\M$ are optimized successively. Experiments show that our graph metric optimization is significantly faster than cone-projection schemes, and produces competitive binary classification performance., Comment: code available: https://github.com/bobchengyang/SGML

Suppose that the vertex set of a connected graph G is $$V(G)=\{v_1,\ldots ,v_n\}$$ . Then we denote by $$Tr_{G}(v_i)$$ the sum of distances between $$v_i$$ and all other vertices of G. Let Tr(G) be the $$n\times n$$ diagonal matrix with its (i, i)-entry equal to $$Tr_{G}(v_{i})$$ and D(G) be the distance matrix of G. Then $$Q_{D}(G)=Tr(G)+D(G)$$ is the distance signless Laplacian matrix of G. The largest eigenvalues of $$Q_D(G)$$ is called distance signless Laplacian spectral radius of G. In this paper we give some graft transformations on distance signless Laplacian spectral radius of the graphs and use them to characterize the graphs with the minimum and maximal distance signless Laplacian spectral radius among non-starlike and non-caterpillar trees.

In this paper, we present a derivative-free multivariate spectral projection algorithm for convex constrained nonlinear monotone equations. The search direction is a product of a convex combination of two different spectral diagonal matrices and the residual vector. Moreover, to ensure positive definiteness of the diagonal matrix associated with the search direction, suitable safeguard is formulated. Some of the remarkable properties of the algorithm include: Jacobian free approach, capacity to solve large-scale problems and the search direction generated by the algorithm, satisfy the descent property independent on the line search employed. Under appropriate assumptions, the global convergence of the algorithm is given. Numerical experiments show that the algorithm has advantages over the recently proposed multivariate derivative-free projection algorithm by Liu and Li (J Ind Manag Optim 13(1):283–295, 2017) and also compete with another algorithm having the standard choice of the Barzilai-Borwein step as the search direction.

In this paper. a matrix-free method for solving large-scale system of nonlinear equations is presented. The method is derived via quasi-Newton approach, where the approximation to the Broyden's update is sufficiently done by constructing diagonal matrix using acceleration parameter. A fascinating feature of the method is that it is a matrix-free, so is suitable for solving large-scale problems. Furthermore, the convergence analysis and preliminary numerical results that is reported using a benchmark test problems, shows that the method is promising.

In this paper, we present a method to solve the quantum marginal problem for symmetric $d$-level systems. The method is built upon an efficient semi-definite program that determines the compatibility conditions of an $m$-body reduced density with a global $n$-body density matrix supported on the symmetric space. We illustrate the applicability of the method in central quantum information problems with several exemplary case studies. Namely, (i) a fast variational ansatz to optimize local Hamiltonians over symmetric states, (ii) a method to optimize symmetric, few-body Bell operators over symmetric states and (iii) a set of sufficient conditions to determine which symmetric states cannot be self-tested from few-body observables. As a by-product of our findings, we also provide a generic, analytical correspondence between arbitrary superpositions of $n$-qubit Dicke states and translationally-invariant diagonal matrix product states of bond dimension $n$., Comment: 27 pages (21 + appendices), 12 figures

A novel quantum algorithm for solving advection–diffusion equation by the lattice Boltzmann method is proposed. The presented quantum algorithm is composed of two major segments. In the first segment, equilibrium distribution function, presented in the form of a non-unitary diagonal matrix, is quantum circuit implemented by using a standard-form encoding approach. For the second segment, the quantum walk procedure as a method for implementing the propagation step is applied. The constructed algorithm is presented as a series of single- and two-qubit gates, as well as multiple-input controlled-NOT gates. In order to demonstrate the validity of the proposed quantum algorithm, the unsteady one-dimensional (1D) and two-dimensional (2D) advection–diffusion equations are solved by using the IBM’s quantum computing software development framework Qiskit, while the analytic solution and the classic code are used for verification. Finally, the complexity analysis and directions for future work are discussed.

The iterative Gauss-Seidel method is an effective and practical method for solving the absolute value equations. However, the solution efficiency of this method usually decreases, and even the equation cannot be solved even when the problem reaches a certain large scale. To improve the efficiency of the Gauss-Seidel method for solving absolute value equations, a modified Gauss-Seidel (MGS) iteration method is presented in this paper. In the our method, we create a diagonal matrix \(\varOmega \) with nonnegative diagonal elements in the Gauss-Seidel matrix splitting. Under the given constraints the convergence theory of the MGS method have been studied. The numerical results show that the method is effective. It can be noted that with the increase in the scale of the problem, the setting effect of the matrix \(\varOmega \) is more obvious.

This paper investigates the optimal estimation for continuous-time linear system subject to delayed measurement and multiplicative noise, in which the multiplicative noise is characterized by a diagonal matrix. The reorganized innovation analysis approach is employed to transform the measurement-delay system into an equivalent delay-free one, and an optimal filter is derived by solving two generalized differential Riccati equations based on a differential Lyapunov equation. Then a stationary filter is developed, where the filter gains are derived in terms of the stabilizing solutions of two generalized algebraic Riccati equations and a algebraic Lyapunov equation. Finally, a numerical example is presented to demonstrate the efficiency of the proposed design method.

The Terwilliger algebra $T(x)$ of a finite connected simple graph $\Gamma$ with respect to a vertex $x$ is the complex semisimple matrix algebra generated by the adjacency matrix $A$ of $\Gamma$ and the diagonal matrices $E_i^*(x)=\operatorname{diag}(v_i)$ $(i=0,1,2,\dots)$, where $v_i$ denotes the characteristic vector of the set of vertices at distance $i$ from $x$. The twisted Grassmann graph $\tilde{J}_q(2D+1,D)$ discovered by Van Dam and Koolen in 2005 has two orbits of the automorphism group on its vertex set, and it is known that one of the orbits has the property that $T(x)$ is thin whenever $x$ is chosen from it, i.e., every irreducible $T(x)$-module $W$ satisfies $\dim E_i^*(x)W\leqslant 1$ for all $i$. In this paper, we determine all the irreducible $T(x)$-modules of $\tilde{J}_q(2D+1,D)$ for this "thin" case., Comment: 22 pages

For linear continuous-time positive systems the paper proposes an approach, reflecting structural constraints and positiveness in the problem of Metzlerian and strictly Metzlerian interval observers design. Every interval matrix boundary is represented by a set of linear matrix inequalities representing Metzler matrix parameter constraints and reflecting potential zero elements in a Metzler matrix structure by structural diagonal matrix variables. Combined with couple of Lyapunov inequalities, the observer Metzler matrix parameters are guaranteed and interval stability is attained. A numerical example is included to assess the feasibility technique.