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This paper addresses the problem of event-triggered $\mathcal {H}_{\infty }$ control for continuous Takagi-Sugeno fuzzy systems with repeated scalar nonlinearities. A feasible stability solution is first proposed based on the fuzzy-rule-dependent Lyapunov functional methods and positive definite diagonally dominant matrix techniques, which not only reduces the conservativeness of the resulting closed-loop dynamic system, but also ensures the concerned fuzzy system is asymptotically stable with a specified $\mathcal {H}_{\infty }$ disturbance attenuation performance. Then, sufficient conditions are presented for the existence of admissible state-feedback controller, and the cone complementarity linearization approach is employed to convert the non-convex feasibility problem into a sequential minimization one subject to linear matrix inequalities, which can be validly solved by employing standard numerical software. In the end, a numerical example and a Chua’s chaotic circuit system are provided to show the applicability of the proposed theories.

In the paper the two-dimensional elliptic equation with integral boundary conditions is solved by finite difference method. The main aim of the paper is to investigate the conditions for the convergence of the iterative methods for the solution of system of nonlinear difference equations. With this purpose, we investigated the structure of the spectrum of the difference eigenvalue problem. Some sufficient conditions are proposed such that the real parts of all eigenvalues of the corresponding difference eigenvalue problem are positive. The proof of convergence of iterative method is based on the properties of the M-matrices not requiring the symmetry or diagonal dominance of the matrices. The theoretical statements are supported by the results of the numerical experiment.

Strong H -tensors play an important role in the theories and applications of numerical linear algebra. It is necessary to identify whether a given tensor is a strong H -tensor or not. In this paper, we establish some iterative criteria for identifying strong H -tensors. These criteria depend on the elements of the tensors; therefore, they are easy to be verified. The results obtained in this paper extend the corresponding conclusions for strictly generalized diagonally dominant matrices. As an application, some sufficient conditions for the positive definiteness of an even-order real symmetric tensor are presented. Some numerical experiments show the feasibility and efficiency of the results which are obtained in this paper.

The dynamic characteristics of the power system have been greatly influenced as the grid-integration of large-scale power electronic devices, such as renewable power generation and direct current transmission. The wide-frequency oscillation dynamic stability problem is prone to occur. In this paper, the Nyquist array theory in the multivariate frequency-domain analysis theory is introduced into the dynamic stability analysis and control of power systems. The power system integrated with doubly-fed induction generator (DFIG) is illustrated in the paper. The forward transfer function matrix and the feedback gain transfer function matrix of the system are derived respectively. Based on the Nyquist array theory, the diagonal dominance characteristics of the system are discriminated. For the diagonal dominance system, the Gershgorin band can be drawn to display the stability characteristics of the system visually. For the non-diagonal dominance system, the system can be transformed into a diagonal dominance system by pseudo-diagonalization, and then the stability analysis and control can be discriminated with Gershgorin band. Finally, the effectiveness of the proposed method is verified by comparison with the eigenvalue calculation results.

A real interval vector/matrix is an array whose entries are real intervals. In this paper, we consider the real linear interval equations \begin{document}$ \bf{Ax} = \bf{b} $\end{document} with \begin{document}$ {{\bf{A}} }$\end{document}, \begin{document}$ \bf{b} $\end{document} respectively, denote an interval matrix and an interval vector. The aim of the paper is to study the numerical solution of the linear interval equations for various classes of coefficient interval matrices. In particular, we study the convergence of interval AOR method when the coefficient interval matrix is either interval strictly diagonally dominant matrices, interval \begin{document}$ L $\end{document}-matrices, interval \begin{document}$ M $\end{document}-matrices, or interval \begin{document}$ H $\end{document}-matrices.

In this paper, we establish some criteria for strong H -tensors. The results obtained in this paper extend the corresponding conclusions for strong H -matrices and improve the existing results for ...

In the paper the author presents a novel kind of the WZ factorization algorithm, namely a block WZ factorization algorithm. The aim of this new algorithm is to utilize the computational power of contemporary computers with hierarchical memory. In the paper, some properties of the matrix Z are given and analyzed. Next, a version of the block WZ factorization is presented. The author shows that such a block WZ factorization exists for strictly diagonally dominant matrices. The computational cost of this block algorithm is presented. The time and the accuracy of proposed block WZ factorization algorithm for random dense square diagonally dominant matrices are reported. The block algorithm turned out to be faster even up to 300 times than the original WZ factorization.

Analysis of large-scale sequential data has been one of the most crucial tasks in areas such as bioinformatics, text, and audio mining. Existing string kernels, however, either (i) rely on local features of short substructures in the string, which hardly capture long discriminative patterns, (ii) sum over too many substructures, such as all possible subsequences, which leads to diagonal dominance of the kernel matrix, or (iii) rely on non-positive-definite similarity measures derived from the edit distance. Furthermore, while there have been works addressing the computational challenge with respect to the length of string, most of them still experience quadratic complexity in terms of the number of training samples when used in a kernel-based classifier. In this paper, we present a new class of global string kernels that aims to (i) discover global properties hidden in the strings through global alignments, (ii) maintain positive-definiteness of the kernel, without introducing a diagonal dominant kernel matrix, and (iii) have a training cost linear with respect to not only the length of the string but also the number of training string samples. To this end, the proposed kernels are explicitly defined through a series of different random feature maps, each corresponding to a distribution of random strings. We show that kernels defined this way are always positive-definite, and exhibit computational benefits as they always produce \emph{Random String Embeddings (RSE)} that can be directly used in any linear classification models. Our extensive experiments on nine benchmark datasets corroborate that RSE achieves better or comparable accuracy in comparison to state-of-the-art baselines, especially with the strings of longer lengths. In addition, we empirically show that RSE scales linearly with the increase of the number and the length of string., KDD'19 Oral Paper, Data and Code link available in the paper

For the traditional diagonal dominance algorithms unconstrained the non-diagonal elements, a strict constraint of diagonal dominance is proposed in the paper. Contrary to the normal definition, the novel definition of strict diagonal dominance restricts the max non-diagonal elements, and the definition of strict Gershgorin discs is also discussed. To achieve diagonal dominance the compensating matrix search algorithm based on trial and error is introduced in the paper. Example of compensating search algorithm is applied to support the proposed method, and the results obtained are comparable with normal method. It gives a novel way to define the diagonal dominance.

This paper addresses the problem of coexistence and dynamical behaviors of multiple equilibria for competitive neural networks. First, a general class of discontinuous nonmonotonic piecewise linear activation functions is introduced for competitive neural networks. Then based on the fixed point theorem and theory of strict diagonal dominance matrix, it is shown that under some conditions, such $\boldsymbol {n}$ -neuron competitive neural networks can have $5^{\boldsymbol n}$ equilibria, among which $3^{\boldsymbol n}$ equilibria are locally stable and the others are unstable. More importantly, it is revealed that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibria and locally stable equilibria than the ones with other activation functions, such as the continuous Mexican-hat-type activation function and discontinuous two-level activation function. Furthermore, the $3^{\boldsymbol n}$ locally stable equilibria given in this paper are located in not only saturated regions, but also unsaturated regions, which is different from the existing results on multistability of neural networks with multiple level activation functions. A simulation example is provided to illustrate and validate the theoretical findings.

In this paper we present an efficient framework for the evaluation of subdivision schemes with polynomial reproduction property. For all interested rational parameters between 0 and 1 with the same denominator, their exact limit positions on the subdivision curve can be obtained by solving a system of linear equations. When the framework is applied to binary and ternary 4-point interpolatory subdivision schemes, we find that the corresponding coefficient matrices are strictly diagonally dominant, and so the evaluation processes are robust. For any individual irrational parameters between 0 and 1, its approximate value is computed by a recursive algorithm which can attain an arbitrary error bound. For surface schemes generalizing univariate subdivision schemes with polynomial reproduction property, exact evaluation methods can also be derived by combining Stam's method with that of this paper. The method is applicable to all subdivision schemes with polynomial reproduction.It performs exact evaluation at rational parameters and approximate evaluation at other arbitrary parameters with tolerance control.It is efficient and robust for the presented schemes with corresponding coefficient matrix being strictly and diagonally dominant.It can also evaluate derivatives under the same framework.Extension of the method to surface cases is straightforward.

The location of the control devices that produce forced oscillation sources is crucial to the elimination of forced oscillations. This paper proposes two frequency domain approaches to reliably locate oscillation sources from mechanical parts and excitation systems of generators to their control devices. Firstly, the transfer function matrix from mechanical disturbances to mechanical powers is demonstrated to be inherently diagonally dominant. The property is used to monitor and preliminarily locate oscillation sources from mechanical parts. Moreover, this paper proposes a systematic method of power spectral density prediction to further locate oscillation source to control device by using the generator terminal voltage (or generator current) to predict the power spectral density of generator responses. It is shown that for non-source generators, the responses are determined by the generator terminal voltage. The generator with distinct mismatch between predicted and measured power spectral density is identified as the oscillation source. The mismatches between predicted and measured power spectral density of different responses are different, further enabling the method to identify the control device producing the oscillation source. The method of power spectral density prediction is used as a general method to provide further and detailed location results. Demonstrative examples of a 39-bus system and the 547-machine 8647-bus North China Power Grid show the effectiveness of the proposed methods.

International audience; We investigate a mixed-integer variant of the alternating current optimal flow problem. The binary variables activate and deactivate power generators installed at a subset of nodes of the electrical grid. We propose some formulations and a mixed-integer semidefinite programming relaxation, from which we derive two mixed-integer diagonally dominant programming approximation (inner and outer, the latter providing a relaxation). We discuss dimensionality reduction methods to extract solution vectors from solution matrices, and present some computational results showing how both our approximations provide tight bounds.

The paper considers upper bounds for the infinity norm of the inverse for matrices in two subclasses of the class of (nonsingular) H-matrices, both of which contain the class of Nekrasov matrices. The first one has been introduced recently and consists of the so-called S-Nekrasov matrices. For S-Nekrasov matrices, the known bounds are improved. The second subclass consists of the socalled QN- (quasi-Nekrasov) matrices, which are defined in the present paper. For QN-matrices, an upper bound on the infinity norm of the inverses is established. It is shown that in application to Nekrasov matrices the new bounds are generally better than the known ones. Bibliography: 15 titles.

The paper first introduces two-dimensional convection-diffusion equation with boundary value condition, later uses the finite difference method to discretize the equation and analyzes positive definite, diagonally dominant and symmetric properties of the discretization matrix. Finally, the paper uses fixed point methods and Krylov subspace methods to solve the linear system and compare the convergence speed of these two methods.

This paper suggests a unifying framework for matrix spectra localizations that originate from different generalizations of strictly diagonally dominant matrices. Although a lot of results of this kind have been published over the years, in many papers same properties were proven for every specific localization area using basically the same techniques. For that reason, here, we introduce a concept of DD-type classes of matrices and show how to construct eigenvalue localization sets. For such sets we then prove some general principles and obtain as corollaries many singular results that occur in the literature. Moreover, obtained principles can be used to construct and use novel Gersgorin-like localization areas. To illustrate this, we first prove a new nonsingularity result and then use established principles to obtain the corresponding localization set and its several properties. In addition, some new results on eigenvalue separation lines and upper bounds for spectral radius are obtained, too.

This paper presents a numerical investigation for the computation of wind or marine current turbines in a farm. A 3D unsteady Lagrangian vortex method is used together with a panel method in order to take into account for the turbines. In order to enforce the boundary condition onto the panel elements, a linear matrix system is defined. Solving general linear matrix systems is a topic with important scientific literature. But the main concern here is the application to a dedicated matrix which is non-sparse, non-symmetric, neither diagonally dominant nor positive-definite. Several iterative approaches were tested and compared. But after some numerical tests, a Bi-CGSTAB method was finally chosen. The main advantage of the presented method is the use of a specific preconditioner well suited for the desired application. The chosen implementation proved to be very efficient with only 3 iterations of our preconditioned Bi-CGSTAB algorithm whatever the turbine geometrical configuration. Although developed for wind or marine turbines, the proposed algorithm is absolutely not restricted to these cases, and can be applied to many others. At the end of the paper, some applications (specifically, wake computations) in a farm are presented, along with a quantitative assessment of the computational time savings brought by the iterative approach.

In this paper, strong relative perturbation bounds are developed for a number of linear algebra problems involving diagonally dominant matrices. The key point is to parameterize diagonally dominant matrices using their off-diagonal entries and diagonally dominant parts and to consider small relative componentwise perturbations of these parameters. This allows us to obtain new relative perturbation bounds for the inverse, the solution to linear systems, the symmetric indefinite eigenvalue problem, the singular value problem, and the nonsymmetric eigenvalue problem. These bounds are much stronger than traditional perturbation results, since they are independent of either the standard condition number or the magnitude of eigenvalues/singular values. Together with previously derived perturbation bounds for the LDU factorization and the symmetric positive definite eigenvalue problem, this paper presents a complete and detailed account of relative structured perturbation theory for diagonally dominant matrices. This research was partially supported by the Ministerio de Economía y Competitividad of Spain under grant MTM2012-32542. Publicado

A real square matrix whose non-diagonal elements are non-positive is called a Z-matrix. This paper shows a necessary and sufficient condition for non-singularity of two types of Z-matrices. The first is for the Z-matrix whose row sums are all non-negative. The non-singularity condition for this matrix is that at least one positive row sum exists in any principal submatrix of the matrix. The second is for the Z-matrix which satisfies where . Let be the ith row and the jth column element of , and be the jth element of . Let be a subset of which is not empty, and be the complement of if is a proper subset. The non-singularity condition for this matrix is such that or such that for . Robert Beauwens and Michael Neumann previously presented conditions similar to these conditions. In this paper, we present a different proof and show that these conditions can be also derived from theirs.

This paper presents numerical methodologies for the robust, accurate and faster solution of the steady state free surface flows. Numerical solution of free surface flow problems using pressure based coupled solver is emerging trend in CFD community that allows implicit coupling of pressure, velocity and volume fractions and provides quiet robust and faster solution. Due to strong presence of body forces in free surface flow applications, implicit treatment of body forces is also a critical aspect for faster reduction of modulations in converged quantities. Free surface problems in steady state often suffer from poor diagonal dominance in system of equations, which could be improved by the usage of pseudo-transient solver, which not only improves the diagonal dominance but also inherently facilitates the local under-relaxation of solution variables. Pressure based coupled algorithm in conjunction with pseudo-transient solver overcomes most of the difficulties associated with robustness, stability and speed-up of free surface steady state simulations. Various case studies related to steady state open channel flow are considered in this paper, which clearly demonstrate the superiority of solving the volume fraction equation in coupled manner compared to solving it in a segregated manner. Convergence behavior of solution variables along with drag monitors are emphasized in the analysis and results have been compared with analytical or experimental data, which are in good agreement.