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In this paper, we introduce the class of diagonally dominant with respect to a given LMI region D ⊂ C matrices. They are shown to possess the analogues of well-known properties of (classical) diagonally dominant matrices, e.g. their spectra are localized inside the region D . Moreover, we show that in some cases, diagonal D -dominance implies ( D , D ) -stability (i.e. the preservation of matrix spectra localization under multiplication by a positive diagonal matrix).

In this paper, we present a direct parallel block W Z algorithm, named DPBWZA, for the solution of block tridiagonal toeplitz–block–toeplitz (TBT) linear system A x = f . The algorithm is based on the proposed block W Z factorization of the coefficient matrix A . Existence of the block W Z factorization for block tridiagonal TBT block diagonally dominant matrix is proved. Error analysis of the parallel algorithm DPBWZA is presented and numerical stability of the algorithm is established. Numerical experiments are conducted to demonstrate the efficiency, stability and accuracy of the Direct Parallel Block W Z Algorithm on GPU platform. Forward and backward errors are computed and DPBWZA is found to be highly accurate, numerically stable and as efficient as the subroutine csrlsvlu of GPU accelerated cuSolverSP library.

Distributed learning can enable scalable and effective decision making in numerous complex cyber-physical systems such as smart transportation, robotics swarm, power systems, etc. However, stability of the system is usually not guaranteed in most existing learning paradigms; and this limitation can hinder the wide deployment of machine learning in decision making of safety-critical systems. This letter presents a stability-guaranteed distributed reinforcement learning (SGDRL) framework for interconnected linear subsystems, without knowing the subsystem models. While the learning process requires data from a peer-to-peer (p2p) communication architecture, the control implementation of each subsystem is only based on its local states. The stability of the interconnected subsystems will be ensured by a diagonally dominant eigenvalue condition, which will then be used in a model-free RL algorithm to learn the stabilizing control gains. The RL algorithm structure follows an off-policy iterative framework, with interleaved policy evaluation and policy update steps. We numerically validate our theoretical results by performing simulations on four interconnected sub-systems.

In this paper, a class of discontinuous sawtooth-type activation function is designed and the multistability of Hopfield neural networks (HNNs) with such kind of activation function is studied. By virtue of the Brouwer’s fixed point theorem and the property of strictly diagonally dominant matrix (SDDM), some sufficient conditions are presented to ensure that the n -neuron HNN can have at least 7 n equilibria, among which 4 n equilibria are locally exponentially stable and the remaining 7 n - 4 n equilibria are unstable. Then, the obtained results are extended to a more general case. We continue to increase the number of the peaks of the sawtooth-type activation function and we find that the n -neuron HNN can have ( 2 k + 3 ) n equilibria, ( k + 2 ) n of them are locally exponentially stable and the remaining equilibria are unstable. Therein, k denotes the total number of the peaks in the designed activation function. That is to say, there is a quantitative relationship between the number of the peaks and the number of the equilibria. It implies that one can improve the storage capacity of a HNN by increasing the number of the peaks of the activation function in theory and in practice. To some extent, this method is convenient and flexible. Compared with the existing results, HNN with the designed sawtooth-type activation function can have more total equilibria as well as more locally stable equilibria. Finally, two examples are presented to demonstrate the validity of the obtained results.

In this paper, we generalize the projected Jacobi and the projected Gauss–Seidel methods to vertical linear complementarity problems (VLCPs) characterized by matrices with positive diagonal entries. First, we formulate the methods and show that the subproblems that must be solved at each iteration have an explicit solution, which is easy to compute. Then, we prove the convergence of the proposed procedures when the matrices of the problem satisfy some assumptions of strict or irreducible diagonal dominance. In this context, for simplicity, we first analyze the convergence in the special case of VLCPs of dimension $$2n\times n$$ , and we then generalize the results to VLCPs of an arbitrary dimension $$\ell n\times n$$ . Finally, we provide several numerical experiments (involving both full and sparse matrices) that show the effectiveness of the proposed approaches. In this context, our methods are compared with existing solution methods for VLCPs. A parallel implementation of the projected Jacobi method in CUDA is also presented and analyzed.

Given any linear stationary iterative methods in the form z^(i+1)=Jz^(i)+f, where J is the iteration matrix, a significant improvements of the iteration matrix will decrease the spectral radius and enhances the rate of convergence of the particular method while solving system of linear equations in the form Az=b. This motivates us to refine the Extended Accelerated Over-Relaxation (EAOR) method called Refinement of Extended Accelerated Over-Relaxation (REAOR) so as to accelerate the convergence rate of the method. In this paper, a refinement of Extended Accelerated Over-Relaxation method that would minimize the spectral radius, when compared to EAOR method, is proposed. The method is a 3-parameter generalization of the refinement of Accelerated Over-Relaxation (RAOR) method, refinement of Successive Over-Relaxation (RSOR) method, refinement of Gauss-Seidel (RGS) method and refinement of Jacobi (RJ) method. We investigated the convergence of the method for weak irreducible diagonally dominant matrix, matrix or matrix and presented some numerical examples to check the performance of the method. The results indicate the superiority of the method over some existing methods.

In this paper, we generalize and analyze the concepts of diagonal dominance and irreducibility in the framework of column and row representative matrices of a set. Our analysis includes the definition of particular sets of M- and H-matrices. We also analyze the form that the matrices of the introduced irreducible sets must have and the implications of the obtained results on the solution of vertical and horizontal linear complementarity problems. In this context, we prove that the projected Jacobi and the projected Gauss-Seidel methods for horizontal linear complementarity problems converge when the matrices of the problem satisfy one of the introduced generalizations of strict diagonal dominance or of irreducible diagonal dominance. This extends current convergence results.

A broad range of scientific simulations involve solving large-scale computationally expensive linear systems of equations. Iterative solvers are typically preferred over direct methods when it comes to large systems due to their lower memory requirements and shorter execution times. However, selecting the appropriate iterative solver is problem-specific and dependent on the type and symmetry of the coefficient matrix. Gauss-Seidel (GS) is an iterative method for solving linear systems that are either strictly diagonally dominant or symmetric positive definite. This technique is an improved version of Jacobi and typically converges in fewer iterations. However, the sequential nature of this algorithm complicates the parallel extraction. In fact, most parallel derivatives of GS rely on the sparsity pattern of the coefficient matrix and require matrix reordering or domain decomposition. In this article, we introduce a new algorithm that exploits the convergence property of GS and adapts the parallel structure of Jacobi. The proposed method works for both dense and sparse systems and is straightforward to implement. We have examined the performance of our method on multicore and many-core architectures. Experimental results demonstrate the superior performance of the proposed algorithm compared with GS and Jacobi. Additionally, performance comparison with built-in Krylov solvers in MATLAB showed that in terms of time per iteration, Krylov methods perform faster on CPUs, but our approach is significantly better when executed on GPUs. Lastly, we apply our method to solve the power flow problem, and the results indicate a significant improvement in runtime, reaching up to 87 times faster speed compared with GS.

The relative state between neighbors represents the difference of two connected agents' states, and it possesses specific physical meanings in practice. Under this background, the saturation constraints in the relative state inevitably occur. This article studies the consensus problems under the relative state saturation constraints. Novel adaptive proportional-integral (PI) protocols are designed to solve the constrained consensus problem. Specifically, the adaptive coupling weights and the saturation functions are embedded into the proposed protocols, and the former can render the protocols independent of any global topology graph information, while the latter can confine the relative state to stay in its constrained set. Sufficient conditions are identified under which the constrained consensus can be achieved. Considering that the solution matrix is required to be diagonally dominant, an iterative learning-based heuristic algorithm is proposed to seek the diagonally dominant positive-definite solution matrix. For the special case that the input matrix is row full rank, more stringent saturation functions are constructed, and it not only achieves the constrained consensus but also realizes the nonovershoot and shorter settling time associated with edge states. Besides, this result can be applied to preserve connectivity of the communication network. The theoretical analyses are validated by a simulation example.

The integration of more components into modern integrated circuits (ICs) has led to very large RLC parasitic networks consisting of millions of nodes that have to be simulated in many times or frequencies to verify the proper operation of the chip. Model order reduction (MOR) techniques have been employed routinely to substitute the large-scale parasitic model with a model of lower order with a similar response at the input–output ports. However, established MOR techniques generally result in dense system matrices that render their simulation impractical. To this end, in this article, we propose a methodology for the sparsification of the dense circuit matrices resulting from MOR of general RLC circuits, which employs a sequence of algorithms based on the computation of the nearest diagonally dominant matrix and the sparsification of the corresponding graph. In addition, we describe a procedure for synthesizing the sparsified reduced-order model into an RLC circuit with only positive elements. Experimental results indicate that a high sparsity ratio of the reduced system matrices can be achieved with very small loss of accuracy.

We use the algebraic orthogonality of rotation-free and divergence-free fields in the Fourier space to derive the solution of a class of linear homogenization problems as the solution of a large linear system. The effective constitutive tensor constitutes only a small part of the solution vector. Therefore, we propose to use a synchronous and local iterative method that is capable to efficiently compute only a single component of the solution vector. If the convergence of the iterative solver is ensured, i.e., the system matrix is positive definite and diagonally dominant, it outperforms standard direct and iterative solvers that compute the complete solution. It has been found that for larger phase contrasts in the homogenization problem, the convergence is lost, and one needs to resort to other linear system solvers. Therefore, we discuss the linear system’s properties and the advantages as well as drawbacks of the presented homogenization approach.

In this paper, the question of when the subdirect sum of two doubly strictly diagonally dominant (DSDDs) matrices is addressed. Some sufficient conditions are given, and these sufficient conditions only depend on the elements of the given matrices. Moreover, examples are presented to illustrate the corresponding results.

The positive definiteness of elasticity tensors plays an important role in the elasticity theory. In this paper, we consider the bi-block symmetric tensors, which contain elasticity tensors as a subclass. First, we define the bi-block M-eigenvalue of a bi-block symmetric tensor, and show that a bi-block symmetric tensor is bi-block positive (semi)definite if and only if its smallest bi-block M-eigenvalue is (nonnegative) positive. Then, we discuss the distribution of bi-block M-eigenvalues, by which we get a sufficient condition for judging bi-block positive (semi)definiteness of the bi-block symmetric tensor involved. Particularly, we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite, including bi-block (strictly) diagonally dominant symmetric tensors and bi-block symmetric (B)B0-tensors. These give easily checkable sufficient conditions for judging bi-block positive (semi)definiteness of a bi-block symmetric tensor. As a byproduct, we also obtain two easily checkable sufficient conditions for the strong ellipticity of elasticity tensors.

Convective transport in convection–diffusion problems can be formulated differently. Convective terms are commonly written in nondivergent or divergent form. For problems of this type, monotone and stable schemes in Banach spaces are constructed in uniform and integral norms, respectively. Monotonicity is related to row or column diagonal dominance. When convective terms are written in symmetric form (the half-sum of the nondivergent and divergent forms), the stability is established in Hilbert spaces of grid functions. Diagonal dominance conditions are given that ensure the monotonicity of two-level schemes for time-dependent convection–diffusion equations and the stability in corresponding spaces.

Analytical solutions to heat conduction problems are not straight forward and it becomes more difficult for the difficult – to – machine materials like Ti6Al4V. Computational methods for analysis of such kind of problems using specific numerical techniques helps to solve engineering heat conduction problems with accuracy and greater speed. As compared to one dimensional heat conduction problem, two dimensional and three-dimensional cases offer great difficulty in analytically solving such kind of problems. The present work covers the 2D rectangular heat conduction problem being solved utilizing the finite-difference scheme. The results from analytical solutions were utilized as the guideline for comparison with the computational scheme. The results of the present scheme were validated for the accuracy of the output results. The diagonal dominance of the solution matrix helped us to use Gauss-Seidel iterative scheme for the present study. The computational work has been carried out using in house code developed in Python release 3.7.0.

The study of thermal stresses developed during machining of implants is essential but it is not a straight forward problem due to unsteady heat diffusion and it becomes more critical as the implants are usually of Titanium alloy which comes under the class of difficult – to – machine materials. In this paper computational methods for analysis have been adopted for such problems with the aid of numerical techniques. This would provide an accurate solution to the heat conduction problems. To provide a clearer and real picture, in the present work, 2D rectangular heat conduction problem has been solved utilizing the finite-difference scheme. The outputs from analytical solutions were utilized as the guideline for comparison with the computational scheme. The results of the present scheme were validated for the accuracy of the output results. The diagonal dominance of the solution matrix helped to use Gauss-Seidel iterative scheme for the present study. The computational work has been carried out using in house code developed in Python.

This paper studies the convergence properties of the well-known message-passing algorithm for convex optimisation. Under the assumption of pairwise separability and scaled diagonal dominance, asymptotic convergence is established and a simple bound for the convergence rate is provided for message-passing. In comparison with previous results, our results do not require the given convex program to have known convex pairwise components and that our bound for the convergence rate is tighter and simpler. When specialised to quadratic optimisation, we generalise known results by providing a very simple bound for the convergence rate.

We develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly., Comment: To appear in Computational Optimization and Applications

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 this article, a local coupling multitrace domain decomposition method (LCMT-DDM) based on surface integral equation (SIE) formulations is proposed to analyze electromagnetic scattering from multilayered dielectric objects. Different from the traditional SIE-DDM, where the interactions between subdomains are accounted for using global radiation coupling, LCMT-DDM uses a local coupling scheme. The original multilayered object is decomposed into several independent domains, i.e., the exterior region (free space) and many homogeneous interior regions (dielectrics). The boundaries of subdomains are all touching faces, where only the Robin transmission conditions (RTCs) are enforced to ensure the field continuity. Hence, each subdomain only couples with its neighboring regions, which makes the DDM system a highly sparse matrix, especially when the number of subdomains is large. In each subdomain, the electric-field integral equation (EFIE) and the magnetic-field integral equation (MFIE) for dielectrics are used as the governing equations. By imposing RTCs, well-conditioned equations are formed in each subdomain without invoking the combined-field integral equation (CFIE), which usually causes accuracy issues in dielectric modeling. Since the subdomain matrices are diagonally dominant, the flexible generalized minimal residual (FGMRES) technique is used to accelerate the iterative solution of the whole DDM system. Moreover, an effective preconditioner that can be recursively constructed is proposed.

Some sufficient conditions ensuring that the subdirect sum of p-norm strictly diagonally dominant [or, for short, SDD(p)] matrices is in the class of SDD(p) matrices, are given. In particular, it i...

Based on the Schur complements, two upper bounds for the infinity norm of the inverse of doubly strictly diagonally dominant (DSDD) matrices are presented. As applications, an error bound for linear complementarity problems of DB-matrices and a lower bound for the smallest singular value of matrices are given.

The paper considers the so-called P-Nekrasov and {P1, P2}-Nekrasov matrices, defined in terms of permutation matrices P, P1, P2, which generalize the well-known notion of Nekrasov matrices. For such matrices A, available upper bounds on ‖A−1‖∞ are recalled, and new upper bounds for the P-Nekrasov and {P1, P2}-Nekrasov matrices are suggested. It is shown that the latter bound generally improves the earlier bounds, as well as the bound for the inverse of a P-Nekrasov matrix and the classical bound for the inverse of a strictly diagonally dominant matrix. Bibliography: 12 titles.

The aim of this study is create an analysis media for calculating the electric current in a closed circuit with one or more loops. Gauss-Jordan, Gauss-Seidel, and Cramer methods were used in this study. This media is packaged into Graphic User Interface (GUI) with matlab language program assisting. In this study, Linear Equation System (SPL) was obtained from kirchhoff current law and kirchhoff voltage law concepts. Gauss-Seidel method is not always convergent for each formed SPL, because it can only be applied when coefficient matrix A was diagonally dominant. The application of this analysis media made the calculation of closed circuit electric current with one or more loops became accurate and time saving.

Gaussian belief propagation algorithm &#x0028 GaBP &#x0029 is one of the most important distributed algorithms in signal processing and statistical learning involving Markov networks. It is well known that the algorithm correctly computes marginal density functions from a high dimensional joint density function over a Markov network in a finite number of iterations when the underlying Gaussian graph is acyclic. It is also known more recently that the algorithm produces correct marginal means asymptotically for cyclic Gaussian graphs under the condition of walk summability &#x0028 or generalised diagonal dominance &#x0029 . This paper extends this convergence result further by showing that the convergence is exponential under the generalised diagonal dominance condition, and provides a simple bound for the convergence rate. Our results are derived by combining the known walk summability approach for asymptotic convergence analysis with the control systems approach for stability analysis.

Although large and sparse linear systems can be solved using iterative methods, its number of iterations is relatively large. In this case, we need to modify the existing methods in order to get approximate solutions in a small number of iterations. In this paper, the modified method called second-refinement of Gauss-Seidel method for solving linear system of equations is proposed. The main aim of this study was to minimize the number of iterations, spectral radius and to increase rate of convergence. The method can also be used to solve differential equations where the problem is transformed to system of linear equations with coefficient matrices that are strictly diagonally dominant matrices, symmetric positive definite matrices or M-matrices by using finite difference method. As we have seen in theorem 1and we assured that, if A is strictly diagonally dominant matrix, then the modified method converges to the exact solution. Similarly, in theorem 2 and 3 we proved that, if the coefficient matrices are symmetric positive definite or M-matrices, then the modified method converges. And moreover in theorem 4 we observed that, the convergence of second-refinement of Gauss-Seidel method is faster than Gauss-Seidel and refinement of Gauss-Seidel methods. As indicated in the examples, we demonstrated the efficiency of second-refinement of Gauss-Seidel method better than Gauss-Seidel and refinement of Gauss-Seidel methods.

This study investigates the asymptotic properties of mixed-order fractional systems. By using the variation of constants formula, properties of real Mittag-Leffler functions, and Banach fixed-point theorem, the authors first propose an explicit criterion guaranteeing global attractivity for a class of mixed-order linear fractional systems. The criterion is easy to check requiring the system's matrix to be strictly diagonally dominant (C1) and elements on its main diagonal to be negative (C2). The authors then show the asymptotic stability of the trivial solution to a non-linear mixed-order fractional system linearised along with its equilibrium point such that its linear part satisfies the conditions (C1) and (C2). Two numerical examples with simulations are given to illustrate the effectiveness of the results over existing ones in the literature.

This paper is concerned with a multiobjective convex polynomial problem, where the objective and constraint functions are first-order scaled diagonally dominant sums-of-squares convex polynomials. We first establish necessary and sufficient optimality criteria in terms of second-order cone (SOC) conditions for (weak) efficiencies of the underlying multiobjective optimization problem. We then show that the obtained result provides us a way to find (weak) efficient solutions of the multiobjective program by solving a scalar second-order cone programming relaxation problem of a given weighted-sum optimization problem. In addition, we propose a dual multiobjective problem by means of SOC conditions to the multiobjective optimization problem and examine weak, strong and converse duality relations.

New bounds for the doubly diagonally dominant degree of the Schur complement of strictly doubly diagonally dominant (SDD) matrices are derived and proved to be better than those in Liu et al. (Linear Algebra Appl 437:168–183, 2012). As applications, a new distribution of the eigenvalues and two new infinity norm bounds for the Schur complements of SDD matrices are obtained. Finally, numerical examples are given to verify the theoretical results.

We consider the problem of decomposing a large covariance matrix into the sum of a low-rank matrix and a diagonally dominant matrix, and we call this problem the “diagonally dominant principal comp...

This paper presents a new fast power series solution method to solve the Hierarchal Method of Moment (MoM) matrix for a large complex, perfectly electric conducting (PEC) 3D structures. The proposed power series solution converges in just two (2) iterations which is faster than the conventional fast solver–based iterative solution. The method is purely algebraic in nature and, as such applicable to existing conventional methods. The method uses regular fast solver Hierarchal Matrix (H-Matrix) and can also be applied to Multilevel Fast Multipole Method Algorithm (MLFMA). In the proposed method, we use the scaling of the symmetric near-field matrix to develop a diagonally dominant overall matrix to enable a power series solution. Left and right block scaling coefficients are required for scaling near-field blocks to diagonal blocks using Schur’s complement method. However, only the right-hand scaling coefficients are computed for symmetric near-field matrix leading to saving of computation time and memory. Due to symmetric property, the left side-block scaling coefficients are just the transpose of the right-scaling blocks. Next, the near-field blocks are replaced by scaled near-field diagonal blocks. Now the scaled near-field blocks in combination with far-field and scaling coefficients are subjected to power series solution terminating after only two terms. As all the operations are performed on the near-field blocks, the complexity of scaling coefficient computation is retained as O(N)O⁢(N). The power series solution only involves the matrix-vector product of the far-field, scaling coefficients blocks, and inverse of scaled near-field blocks. Hence, the solution cost remains O(NlogN)O⁢(N⁢l⁢o⁢g⁢N). Several numerical results are presented to validate the efficiency and robustness of the proposed numerical method.