Left Eigenvectors Research Articles

Synchronization of directly coupled complex networks with multiweights and multiple delays

This paper investigates the synchronization of directed and multiweighted dynamic networks (DMDNs) with different delay couplings. Time delays in this paper are assumed to exist between the communication process, so communication between different nodes can generate time delays, while communications between the same nodes do not need time delays. Using the normalized left eigenvectors (NLEigs) corresponding to the zero eigenvalue of the coupling matrix, we prove that if both the time delay and the Chebyshev distance between NLEigs are small enough, all coupling matrices can accelerate the synchronization. Moreover, we also investigate the consensus for directed and multiweighted multi-agents with arbitrary bounded/unbounded time delays. Furthermore, the corresponding pinning control synchronization and consensus problem are also investigated, and sufficient synchronization criteria are also obtained. Numerical simulations are presented to show the validity of obtained theoretical results.

Reinforcement topology optimization considering the dynamic instability

AbstractThe present study develops a new topology optimization scheme considering the dynamic instability caused by the unsymmetrical properties of system. From a mathematical point of view, the left and right eigenvectors of asymmetric system are observed with the complex eigenvalues. With the dynamic instability, the magnitudes of structural responses are increasing with respect to time and this phenomenon causes many engineering issues. As the dynamic instability is one of the serious problems, the suppression is desired from an engineering point of view. To systematically reduce this dynamic instability, the present study develops a new topology optimization scheme for the reinforcement design. To overcome the numerical difficulties of the mode conversion and the highly nonlinear behavior, this research proposes the summation of the first several complex eigenvalues. To show the issues of the dynamic instability and the validity of the present approach, several topological reinforcement problems are solved.

Reduced basis method for non-symmetric eigenvalue problems: application to the multigroup neutron diffusion equations

In this article, we propose a reduced basis method for parametrized non-symmetric eigenvalue problems arising in the loading pattern optimization of a nuclear core in neutronics. To this end, we derive a posteriori error estimates for the smallest eigenvalue which is assumed to be simple and the associated left and right eigenvectors. The practical computation of these estimators requires the estimation of a constant called prefactor, which we can express as the spectral norm of some operator. We provide some elements of theoretical analysis which illustrate the link between the expression of the prefactor we obtain here and its well-known expression in the case of symmetric eigenvalue problems, either using the notion of numerical range of the operator, or via a perturbative analysis. Lastly, we propose a practical method in order to estimate this prefactor which yields interesting numerical results on actual test cases. We provide detailed numerical simulations on two-dimensional examples including a multigroup neutron diffusion equation.

Prescribed-time fully distributed Nash equilibrium seeking of nonlinear multi-agent systems over unbalanced digraphs

This paper investigates the prescribed-time fully distributed Nash equilibrium seeking (PT-FDNES) problem for nonlinear multi-agent systems (MASs) over weight-unbalanced directed graphs (digraphs). To counterbalance unweighted communication flows caused by the unbalanced network, the temporal transformation technique is first introduced to derive a consensus-based algorithm for calculating the left eigenvector of the Laplacian matrix, based on which a prescribed-time fully distributed estimator with adaptive gains is developed to obtain the NE without requiring any global information. By utilizing this NE estimator to formulate an intermediate control law, the NE seeking problem is then effectively transformed into a local reference-tracking problem. To deal with the non-differentiability issue caused by the NE estimator and ensure the exact tracking, the dynamic gain technique and novel scaling functions are employed to derive a prescribed-time tracking controller which completely compensates for the mismatched uncertainties while reducing the computational burden. The proposed control framework enables the actions of all agents to reach the NE within a prescribed time. In addition, the obtained results are further re-designed to exhibit the robustness by introducing an improved σ-modification scheme without sacrificing the convergence performance. Numerical examples are provided to demonstrate the validity of the theoretical results.

Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge

Sensitivity of an eigenvalue λ i to the perturbation of matrix elements is controlled by the eigenvalue condition number defined as κi=⟨Li|Li⟩⟨Ri|Ri⟩ , where ⟨Li| and |Ri⟩ are left and right eigenvectors to the eigenvalue λ i . In random matrix theory the squared eigenvalue condition number is also known as the eigenvector self-overlap. In this work we calculate the asymptotics of the joint probability density function of the real eigenvalue and the square of the corresponding eigenvalue condition number for the real elliptic ensemble in the double scaling regime of almost Hermiticity and close to the edge of the spectrum. As a byproduct, we also calculate the one-parameter deformation of the Scorer’s function.

Quantum metric of non-Hermitian Su-Schrieffer-Heeger systems

Topological insulators have been studied intensively over the last decades. Earlier research focused on Hermitian Hamiltonians, but recently, peculiar and interesting properties were found by introducing non-Hermiticity. In this work, we apply a quantum geometric approach to various Hermitian and non-Hermitian versions of the Su-Schrieffer-Heeger (SSH) model. We find that this method allows one to correctly identify different topological phases and topological phase transitions for all SSH models, but only when using the metric tensor containing both left and right eigenvectors. Whereas the quantum geometry of Hermitian systems is Riemannian, introducing non-Hermiticity leads to pseudo-Riemannian and complex geometries, thus significantly generalizing from the quantum geometries studied thus far. One remarkable example of this is the mathematical agreement between topological phase transition curves and lightlike paths in general relativity, suggesting a possibility of simulating space-time in non-Hermitian systems. We find that the metric in non-Hermitian phases degenerates in such a way that it effectively reduces the dimensionality of the quantum geometry by one. This implies that within linear response theory, one can perturb the system by a particular change of parameters while maintaining a zero excitation rate. Published by the American Physical Society 2024

A four parameter extension to the Clement matrix and its role in numerical software testing

The Sylvester–Kac matrix (also known as the Clement matrix) and its generalizations are of interest to researchers in diverse fields. A new parameterization of the matrix has recently been presented with closed forms for the eigenvalues and Oste and Van der Jeugt (2017) have proposed their family of matrices as a source of test problems for numerical eigensolvers. In this article we extend their generalization by adding a further two parameters to the matrix definition and, for this new extension, we obtain closed forms for the eigenvalues, determinant and, the left and right eigenvectors. We show that, for certain values of the free parameters and relatively low order, these matrices may become very ill-conditioned w.r.t. both eigenvalues and inversion. In light of this we re-assess the numerical results presented by Oste and Van der Jeugt (2017) and propose a possible new testing role for parameterized Clement matrices.

Design sensitivity analysis of modal frequencies of elastic structures submerged in an infinite fluid domain

SummaryThis paper presents a numerical approach for the design sensitivity analysis of modal frequencies of elastic structures submerged in an unbounded heavy fluid domain. Because the feedback of sound pressure onto the fluid‐loaded structures has to be taken into account in this case, a fully coupled scheme which combines the structural finite element method (FEM) and the acoustic boundary element method (BEM) is employed in the numerical simulation. The resulting nonlinear eigenvalue problem created by the coupled finite element and boundary element (FE‐BE) method is converted into a generalized eigenvalue problem (GEP) of reduced dimension by using a contour integral method. The design sensitivity formulation for non‐repeated eigenvalues is derived based on an adjoint method which uses both the left and right eigenvectors, and a small GEP is formed to calculate the gradients of repeated eigenvalues. The Burton‐Miller formulation is employed to shift the fictitious eigenfrequencies and their derivatives, and the parameter setting which is beneficial to the filtering of fictitious eigenfrequencies is investigated. Numerical examples are given to verify the accuracy and effectiveness of the developed method.

Robust Passivity and Control for Directed and Multiweighted Coupled Dynamical Networks.

In this article, the passivity and control issues for directed uncertain coupled dynamical networks are solved. The presented model is directly coupled with multiple coupling matrices and parametric uncertainty, while previous literatures of multiweighted networks usually suppose that outer coupling matrices (OMs) are connected, undirected, and certain. The viewpoint of inner coupling matrices (IMs) in this article is added and OMs can be directed and not connected, which is a great improvement on the existing results. First, for all diagonal IMs, considering each dimension separately, we can derive if the weighted combination of multiple OMs for each dimension is strongly connected, then passivity and pinning control rules can be established. In addition, we also discuss the situation that IMs are positive definite but not diagonal. By means of the weighted combination of normalized left eigenvectors (NLEVec) corresponding to zero eigenvalue for multiple coupling matrices, we prove if the Chebyshev distance (Cheb-Dist) among these NLEVec is less than a tolerant deviation interval, then passivity, synchronization, and pinning control criteria are acquired. Moreover, a matter of adaptive coupling strengths is also settled. Examples are provided to verify the validity of established results.

Distributed online primal-dual subgradient method on unbalanced directed networks

In this paper, we investigate a distributed online optimization method on multiagent communication networks. We consider a distributed prime-dual subgradient algorithm for an online convex optimization problem with time-varying coupled constraints. Each agent updates the estimations for the primal and dual optimizers by a consensus-based online algorithm. In the proposed algorithm, the gradient direction is scaled by estimating the left eigenvector of a weight matrix associated with the directed communication network. The scaling procedure enables agents in the network to estimate the time-varying optimal solutions by counterbalancing the unbalanced communication flows. The performance of the proposed algorithm is examined by a dynamic regret and a fit, which evaluate the cumulative error against the time-varying optimal cost function and the constraint function, respectively. We provide a sufficient condition under which both the dynamic regret and the fit are sublinear. A numerical example of an online economic dispatch problem confirms the validity of the proposed method.

Distributed algorithm for social cost minimization with second-order nonlinear dynamics over weight-unbalanced digraphs

In this paper, the emerging social cost minimization framework of second-order nonlinear systems over weight-unbalanced digraphs is studied. It provides a general framework containing several important classes of problems. Therein each local cost function depends on a global decision variable consisting of all the agents’ decisions not limited to the form of aggregative terms. Under the partial decision and cost function information setting, distributed algorithms are designed based on the state feedback and left eigenvector estimation mechanism to overcome the challenges posed by second-order nonlinear systems and weight-unbalanced digraphs. The asymptotic convergence of the algorithm is demonstrated via the Lyapunov stability theory. Finally, numerical simulations of the vehicle monitoring problem are provided to support the algorithm design.

Distributed Optimization Algorithms for Heterogeneous Linear Multi-agent Systems With Inequality Constraints

In this paper, for heterogeneous linear multi-agent systems, a distributed constrained optimization problem about digraphs is studied. Every agent only utilizes local interaction and information such that all agents can achieve the global objective function. The state of each agent is limited to a local inequality constraint set. First, this paper proposes a distributed continuous-time optimization algorithm by designing a left eigenvector corresponding to the zero eigenvalue of the Laplacian matrix, which removes the imbalance of the communication graph. Next, the asymptotical convergence about the algorithm is demonstrated using Lyapunov stability. Finally, two numerical examples are given to illustrate the effectiveness of the algorithm.

Right-left asymmetry of the eigenvector method: A simulation study

The eigenvalue method, suggested by the developer of the extensively used Analytic Hierarchy Process methodology, exhibits right-left asymmetry: the priorities derived from the right eigenvector do not necessarily coincide with the priorities derived from the reciprocal left eigenvector. This paper offers a comprehensive numerical experiment to compare the two eigenvector-based weighting procedures and their reasonable alternative of the row geometric mean with respect to four measures. The underlying pairwise comparison matrices are constructed randomly with different dimensions and levels of inconsistency. The disagreement between the two eigenvectors turns out to be not always a monotonic function of these important characteristics of the matrix. The ranking contradictions can affect alternatives with relatively distant priorities. The row geometric mean is found to be almost at the midpoint between the right and inverse left eigenvectors, making it a straightforward compromise between them.

Oscillatory banded Hessenberg matrices, multiple orthogonal polynomials and Markov chains

A spectral Favard theorem is derived for bounded banded lower Hessenberg matrices that possess a positive bidiagonal factorization. The rich knowledge concerning the spectral and factorization properties of oscillatory matrices forms the basis of this theorem, which is formulated in terms of sequences of multiple orthogonal polynomials of types I and II, associated with a set of positive Lebesgue-Stieltjes measures. Additionally, a multiple Gauss quadrature is established, and the corresponding degrees of precision are determined. The spectral Favard theorem finds application in the context of Markov chains with transition matrices having (p + 2) diagonals, extending beyond the birth and death scenario, while still maintaining a positive stochastic bidiagonal factorization. In the finite case, the Karlin–McGregor spectral representation is provided, along with the demonstration of recurrent behavior and explicit expressions for the stationary distributions in terms of the orthogonal polynomials. Analogous results are obtained for countably infinite Markov chains. In this case, the Markov chain may not be recurrent, and its characterization is expressed in relation to the first measure. The ergodicity of the Markov chain is explored, taking into consideration the presence of a mass at 1, which corresponds to eigenvalues associated with the right and left eigenvectors.

Definition and calculation method of modal effective mass of asymmetric fluid-structure interaction system for seismic analysis

In this paper, modal effective mass for asymmetric fluid-structure interaction system is defined and equations for its calculation is derived. To establish consistency, modal effective mass in symmetric structure only system is briefly reviewed, followed by a definition of the modal effective mass in asymmetric system. The equations for calculating modal effective mass in asymmetric system are derived by utilizing the properties of left and right eigenvectors. To simplify the equations, the assumption is made that the mass matrix is only affected by the fluid. The simplified equation is then compared to the equation already used in ANSYS. Finally, the validity of the modal effective mass definition and derivation in this paper is demonstrated through a simple example.

Eigensensitivity of damped system with distinct and repeated eigenvalues by chain rule

AbstractIn this paper, a novel algorithm for computing the derivatives of eigensolutions of asymmetric damped systems with distinct and repeated eigenvalues is developed without using second‐order derivatives of the eigenequations, which has a significant benefit over the existing published methods. To achieve this, the algorithm is proposed by (1) imposing consistent normalization conditions for both left and right eigenvectors throughout the computational system, and (2) solving sensitivity problems using the chain rule. These contributions add significant value to the proposed method. Additionally, the numerical difficulty is overcome by suggesting justifying Nelson (1976)'s technique based on the normalization conditions. Even though the proposed algorithm is simple and easy to implement, the employment of chain rule with new normalization in this note has never been reported in the literature, especially for the problem of eigensensitivity. Because this new algorithm does not require the second‐order derivatives of the eigenequations, the analytical and numerical solutions for eigensensitivity offered by this method are less complicated than existing available solutions. The validity and applicability of the method are demonstrated through five numerical examples, showcasing its implementation in displacement‐ and state‐space equations (rotordynamic system). Also, verifications are found to be in reasonable agreement with the approximated method.

LEFT AND RIGHT EIGENVECTORS OF A VARIANT OF THE SYLVESTER–KAC MATRIX

AbstractAs an extension of Sylvester’s matrix, a tridiagonal matrix is investigated by determining both left and right eigenvectors. Orthogonality relations between left and right eigenvectors are derived. Two determinants of the matrices constructed by the left and right eigenvectors are evaluated in closed form.

Synchronization and Control for Multiweighted and Directed Complex Networks.

The study of complex networks with multiweights (CNMWs) has been a hot topic recently. For a network with a single weight, previous studies have shown that they can promote synchronization, but for CNMWs, there are no rigorous analyses about the role of coupling matrices. In this brief, the complex network is allowed to be directed, which is the main difference with previous studies and may make the synchronization analysis difficult for multiple couplings. At first, we prove that if the inner coupling matrices are all diagonal, then synchronization can be realized only if the weighted sum (or union) of multiple coupling matrices is strongly connected, which bridges the gap between single-weighted and multiweighted networks. Moreover, we also consider the case that inner coupling matrices are positive definite but not diagonal. We design two techniques for this hard problem. One technique is to decompose inner coupling matrices into diagonal matrices and residual matrices. The other one is to measure the similarity between outer coupling matrices. In virtue of the normalized left eigenvectors (NLEVecs) corresponding to the zero eigenvalue of coupling matrices, we prove that if the Chebyshev distance between NLEVec is less than some value, defined as the allowable deviation bound, then the synchronization and control will be realized with sufficiently large coupling strengths. Furthermore, adaptive rules are also designed for coupling strength.

Distributed primal-dual method on unbalanced digraphs with row stochasticity

This study investigates distributed convex optimisation that accounts for the inequality constraints over unbalanced directed graphs. In distributed optimisation, agents exchange information on a network to obtain an optimal solution when they only know their own cost function. We propose a distributed primal-dual subgradient method based on a row stochastic weight matrix that is associated with a communication network. In the proposed method, the normalised left eigenvector of the weight matrix is estimated by a consensus algorithm. Then, the subgradient of the Lagrange function is scaled by the estimated values of the left eigenvector to compensate for the imbalance of the information flow in the network. We show that the pair of estimations for the primal and dual problems converge to an optimal primal-dual solution. We also show the relation between the convergence rate of the proposed algorithm and the step-size rule. A numerical example confirms the validity of the proposed method.

Efficient calculation of three-dimensional tensor networks

We have proposed an efficient algorithm to calculate physical quantities in the translational invariant three-dimensional tensor networks, which is particularly relevant to the study of the three-dimensional classical statistical models and the ($2+1$)-dimensional quantum lattice models. In the context of a classical model, we determine the partition function by solving the dominant eigenvalue problem of the transfer matrix, whose left and right dominant eigenvectors are represented by two projected entangled simplex states. These two projected entangled simplex states are not Hermitian conjugate to each other but are appropriately arranged so that their inner product can be computed much more efficiently than in the usual prescription. For the three-dimensional Ising model, the calculated internal energy and spontaneous magnetization agree with the published results in the literature. The possible improvement and extension to other models are also discussed.