Diagonal Matrix Research Articles

Approximate GCD of several multivariate sparse polynomials based on SLRA interpolation

To compute the greatest common divisor (GCD) of a set of multivariate polynomials, modular algorithms are typically employed to prevent any growth in the coefficient polynomials in the intermediate expressions. However, when dealing with multivariate polynomials with a priori errors on their coefficients, using such modular algorithms becomes challenging. This is because any resulting approximate GCD computed in one variable may have perturbations depending on the evaluation point and may not be an image of the same desired multivariate approximate GCD. This necessitates computing it as given multivariate polynomials, and operating with large matrices whose size is exponential in the number of variables. In this paper, we present a new modular algorithm, suitable for dense cases and effective for sparse ones, called “SLRA interpolation”. This algorithm uses the multidimensional fast Fourier transform (FFT) and the structured low-rank approximation (SLRA) of non-square block diagonal matrices. The SLRA interpolation technique may reduce the time-complexity for one iteration in the computation of approximate GCD of several multivariate polynomials, especially for the sparse case.

Some results involving the A α -eigenvalues for graphs and line graphs

Abstract Let G G be a simple graph with adjacency matrix A ( G ) A\left(G) , degree diagonal matrix D ( G ) , D\left(G), and let l ( G ) l\left(G) be the line graph of G G . In 2017, Nikiforov defined the A α {A}_{\alpha } -matrix of G G , A α ( G ) {A}_{\alpha }\left(G) , as a linear convex combination of A ( G ) A\left(G) and D ( G ) D\left(G) , in the following way, A α ( G ) ≔ α A ( G ) + ( 1 − α ) D ( G ) , {A}_{\alpha }\left(G):= \alpha A\left(G)+\left(1-\alpha )D\left(G), where α ∈ [ 0 , 1 ] \alpha \in \left[0,1] . In this study, we present some bounds for the largest eigenvalue of A α ( G ) , {A}_{\alpha }\left(G), and for some eigenvalues of A α ( l ( G ) ) {A}_{\alpha }\left(l\left(G)) . Extremal graphs attaining some of these bounds are also characterized. Furthermore, some comparisons between the new bounds obtained in this study, and between these and some bounds presented by Nikiforov are made.

Receding horizon estimation for multi‐rate sampled data systems under component‐based event‐triggered mechanisms: Handling delayed and degraded measurements

AbstractThe purpose of this article is to develop the receding horizon (RH) estimation for multi‐rate sampled systems with measurement delays and packet losses in the measurements. An event‐triggered transmission scheme is introduced to remove measurements that are unnecessary for the design of the estimator. The stochastic diagonal matrixes are introduced to represent the phenomenon of packet losses, where each component is subject to an individual Bernoulli process. The original system is firstly transformed into a delay‐free one by using the reorganized observation method. Further, a batch form and an iterative form of RH estimation are proposed by minimizing a given cost function that includes some terminal weighting terms based on the new system. The stability of the proposed RH estimation is guaranteed by the natural assumptions and a simulation instance is given to verify the effectiveness of the proposed method.

Specific target pinning control of complex dynamical networks based on semidefinite programming strategy

Pinning control of complex networks aims to force the states of all nodes to reach the desired goal by controlling a few nodes in the network. The easiness of a pinning strategy can be measured by the minimum eigenvalue λ1 of the specific matrix L+Γ, where L is the Laplacian matrix of the underlying network and Γ is a diagonal matrix with Γi,i=1 if node i is pinned. A larger value λ1 indicates a smaller coupling strength. We here are interested in two key questions on this issue. The first one is, what is the minimum number of pinned nodes to achieve system synchronization when individual node dynamics and the coupling strength of the network are given? We show an upper bound of the minimum eigenvalue λ1, by which a lower bound of the number of pinned nodes is presented. The second one is, which nodes should be pinned such that the system achieves synchronization much easier if a potential pinning target node set S0 (instead of the whole node set) and the number of pinned nodes l (l<|S0| ) are both restricted beforehand? This specific target pinning control problem is raised for the first time. By transforming it into a semi-definite programming problem, we then use a convex optimization tool to solve this problem. Experimental results show that the strategy is more effective than other classical node selection indices.

A Novel Weak-Form Plane Frame Quadrature Element and its Application to Bridge Analysis

A novel weak-form plane frame quadrature element (WPFQE) is developed. A complex structure can be divided into several non-homogeneous (or homogeneous) elements with large size. The Chebyshev–Lobatto differential quadrature dealing with differentiation and the Gauss–Lobatto quadrature dealing with integration have the same integration points including the ends of each element. Based on the energy variational principle, the diagonal mass matrix and positive definite real symmetric stiffness matrix for the element can be obtained. The unknown quantities at the internal quadrature points of the element can be represented by those at the endpoints of the element based on the principle of static equal effect anterior to the development of global matrices, which significantly reduces the sizes of modified element matrices while maintaining sufficient accuracy, thereby obtaining the small-sized global matrices. Assembly of elements can be performed efficiently just as that in the finite element (FE) method. In the case study, the static and dynamic performances of a three-span high-pier rigid-frame bridge with variable cross-section are studied by applying the proposed method. The results indicate that the orders of structural matrices and calculation time obtained by the WPFQE method are much smaller than those of the FE method.

Distributed delayed impulsive control for [formula omitted]-synchronization of multi-link structure networks with bounded uncertainties and time-varying delays of unmeasured bounds: A novel Halanay impulsive inequality approach

This study discusses the μ−synchronization of multi-link structure networks with parameter uncertainties and nonlinear couplings. Three types of variable delays, including internal, coupling, and distributed delays at sampling moments are deliberated in the synchronization control process. Different from existing multi-link models, the bulk of delays can be non-differentiable and unmeasured, which also brings new challenges to our research. To conquer the difficulties caused by such multiple generalized time delays, a novel Halanay-like impulsive inequality is established by utilizing the properties of μ−function and mathematical induction recipes. For letting the multi-link networks achieve μ−synchronization, a distributed-delay impulsive controller is designed, and multiple coupling matrices are decomposed into diagonal matrices and residual matrices. Based on the stability function method, the derived impulsive inequality, and matrix decomposition techniques, important synchronization conditions for the concerning multi-link networks are obtained, which releases the traditional constraints that delays should be smaller than impulsive intervals, and popularizes the existing synchronization results respecting multi-link systems with unbounded delays.

Correction of nonmonotone trust region algorithm based on a modified diagonal regularized quasi-Newton method

In this paper, a new appropriate diagonal matrix estimation of the Hessian is introduced by minimizing the Byrd and Nocedal function subject to the weak secant equation. The Hessian estimate is used to correct the framework of a nonmonotone trust region algorithm with the regularized quasi-Newton method. Moreover, to counteract the adverse effect of monotonicity, we introduce a new nonmonotone strategy. The global and superlinear convergence of the suggested algorithm is established under some standard conditions. The numerical experiments on unconstrained optimization test functions show that the new algorithm is efficient and robust.

On automorphisms of tame polynomial automorphism Ind-Schemes in positive characteristic

In this paper we study certain combinatorial attributes of Ind -schemes of polynomial automorphisms in positive characteristic. In particular, we prove that over an algebraically closed field K of positive characteristic ≠ 2 every automorphism of the group of origin-preserving automorphisms of the polynomial algebra K [ x 1 , … , x n ] ( n ≥ 3 ), which fixes every diagonal matrix, preserves, up to composition with a linear inner automorphism, every tame automorphism.

On diagonally structured scheme for nonlinear least squares and data-fitting problems

Recently, structured nonlinear least-squares (NLS) based algorithms gained considerable emphasis from researchers; this attention may result from increasingly applicable areas of these algorithms in different science and engineering domains. In this article, we coined a new efficient structured-based NLS algorithm. We developed a diagonal Hessian-based formulation for solving NLS problems. We derived the quasi-Newton update based on a diagonal matrix scheme subject to a modified structured secant condition. Also, we show that the algorithm’s search direction satisfies a sufficient descent condition under some standard assumptions. Subsequently, we also prove the global convergence of the algorithm and then eventually show its linear convergence rate for strongly convex functions. Furthermore, to show case the proposed algorithm’s performance, we experimented numerically by comparing it with other approaches on some benchmark test functions available in the literature. Finally, the introduced scheme is applied to solve some data-fitting problems

Generalized Quantum Signal Processing

Quantum signal processing (QSP) and quantum singular value transformation (QSVT) currently stand as the most efficient techniques for implementing functions of block-encoded matrices, a central task that lies at the heart of most prominent quantum algorithms. However, current QSP approaches face several challenges, such as the restrictions imposed on the family of achievable polynomials and the difficulty of calculating the required phase angles for specific transformations. In this paper, we present a generalized quantum signal processing (GQSP) approach, employing general SU(2) rotations as our signal-processing operators, rather than relying solely on rotations in a single basis. Our approach lifts all practical restrictions on the family of achievable transformations, with the sole remaining condition being that |P|≤1, a restriction necessary due to the unitary nature of quantum computation. Furthermore, GQSP provides a straightforward recursive formula for determining the rotation angles needed to construct the polynomials in cases where P and Q are known. In cases where only P is known, we provide an efficient optimization algorithm capable of identifying in under a minute of GPU time, a corresponding Q for polynomials of degree on the order of 107. We further illustrate GQSP simplifies QSP-based strategies for Hamiltonian simulation, offer an optimal solution to the ϵ-approximate fractional query problem that requires O((1/δ)+log(1/ϵ)) queries to perform where O(1/δ) is a proved lower bound, and introduces novel approaches for implementing bosonic operators. Moreover, we propose a novel framework for the implementation of normal matrices, demonstrating its applicability through synthesis of diagonal matrices, as well as the development of a new algorithm for convolution through synthesis of circulant matrices using only O(dlogN+log2N) 1 and 2-qubit gates for a filter of lengths d. Published by the American Physical Society 2024

On non-bipartite graphs with strong reciprocal eigenvalue property

Let G be a simple connected graph and A(G) be the adjacency matrix of G. A diagonal matrix with diagonal entries ±1 is called a signature matrix. If A(G) is nonsingular and X=SA(G)−1S−1 is entrywise nonnegative for some signature matrix S, then X can be viewed as the adjacency matrix of a unique weighted graph. It is called the inverse of G, denoted by G+. A graph G is said to have the reciprocal eigenvalue property (property(R)) if A(G) is nonsingular, and 1λ is an eigenvalue of A(G) whenever λ is an eigenvalue of A(G). Further, if λ and 1λ have the same multiplicity for each eigenvalue λ, then G is said to have the strong reciprocal eigenvalue property (property (SR)). It is known that for a tree T, the following conditions are equivalent: a) T+ is isomorphic to T, b) T has property (R), c) T has property (SR) and d) T is a corona tree (it is a tree which is obtained from another tree by adding a new pendant at each vertex).Studies on the inverses, property (R) and property (SR) of bipartite graphs are available in the literature. However, their studies for the non-bipartite graphs are rarely done. In this article, we study the inverse and property (SR) for non-bipartite graphs. We first introduce an operation, which helps us to study the inverses of non-bipartite graphs. As a consequence, we supply a class of non-bipartite graphs for which the inverse graph G+ exists and G+ is isomorphic to G. It follows that each graph G in this class has property (SR).

Structured prior distributions for the covariance matrix in latent factor models

Factor models are widely used for dimension reduction in the analysis of multivariate data. This is achieved through decomposition of a p×p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \ imes p$$\\end{document} covariance matrix into the sum of two components. Through a latent factor representation, they can be interpreted as a diagonal matrix of idiosyncratic variances and a shared variation matrix, that is, the product of a p×k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \ imes k$$\\end{document} factor loadings matrix and its transpose. If k≪p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k \\ll p$$\\end{document}, this defines a parsimonious factorisation of the covariance matrix. Historically, little attention has been paid to incorporating prior information in Bayesian analyses using factor models where, at best, the prior for the factor loadings is order invariant. In this work, a class of structured priors is developed that can encode ideas of dependence structure about the shared variation matrix. The construction allows data-informed shrinkage towards sensible parametric structures while also facilitating inference over the number of factors. Using an unconstrained reparameterisation of stationary vector autoregressions, the methodology is extended to stationary dynamic factor models. For computational inference, parameter-expanded Markov chain Monte Carlo samplers are proposed, including an efficient adaptive Gibbs sampler. Two substantive applications showcase the scope of the methodology and its inferential benefits.

Distributed fusion filtering for multi-sensor nonlinear networked systems with multiple fading measurements via stochastic communication protocol

This paper studies the distributed fusion filtering (DFF) issue for a class of nonlinear delayed multi-sensor networked systems (MSNSs) subject to multiple fading measurements (MFMs) under stochastic communication protocol (SCP). The phenomenon of MFMs occurs randomly in the network communication channels and is characterized by a diagonal matrix with certain statistical information. In order to decrease the overload of communication network and save network resources, the SCP that can regulate the information transmission between sensors and estimators is adopted. The primary aim of the tackled problem is to develop the DFF method for nonlinear delayed MSNSs in the presence of MFMs and SCP based on the inverse covariance intersection fusion rule. In addition, the local upper bound (UB) of the filtering error covariance (FEC) is derived and minimized by means of suitably designing the local filter gain. Moreover, the boundedness analysis regarding the local UB is proposed with corresponding theoretical proof. Finally, two simulation examples with comparative illustrations are given to display the usefulness and feasibility of the derived theoretical results.

Stability and synchronization of octonion-valued neural networks with leakage and mixed delays on time scales

There has been a great deal of interest in the last few years for neural networks (NNs) with values in multidimensional domains. The most popular models are complex-valued neural networks (CVNNs), followed by quaternion-valued neural networks (QVNNs), and, more recently, by Clifford-valued neural networks (ClVNNs). However, also very recently, a different type of NNs were put forward, namely octonion-valued neural networks (OVNNs). OVNNs are defined on the 8D octonion algebra, and they are not a special type of ClVNNs, because Clifford numbers are associative, whereas octonions are not. Moreover, beside the complex and quaternion algebras, the only other normed division algebra over the reals is the algebra of octonions, which makes OVNNs a direct generalization of CVNNs and QVNNs from this point of view, raising interest for applications handling high-dimensional data. On the other hand, systems defined on time scales were proposed as a generalization of both discrete time and continuous time systems, or any type of hybrid combination between the two. Finally, time delays appear as a consequence of implementing NNs in real life circuits. Taking all these into consideration, this paper studies the fundamental properties of exponential stability and exponential synchronization for OVNNs with leakage and mixed delays defined on time scales. In order to avoid the problems raised by the non-associativity of the octonion algebra, the OVNN model is decomposed into a real-valued one. Then, two different Lyapunov-type functionals are defined and the particularities of time scale calculus are used in order to deduce sufficient conditions expressed as scalar and linear matrix inequalities (LMIs) for the exponential stability of the proposed models, based on Halanay-type inequalities suitable for time scale systems. Afterwards, a state feedback controller is used to deduce sufficient criteria given as scalar inequalities and LMIs for the exponential synchronization of the same type of models. The generality of the model, given by the definition on the algebra of octonions, the use of different types of delays, and the definition on time scales represent an important advantage. Also, for less general models, it is possible to particularize the obtained results. With the aim to demonstrate each of the paper’s four theorems, four numerical examples are provided.

Proximate Tomonaga-Luttinger liquid in an anisotropic Kitaev-Gamma model

The characterization of quantum spin liquid phases in Kitaev materials has been a subject of intensive studies over the recent years, both theoretically and experimentally. Most theoretical studies have focused on an isotropically interacting model with its coupling strength being equivalent on each bond in an attempt to simplify the problem. In this Letter, we study an extended spin-1/2 Kitaev-Γ model on a honeycomb lattice with an additional tuning parameter that controls the coupling strength on one of the bonds: We connect the limit of isolated Kitaev-Γ chains, which is known to exhibit an emergent SU(2)1 Tomonaga-Luttinger liquid phase [Yang , ], to the two-dimensional model. We report on an instance, in which properties of the Tomonaga-Luttinger liquid persist for finite interchain coupling forming a phase we call the proximate Tomonaga-Luttinger liquid (pTLL). This pTLL is strongly anisotropic in character analogous to , and features spinon-like excitations similar to those of the antiferromagnetic Heisenberg chain. We use numerical exact diagonalization and density matrix renormalization group on various cluster geometries in a complementary way to overcome finite-size limitations. Published by the American Physical Society 2024

A single-sided all-at-once preconditioning for linear system from a non-local evolutionary equation with weakly singular kernels

In Lin et al. (2021) [21] and Zhao et al. (2023) [37], two-sided preconditioning techniques are proposed for non-local evolutionary equations, which possesses (i) mesh-size independent theoretical bound of condition number of the two-sided preconditioned matrix; (ii) small and stable iteration numbers in numerical tests. In this paper, we modify the two-sided preconditioning by multiplying the left-sided and the right-sided preconditioners together as a single-sided preconditioner. Such a single-sided preconditioner essentially derives from approximating the spatial matrix with a fast diagonalizable matrix and keeping the temporal matrix unchanged. Clearly, the matrix-vector multiplication of the single-sided preconditioning is faster to compute than that of the two-sided one, since the single-sided preconditioned matrix has a simpler structure. More importantly, we show theoretically that the single-sided preconditioned generalized minimal residual (GMRES) method has a convergence rate no worse than the two-sided preconditioned one. As a result, the one-sided preconditioned GMRES solver requires less computational time than the two-sided preconditioned GMRES solver in total. Numerical results are reported to show the efficiency of the proposed single-sided preconditioning technique.

Differentiable microstructures design via anisotropic thermal diffusion

We propose a novel method to design differentiable microstructures. Central to our algorithm is a new representation of the mapping from the parameters to microstructures, formulated as the anisotropic thermal diffusion. A metric field governs the anisotropic diffusion. The metric associated with each point is represented as a 2 × 2 symmetric positive definite matrix that becomes the design variable. To alleviate the difficulties caused by symmetric positive definite constraints, we perform the singular value decomposition of the metric matrix so that the design variable includes a rotation angle and a diagonal matrix. Then, the positive definiteness is converted to requiring the two diagonal entries of the diagonal matrix to be positive, which is easier to deal with. The effectiveness of our algorithm is demonstrated through evaluations and comparisons over various examples.

Mass lumping the dual cell method to arbitrary polynomial degree for acoustic and electromagnetic waves

We present a fundamental improvement of a high polynomial degree time domain cell method recently introduced by the last three authors. The published work introduced a method featuring block-diagonal system matrices where the block size and conditioning scaled poorly with respect to polynomial degree. The issue is herein bypassed by the construction of new basis functions exploiting quadrature rule based mass lumping techniques for arbitrary polynomial degrees in two dimensions for the Maxwell equations and the acoustic wave equation in the first order velocity pressure formulation. We characterize the degrees of freedom of all new discrete approximation spaces we employ for differential forms and show that the resulting block diagonal (inverse) mass matrices have block sizes independent of the polynomial degree. We demonstrate on an extensive number of examples how the new technique is applicable and efficient for large scale computations.

Holography of a single free matrix

In this paper we consider the collective field theory description of a single free massless scalar matrix theory in 2+1 dimensions. The collective fields are given by k-local operators obtained by tracing a product of k-matrices. For k = 2 and k = 3 we argue that the collective field packages the fields associated to a single and two Regge trajectories respectively. We also determine the coordinate transformation between the coordinates of the collective field theory and the bulk AdS space time. This is used to verify that the bulk equations of motion holds in the collective field theory description.

On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators

In this paper, we derive novel families of inclusion sets for the spectra and pseudospectra of large classes of bounded linear operators, and establish convergence of particular sequences of these inclusion sets to the spectrum or pseudospectrum, as appropriate. Our results apply, in particular, to bounded linear operators on a separable Hilbert space that, with respect to some orthonormal basis, have a representation as a bi-infinite matrix that is banded or band-dominated. More generally, our results apply in cases where the matrix entries themselves are bounded linear operators on some Banach space. In the scalar matrix entry case, we show that our methods, given the input information we assume, lead to a sequence of approximations to the spectrum, each element of which can be computed in finitely many arithmetic operations, so that, with our assumed inputs, the problem of determining the spectrum of a band-dominated operator has solvability complexity index one in the sense of Ben-Artzi et al. (2020). As a concrete and substantial application, we apply our methods to the determination of the spectra of non-self-adjoint bi-infinite tridiagonal matrices that are pseudoergodic in the sense of Davies [Commun. Math. Phys. 216 (2001), 687–704].