Sylvester Matrix Research Articles

Matrix-oriented FEM formulation for reaction-diffusion PDEs on a large class of 2D domains

For the spatial discretization of elliptic and parabolic partial differential equations (PDEs), we provide a Matrix-Oriented formulation of the classical Finite Element Method, called MO-FEM, of arbitrary order k∈N. On a quite general class of 2D domains, namely separable domains, and even on special surfaces, the discrete problem is then reformulated as a multiterm Sylvester matrix equation where the additional terms account for the geometric contribution of the domain shape. By considering the IMEX Euler method for the PDE time discretization, we obtain a sequence of these equations. To solve each multiterm Sylvester equation, we apply the matrix-oriented form of the Preconditioned Conjugate Gradient (MO-PCG) method with a matrix-oriented preconditioner that captures the spectral properties of the Sylvester operator. Solving the Poisson problem and the heat equation on some separable domains by MO-FEM-PCG, we show a gain in computational time and memory occupation wrt the classical vector PCG with same preconditioning or wrt a LU based direct method. As an application, we show the advantages of the MO-FEM-PCG to approximate Turing patterns on some separable domains and cylindrical surfaces for a morphochemical reaction-diffusion PDE system for battery modelling.

A numerical approach based on Bernstein collocation method: Application to differential Lyapunov and Sylvester matrix equations

In this paper, we apply the Bernstein collocation method to construct the solution set of the Sylvester matrix differential equation (Sy-MDE) which involves the Lyapunov matrix differential equation. The method depends on the collocation method and Bernstein polynomials. The main advantage of the proposed method is that by using this method Sy-MDE reduces to a linear system of algebraic equations which can be solved by using an appropriate iterative method. We analyze the error and give a theorem that bounds the error. We also give the residual correction procedure to estimate the error. By using the procedure, we obtain a new approximate solution, namely a corrected Bernstein collocation solution. To illustrate how the proposed method is applied, several examples are given. Numerical experiments show the effectiveness and accuracy of the method for solving such types of Sy-MDE.

Numerical solutions for generalized trapezoidal fully fuzzy Sylvester matrix equation with sufficient conditions to have a positive solution

Numerical solutions for generalized trapezoidal fully fuzzy Sylvester matrix equation with sufficient conditions to have a positive solution

Malcolmson semigroups

Inspired by the construction of the Cuntz semigroup for a C⁎-algebra, we introduce the matrix Malcolmson semigroup and the finitely presented module Malcolmson semigroup for a unital ring. These two semigroups are shown to have isomorphic Grothendieck group in general and be isomorphic for von Neumann regular rings. For unital C⁎-algebras, it is shown that the matrix Malcolmson semigroup has a natural surjective order-preserving homomorphism to the Cuntz semigroup, every dimension function is a Sylvester matrix rank function, and there exist Sylvester matrix rank functions which are not dimension functions.

Analysis and design of control systems via parameter‐based approach

Analysis and design of control systems via parameter‐based approach

An improved gradient neural network for solving periodic Sylvester matrix equations

The gradient neural network (GNN) method is a novel approach to solving matrices. Based on this method, this paper improves the gradient neural network (IGNN) model with a better effect. The convergence speed is increased by replacing the Xi−1(k) matrix in the original gradient neural network with the current matrix Xi−1(k+1). According to theoretical study, IGNN is globally convergent to the solution of periodic matrix equations. The computer simulation results further confirm the feasibility and effectiveness of IGNN.

Finite-time control for systems with multiple disturbances based on adaptive disturbance observer

For a type of nonlinear systems with multiple disturbances, finite-time stability is researched in this paper. The disturbances include unknown frequency disturbance and unknown bounded disturbances. The unknown frequency disturbances are transformed into the model with unknown vectors by Sylvester matrix transformation, which is estimated by designing an adaptive disturbance observer. Then, a finite-time anti-disturbance controller is constructed to keep the system globally stable in finite time. Finally, the validity of the finite-time anti-disturbance controller is proved by simulations, including a numerical example and a satellite attitude control system.

A noise tolerant parameter-variable zeroing neural network and its applications

Time-varying problems frequently arise in the territories of science and engineering, and most of the time-varying problems can be described by dynamic matrix equations. As a powerful tool for solving dynamic matrix equations, the zeroing neural network (ZNN) develops fast in recent years. Convergence and robustness are two main performance indicators of the ZNN model. However, the development of the ZNN is focused on the improvement of its convergence in the past, and its robustness to noises is rarely considered. In order to achieve fast convergence and robustness of the ZNN model, a novel activation function (NAF) is presented in this paper. Based on the NAF, a noise-tolerant parameter-variable ZNN (NTPVZNN) model for solving dynamic Sylvester matrix equations (DSME) is realized, and its fixed-time convergence and robustness to noises are verified by rigorous mathematical analysis and numerical simulation results. Besides, two examples of electrical circuit currents computing and robotic manipulator trajectory tracking using the proposed NTPVZNN model in noisy environment further demonstrates its practical application ability.

Emergent non-Hermitian localization phenomena in the synthetic space of zero-dimensional bosonic systems

Phase transitions in non-Hermitian systems are at the focus of cutting edge theoretical and experimental research. On the one hand, parity-time- $(\mathcal{PT}\ensuremath{-})$ and $\mathrm{anti}\ensuremath{-}\mathcal{PT}\ensuremath{-}\mathrm{symmetric}$ physics have gained ever-growing interest, due to the existence of non-Hermitian spectral singularities called exceptional points (EPs). On the other hand, topological and localization transitions in non-Hermitian systems reveal new phenomena, e.g., the non-Hermitian skin effect and the absence of conventional bulk-boundary correspondence. The great majority of previous studies exclusively focus on non-Hermitian Hamiltonians, whose realization requires an a priori fine-tuned extended lattice to exhibit topological and localization transition phenomena. In this work, we show how the non-Hermitian localization phenomena can naturally emerge in the synthetic field moment space of zero-dimensional bosonic systems, e.g., in $\mathrm{anti}\ensuremath{-}\mathcal{PT}$- and $\mathcal{PT}\ensuremath{-}\mathrm{symmetric}$ quantum dimers. This offers an opportunity to simulate localization transitions in low-dimensional systems, without the need to construct complex arrays of, e.g., coupled cavities or waveguides. Indeed, the field moment equations of motion can describe an equivalent (quasi)particle moving in a one-dimensional (1D) synthetic lattice. This synthetic field moment space can exhibit nontrivial localization phenomena, such as non-Hermitian skin effect, induced by the presence of highly degenerate EPs. We demonstrate our findings on the example of an $\mathrm{anti}\ensuremath{-}\mathcal{PT}\ensuremath{-}\mathrm{symmetric}$ two-mode system, whose higher-order field moment eigenspace is emulated by a synthetic 1D non-Hermitian Hamiltonian having a Sylvester matrix shape. Our results can be directly verified in state-of-the-art optical setups, such as superconducting circuits and toroidal resonators, by measuring photon moments or correlation functions.

Numerical solution of dual fuzzy Sylvester matrix equations by an iteration method

Numerical solution of dual fuzzy Sylvester matrix equations by an iteration method

Structured condition numbers for Sylvester matrix equation with parameterized quasiseparable matrices

Structured condition numbers for Sylvester matrix equation with parameterized quasiseparable matrices

The BiCR Algorithm for Solving the Perhermitian Solutions of Sylvester Matrix Equations

The BiCR Algorithm for Solving the Perhermitian Solutions of Sylvester Matrix Equations

AN ALTERNATING DIRECTION METHOD OF MULTIPLIERS WITH THE CONDITIONAL GRADIENT TOTAL VARIATION METHOD FOR LINEAR INVERSE PROBLEMS

Abstract In this paper, we study the ill-posed problem using total variation regularization. To solve such a problem, we use an alternating direction method of multipliers to split our problem into two sub-problems. The novelty of our paper is in the use of the conditional gradient total variation method (CGTV) we have recently introduced. The second split- ting sub-problem is solved by transforming the obtained optimization problem to a general Sylvester matrix equation and then an orthogonal projection method is used to solve the obtained matrix equation. We give proof of the convergence of this method. Some numerical examples and applications to image restoration are given to illustrate the effectiveness of the proposed method.

Simple Algorithm for GCD of Polynomials

Based on the Bezout approach we propose a simple algorithm to determine the gcd of two polynomials which doesn’t need division, like the Euclidean algorithm, or determinant calculations, like the Sylvester matrix algorithm. The algorithm needs only n steps for polynomials of degree n. Formal manipulations give the discriminant or the resultant for any degree without needing division nor determinant calculation.

A simple algorithm for GCD of polynomials

Based on the Bezout approach we propose a simple algorithm to determine the gcd of two polynomials that don't need division, like the Euclidean algorithm, or determinant calculations, like the Sylvester matrix algorithm. The algorithm needs only n steps for polynomials of degree n. Formal manipulations give the discriminant or the resultant for any degree without needing division or determinant calculation.

Network Flows That Solve Sylvester Matrix Equations

In this article, we study methods to solve a Sylvester equation in the form of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbf {A}\mathbf {X}+\mathbf {X}\mathbf {B}=\mathbf {C}$</tex-math></inline-formula> for given matrices <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbf {A}, \mathbf {B}, \mathbf {C}\in \mathbb {R}^{n\times n}$</tex-math></inline-formula> , inspired by the distributed linear equation flows. The entries of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbf {A}, \mathbf {B},$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbf {C}$</tex-math></inline-formula> are separately partitioned into a number of pieces (or sometimes we permit these pieces to overlap), which are allocated to nodes in a network. Nodes hold a dynamic state shared among their neighbors defined from the network structure. Natural partial or full row/column partitions and block partitions of the data <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbf {A}, \mathbf {B},$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbf {C}$</tex-math></inline-formula> are formulated by use of the vectorized matrix equation. We show that existing network flows for distributed linear algebraic equations can be extended to solve this special form of matrix equations over networks. A “consensus <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$+$</tex-math></inline-formula> projection <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$+$</tex-math></inline-formula> symmetrization” flow is also developed for equations with symmetry constraints on the matrix variables. We prove the convergence of these flows and obtain the fastest convergence rates that these flows can achieve regardless of the choices of node interaction strengths and network structures.

Enumeration of maximal cycles generated by orthogonal cellular automata

Cellular automata (CA) are an interesting computational model for designing pseudorandom number generators (PRNG), due to the complex dynamical behavior they can exhibit depending on the underlying local rule. Most of the CA-based PRNGs proposed in the literature, however, suffer from poor diffusion since a change in a single cell can propagate only within its neighborhood during a single time step. This might pose a problem especially when such PRNGs are used for cryptographic purposes. In this paper, we consider an alternative approach to generate pseudorandom sequences through orthogonal CA (OCA), which guarantees a better amount of diffusion. After defining the related PRNG, we perform an empirical investigation of the maximal cycles in OCA pairs up to diameter $$d=8$$ . Next, we focus on OCA induced by linear rules, giving a characterization of their cycle structure based on the rational canonical form of the associated Sylvester matrix. Finally, we devise an algorithm to enumerate all linear OCA pairs characterized by a single maximal cycle, and apply it up to diameter $$d=16$$ and $$d=13$$ for OCA respectively over the binary and ternary alphabets.

On the minimum-norm least squares solution of the complex generalized coupled sylvester matrix equations

By means of the real linear operator, we establish an iterative algorithm for solving a class of complex generalized coupled Sylvester matrix equations. The finite termination of the proposed algorithm is proved. By representing a complex matrix as a larger real matrix, we present a new method to prove that the minimum-norm solution or minimum-norm least squares solution of the complex generalized coupled Sylvester matrix equations can be obtained by an appropriate selection for the initial matrices, which has not been found in the existing work. Numerical experiments on some randomly generated data and practical image restoration problem show that the proposed algorithm is feasible and effective.

Robust model reference tracking for uncertain second‐order nonlinear systems with application to robot manipulator

AbstractIn this article, robust model reference control for uncertain second‐order nonlinear systems is investigated by applying fully actuated system approaches. A robust stabilizing control law is constructed for the uncertain systems based on the Lyapunov stability theory. With the obtained robust control results, a robust model reference tracking (RMRT) control scheme is proposed to ensure the tracking error finally converges globally into a bounded ellipsoid. The established RMRT controller is composed of three parts, the basic part cancels the known nonlinearities in the system and simultaneously assigns the linear dominant term in the closed‐loop system, the robustness part overcomes the effects of the nonlinear uncertainties in the system, the compensator part compensates the effect of the reference model state and reference control input to the tracking error. Furthermore, based on a general parametric solution to the type of Sylvester matrix equations, simple and complete parameterization of the RMRT control designs is provided. An application to robot manipulator indicates satisfactory control system performances with the proposed RMRT control scheme.

A Jordan-Schur Algorithm for Solving Sylvester and Lyapunov Matrix Equations

This paper presents a version of the Bartels-Stewart algorithm for solving the Sylvester and Lyapunov equations that utilizes the Jordan-Schur form of the equation matrices. The Jordan-Schur form is a type of Schur form which contains additional information about the Jordan structure of the corresponding matrix. This information can be used to solve more efficiently the Sylvester and Lyapunov equations in some cases. A two-level algorithm is implemented which allows us to find directly non-scalar blocks of the solution matrix. These blocks have sizes that are determined by the Weyr characteristics associated with the eigenvalues of the equation matrices. In the case of large elements of the Weyr characteristics associated with multiple eigenvalues, the determination of the solution blocks can be done more efficiency. Also, the blocks equations can be more appropriate in solving the Sylvester and Lyapunov equations in the case of parallel computations. Results obtained from numerical experiments confirm that the accuracy of the new algorithm is comparable with the accuracy of the Bartels-Stewart algorithm.