Sylvester Matrix Equations Research Articles

Global quaternion generalized minimal residual method for generalized coupled Sylvester quaternion matrix equations with application to colour image encryption and decryption

ABSTRACT Sylvester matrix equations play important roles in control and system theory. We consider a class of the generalized coupled Sylvester quaternion matrix equations, which includes a lot of special matrix equations such as Stein matrix equation, Lyapunov matrix equation and Sylvester matrix equation on the quaternion skew field. First, the quaternion Krylov subspace and its F-orthogonal basis are constructed via a new quaternion matrix product and the global quaternion Arnoldi process. Then a global quaternion generalized minimal residual (Gl-QGMRES) method is proposed for solving the generalized coupled Sylvester quaternion matrix equations. The convergence of the proposed method is theoretically analysed. Finally, the effectiveness and feasibility of the proposal are verified by some numerical experiments. As an application of Gl-QGMRES method, we establish a kind of colour image encryption and decryption by embedding watermark images.

A System of Four Generalized Sylvester Matrix Equations over the Quaternion Algebra

A System of Four Generalized Sylvester Matrix Equations over the Quaternion Algebra

Tikhonov regularization with conjugate gradient least squares method for large-scale discrete ill-posed problem in image restoration

Image restoration is a large-scale discrete ill-posed problem, which can be transformed into a Tikhonov regularization problem that can approximate the original image. Kronecker product approximation is introduced into the Tikhonov regularization problem to produce an alternative problem of solving the generalized Sylvester matrix equation, reducing the scale of the image restoration problem. This paper considers solving this alternative problem by applying the conjugate gradient least squares (CGLS) method which has been demonstrated to be efficient and concise. The convergence of the CGLS method is analyzed, and it is demonstrated that the CGLS method converges to the least squares solution within the finite number of iteration steps. The effectiveness and superiority of the CGLS method are verified by numerical tests.

The relaxed gradient based iterative algorithm for solving the generalized coupled complex conjugate and transpose Sylvester matrix equations

Inspired by the idea of Ma et al. (Journal of the Franklin Institute, 2018), we adopt relaxation technique and introduce relaxation factors into the gradient based iterative (GI) algorithm, and the relaxed based iterative (RGI) algorithm is established to solve the generalized coupled complex conjugate and transpose Sylvester matrix equations. By applying the real representation and straighten operation, we contain the sufficient and necessary condition for convergence of the RGI method. In order to effectively utilize this algorithm, we further derive the optimal convergence parameter and some related conclusions. Moreover, to overcome the high dimension calculation problem, a sufficient condition for convergence with less computational complexity is determined. Finally, numerical examples are reported to demonstrate the availability and superiority of the constructed iterative algorithm.

Control theory for fractional differential Sylvester matrix equations with Caputo fractional derivative

In this manuscript, we introduce the concept of first-time controllability and observability concerning fractional differential Sylvester matrix equations employing the Caputo fractional derivative. Our work establishes necessary and sufficient conditions for controllability and observability, wherein controllability equates to having a controllability matrix with full rank and observability aligns with an observability Gramian matrix that is nonsingular. Furthermore, we provide several theorems addressing the observability of fractional differential Sylvester matrix equations. Finally, we offer two illustrative examples to demonstrate the efficacy and application of our established results.

A General Approach to Sylvester-Polynomial-Conjugate Matrix Equations

Sylvester-polynomial-conjugate matrix equations unify many well-known versions and generalizations of the Sylvester matrix equation AX−XB=C which have a wide range of applications. In this paper, we present a general approach to Sylvester-polynomial-conjugate matrix equations via groupoids, vector spaces, and matrices over skew polynomial rings. The obtained results are applied to Sylvester-polynomial-conjugate matrix equations over complex numbers and quaternions. The main role in our approach is played by skew polynomial rings, which are well-known tools in algebra to provide examples of asymmetry between left-sided and right-sided versions of many ring objects.

The shifted inner–outer iteration methods for solving Sylvester matrix equations

In this paper, based on the iteration frameworks (Tian et al., 2018, 2020), by introducing a tunable parameter γ to improve their convergence performances, the shifted inner–outer (IO) and multi-step Smith–inner–outer (MSIO) iteration methods are proposed for solving the Sylvester matrix equations. Their convergence properties are analyzed in detail, and the choices of the shifted parameter γ in these algorithms are also investigated. Finally, four numerical examples are implemented to verify the effectiveness of the proposed shifted iteration methods.

Trapezoidal fully fuzzy Sylvester matrix equation with arbitrary coefficients

Trapezoidal fully fuzzy Sylvester matrix equation with arbitrary coefficients

SOLVING BIPOLAR FULLY FUZZY SYLVESTER MATRIX EQUATIONS WITH NEGATIVE FUZZY NUMBERS

Sylvester matrix equations play a crucial role in control theory for controller design. Bipolar Fully Fuzzy Sylvester Matrix Equations (FFSME), incorporating both positive and negative components, are employed in controller design to address uncertainties that may affect a system’s performance and stability. However, there is not much existing research on combining bipolar fuzzy numbers and FFSME,and most of them mainly deal with positive coefficients. Thus, this paper presents a method that enables solving the negative coefficient of bipolar FFSME in the form of Left-Right (LR) triangular fuzzynumbers using an Associated Bipolar Linear System (ABLS). To obtain the ABLS, bipolar FFSME is transformed into a bipolar Fully Fuzzy Linear System (FFLS) using the Kronecker product and Vecoperator. Subsequently, the solution is derived through the inverse method, and the equation of the ABLS is rearranged as a bipolar fuzzy matrix. Additionally, this paper provides two numerical examples to illustrate the applicability of the constructed method.

General strong fuzzy solutions of fuzzy Sylvester matrix equations involving the BT inverse

This paper aims to solve the fuzzy Sylvester matrix equation (FSME) AX˜+X˜B=C˜, in which A, B are n×n and m×m crisp matrices, and C˜ is a n×m fuzzy matrix. Using the Kronecker product, we transform the FSME into an mn×mn fuzzy linear system. Firstly, we obtain the necessary and sufficient conditions for the existence of strong fuzzy solutions to the FSME by using the BT inverse of the coefficient matrix. Secondly, we investigate the existence of a nonnegative BT inverse by using its block structure. Next, we derive general strong fuzzy solutions to a class of FSME and establish an algorithm to obtain general strong fuzzy solutions to the FSME by the BT inverse. Finally, we give three numerical examples and an applied example in the economy to illustrate the proposed methods.

Improved Numerical Calculation Method for Second-Order Scale Splitting Based on the Complex Sylvester Matrix Equation

Improved Numerical Calculation Method for Second-Order Scale Splitting Based on the Complex Sylvester Matrix Equation

Intensive Noise-Tolerant Zeroing Neural Network Based on a Novel Fuzzy Control Approach

To overcome the disadvantages of the current zeroing neural network (ZNN) in noise tolerance, this article first proposes an intensive noise-tolerant ZNN (INT-ZNN) by introducing a novel fuzzy control approach (FCA). This FCA is designed dexterously according to the variation of two errors related to the INT-ZNN. Thus, the most feature of the INT-ZNN is that the added fuzzy control can inherently restrain the various noises. Compared with the previous noise-tolerant ZNN derived by the integral design formula, the INT-ZNN with a much simpler structure can tolerate the noise in finite/fixed time. That is, the INT-ZNN activated by nonlinear functions possesses finite/fixed-time convergence while suppressing the noise, which is guaranteed by the presented theorems. Besides, it also theoretically proves that the INT-ZNN has global stability under the interference of noise. In the simulative experiment, the INT-ZNN is used to solve the time-varying Sylvester matrix equation problem and the experimental results verify the excellent noise-tolerance of the INT-ZNN. Meanwhile, the INT-ZNN is successfully applied to image processing.

The Weighted, Relaxed Gradient-Based Iterative Algorithm for the Generalized Coupled Conjugate and Transpose Sylvester Matrix Equations

By applying the weighted relaxation technique to the gradient-based iterative (GI) algorithm and taking proper weighted combinations of the solutions, this paper proposes the weighted, relaxed gradient-based iterative (WRGI) algorithm to solve the generalized coupled conjugate and transpose Sylvester matrix equations. With the real representation of a complex matrix as a tool, the necessary and sufficient conditions for the convergence of the WRGI algorithm are determined. Also, some sufficient convergence conditions of the WRGI algorithm are presented. Moreover, the optimal step size and the corresponding optimal convergence factor of the WRGI algorithm are given. Lastly, some numerical examples are provided to demonstrate the effectiveness, feasibility and superiority of the proposed algorithm.

The constant solution method for solving large-scale differential Sylvester matrix equations with time invariant coefficients

The constant solution method for solving large-scale differential Sylvester matrix equations with time invariant coefficients

A parametric poles assignment algorithm for high-order linear discrete periodic systems

The periodic state feedback pole assignment problem of high-order periodic discrete systems is investigated, and the pole assignment problem for such systems is transformed into a class of problems for resolving periodic Sylvester matrix equations with constraints. Using the technique of cyclic lifting, such equations can be transformed into high-order time-invariant Sylvester matrix equations and solved using a parametric method. Lastly, a numerical example is provided to demonstrate the validity of the procedure.

Generalized conjugate direction algorithm for solving general coupled Sylvester matrix equations

In this paper, a generalized conjugate direction algorithm (GCD) is proposed for solving general coupled Sylvester matrix equations. The GCD algorithm is an improved gradient algorithm, which can realize gradient descent by introducing matrices Pj(k) and Tj(k) to construct parameters α(k) and β(k). The matrix Pj(k) and Tj(k) are iterated in a cross way to accelerate the convergence rate. In addition, it is further proved that the algorithm converges to the exact solution in finite iteration steps in the absence of round-off errors if the system is consistent. Also, the sufficient conditions for least squares solutions and the minimum F-norm solutions are obtained. Finally, numerical examples are given to demonstrate the effectiveness of the GCD algorithm.

Compact formula for skew-symmetric system of matrix equations

In this paper, we consider skew-Hermitian solution of coupled generalized Sylvester matrix equations encompassing *-hermicity over complex field. The compact formula of the general solution of this system is presented in terms of generalized inverses when some necessary and sufficient conditions are fulfilled. An algorithm and a numerical example are provided to validate our findings. A numerical example is carried out using determinantal representations of the Moore–Penrose inverse.

An efficient approach for enclosing the solution set of the interval coupled Sylvester matrix equations

In this paper, we investigate the interval coupled Sylvester matrix equations which include the well-known (generalized) Sylvester and Lyapunov matrix equations in both real and interval forms. Assuming that certain matrices are simultaneously diagonalizable, we present a fast and efficient approach for enclosing the solution set of the interval coupled Sylvester matrix equations. Our approach, which is a modification of the Krawczyk operator, enables us to reduce the computational complexity considerably. Some numerical tests are given to illustrate the effectiveness of the proposed approach.

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.