Solution Of Matrix Equation Research Articles

On the classic solution of fuzzy linear matrix equations

On the classic solution of fuzzy linear matrix equations

One step inversion free iterative algorithm for computing generalised bisymmetric solution for nonlinear matrix equations

In this paper, we investigate one step inversion free iterative algorithm for computing generalised bisymmetric solution to the nonlinear matrix equations (NLME) X + A T X − q A + B T X − q B = H ( q ≥ 2 ) , where A , B , H are given matrices, while X is a generalised bisymmetric matrix to be computed. The necessary conditions for the existence of a generalised bisymmetric solution and the convergence of the suggested iterative method are derived. Some lemmas and theorems are provided and proved where the iterative solutions are obtained. Four numerical examples are presented to exhibit the effectiveness of the suggested method and to verify the theoretical findings of this paper.

The general algebraic solution of dual fuzzy linear systems and fuzzy Stein matrix equations

In this paper, we present a new method for solving a dual fuzzy linear system (DFLS), AX˜+C˜=BX˜+D˜, where the coefficient matrices A and B are arbitrary real m×n matrices and C˜ and D˜ are given fuzzy number vectors. A necessary and sufficient condition for the R-consistency of the associated system of linear equations is obtained. The straightforward method for solving m×n DFLS based on an arbitrary {1}-inverse of A−B is introduced. Also, as an application, we present the first algorithm for solving the fuzzy Stein matrix equations, based on {1}-inverses. Finally, these results are illustrated by examples.

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.

The numerical solution of fuzzy singular Lyapunov matrix equations

Fuzzy singular Lyapunov matrix equations have many applications, but feasible numerical methods to solve them are absent. In this paper, we propose an efficient numerical method for fuzzy singular Lyapunov matrix equations, where A is crisp and semi-stable. In our method, we transform fuzzy singular Lyapunov matrix equation into two crisp Lyapunov matrix equations. Then we solve the least squares solutions of the two crisp Lyapunov matrix equations, respectively. The existence of fuzzy solution is also considered. At last, two small examples are presented to illustrate the validate of the method and two large scale examples that the existing method fails to slove are presented to show the efficiency of the method.

Least squares solutions of matrix equation $ AXB = C $ under semi-tensor product

<abstract><p>This paper mainly studies the least-squares solutions of matrix equation $ AXB = C $ under a semi-tensor product. According to the definition of the semi-tensor product, the equation is transformed into an ordinary matrix equation. Then, the least-squares solutions of matrix-vector and matrix equations respectively investigated by applying the derivation of matrix operations. Finally, the specific form of the least-squares solutions is given.</p></abstract>

Fixed point results for generalized almost contractions and application to a nonlinear matrix equation

<abstract><p>The goal of this paper was to improve some known results of fixed points by using $ w $-distances and properties of locally symmetric $ \mathcal{H} $-transitivity of binary relations. Also, we gave the application of the obtained results for finding the solution of nonlinear matrix equations. Finally, we gave a numerical example to demonstrate the applicability of our results.</p></abstract>

Some new characterizations of normal matrices

This paper mainly studies some properties of normal matrix and gives the relation between the general solution of related matrix equations and normal matrices.

Inter-robot management via neighboring robot sensing and measurement using a zeroing neural dynamics approach

This paper proposes a complex number representation method for dynamically recording robot positions and develops an optimization strategy for measuring and minimizing inter-robot distances in real-time. Based on this, the optimization strategy is transformed into finding the theoretical solution of the complex time-varying matrix equation by combining the leader–follower frameworks, and a complex zeroing neural dynamics (CZND) model is proposed for this objective. Meanwhile, the effectiveness of the proposed CZND model and the ideal trajectories of the follower robots are ensured and given through rigorous theoretical analyses. Subsequently, these theoretical findings are validated by two numerical experiments (four cases). In addition, the superiority of the CZND model for inter-robot management tasks is comparatively analyzed from three perspectives: initial positions, number of follower robots, and design parameters. Compared to conventional gradient-based neural network (CGNN) methods, the proposed CZND model, based on a dynamic error function (which is not required to be non-negative as in CGNN), demonstrates improved handling of the dynamic behavior of the system by utilizing the first-order time derivatives of matrices, resulting in lower residual errors in the implementation of inter-robot management. Specifically, for the same inter-robot management task, the steady-state error implemented by CZND is on the order of 10−4, whereas for CGNN, it is on the order of 10−1, indicating a significantly larger lagging error.

Design, simulation, and measurement of near-zero-index shells for electromagnetic beam generation

We present design and simulation of three-dimensional (3D) shell structures, which generate directional radiation patterns from isotropic sources thanks to their near-zero-index (NZI) characteristics, as well as realizations of these shells via low-cost 3D printing. Throughout the design process of NZI beam generators, both homogenized structures, for which near-zero relative permittivity and/or permeability values are enforced, and actual models involving periodic arrangements of dielectric rods are considered. Solutions of the electromagnetic problems are obtained by using rigorous implementations of the state-of-the-art surface-integral-equation (SIE) formulations in frequency domain. Iterative solutions of matrix equations derived from SIEs are accelerated by different forms of the multilevel fast multipole algorithm (MLFMA) and suitable preconditioners, when necessary. In the design process of NZI shells, alternative strategies are employed to obtain customized radiation patterns. In this context, various cavities with strong resonance behaviors are designed as source regions. At the same time, outer surfaces are modified to either enhance or suppress outgoing electromagnetic fields. In addition to comprehensive simulations and analyses of NZI beam generators, their capabilities are verified by measurements, specifically at 10.3 GHz, on different prototypes fabricated via 3D printing. Measurements of diverse NZI shell structures are presented to demonstrate that NZI properties can successfully be achieved by well-designed arrangements of dielectric rods with proper materials. The results demonstrate the feasibility of efficient, effective, low-cost, and reconfigurable NZI shells to generate alternative beam configurations that can be useful in a plethora of microwave applications.

General strong fuzzy solutions of complex fuzzy matrix equations involving the Moore-Penrose weak group inverse

In this paper, we investigate the fuzzy solutions of the complex fuzzy matrix equation (CFME) CZ˜=W˜, in which C is a complex crisp matrix, and W˜ is a complex fuzzy matrix. The purpose of this paper is three-fold. Firstly, the necessary and sufficient conditions for the existence of strong fuzzy solutions to the CFME are obtained using the MPWG inverse of the coefficient matrix. Secondly, we obtain a necessary and sufficient condition for the existence of S†,WG≥0 (S†,WG is called the Moore-Penrose weak group inverse of the coefficient matrix S) in order to obtain a strong fuzzy matrix solution of CFME. Moreover, general strong fuzzy solutions of the CFME are derived and an algorithm for obtaining general strong fuzzy solutions of CFME by the MPWG inverse is also established. Finally, some numerical examples are given to illustrate the main results.

Investigation of surface-inset Halbach machines with trapezoidal mixed grade magnets

In order to increase the torque performance of the conventional surface-inset Halbach machines, a surface-inset Halbach machine with trapezoidal mixed grade magnets is proposed and investigated. Based on a layered and linear superposition method, the air-gap magnetic fields are analytically obtained from solution of matrix equations. In addition, the dimensions of mid-magnet and the remanences of Halbach magnets are analytically optimized. In order to show its advantages, the optimized performances of the proposed machine model are compared with those of the conventional Halbach and trapezoidal equal grade Halbach machines. The prediction shows that the proposed optimized machine model has the largest electromagnetic torque and the lowest torque ripple. The demagnetization performance is also investigated. Finally, the finite element analysis (FEA) results verify the analytical predictions.

Best Proximity Point and Existence of the Positive Definite Solution for Matrix Equations

In this research, α − ψ − θ contraction has been defined to find the best proximity point in partially ordered metric spaces. Proper support for the result has been given in the form of a suitable example. The third part is fully devoted to the positive definite solution of matrix equations.

A dynamic matrix equation solution method based on NCBC-ZNN and its application on hyperspectral image multi-target detection

A dynamic matrix equation solution method based on NCBC-ZNN and its application on hyperspectral image multi-target detection

Stein-based preconditioners for weak-constraint 4D-var

Algorithms for data assimilation try to predict the most likely state of a dynamical system by combining information from observations and prior models. Variational approaches, such as the weak-constraint four-dimensional variational data assimilation formulation considered in this problem, can ultimately be interpreted as a minimization problem. One of the main challenges of such a formulation is the solution of large linear systems of equations which arise within the inner linear step of the adopted nonlinear solver. Depending on the adopted approach, these linear algebraic problems amount to either a saddle point linear system or a symmetric positive definite (SPD) one. Both formulations can be solved by means of a Krylov method, like GMRES or CG, that needs to be preconditioned to ensure fast convergence in terms of the number of iterations. In this paper we illustrate novel, efficient preconditioning operators which involve the solution of certain Stein matrix equations. In addition to achieving better computational performance, the latter machinery allows us to derive tighter bounds for the eigenvalue distribution of the preconditioned linear system for certain problem settings. A panel of diverse numerical results displays the effectiveness of the proposed methodology compared to current state-of-the-art approaches.

Nonlinear function activated GNN versus ZNN for online solution of general linear matrix equations

In this work, the gradient-based design method is generalized to establish a finite-/fixed-time GNN (gradient-based neural network) to obtain an online solution of the general linear matrix equation (LME): P1X(t)Q1+P2X(t)Q2=G. Two nonlinear activation functions (AFs), which are broadly investigated in ZNN (zeroing neural network), are introduced and put to use for the construction of the GNN model. Theoretical analyses and comparisons are given to illustrate their impacts on the GNN and the corresponding ZNN models in terms of convergence. It is shown that both the GNN model and the ZNN model exhibit the same convergence property (i.e., finite-/fixed-time convergence) under the action of the same AF. A tighter estimate of the upper bound of fixed convergence time is also obtained for the ZNN model. Comparative verification is conducted to confirm the validation of theoretical analysis.

Iterative optimal solutions of linear matrix equations for hyperspectral and multispectral image fusing

For a linear matrix function f in X in {mathbb {R}}^{mtimes n} we consider inhomogeneous linear matrix equations f(X) = E for E ne 0 that have or do not have solutions. For such systems we compute optimal norm constrained solutions iteratively using the Conjugate Gradient and Lanczos’ methods in combination with the More–Sorensen optimizer. We build codes for ten linear matrix equations, of Sylvester, Lyapunov, Stein and structured types and their T-versions, that differ only in two five times repeated equation specific code lines. Numerical experiments with linear matrix equations are performed that illustrate universality and efficiency of our method for dense and small data matrices, as well as for sparse and certain structured input matrices. Specifically we show how to adapt our universal method for sparse inputs and for structured data such as encountered when fusing image data sets via a Sylvester equation algorithm to obtain an image of higher resolution.

A new matrix equation expression for the solution of non‐autonomous linear systems of ODEs

AbstractThe solution of systems of non‐autonomous linear ordinary differential equations is crucial in a variety of applications, such us nuclear magnetic resonance spectroscopy. A new method with spectral accuracy has been recently introduced in the scalar case. The method is based on a product that generalizes the convolution. In this work, we show that it is possible to extend the method to solve systems of non‐autonomous linear ordinary differential equations (ODEs). In this new approach, the ODE solution can be expressed through a linear system that can be equivalently rewritten as a matrix equation. Numerical examples illustrate the method's efficacy and the low‐rank property of the matrix equation solution.

Solving a nonlinear matrix equation by Newton's method

Nonlinear matrix equations are often used in the quality theory of ordinary differential, functional differential, differential-algebraic and integro-differential equations, in the theory of motion stability, control theory, and in image reconstruction problems. In this paper, we study a nonlinear matrix equation with respect to an unknown rectangular matrix. In general, the linearization of a nonlinear matrix equation with respect to an unknown rectangular matrix defines a linear matrix operator that has no inverse. For such a nonlinear matrix equation, it is not possible to use the classical Newton method, but the Newton-Kantorovich method is applicable. The paper proposes original conditions for solvability and a scheme for finding solutions to a nonlinear matrix equation. To find approximations to solutions of nonlinear matrix equations in the case of an unknown rectangular matrix and to verify the convergence of the constructed iterative scheme, the paper uses the Newton method. To verify the effectiveness of the constructed iterative scheme, we find the nonconformities of the obtained approximations in the solution of a nonlinear matrix algebraic equation.

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.