Linear Matrix Equations Research Articles

On applying deflation and flexible preconditioning to the adaptive Simpler GMRES method for Sylvester tensor equations

Due to their multidimensional structure, the use of tensors when solving linear matrix equations has been one of the most explored topics over the last decade. In particular, the development of Krylov subspace methods for solving tensor equations is currently attracting a great deal of interest. In this work, we develop the adaptive simpler GMRES method based on tensor format (Ad-SGMRES−BTF) for solving Sylvester tensor equations, in order to address the numerical instability of the simpler version of GMRES based on tensor format (SGMRES−BTF). In addition, an efficient Krylov subspace method, formed by the integration of adaptive deflation and preconditioning and called FAd-SGMRES-D−BTF(m,k) -in which m is the restart parameter and k is the number of harmonic Ritz pairs-, is proposed. Note that the idea behind the FAd-SGMRES-D−BTF(m,k) method is to reuse certain key information, which can be obtained from the current tensor Krylov subspace in the next cycle of the FAd-SGMRES-D−BTF(m) method. It is also important to note that FAd-SGMRES-D−BTF(m,k) represents the first attempt to synchronously combine flexible preconditioning and deflation techniques, with the aim of improving the convergence rate and reducing the computational cost of Ad-SGMRES−BTF(m). Finally, the computational cost of each cycle of the proposed methods when applied to a three-dimensional Sylvester tensor equation is estimated. Moreover, numerical experiments on synthetic and real-world applications demonstrate the potential of Ad-SGMRES−BTF and FAd-SGMRES-D−BTF(m) to solve Sylvester tensor equations.

Multiscale Damage Identification Method of Beam-Type Structures Based on Node Curvature

This paper proposes a multiscale damage identification method for beam-type structures based on node curvature. Firstly, based on the assumption that micro-damage has little effect on stress redistribution and the basic relationship between structural bending moment and curvature, combined with the denoising function of wavelet analysis, the linear matrix equation before and after node curvature damage is solved using the singular value decomposition (SVD) method. Then, the theoretical feasibility of this method is verified with laboratory tests of a simply supported beam. Finally, the damage sensitivity and noise resistance of this method are verified using field measurements of a beam bridge. The results show that the nodal curvature serves as an indicator parameter for damage identification in beam-type structures, enabling the precise localization of damage within these structures. When utilizing a multiscale finite element model for analysis, the nodal curvature enhances the ability to identify both the location and severity of damage within small-scale elements. Furthermore, this method can provide a reference for the damage identification and health monitoring of other types of bridges.

Minimal rank weighted weak Drazin inverses

The concept of a minimal rank weak Drazin inverse for square matrices is extended to rectangular matrices. Precisely, a minimal rank weighted weak Drazin inverse is introduced and its properties are investigated. Some known generalized inverses such as the weighted Drazin inverse, the weighted core-EP inverse, and the weighted $p$-WGI are particular cases of a minimal rank weighted weak Drazin inverse. Thus, a wider class of generalized inverses is proposed. General representation forms of a minimal rank weighted weak Drazin inverse are presented as well as its canonical form. Applying the minimal rank weighted weak Drazin inverse, corresponding systems of linear matrix equations are solved and their solutions are expressed. As consequences of our results, new properties of minimal rank weak Drazin inverse are obtained.

Extended Special Linear group ESL2(F) and matrix equations in SL2(F), SL2(Z) and GL2(Fp)

The problem of roots existence for different classes of matrix such as simple and semisimple matrices from SL2(F), SL2(Z) and GL2(F) are solved. For this purpose, we first introduced the concept of an extended special linear group ESL2(F), which is generalisation of the matrix group SL2(F), where F is arbitrary perfect field. The group of unimodular matrices and extended symplectic group ESp2(R) are generalised by us, their structures are found. Our criterion oriented on a general class of matrix depending of the form of minimal and characteristic polynomials, moreover a proposed criterion holds in GL2(F), where F is an arbitrary field. The method of matrix factorisation is outlined. We show that ESL2(F) is a set of all square matrix roots from SL2(F) except of that established in our root existence criterion.

Weighted index-MP matrices

Solving certain systems of matrix equations, we generalize notions of index-MP, MP-index and MP-index-MP matrices defined for square matrices and we introduce novel types of rectangular matrices, the so-called weighted index-MP, weighted MP-index and weighted MP-index-MP matrices. Many characterizations and expressions for new kinds of matrices are developed. The canonical forms of weighted index-MP, weighted MP-index and weighted MP-index-MP matrices are presented too. We apply the weighted index-MP, weighted MP-index and weighted MP-index-MP matrices to solve systems of linear matrix equations and represent their solutions.

A transparent experimental modelling method for linear multiple-input multiple-output systems

This paper presents a transparent technique that simulates processes using input and output measurement data without the use of any complex optimization or iterative algorithms. The method is very simple and easy to understand as it comprises only of linear ordinary differential equations, Laplace transformations and least squares technique. The solution can be expressed only in two linear matrix equations. The main aim is to provide bachelor students of engineering with a tool to formulate transfer functions.The method is validated with artificially generated erroneous measurement data as well as using measurements obtained from a practical application at the university. Inclusion of dead times and estimation of initial conditions make it ideal to be used for various range of applications such as stable and unstable systems with and without damping, open and closed loop systems. Over-integration, normalizing and/or zeroing of different coefficients add more degrees of freedom to improve the model quality.

On the classic solution of fuzzy linear matrix equations

On the classic solution of fuzzy linear matrix equations

The Solutions of a Class of Sylvester-like Linear Matrix Equations and the Estimation of the Associated Measurements of Their Solutions

The Solutions of a Class of Sylvester-like Linear Matrix Equations and the Estimation of the Associated Measurements of Their Solutions

Gradient neural network model for the system of two linear matrix equations and applications

In this paper, the new type of Gradient Neural Network (GNN) model is proposed for the following linear system of matrix equations: AX=C,XB=D. The convergence analysis of given models is shown. The model is applied for the computation of the regular matrix inverse, as well as Moore-Penrose and Drazin generalized inverses. Some illustrative examples and simulations are given to verify theoretical results.

A randomized block Douglas–Rachford method for solving linear matrix equation

A randomized block Douglas–Rachford method for solving linear matrix equation

Matrix Linear Bilateral Equations Over Different Domains, Methods for the Construction of Solutions, and Description of Their Structure

Matrix Linear Bilateral Equations Over Different Domains, Methods for the Construction of Solutions, and Description of Their Structure

Orthogonality and unitary conditions for solutions to some consistent linear matrix equations

The nonlinear matrix equation (A1 + B1X1D1)(A2 + B2X2D2) = A, presents a matrix identity, where A, Ai, Bi, and Di are known matrices of suitable sizes and Xi are unknown matrices for i = 1,2, over the field of complex numbers ℂ. Several known simple methods in linear algebra can handle the orthogonality problem. In this paper, we select some linear matrix equations as illustrative examples to discuss some properties of pairs of identically dimensional consistent linear matrix equations, then apply specific matrix analytic tools, among them is the matrix rank method, the rank of a matrix is one of the most basic quantities and useful methods and tools that are widely used in linear algebra specifically, in matrix theory and its applications. We consider certain forms of linear matrix equations such as T1X1 = N1 and the system A3X3 = C3, X3B3 = D3, to derive novel conditions dictating the orthogonality of all solutions to two consistent matrix equations, as well as the necessary and sufficient conditions for all solutions to be unitary, further we give an illustrative example.

Matrix Pencil Optimal Iterative Algorithms and Restarted Versions for Linear Matrix Equation and Pseudoinverse

We derive a double-optimal iterative algorithm (DOIA) in an m-degree matrix pencil Krylov subspace to solve a rectangular linear matrix equation. Expressing the iterative solution in a matrix pencil and using two optimization techniques, we determine the expansion coefficients explicitly, by inverting an m×m positive definite matrix. The DOIA is a fast, convergent, iterative algorithm. Some properties and the estimation of residual error of the DOIA are given to prove the absolute convergence. Numerical tests demonstrate the usefulness of the double-optimal solution (DOS) and DOIA in solving square or nonsquare linear matrix equations and in inverting nonsingular square matrices. To speed up the convergence, a restarted technique with frequency m is proposed, namely, DOIA(m); it outperforms the DOIA. The pseudoinverse of a rectangular matrix can be sought using the DOIA and DOIA(m). The Moore–Penrose iterative algorithm (MPIA) and MPIA(m) based on the polynomial-type matrix pencil and the optimized hyperpower iterative algorithm OHPIA(m) are developed. They are efficient and accurate iterative methods for finding the pseudoinverse, especially the MPIA(m) and OHPIA(m).

A novel hybrid finite element – Finite difference approach for 3D Magnetotelluric modeling

A new method is developed to solve the Magnetotelluric (MT) forward problem in three dimensions (3D). The finite-differences (FD) and the finite-elements (FE) methods are considered to be popular numerical techniques to solve the partial differential equations in electromagnetics. The FD approach is known to be easily applicable to a given problem while the FE method is much more flexible when a topography is present, however, it is also computationally more expensive to solve the linear matrix equations for the FE due to the number of non-zero entries it populates when compared to the FD’s. For that reason, a new algorithm is offered to mix the FE and FD in a single mesh using hexahedra and rectangular prisms in 3D. By doing that, it is possible to take advantage of the beneficial aspects of these two methods. It was found out that the newly developed hybrid approach (HYB) is almost as fast as the FD method while being very memory efficient which is an important aspect due to the limited memory space of a GPU which is employed to solve the systems iteratively. It was found that the HYB method is as accurate as the FE method while it is a more reasonable choice for larger models due to the solution speed of the FE method as it does not scale well with the increased model size. On average, a speed-up of 1.6 is found between the FE and HYB methods with the three models tested and it increases with the increasing model size. Using a GPU during the iterative solution made it possible to solve the linear systems in seconds and with the help of the HYB method, the memory consumption was found to be less than half of what the FE method needed. This ultimately means that much larger models can be simulated on the limited memory that a GPU has. Finally, the technique shown to implement the HYB approach makes the deployment of it to existing FE codes so much easier.

Advances on similarity via transversal intersection of manifolds

Let A be an n×n real matrix. As shown in the recent paper S.M. Fallat, H.T. Hall, J.C.-H. Lin, and B.L. Shader (2022) [5], if the manifolds MA={G−1AG:G∈GL(n,R)} and Q(sgn(A)) (consisting of all real matrices having the same sign pattern as A), both considered as embedded submanifolds of Rn×n, intersect transversally at A, then every superpattern of sgn(A) also allows a matrix similar to A. Those authors introduced a condition on A (in terms of certain linear matrix equations) equivalent to the above transversality, called the nonsymmetric strong spectral property (nSSP). In this paper, this transversality property of A is characterized using an alternative, more direct and convenient condition, called the similarity-transversality property (STP). Let X=[xij] be a generic matrix of order n whose entries are independent variables. The STP of A is defined as the full row rank property of the Jacobian matrix of the entries of AX−XA at the zero entry positions of A with respect to the nondiagonal entries of X. This new approach makes it possible to take better advantage of the combinatorial structure of the matrix A, and provides theoretical foundation for constructing matrices similar to a given matrix while the entries have certain desired signs. In particular, several important classes of zero-nonzero patterns and sign patterns that require or allow this transversality property are identified. Examples illustrating many possible applications (such as diagonalizability, number of distinct eigenvalues, nilpotence, idempotence, semi-stability, eigenvalues and their algebraic and geometric multiplicities, Jordan canonical form, minimal polynomial, and rank) are provided.

Gradient-based iterative approach for solving constrained systems of linear matrix equations

Gradient-based iterative approach for solving constrained systems of linear matrix equations

Distributed algorithms of solving linear matrix equations via double-layered networks

This paper focuses on designing distributed algorithms for solving the linear matrix equation in the form of AXB=F via a double-layered multi-agent network. Firstly, we equivalently transform the least squares (LS) problem of AXB=F into two subproblems. We introduce a double-layered multi-agent network, in which each agent only has access to partial information of matrices A, B and F, to solve these two subproblems in a distributed structure. Subsequently, we develop continuous-time and discrete-time distributed algorithms to obtain an LS solution of AXB=F. In particular, the sufficient and necessary conditions about the step-sizes are established, under which the discrete-time algorithm can linearly converge to an LS solution. Compared with the previous results in which each agent needs to communicate multiple matrix variables with the neighbors, the proposed algorithms effectively reduce communication burden for each agent since only one matrix variable is communicated in the upper-layer network and the communication variables in the lower-layered network are vectors with lower dimensions than matrices. Finally, an application example is provided to demonstrate the effectiveness of the proposed algorithms.

Low-Rank Methods for Solving Discrete-Time Projected Lyapunov Equations

In this paper, we consider the numerical solution of large-scale discrete-time projected Lyapunov equations. We provide some reasonable extensions of the most frequently used low-rank iterative methods for linear matrix equations, such as the low-rank Smith method and the low-rank alternating-direction implicit (ADI) method. We also consider how to reduce complex arithmetic operations and storage when shift parameters are complex and propose a partially real version of the low-rank ADI method. Through two standard numerical examples from discrete-time descriptor systems, we will show that the proposed low-rank alternating-direction implicit method is efficient.

A pre-defined finite time neural solver for the time-variant matrix equation [formula omitted

During the process of the general practical applications and scientific researches, time varying issues are an inescapable challenge to be tackled. In this article, a pre-defined finite-time zeroing neural networks (PDFT-ZNN) is devoted to solve the linear time-variant matrix equation (TVME) E(t)X(t)G(t)=D(t) within a finite time (FT), which contrasts with the general ZNN (G-ZNN) models with relatively long global convergence time. Moreover, the proposed PDFT-ZNN model’s convergence time could be calculated in advance by adhering to the designed system parameters; this has nothing to do with the model’s initial state. Additionally, after the convergence analysis, if the solution error is relative small, the simple and effective method is to introduce a linear item to accelerate the convergence speed, in comparison with only the power-type ZNN models. Theory-based analysis and simulation-based results further validate that, the neural state solved by the presented FT-ZNN model can reach the theory-based solution of E(t)X(t)G(t)=D(t) within the pre-defined finite time.

Publisher's Note

Let A be an n×n real matrix. As shown in the recent paper S.M. Fallat, H.T. Hall, J.C.-H. Lin, and B.L. Shader (2022) [5], if the manifolds MA={G−1AG:G∈GL(n,R)} and Q(sgn(A)) (consisting of all real matrices having the same sign pattern as A), both considered as embedded submanifolds of Rn×n, intersect transversally at A, then every superpattern of sgn(A) also allows a matrix similar to A. Those authors introduced a condition on A (in terms of certain linear matrix equations) equivalent to the above transversality, called the nonsymmetric strong spectral property (nSSP). In this paper, this transversality property of A is characterized using an alternative, more direct and convenient condition, called the similarity-transversality property (STP). Let X=[xij] be a generic matrix of order n whose entries are independent variables. The STP of A is defined as the full row rank property of the Jacobian matrix of the entries of AX−XA at the zero entry positions of A with respect to the nondiagonal entries of X. This new approach makes it possible to take better advantage of the combinatorial structure of the matrix A, and provides theoretical foundation for constructing matrices similar to a given matrix while the entries have certain desired signs. In particular, several important classes of zero-nonzero patterns and sign patterns that require or allow this transversality property are identified. Examples illustrating many possible applications (such as diagonalizability, number of distinct eigenvalues, nilpotence, idempotence, semi-stability, eigenvalues and their algebraic and geometric multiplicities, Jordan canonical form, minimal polynomial, and rank) are provided.