Generalized Sylvester Matrix Equation Research Articles

Convergence of HS version of BCR algorithm to solve the generalized Sylvester matrix equation over generalized reflexive matrices

In recent years, several linear matrix equations such as Lyapunov and Sylvester matrix equations have received considerable attention due to their important applications in engineering and applied mathematics. In this work, an iterative technique based on the Hestenes–Stiefel (HS) version of biconjugate residual (BCR) algorithm is introduced to solve the generalized Sylvester matrix equation∑i=1f(AiXBi)+∑j=1g(CjYDj)=E,over the generalized reflexive matrices X and Y. We show that the proposed iterative technique converges to the generalized reflexive solutions within a finite number of iterations in the absence of round-off errors. Numerical examples confirm the efficiency and accuracy of the proposed iterative technique and also comparisons with other algorithms are made.

Convergence Results of the Biconjugate Residual Algorithm for Solving Generalized Sylvester Matrix Equation

AbstractIn this article, we investigate a variant of the biconjugate residual (BCR) algorithm to solve the generalized Sylvester matrix equation which includes the well‐known Lyapunov, Stein and Sylvester matrix equations. We show that the BCR algorithm with any (special) initial matrix pair can smoothly compute the (least Frobenius norm) solution pair of the generalized Sylvester matrix equation within a finite number of iterations in the absence of round‐off errors. Finally the accuracy and effectiveness of the BCR algorithm in comparison to some existing algorithms are demonstrated by two numerical examples.

R-FUSE: Robust Fast Fusion of Multiband Images Based on Solving a Sylvester Equation

This paper proposes a robust fast multi-band image fusion method to merge a high-spatial low-spectral resolution image and a low-spatial high-spectral resolution image. Following the method recently developed in [1], the generalized Sylvester matrix equation associated with the multi-band image fusion problem is solved in a more robust and efficient way by exploiting the Woodbury formula, avoiding any permutation operation in the frequency domain as well as the blurring kernel invertibility assumption required in [1]. Thanks to this improvement, the proposed algorithm requires fewer computational operations and is also more robust with respect to the blurring kernel compared with the one in [1]. The proposed new algorithm is tested with different priors considered in [1]. Our conclusion is that the proposed fusion algorithm is more robust than the one in [1] with a reduced computational cost.

Wreath product of matrices

We introduce a new matrix product, that we call the wreath product of matrices. The name is inspired by the analogous product for graphs, and the following important correspondence is proven: the wreath product of the adjacency matrices of two graphs provides the adjacency matrix of the wreath product of the graphs. This correspondence is exploited in order to study the spectral properties of the famous Lamplighter random walk: the spectrum is explicitly determined for the “Walk or switch” model on a complete graph of any size, with two lamp colors. The investigation of the spectrum of the matrix wreath product is actually developed for the more general case where the second factor is a circulant matrix. Finally, an application to the study of the uniqueness of the solution of generalized Sylvester matrix equations is treated.

Symmetric solutions of the coupled generalized Sylvester matrix equations via BCR algorithm

The symmetric solutions of linear matrix equations are extensively required in mathematics and engineering problems. The purpose of this paper is on deriving the biconjugate residual (BCR) algorithm for finding the least Frobenius norm symmetric solution pair (X,Y) of the coupled generalized Sylvester matrix equations{A1XB1+C1XD1+E1YF1=G1,A2XB2+C2XD2+E2YF2=G2.The convergence analysis shows that the BCR algorithm can compute the least Frobenius norm symmetric solution pair of the coupled generalized Sylvester matrix equations within a finite number of iterations in the absence of round-off errors. Finally, we give three numerical examples to illustrate the performance of the BCR algorithm.

Constraint generalized Sylvester matrix equations

In this paper, some necessary and sufficient conditions are established for the constraint generalized Sylvester matrix equations to have a common solution. The expression of the general common solution is also given under the solvable conditions. In addition, a numerical example is presented to illustrate the presented theory.

Two-sided coupled generalized Sylvester matrix equations solving using a simultaneous decomposition for fifteen matrices

In this paper, we investigate and analyze in detail the structure and properties of a simultaneous decomposition for fifteen matrices: Ai∈Cpi×ti, Bi∈Csi×qi, Ci∈Cpi×ti+1, Di∈Csi+1×qi, and Ei∈Cpi×qi (i=1,2,3). We show that from this simultaneous decomposition we can derive some necessary and sufficient conditions for the existence of a solution to the system of two-sided coupled generalized Sylvester matrix equations with four unknowns AiXiBi+CiXi+1Di=Ei (i=1,2,3). Apart from proving an expression for the general solutions to this system, we derive the range of ranks of these solutions using the ranks of the given matrices Ai, Bi, Ci, Di, and Ei. We provide some numerical examples to illustrate our results. Moreover, we present a similar approach to consider the simultaneous decomposition for 5k matrices and the system of k two-sided coupled generalized Sylvester matrix equations with k+1 unknowns AiXiBi+CiXi+1Di=Ei (i=1,…,k, k≥4). The main results are also valid over the real number field and the real quaternion algebra.

Parametric stabilization for descriptor linear systems via state-proportional and -derivative feedback

This paper considers eigenstructure assignment problem for descriptor linear systems via state-proportional and -derivative feedback. New complete parametric approaches for the problem are proposed in two cases. General complete parametric expressions in direct closed forms for the closed-loop eigenvectors associated with the finite closed-loop eigenvalues are presented. Based on a recently proposed complete parametric solution to the generalized Sylvester matrix equation AV−EVJ=BWDJ+BWP, very simple complete parametric solutions to the feedback gains are also established. The approach guarantees arbitrary assignment of rank[EB] number finite closed-loop eigenvalues with arbitrary given algebraic and geometric multiplicities. It also guarantees the closed-loop regularity and impulse-freeness when several simple constraints on the design parameters are added. A numerical example shows the effectiveness of the proposed approaches.

Minimal-Order Functional Observer-Based Residual Generators for Fault Detection and Isolation of Dynamical Systems

This paper examines the design of minimal-order residual generators for the purpose of detecting and isolating actuator and/or component faults in dynamical systems. We first derive existence conditions and design residual generators using only first-order observers to detect and identify the faults. When the first-order functional observers do not exist, then based on a parametric approach to the solution of a generalized Sylvester matrix equation, we develop systematic procedures for designing residual generators utilizing minimal-order functional observers. Our design approach gives lower-order residual generators than existing results in the literature. The advantages for having such lower-order residual generators are obvious from the economical and practical points of view as cost saving and simplicity in implementation can be achieved, particularly when dealing with high-order complex systems. Numerical examples are given to illustrate the proposed fault detection and isolation schemes. In all of the numerical examples, we design minimum-order residual generators to effectively detect and isolate actuator and/or component faults in the system.

The scaling conjugate gradient iterative method for two types of linear matrix equations

In this paper, we propose a new iteration method which is based on the conjugate gradient method for solving the linear matrix equations of the form AiXBi=Fi,(i=1,2,…,N) and the generalized Sylvester matrix equation AXB+CXD=E. This method is compared with some existing methods in detail, such as gradient based iterative (GI) method and least squares iterative (LSI) method (Ding et al. 2010) proposed by Ding et al., also cyclic and simultaneous methods (Tang et al. 2014) given by Tang et al. Some numerical experiments demonstrate that the introduced iterative method is more efficient than the four existing methods.

The automatic solution of partial differential equations using a global spectral method

A spectral method for solving linear partial differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of partial differential operators and the one-dimensional ultraspherical spectral method. If a partial differential operator is of splitting rank 2, such as the operator associated with Poisson or Helmholtz, the corresponding PDE is solved via a generalized Sylvester matrix equation, and a bivariate polynomial approximation of the solution of degree (nx,ny) is computed in O((nxny)3/2) operations. Partial differential operators of splitting rank ≥3 are solved via a linear system involving a block-banded matrix in O(min⁡(nx3ny,nxny3)) operations. Numerical examples demonstrate the applicability of our 2D spectral method to a broad class of PDEs, which includes elliptic and dispersive time-evolution equations. The resulting PDE solver is written in Matlab and is publicly available as part of Chebfun. It can resolve solutions requiring over a million degrees of freedom in under 60 seconds. An experimental implementation in the Julia language can currently perform the same solve in 10 seconds.

Least-squares symmetric and skew-symmetric solutions of the generalized Sylvester matrix equation [formula omitted

The generalized Sylvester matrix equation ∑i=1sAiXBi+∑j=1tCjYDj=E with unknown matrices X and Y is encountered in many system and control applications. In this paper, a direct method is established to solve the least-squares symmetric and skew-symmetric solutions of the equation by using the Kronecker product and the generalized inverses and, the expression of the solution set S are provided. Moreover, an optimal approximation between a given matrix pair and the affine subspace S is discussed, and an explicit formula for the unique optimal approximation solution is presented. Finally, two numerical examples are given which demonstrate that the introduced algorithm is quite efficient.

Alternating direction method for generalized Sylvester matrix equation AXB + CYD = E

This paper presents alternating direction methods of multipliers for finding the solution, the best approximate solution and the nonnegative solution of the generalized Sylvester matrix equation AXB + CYD = E, where A, B, C, D and E are given matrices of suitable sizes. Preliminary convergence properties of the proposed algorithms are given. Numerical experiments show that the proposed algorithms tend to deliver higher quality solutions with less iteration steps and less computing times than recent algorithms on the tested problems.

Generalized solution sets of the interval generalized Sylvester matrix equation [formula omitted] and some approaches for inner and outer estimations

In this work, we extend the concept of generalized solution sets that is introduced for the first time by Shary (1996, 1999), to the interval generalized Sylvester matrix equation ∑i=1pAiXi+∑j=1qYjBj=C. Then AE-solution sets for this equation will be characterized and we develop some approaches (called algebraic approaches) in which the inner and outer estimation problems for some special cases of AE-solution sets reduce to problems of computing algebraic solutions of some auxiliary interval equations. Also we present a numerical technique that is an extension of the well-known interval Gauss–Seidel method for outer estimation of the solution set.

Systems of coupled generalized Sylvester matrix equations

This paper studies some systems of coupled generalized Sylvester matrix equations. We present some necessary and sufficient conditions for the solvability to these systems. We give the expressions of the general solutions to the systems when their solvability conditions are satisfied.

Explicit and Iterative Methods for Solving the Matrix Equation AV + BW = EVF + C

AbstractIn this paper, we consider explicit and iterative methods for solving the Generalized Sylvester matrix equation AV + BW = EVF + C. Based on the use of Kronecker map and Sylvester sum some lemmas and theorems are stated and proved where explicit and iterative solutions are obtained. The proposed methods are illustrated by numerical example. The obtained results show that the methods are very neat and efficient.

A modified iterative algorithm for the (Hermitian) reflexive solution of the generalized Sylvester matrix equation

Recently, Ramadan et al. have focused on the following matrix equation: [Formula: see text] and propounded two gradient-based iterative algorithms for solving the above matrix equation over reflexive and Hermitian reflexive matrices, respectively. In this paper, we develop two new iterative algorithms based on a two-dimensional projection technique for solving the mentioned matrix equation over reflexive and Hermitian reflexive matrices. The performance of our proposed algorithms is collated with the gradient-based iterative algorithms. It is both theoretically and experimentally demonstrated that the approaches handled surpass the offered algorithms in the earlier referred work in solving the mentioned matrix equation over reflexive and Hermitian reflexive matrices. In addition, it is briefly discussed that a one-dimensional projection technique can accelerate the speed of convergence of the gradient-based iterative algorithm for solving general coupled Sylvester matrix equations over reflexive matrices without assuming the restriction of the existence of a unique solution.

Finite iterative method for solving coupled Sylvester-transpose matrix equations

This paper is concerned with solutions to the so-called coupled Sylvester-transpose matrix equations, which include the generalized Sylvester matrix equation and Lyapunov matrix equation as special cases. By extending the idea of conjugate gradient method, an iterative algorithm is constructed to solve this kind of coupled matrix equations. When the considered matrix equations are consistent, for any initial matrix group, a solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm solution group of the coupled Sylvester-transpose matrix equations can be derived when a suitable initial matrix group is chosen. By applying the proposed algorithm, the optimal approximation solution group to a given matrix group can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, a numerical example is given to illustrate that the algorithm is effective.

Stabilization of discrete-time chaotic systems via improved periodic delayed feedback control based on polynomial matrix right coprime factorization

Delayed feedback control, proposed by Pyragas,is one of useful control methods of stabilizing unstable periodic orbits of chaotic systems without their exact information. This paper discusses an improved periodic delayed feedback control for the stabilization of the discrete-time chaotic systems. A technique to solve the generalized Sylvester matrix equation, based on polynomial matrix right coprime factorization, is introduced in order to give an explicit solution to the periodic gains of the closed-loop system. Moreover, some examples are demonstrated to show the effectiveness of the proposed method.

Two iterative algorithms for the reflexive and Hermitian reflexive solutions of the generalized Sylvester matrix equation

This paper is concerned with iterative solutions to the generalized Sylvester matrix equation . Two iterative algorithms are presented to obtain the reflexive and Hermitian reflexive solutions. With these iterative algorithms, for any initial reflexive and Hermitian reflexive matrices the solutions can be obtained. Some needed lemmas are first stated, then two theorems are stated and proved where the iterative solutions are obtained. Finally, we report two numerical examples to verify the theoretical results.