Sylvester-transpose Matrix Equations Research Articles

Exact and least-squares solutions of a generalized Sylvester-transpose matrix equation over generalized quaternions

<abstract><p>We have considered a generalized Sylvester-transpose matrix equation $ AXB + CX^TD = E, $ where $ A, B, C, D, $ and $ E $ are given rectangular matrices over a generalized quaternion skew-field, and $ X $ is an unknown matrix. We have applied certain vectorizations and real representations to transform the matrix equation into a matrix equation over the real numbers. Thus, we have investigated a solvability condition, general exact/least-squares solutions, minimal-norm solutions, and the exact/least-squares solution closest to a given matrix. The main equation included the equation $ AXB = E $ and the Sylvester-transpose equation. Our results also covered such matrix equations over the quaternions, and quaternionic linear systems.</p></abstract>

Adaptive Parameter Alternating Direction Algorithm for Centrosymmetric Solutions of a Class of Generalized Coupled Sylvester-transpose Matrix Equations

Adaptive Parameter Alternating Direction Algorithm for Centrosymmetric Solutions of a Class of Generalized Coupled Sylvester-transpose Matrix Equations

Conjugate Gradient Algorithm for Least-Squares Solutions of a Generalized Sylvester-Transpose Matrix Equation

We derive a conjugate-gradient type algorithm to produce approximate least-squares (LS) solutions for an inconsistent generalized Sylvester-transpose matrix equation. The algorithm is always applicable for any given initial matrix and will arrive at an LS solution within finite steps. When the matrix equation has many LS solutions, the algorithm can search for the one with minimal Frobenius-norm. Moreover, given a matrix Y, the algorithm can find a unique LS solution closest to Y. Numerical experiments show the relevance of the algorithm for square/non-square dense/sparse matrices of medium/large sizes. The algorithm works well in both the number of iterations and the computation time, compared to the direct Kronecker linearization and well-known iterative methods.

On the relaxed gradient-based iterative methods for the generalized coupled Sylvester-transpose matrix equations

In this paper, we propose the full-rank and reduced-rank relaxed gradient-based iterative algorithms for solving the generalized coupled Sylvester-transpose matrix equations. We provide analytically the necessary and sufficient condition for the convergence of the proposed iterative algorithm and give explicitly the optimal step size such that the convergence rate of the algorithm is maximized. Some numerical examples are examined to confirm the feasibility and efficiency of the proposed algorithms.

Solving the Sylvester-Transpose Matrix Equation under the Semi-Tensor Product

This paper investigates the Sylvester-transpose matrix equation A⋉X+XT⋉B=C, where all mentioned matrices are over an arbitrary field. Here, ⋉ is the semi-tensor product, which is a generalization of the usual matrix product defined for matrices of arbitrary dimensions. For matrices of compatible dimensions, we investigate criteria for the equation to have a solution, a unique solution, or infinitely many solutions. These conditions rely on ranks and linear dependence. Moreover, we find suitable matrix partitions so that the matrix equation can be transformed into a linear system involving the usual matrix product. Our work includes the studies of the equation A⋉X=C, the equation X⋉B=C, and the classical Sylvester-transpose matrix equation.

Conjugate gradient algorithm for consistent generalized Sylvester-transpose matrix equations

<abstract><p>We develop an effective algorithm to find a well-approximate solution of a generalized Sylvester-transpose matrix equation where all coefficient matrices and an unknown matrix are rectangular. The algorithm aims to construct a finite sequence of approximated solutions from any given initial matrix. It turns out that the associated residual matrices are orthogonal, and thus, the desire solution comes out in the final step with a satisfactory error. We provide numerical experiments to show the capability and performance of the algorithm.</p></abstract>

Approximated least-squares solutions of a generalized Sylvester-transpose matrix equation via gradient-descent iterative algorithm

This paper proposes an effective gradient-descent iterative algorithm for solving a generalized Sylvester-transpose equation with rectangular matrix coefficients. The algorithm is applicable for the equation and its interesting special cases when the associated matrix has full column-rank. The main idea of the algorithm is to have a minimum error at each iteration. The algorithm produces a sequence of approximated solutions converging to either the unique solution, or the unique least-squares solution when the problem has no solution. The convergence analysis points out that the algorithm converges fast for a small condition number of the associated matrix. Numerical examples demonstrate the efficiency and effectiveness of the algorithm compared to renowned and recent iterative methods.

Convergence analysis of a gradient iterative algorithm with optimal convergence factor for a generalized Sylvester-transpose matrix equation

<abstract> Consider a generalized Sylvester-transpose matrix equation with rectangular coefficient matrices. Based on gradients and hierarchical identification principle, we derive an iterative algorithm to produce a sequence of approximated solutions with a reasonable stopping rule concerning a relative norm-error. A convergence analysis via Banach fixed-point theorem reveals the sequence converges to a unique solution of the matrix equation for any given initial matrix if and only if the convergence factor is chosen appropriately in a certain range. The performance of algorithm is theoretically analysed through the convergence rate and error estimations. The optimal convergence factor is chosen to attain the fastest asymptotic behaviour. Finally, numerical experiments are provided to illustrate the capability and efficiency of the proposed algorithm, compared to recent gradient-based iterative algorithms. </abstract>

FINITE ITERATIVE (R, S)-CONJUGATE SOLUTIONS OF THE GENERALIZED COMPLEX COUPLED SYLVESTER-TRANSPOSE EQUATIONS

The iterative method of generalized complex coupled Sylvester-transpose equations $ AXB+CY^TD = E,\; MX^TN+GYH = F $ over (R, S)-conjugate matrix solution $ (X,Y) $ is proposed. Usually, the type of matrix arises from some physical problems with some form of generalized reflexive symmetry. On the condition that the coupled matrix equations are consistent, we show the solution pair $ (X^*, Y^*) $ can be obtained by generalization of CG iterative method within finite iterative steps in the absence of roundoff-error for any initial guess chosen by the (R, S)-conjugate matrix. Moreover, the optimal approximation (R, S)-conjugate matrix solutions can be derived by searching the least Frobenius norm solution of the novel generalized complex coupled Sylvester-transpose matrix equations. Finally, some numerical examples are given to illustrate the presented iterative algorithm is efficient.

AN ITERATIVE ALGORITHM FOR SOLVING A CLASS OF GENERALIZED COUPLED SYLVESTER-TRANSPOSE MATRIX EQUATIONS OVER BISYMMETRIC OR SKEW-ANTI-SYMMETRIC MATRICES

This paper presents an iterative algorithm to solve a class of generalized coupled Sylvester-transpose matrix equations over bisymmetric or skew-anti-symmetric matrices. When the matrix equations are consistent, the bisymmetric or skew-anti-symmetric solutions can be obtained within finite iteration steps in the absence of round-off errors for any initial bisymmetric or skew-anti-symmetric matrices by the proposed iterative algorithm. In addition, we can obtain the least norm solution by choosing the special initial matrices. Finally, numerical examples are given to demonstrate the iterative algorithm is quite efficient. The merit of our method is that it is easy to implement.

The least squares solution of a class of generalized Sylvester-transpose matrix equations with the norm inequality constraint

In this paper, we present an iterative method for finding the least squares solution of a class of generalized Sylvester-transpose matrix equations with the norm inequality constraint. We prove that if the constrained matrix equations are consistent, the solution can be obtained within finite iterative steps in the absence of round-off errors; if constrained matrix equations are inconsistent, the least squares solution can be obtained within finite iterative steps in the absence of round-off errors. Finally, numerical examples are provided to illustrate the efficiency of the proposed method and testify the conclusions suggested in this paper.

An iterative algorithm for the least Frobenius norm least squares solution of a class of generalized coupled Sylvester-transpose linear matrix equations

The iterative algorithm of a class of generalized coupled Sylvester-transpose matrix equations is presented. We prove that if the system is consistent, a solution can be obtained within finite iterative steps in the absence of round-off errors for any initial matrices; if the system is inconsistent, the least squares solution can be obtained within finite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrices to obtain the least Frobenius norm least squares solution of the problem. Finally, numerical examples are presented to demonstrate that the algorithm is efficient.

Biconjugate residual algorithm for solving general Sylvester-transpose matrix equations

The present paper is concerned with the solution of the coupled generalized Sylvester-transpose matrix equations {A1XB1 + C1XD1 + E1XTF1 = M1, A2XB2 + C2XD2 + E2XTF2 = M2, including the well-known Lyapunov and Sylvester matrix equations. Based on a variant of biconjugate residual (BCR) algorithm, we construct and analyze an efficient algorithm to find the (least Frobenius norm) solution of the general Sylvester-transpose matrix equations within a finite number of iterations in the absence of round-off errors. Two numerical examples are given to examine the performance of the constructed algorithm.

A matrix CRS iterative method for solving a class of coupled Sylvester-transpose matrix equations

In this paper, we apply Kronecker product and vectorization operator to extend the conjugate residual squared (CRS) method for solving a class of coupled Sylvester-transpose matrix equations. Some numerical examples are given to compare the accuracy and efficiency of the new matrix iterative method with other methods presented in the literature. Numerical results validate that the proposed method can be much more efficient than some existing methods.

A cyclic iterative approach and its modified version to solve coupled Sylvester-transpose matrix equations

Recently, Tang et al. [Numer Algorithms. 2014;66(2):379–397] have offered a cyclic iterative method for determining the unique solution of the coupled matrix equationsAnalogues to the gradient-based algorithm, the proposed algorithm relies on a fixed parameter whereas it has wider convergence region. Nevertheless, the application of the algorithm to find the centro-symmetric solution of the mentioned problem has been left as a project to be investigated and the optimal value for the fixed parameter has not been derived. In this paper, we focus on a more general class of the coupled linear matrix equations that incorporate the mentioned ones in the earlier refereed work. More precisely, we first develop the authors’ propounded algorithm to resolve our considered coupled linear matrix equations over centro-symmetric matrices. Afterwards, we disregard the restriction of the existence of the unique (centro-symmetric) solution and also modify the authors’ algorithm by applying an oblique projection technique which allows to produce a sequence of approximate solutions which gratify an optimality property. Numerical results are reported to confirm the validity of the established results and to demonstrate the superior performance of the modified version of the cyclic iterative algorithm.

Corrigendum: An iterative algorithm to solve the generalized coupled Sylvester-transpose matrix equations

Owing to errors made by the authors, Caiqin Song and Jun-e Feng, the article is incorrect: Song C and Feng J An iterative algorithm to solve the generalized coupled Sylvester-transpose matrix equations. Transactions of the Institute of Measurement and Control 38(7): 863–875, doi: 10.1177/0142331215588808 .

The accelerated gradient based iterative algorithm for solving a class of generalized Sylvester-transpose matrix equation

In this paper, we present an accelerated gradient based algorithm by minimizing certain criterion quadratic function for solving the generalized Sylvester-transpose matrix equation AXB+CXTD=F. The idea is from (Ding and Chen, 2005; Niu et al., 2011; Wang et al., 2012) in which some efficient algorithms were developed for solving the Sylvester matrix equation and the Lyapunov matrix equation. On the basis of the information generated in the previous half-step, we further introduce a relaxation factor to obtain the solution of the generalized Sylvester-transpose matrix equation. We show that the iterative solution converges to the exact solution for any initial value provided that some appropriate assumptions. Finally, some numerical examples are given to illustrate that the introduced iterative algorithm is efficient.

An iterative algorithm to solve the generalized coupled Sylvester-transpose matrix equations

This note presents an iterative algorithm to solve the coupled Sylvester-transpose matrix equations (including the generalized coupled Sylvester matrix equations and Lyapunov matrix equations as special cases) over generalized centro-symmetric matrices. When the considered matrix equations are consistent, for any initial generalized centro-symmetric matrix group, a generalized centro-symmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm generalized centro-symmetric solution group of the coupled Sylvester-transpose matrix equations can be derived when a suitable initial generalized centro-symmetric matrix group is chosen. In addition, for a given generalized centro-symmetric matrix group, the optimal approximation generalized centro-symmetric solution group can be obtained by finding the least Frobenius norm generalized centro-symmetric solution group of new coupled Sylvester-transpose matrix equations. Finally, a numerical example is given to demonstrate the efficiency of the introduced iterative algorithm.

Extending the CGLS algorithm for least squares solutions of the generalized Sylvester-transpose matrix equations

This paper deals with the solution to the least squares problemminX‖∑i=1sAiXBi+∑j=1tCjXTDj−E‖,corresponding to the generalized Sylvester-transpose matrix equation. The conjugate gradient least squares (CGLS) method is extended to obtain a matrix algorithm for solving this problem. We show that the matrix algorithm can solve this problem within a finite number of iterations in the absence of roundoff errors. Also the descent property of the norm of residuals is obtained. Finally numerical results demonstrate the accuracy and robustness of the algorithm.

Extending the GPBiCG algorithm for solving the generalized Sylvester-transpose matrix equation

By applying Kronecker product and vectorization operator, we extend the generalized product bi-conjugate gradient (GPBiCG) algorithm for solving the generalized Sylvester-transpose matrix equation $\sum\nolimits_{i = 1}^r {(A_i XB_i + C_i X^T D_i ) = E} $ . By using numerical results, we compare the new method with other popular iterative solvers in use today.