Sylvester-transpose Matrix Equations Research Articles

The modified conjugate gradient methods for solving a class of generalized coupled Sylvester-transpose matrix equations

In this paper, we consider the iteration solutions of generalized coupled Sylvester-transpose matrix equations: A1XB1+C1YTD1=F1, A2YB2+C2XTD2=F2. When the coupled matrix equations are consistent, we propose a modified conjugate gradient method to solve the equations and prove that a solution (X∗,Y∗) can be obtained within finite iterative steps in the absence of roundoff-error for any initial value. Furthermore, we show that the minimum-norm solution can be got by choosing a special kind of initial matrices. When the coupled matrix equations are inconsistent, we present another modified conjugate gradient method to find the least-squares solution with the minimum-norm. Finally, some numerical examples are given to show the behavior of the considered algorithms.

Developing Bi-CG and Bi-CR Methods to Solve Generalized Sylvester-transpose Matrix Equations

The bi-conjugate gradients (Bi-CG) and bi-conjugate residual (Bi-CR) methods are powerful tools for solving nonsymmetric linear systems Ax = b. By using Kronecker product and vectorization operator, this paper develops the Bi-CG and Bi-CR methods for the solution of the generalized Sylvester-transpose matrix equation \(\sum _{i = 1}^p({A_i}X{B_i} + {C_i}{X^{\rm{T}}}{D_i}) = E\) (including Lyapunov, Sylvester and Sylvester-transpose matrix equations as special cases). Numerical results validate that the proposed algorithms are much more efficient than some existing algorithms.

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.

Matrix iterative methods for solving the Sylvester-transpose and periodic Sylvester matrix equations

The problem of solving matrix equations has many applications in control and system theory. This paper is concerned with the iterative solutions of the Sylvester-transpose matrix equation∑i=1k(AiXBi+CiXTDi)=E,and the periodic Sylvester matrix equationA^jX^jB^j+C^jX^j+1D^j=E^jforj=1,2,…,λ.The basic idea is to develop the conjugate gradients squared (CGS) and bi-conjugate gradient stabilized (Bi-CGSTAB) methods for obtaining matrix iterative methods for solving the Sylvester-transpose and periodic Sylvester matrix equations. Numerical test results are given to compare matrix iterative methods with other well-known methods.

Gradient-based iterative algorithm for solving the generalized coupled Sylvester-transpose and conjugate matrix equations over reflexive (anti-reflexive) matrices

Linear matrix equations play an important role in many areas, such as control theory, system theory, stability theory and some other fields of pure and applied mathematics. In the present paper, we consider the generalized coupled Sylvester-transpose and conjugate matrix equations [Formula: see text] where [Formula: see text] is a group of unknown matrices and for [Formula: see text], [Formula: see text] in which [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are given matrices with suitable dimensions defined over complex number field. By using the hierarchical identification principle, an iterative algorithm is proposed for solving the above coupled linear matrix equations over the group of reflexive (anti-reflexive) matrices. Meanwhile, sufficient conditions are established which guarantee the convergence of the presented algorithm. Finally, some numerical examples are given to demonstrate the validity of our theoretical results and the efficiency of the algorithm for solving the mentioned coupled linear matrix equations.

The coupled Sylvester-transpose matrix equations over generalized centro-symmetric matrices

In this paper, we present an iterative algorithm for solving the following coupled Sylvester-transpose matrix equations over the generalized centro-symmetric matrix group (X 1, X 2, …, X q ). The solvability of the problem can be determined by the proposed algorithm, automatically. If the coupled Sylvester-transpose matrix equations are consistent over the generalized centro-symmetric matrices, then a generalized centro-symmetric solution group can be obtained within finite iterative steps for any initial generalized centro-symmetric matrix group in the exact arithmetic. Furthermore, it is shown that the least-norm generalized centro-symmetric solution group of the coupled Sylvester-transpose matrix equations can be computed by choosing an appropriate initial iterative matrix group. Moreover, the optimal approximate generalized centro-symmetric solution group to a given arbitrary matrix group (V 1, V 2, …, V q ) can be derived by finding the least-norm generalized centro-symmetric solution group of a new coupled Sylvester-transpose matrix equations. Finally, some numerical results are given to illustrate the validity and practicability of the theoretical results established in this work.

The generalized QMRCGSTAB algorithm for solving Sylvester-transpose matrix equations

In this work we generalize the Quasi-Minimal Residual Variant of the Bi-CGSTAB algorithm (QMRCGSTAB) to obtain a matrix iterative method for solving the Sylvester-transpose matrix equation ∑ i = 1 l ( A i X B i + C i X T D i ) = E . In order to compare the generalized algorithm with some existing methods, we present two numerical examples.

Iterative solutions to coupled Sylvester-transpose matrix equations

This note studies the iterative solutions to the coupled Sylvester-transpose matrix equation with a unique solution. By using the hierarchical identification principle, an iterative algorithm is presented for solving this class of coupled matrix equations. It is proved that the iterative solution consistently converges to the exact solution for any initial values. Meanwhile, sufficient conditions are derived to guarantee that the iterative solutions given by the proposed algorithm converge to the exact solution for any initial matrices. Finally, a numerical example is given to illustrate the efficiency of the proposed approach.