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1.
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.  相似文献

2.
An iterative algorithm is constructed to give a common solution to a group of complex matrix equations. By using the proposed algorithm, the existence of a common solution can be determined automatically. When a common solution exists for this group of matrix equations, it is proven by using a real inner product in complex matrix spaces as a tool that a solution can be obtained within finite iteration steps for any initial values in the absence of round-off errors. The algorithm is also generalized to solve a more general case. A numerical example is given to illustrate the effectiveness of the proposed method.  相似文献

3.
An iterative method is proposed to solve generalized coupled Sylvester matrix equations, based on a matrix form of the least-squares QR-factorization (LSQR) algorithm. By this iterative method on the selection of special initial matrices, we can obtain the minimum Frobenius norm solutions or the minimum Frobenius norm least-squares solutions over some constrained matrices, such as symmetric, generalized bisymmetric and (RS)-symmetric matrices. Meanwhile, the optimal approximate solutions to the given matrices can be derived by solving the corresponding new generalized coupled Sylvester matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the present method.  相似文献

4.
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.  相似文献

5.
Two iterative algorithms are presented in this paper to solve the minimal norm least squares solution to a general linear matrix equations including the well-known Sylvester matrix equation and Lyapunov matrix equation as special cases. The first algorithm is based on the gradient based searching principle and the other one can be viewed as its dual form. Necessary and sufficient conditions for the step sizes in these two algorithms are proposed to guarantee the convergence of the algorithms for arbitrary initial conditions. Sufficient condition that is easy to compute is also given. Moreover, two methods are proposed to choose the optimal step sizes such that the convergence speeds of the algorithms are maximized. Between these two methods, the first one is to minimize the spectral radius of the iteration matrix and explicit expression for the optimal step size is obtained. The second method is to minimize the square sum of the F-norm of the error matrices produced by the algorithm and it is shown that the optimal step size exits uniquely and lies in an interval. Several numerical examples are given to illustrate the efficiency of the proposed approach.  相似文献

6.
A non-interior point algorithm based on projection for second-order cone programming problems is proposed and analyzed. The main idea of the algorithm is that we cast the complementary equation in the primal-dual optimality conditions as a projection equation. By using this reformulation, we only need to solve a system of linear equations with the same coefficient matrix and compute two simple projections at each iteration, without performing any line search. This algorithm can start from an arbitrary point, and does not require the row vectors of A to be linearly independent. We prove that our algorithm is globally convergent under weak conditions. Preliminary numerical results demonstrate the effectiveness of our algorithm.  相似文献

7.
This paper is concerned with iterative solutions to a class of complex matrix equations. By applying the hierarchical identification principle, an iterative algorithm is constructed to solve this class of complex matrix equations. The range of the convergence factor is given to guarantee that the proposed algorithm is convergent for arbitrary initial matrix by applying a real representation of a complex matrix as a tool. By using some properties of the real representation, a sufficient convergence condition that is easier to compute is also given by original coefficient matrices. Two numerical examples are given to illustrate the effectiveness of the proposed methods.  相似文献

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9.
This paper is concerned with iterative solutions to a class of complex matrix equations, which include some previously investigated matrix equations as special cases. By applying the hierarchical identification principle, an iterative algorithm is constructed to solve this class of matrix equations. A sufficient condition is presented to guarantee that the proposed algorithm is convergent for an arbitrary initial matrix with a real representation of a complex matrix as tools. By using some properties of the real representation, a convergence condition that is easier to compute is also given in terms of original coefficient matrices. A numerical example is employed to illustrate the effectiveness of the proposed methods.  相似文献

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This paper is concerned with weighted least squares solutions to general coupled Sylvester matrix equations. Gradient based iterative algorithms are proposed to solve this problem. This type of iterative algorithm includes a wide class of iterative algorithms, and two special cases of them are studied in detail in this paper. Necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented. Sufficient conditions that are easy to compute are also given. The optimal step sizes such that the convergence rates of the algorithms, which are properly defined in this paper, are maximized and established. Several special cases of the weighted least squares problem, such as a least squares solution to the coupled Sylvester matrix equations problem, solutions to the general coupled Sylvester matrix equations problem, and a weighted least squares solution to the linear matrix equation problem are simultaneously solved. Several numerical examples are given to illustrate the effectiveness of the proposed algorithms.  相似文献

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The need to estimate a positive definite solution to an overdetermined linear system of equations with multiple right hand side vectors arises in several process control contexts. The coefficient and the right hand side matrices are respectively named data and target matrices. A number of optimization methods were proposed for solving such problems, in which the data matrix is unrealistically assumed to be error free. Here, considering error in measured data and target matrices, we present an approach to solve a positive definite constrained linear system of equations based on the use of a newly defined error function. To minimize the defined error function, we derive necessary and sufficient optimality conditions and outline a direct algorithm to compute the solution. We provide a comparison of our proposed approach and two existing methods, the interior point method and a method based on quadratic programming. Two important characteristics of our proposed method as compared to the existing methods are computing the solution directly and considering error both in data and target matrices. Moreover, numerical test results show that the new approach leads to smaller standard deviations of error entries and smaller effective rank as desired by control problems. Furthermore, in a comparative study, using the Dolan-Moré performance profiles, we show the approach to be more efficient.  相似文献

14.
The iterative method of the generalized coupled Sylvester-conjugate matrix equations $$\sum\limits _{j=1}^{l}\left (A_{ij}X_{j}B_{ij}+C_{ij}\overline {X}_{j}D_{ij}\right )=E_{i} (i=1,2,\cdots ,s)$$ over Hermitian and generalized skew Hamiltonian solution is presented. When these systems of matrix equations are consistent, for arbitrary initial Hermitian and generalized skew Hamiltonian matrices X j (1), j = 1,2,? , l, the exact solutions can be obtained by iterative algorithm 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 Hermitian and generalized skew Hamiltonian solution of the problem. Finally, numerical examples are presented to demonstrate the proposed algorithm is efficient.  相似文献

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In this paper, an algorithm for computing a generalized eigenmode of reducible regular matrices over the max-plus algebra is applied to the Metro-bus public transport system in Mexico city. A timed event Petri net model is constructed from the data table that characterizes the transport system. A max-plus recurrence equation, with a reducible and regular matrix, is associated with the transport system timed event Petri net. Next, given the reducible and regular matrix, the problem consists of giving an algorithm which will tell us how to compute its generalized eigenmode over the max plus algebra. The solution to the problem is achieved by studying some type of recurrence equations. In fact, by transforming the reducible regular matrix into its normal form, and considering a very specific recurrence equation, an explicit mathematical characterization is obtained, upon which the algorithm is constructed. The generalized eigenmode obtained sets a timetable for the transport system.  相似文献

17.
Newton’s method for unconstrained optimization problems on the Euclidean space can be generalized to that on Riemannian manifolds. The truncated singular value problem is one particular problem defined on the product of two Stiefel manifolds, and an algorithm of the Riemannian Newton’s method for this problem has been designed. However, this algorithm is not easy to implement in its original form because the Newton equation is expressed by a system of matrix equations which is difficult to solve directly. In the present paper, we propose an effective implementation of the Newton algorithm. A matrix-free Krylov subspace method is used to solve a symmetric linear system into which the Newton equation is rewritten. The presented approach can be used on other problems as well. Numerical experiments demonstrate that the proposed method is effective for the above optimization problem.  相似文献

18.
Recently, Ding and Chen [F. Ding, T. Chen, On iterative solutions of general coupled matrix equations, SIAM J. Control Optim. 44 (2006) 2269-2284] developed a gradient-based iterative method for solving a class of coupled Sylvester matrix equations. The basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified, so that the iterative solutions are obtained by applying hierarchical identification principle. In this note, by considering the coupled Sylvester matrix equation as a linear operator equation we give a natural way to derive this algorithm. We also propose some faster algorithms and present some numerical results.  相似文献

19.
In the course of the numerical approximation of mathematical models there is often a need to solve a system of linear equations with a tridiagonal or a block-tridiagonal matrices. Usually it is efficient to solve these systems using a special algorithm (tridiagonal matrix algorithm or TDMA) which takes advantage of the structure. The main result of this work is to formulate a sufficient condition for the numerical method to preserve the non-negativity for the special algorithm for structured meshes. We show that a different condition can be obtained for such cases where there is no way to fulfill this condition. Moreover, as an example, the numerical solution of the two-dimensional heat conduction equation on a rectangular domain is investigated by applying Dirichlet boundary condition and Neumann boundary condition on different parts of the boundary of the domain. For space discretization, we apply the linear finite element method, and for time discretization, the well-known Θ-method. The theoretical results of the paper are verified by several numerical experiments.  相似文献

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