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.     

Finite iterative algorithms for a common solution to a group of complex matrix equations

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.     

LSQR iterative method for generalized coupled Sylvester matrix equations

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 (R, S)-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.     

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.     

Least squares solution with the minimum-norm to general matrix equations via iteration

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.     

A globally convergent non-interior point algorithm with full Newton step for second-order cone programming

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.     

Iterative solutions to the extended Sylvester-conjugate matrix equations

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.     

循环矩阵及其在结构计算中的应用

本文推广了循环矩阵的概念,讨论了它的一般性质,并提出了一种解系数矩阵为循环矩阵或准循环矩阵的线性代数方程组(这种方程组在一大类常见的结构物的计算中出现)的方法,这种解法比通常解法计算量小而且节省存储,同时还允许应用快速富氏变换以增怏其计算速度。     

Iterative algorithms for solving a class of complex conjugate and transpose matrix equations

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.     

几乎各向同性的高维空间分数阶扩散方程的分块快速正则Hermite分裂预处理方法

一类空间分数阶扩散方程经过有限差分离散后所得到的离散线性方程组的系数矩阵是两个对角矩阵与Toeplitz型矩阵的乘积之和.在本文中,对于几乎各向同性的二维或三维空间分数阶扩散方程的离散线性方程组,采用预处理Krylov子空间迭代方法,我们利用其系数矩阵的特殊结构和具体性质构造了一类分块快速正则Hermite分裂预处理子.通过理论分析,我们证明了所对应的预处理矩阵的特征值大部分都聚集于1的附近.数值实验也表明,这类分块快速正则Hermite分裂预处理子可以明显地加快广义极小残量(GMRES)方法和稳定化的双共轭梯度(BiCGSTAB)方法等Krylov子空间迭代方法的收敛速度.     

Weighted least squares solutions to general coupled Sylvester matrix equations

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.     

基于角点处扭矢构造双三次Coons曲面的一种优化方法

本文针对矩形网格角点处的扭矢采用优化方法构造双三次Coons曲面,提出一种新的优化准则来确定角点处的扭矢.首先,通过变分原理,考虑曲面导矢的极小化问题转化的Euler-Lagrange偏微分方程,将该方程应用于每一个Coons块的角点上,引入一个新的极小化问题,其解是Euler-Lagrange偏微分方程的近似最优解.然后,建立一个具有块三对角系数矩阵的线性方程组来求解新的极小化问题.该系数矩阵可以表示为两个相同的形式特殊的矩阵的Kronnecker积,进而可以证明其非奇异性.最后,数值实验验证本文方法的稳定性和有效性.     

A new error in variables model for solving positive definite linear system using orthogonal matrix decompositions

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.     

An iterative algorithm for the least Frobenius norm Hermitian and generalized skew Hamiltonian solutions of the generalized coupled Sylvester-conjugate matrix equations

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.     

一类非线性矩阵方程对称解的双迭代算法

利用逆矩阵的Neumann级数形式,将在Schur插值问题中遇到的含未知矩阵二次项之逆的非线性矩阵方程转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求非线性矩阵方程的对称解的双迭代算法.双迭代算法仅要求非线性矩阵方程有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定.数值算例表明,双迭代算法是有效的.     

A generalized eigenmode algorithm for reducible regular matrices over the max-plus algebra with applications to the Metro-bus public transport system in Mexico city

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.     

A matrix-free implementation of Riemannian Newton’s method on the Stiefel manifold

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.     

A note on the iterative solutions of general coupled matrix equation

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.     

Some sufficient conditions for the non-negativity preservation property in the discrete heat conduction model

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.     

连续Sylvester矩阵方程求解的分裂迭代算法

有效求解连续的Sylvester矩阵方程对于科学和工程计算有着重要的应用价值,因此该文提出了一种可行的分裂迭代算法.该算法的核心思想是外迭代将连续Sylvester矩阵方程的系数矩阵分裂为对称矩阵和反对称矩阵,内迭代求解复对称矩阵方程.相较于传统的分裂算法,该文所提出的分裂迭代算法有效地避免了最优迭代参数的选取,并利用了复对称方程组高效求解的特点,进而提高了算法的易实现性、易操作性.此外,从理论层面进一步证明了该分裂迭代算法的收敛性.最后,通过数值算例表明分裂迭代算法具有良好的收敛性和鲁棒性,同时也证实了分裂迭代算法的收敛性很大程度依赖于内迭代格式的选取.