A note on combined generalized Sylvester matrix equations

The solution of two combined generalized Sylvester matrix equations is studied. It is first shown that the two combined generalized Sylvester matrix equations can be converted into a normal Sylvester matrix equation through extension, and then with the help of a result for solution to normal Sylvester matrix equations, the complete solution to the two combined generalized Sylvester matrix equations is derived. A demonstrative example shows the effect of the proposed approach.     

Efficient iterative solutions to general coupled matrix equations

Linear matrix equations are encountered in many systems and control applications.In this paper,we consider the general coupled matrix equations（including the generalized coupled Sylvester matrix equations as a special case）l t=1EstYtFst = Gs,s = 1,2,···,l over the generalized reflexive matrix group（Y1,Y2,···,Yl）.We derive an efcient gradient-iterative（GI） algorithm for fnding the generalized reflexive solution group of the general coupled matrix equations.Convergence analysis indicates that the algorithm always converges to the generalized reflexive solution group for any initial generalized reflexive matrix group（Y1（1）,Y2（1）,···,Yl（1））.Finally,numerical results are presented to test and illustrate the performance of the algorithm in terms of convergence,accuracy as well as the efciency.     

On global Hessenberg based methods for solving Sylvester matrix equations

In the first part of this paper, we investigate the use of Hessenberg-based methods for solving the Sylvester matrix equation $AX+XB=C$. To achieve this goal, the Sylvester form of the global generalized Hessenberg process is presented. Using this process, different methods based on a Petrov–Galerkin or on a minimal norm condition are derived. In the second part, we focus on the SGl-CMRH method which is based on the Sylvester form of the Hessenberg process with pivoting strategy combined with a minimal norm condition. In order to accelerate the SGl-CMRH method, a preconditioned framework of this method is also considered. It includes both fixed and flexible variants of the SGl-CMRH method. Moreover, the connection between the flexible preconditioned SGl-CMRH method and the fixed one is studied and some upper bounds for the residual norm are obtained. In particular, application of the obtained theoretical results is investigated for the special case of solving linear systems of equations with several right-hand sides. Finally, some numerical experiments are given in order to evaluate the effectiveness of the proposed methods.     

On the generalized Sylvester mapping and matrix equations

General parametric solution to a family of generalized Sylvester matrix equations arising in linear system theory is presented by using the so-called generalized Sylvester mapping which has some elegant properties. The solution consists of some polynomial matrices satisfying certain conditions and a parametric matrix representing the degree of freedom in the solution. The results provide great convenience to the computation and analysis of the solutions to this family of equations, and can perform important functions in many analysis and design problems in linear system theory. It is also expected that this so-called generalized Sylvester mapping tool may have some other applications in control system theory.     

A new solution to the generalized Sylvester matrix equation

This note deals with the problem of solving the generalized Sylvester matrix equation AV-EVF=BW, with F being an arbitrary matrix, and provides complete general parametric expressions for the matrices V and W satisfying this equation. The primary feature of this solution is that the matrix F does not need to be in any canonical form, and may be even unknown a priori. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in control systems theory.     

Solutions to the generalized Sylvester matrix equations by a singular value decomposition

In this paper,solutions to the generalized Sylvester matrix equations AX-XF=BY and MXN-X=TY with A,M∈Rn×n,B,T∈Rn×n,F,N∈Rp×p and the matrices N,F being in companion form,are established by a singular value decomposition of a matrix with dimensions n×(n pr).The algorithm proposed in this paper for the euqation AX-XF=BY does not require the controllability of matrix pair(A,B)andthe restriction that A,F do not have common eigenvalues.Since singular value decomposition is adopted,the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations,and can perform important functions in many design problems in control systems theory.     

Gradient-based maximal convergence rate iterative method for solving linear matrix equations

This paper is concerned with numerical solutions to general linear matrix equations including the well-known Lyapunov matrix equation and Sylvester matrix equation as special cases. Gradient based iterative algorithm is proposed to approximate the exact solution. A necessary and sufficient condition guaranteeing the convergence of the algorithm is presented. A sufficient condition that is easy to compute is also given. The optimal convergence factor such that the convergence rate of the algorithm is maximized is established. The proposed approach not only gives a complete understanding on gradient based iterative algorithm for solving linear matrix equations, but can also be served as a bridge between linear system theory and numerical computing. Numerical example shows the effectiveness of the proposed approach.     

Gradient-based iterative algorithms for generalized coupled Sylvester-conjugate matrix equations

By applying the hierarchical identification principle, the gradient-based iterative algorithm is suggested to solve a class of complex matrix equations. With the real representation of a complex matrix as a tool, the sufficient and necessary conditions for the convergence factor are determined to guarantee that the iterative solutions given by the proposed algorithm converge to the exact solution for any initial matrices. Also, we solve the problem which is proposed by Wu et al. (2010). Finally, some numerical examples are provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper.     

An efficient method based on the second kind Chebyshev wavelets for solving variable-order fractional convection diffusion equations

In this paper, a class of variable-order fractional convection diffusion equations have been solved with assistance of the second kind Chebyshev wavelets operational matrix. The operational matrix of variable-order fractional derivative is derived for the second kind Chebyshev wavelets. By implementing the second kind Chebyshev wavelets functions and also the associated operational matrix, the considered equations will be reduced to the corresponding Sylvester equation, which can be solved by some appropriate iterative solvers. Also, the convergence analysis of the proposed numerical method to the exact solutions and error estimation are given. A variety of numerical examples are considered to show the efficiency and accuracy of the presented technique.     

Application of Bernoulli matrix method for solving two-dimensional hyperbolic telegraph equations with Dirichlet boundary conditions

The present article is devoted to develop a new approach and methodology to find the approximate solution of second order two-dimensional telegraph equations with the Dirichlet boundary conditions. We first transform the telegraph equations into equivalent partial integro-differential equations (PIDEs) which contain both initial and boundary conditions and therefore can be solved numerically in a more appropriate manner. Operational matrices of integration and differentiation of Bernoulli polynomials together with the completeness of these polynomials are used to reduce the PIDEs into the associated algebraic generalized Sylvester equations which can be solved by an efficient Krylov subspace iterative (i.e., BICGSTAB) method. The efficiency of the proposed method has been confirmed with several test examples and it is clear that the results are acceptable and found to be in good agreement with exact solutions. We have compared the numerical results of the proposed method with radial basis function method and differential quadrature method. Also, the method is simple, efficient and produces very accurate numerical results in considerably small number of basis functions and hence reduces computational effort. Moreover, the technique is easy to apply for multidimensional problems.     

Periodic conjugate direction algorithm for symmetric periodic solutions of general coupled periodic matrix equations

Analysis and design of linear periodic control systems are closely related to the periodic matrix equations. The conjugate direction (CD) method is a famous iterative algorithm to find the solution to nonsymmetric linear systems $Ax=b$. In this work, a new method based on the CD method is proposed for computing the symmetric periodic solutions $(X_1,X_2,\dots ,X_{\lambda })$ and $(Y_1,Y_2,\dots ,Y_{\lambda })$ of general coupled periodic matrix equations

for $i=1,2,\dots ,\lambda$. The key idea of the scheme is to extend the CD method by means of Kronecker product and vectorization operator. In order to assess the convergence properties of the method, some theoretical results are given. Finally two numerical examples are included to illustrate the efficiency and effectiveness of the method.     

Convergence of bpsd method for T (q,r) matrix

In this paper, we obtain a necessary and sufficient condition, when the coefficient matrix A of the equation Ax = f considered is a T(l, 1) matrix, a sufficient condition, when A is a T(l, 2) or T(2, 1) matrix for the convergence of BPSD method. We also obtain the optimum parameters and the optimum rate of convergence of BPSD method, when A is T(l, 1) matrix and a necessary and sufficient condition, when A is positive definite and we point out that the necessary and sufficient condition in [1] and [9] is only sufficient.     

Block constrained versus generalized Jacobi preconditioners for iterative solution of large-scale Biot’s FEM equations

Generalized Jacobi (GJ) diagonal preconditioner coupled with symmetric quasi-minimal residual (SQMR) method has been demonstrated to be efficient for solving the 2 × 2 block linear system of equations arising from discretized Biot’s consolidation equations. However, one may further improve the performance by employing a more sophisticated non-diagonal preconditioner. This paper proposes to employ a block constrained preconditioner P_c that uses the same 2 × 2 block matrix but its (1, 1) block is replaced by a diagonal approximation. Numerical results on a series of 3-D footing problems show that the SQMR method preconditioned by P_c is about 55% more efficient time-wise than the counterpart preconditioned by GJ when the problem size increases to about 180,000 degrees of freedom. Over the range of problem sizes studied, the P_c-preconditioned SQMR method incurs about 20% more memory than the GJ-preconditioned counterpart. The paper also addresses crucial computational and storage issues in constructing and storing P_c efficiently to achieve superior performance over GJ on the commonly available PC platforms.     

A modified hybrid projection method for solving generalized mixed equilibrium problems and fixed point problems in Banach spaces

In this paper, we introduce a modified new hybrid projection method for finding the set of solutions of the generalized mixed equilibrium problems and the convex feasibility problems for an infinite family of closed and uniformly quasi-?-asymptotically nonexpansive mappings. Strong convergence theorems are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Our results improve and extend the corresponding results announced by Qin et al. (2010) and many authors.