Search

Abstract In this study, based on the MHSS (Modified Hermitian and skew-Hermitian splitting) method, we will present a generalized MHSS approach for solving large sparse Sylvester equation with non-Hermitian and complex symmetric positive definite/semi-definite matrices. The new method (GMHSS) is a four-parameter iteration procedure where the iterative sequence is unconditionally convergent to the unique solution of the Sylvester equation. Then to improve the GMHSS method, the inexact version of the GMHSS iterative method (IGMHSS) will be described and will be analyzed. Also, by using a new idea, we try to minimize the upper bound of the spectral radius of iteration matrix. Two test problems are given to illustrate the efficiency of the new approach. [ABSTRACT FROM AUTHOR]

Recent years have seen a renewal of interest in multi-term linear matrix equations, as these have come to play a role in a number of important applications. Here, we consider the solution of such equations by means of the dynamical functional particle method, an iterative technique that relies on the numerical integration of a damped second order dynamical system. We develop a new algorithm for the solution of a large class of these equations, a class that includes, among others, all linear matrix equations with Hermitian positive definite or negative definite coefficients. In numerical experiments, our MATLAB implementation outperforms existing methods for the solution of multi-term Sylvester equations. For the Sylvester equation A X + X B = C , in particular, it can be faster and more accurate than the built-in implementation of the Bartels–Stewart algorithm, when A and B are well conditioned and have very different size. [ABSTRACT FROM AUTHOR]

In this paper, a modified conjugate gradient iterative method for solving Sylvester equation is presented. By using this iterative method, the least squares anti-bisymmetric solution and the optimal approximation solution can be obtained. Here we present the derivation and theoretical analysis of our iterative method. Numerical results illustrate the feasibility and effectiveness of the proposed iterative method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a wavelet operational method based on Haar wavelet is proposed to solve the fractional partial differential equations in the Caputo derivative sense. We give the Haar wavelet operational matrix of fractional order integration. A truncated Haar wavelet series together with the wavelet operational matrix are utilized to reduce the fractional partial differential equations to Sylvester equations. In addition, some examples are presented to show the efficiency and the accuracy of the approach. [Copyright &y& Elsevier]

Abstract: In this paper, we first introduce two dimensional block pulse functions and the block pulse operational matrices of the fractional order integration. Also the block pulse operational matrices of the fractional order differentiation are obtained. Then we present a computational method based on the above results for solving a class of fractional partial differential equations. Transforming the initial equation into a Sylvester equation. The error analysis of the method is given. The method is computationally attractive and applications are demonstrated by some numerical examples. Moreover, comparing the methodology with the known technique shows that our approach is more efficient and more convenient. [Copyright &y& Elsevier]

Abstract: We show that when a symmetric Sinc-Galerkin method is used to solve a Poisson problem, the resulting Sylvester matrix equation is a discrete ADI model problem. We employ a new alternating direction scheme known as the alternating-direction Sinc-Galerkin (ADSG) method on illustrative partial differential equation boundary-value problems to document the exponential convergence rate that can be achieved. Unlike classical ADI schemes, direct numerical application of ADSG avoids the computation of iteration parameters, matrix eigenvalues and eigenvectors, as well as the use of the Kronecker product and sum. [Copyright &y& Elsevier]

Abstract: In this paper we propose minimal residual methods augmented with eigenvectors for solving large Sylvester matrix equations AX + XB = C and generalized Sylvester matrix equations AXB + X = C. The subspace, from which the approximate solution is extracted, is the Kronecker product subspace of two augmented Krylov subspaces. Numerical experiments report the effectiveness of these methods. [Copyright &y& Elsevier]

Abstract: In the present paper, we propose an Arnoldi-based method for solving large and sparse Sylvester matrix equations with low rank right hand sides. We will show how to extract low-rank approximations via a matrix Krylov subspace method. We give some theoretical results such an expression of the exact solution and upper bounds for the norm of the error and for the residual. Numerical experiments will also be given to show the effectiveness of the proposed method. [Copyright &y& Elsevier]

Abstract: This paper presents a new method for obtaining a matrix M which is an approximate inverse preconditioner for a given matrix A, where the eigenvalues of A all either have negative real parts or all have positive real parts. This method is based on the approximate solution of the special Sylvester equation AX + XA =2I. We use a Krylov subspace method for obtaining an approximate solution of this Sylvester matrix equation which is based on the Arnoldi algorithm and on an integral formula. The computation of the preconditioner can be carried out in parallel and its implementation requires only the solution of very simple and small Sylvester equations. The sparsity of the preconditioner is preserved by using a proper dropping strategy. Some numerical experiments on test matrices from Harwell–Boing collection for comparing the numerical performance of the new method with an available well-known algorithm are presented. [Copyright &y& Elsevier]

Abstract: The global GMRES and global FOM algorithm are recently proposed by Jbilou et al. for solving the linear equations with multiple right-hand sides [K. Jbilou, A. Messaoudi, H. Sados, Global FOM and GMRES algorithms for matrix equations, Appl. Numer. Math. 31 (1999) 49–63]. Like GMRES for the linear equations, they generally uses restarting, which slows the convergence. However, some information can be retained at the time of the restart and used in the next cycle. We present algorithms that use implicit restarting in order to retain this information as Morgan have proposed recently [R.B. Morgan, Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations, SIAM J. Matrix anal. Appl. 21 (2000) 1112–1135]. At the same time, we prove that global GMRES and global FOM methods for matrix equations are equivalent with the corresponding methods for linear equations and propose implicitly restarted global FOM and GMRES. Numerical examples show that our methods are efficient. [Copyright &y& Elsevier]

• The relative errors are considered in numerical tests. • Numerical tests provided in Sections 5.1 and 5.2 show that for certain random matrices, the newly obtained upper bound (1.9) is much better than upper bound (5.2). • A comparison between the newly obtained lower bound (3.25) and lower bound (5.3) is added at the end of Example 5.2. This paper studies the Frobenius norm upper bounds and lower bounds of the unique solution to A X + X B = A C + D B , where A ∈ C m × m and B ∈ C n × n are Hermitian positive definite, and C , D ∈ C m × n are arbitrary. Some theoretical improvements of the known results are made. Numerical tests to illustrate the sharpness of the newly obtained upper bounds are dealt with, and numerical examples associated with the positivity of lower bounds are also provided. [ABSTRACT FROM AUTHOR]

The key purpose of this study is to present two schemes based on Lagrange polynomials to deal with the numerical solution of second order two-dimensional telegraph equation (TDTE) with the Dirichlet boundary conditions. First, we convert the main equation into partial integro-differential equations (PIDEs) with the help of initial and boundary conditions. The operational matrices of differentiation and integration are then used to transform the PIDEs into algebraic generalized Sylvester equation. We compared the results obtained by the proposed schemes with Bernoulli matrix method and B-spline differential quadrature method which shows that the proposed schemes are accurate for small number of basis functions. [ABSTRACT FROM AUTHOR]