Sylvester Equation AX Research Articles

Retracted: The MGHSS for Solving Continuous Sylvester Equation AX + XB = C

Retracted: The MGHSS for Solving Continuous Sylvester Equation AX + XB = C

On the generalized Sylvester operator equation AX−Y B = C

In this paper, we study the necessary and sufficient conditions to have a solution pair ( X , Y ) to the generalized Sylvester equation AX−Y B = C where A , B , C ∈ L ( H ) are given. Moreover, we give some connections the generalized Fuglede-Putnam property and the solution pair ( X , Y ) of the operator equation AX−Y B = C. Finally, we consider some properties of the operator equation AX−Y B = C when A and B belong to the special classes of operators.

Homogeneous Sylvester equations of lower triangular matrices

We study the homogeneous Sylvester equation AX=XB of lower triangular matrices with 1's along the diagonal. A necessary and sufficient condition for existence of solution is presented in terms of computable linear systems.

Numerical Solution of Sylvester Equation Based on Iterative Predictor‐Corrector Method

The inspiration of the study concerns an iterative predictor‐corrector method with order of convergence p = 45 for computing the inverse of the coefficient matrix Λ = (In ⊗ A) + (BT ⊗ Im), which is obtained by the Sylvester equation AX + XB = C. The numerical solutions of three examples by predictor‐corrector algorithm are given. The final numerical results also support the applicability, fast convergency, and high accuracy of the method for finding matrix inverses.

The Sylvester equation in Banach algebras

Let A be a unital complex semisimple Banach algebra, and MA denote its maximal ideal space. For a matrix M∈An×n, Mˆ denotes the matrix obtained by taking entry-wise Gelfand transforms. For a matrix M∈Cn×n, σ(M)⊂C denotes the set of eigenvalues of M. It is shown that if A∈An×n and B∈Am×m are such that for all φ∈MA, σ(Aˆ(φ))∩σ(Bˆ(φ))=∅, then for all C∈An×m, the Sylvester equation AX−XB=C has a unique solution X∈An×m. As an application, Roth's removal rule is proved in the context of matrices over a Banach algebra.

On inexact alternating direction implicit iteration for continuous Sylvester equations

AbstractIn this paper, we study the alternating direction implicit (ADI) iteration for solving the continuous Sylvester equation AX + XB = C, where the coefficient matrices A and B are assumed to be positive semi‐definite matrices (not necessarily Hermitian), and at least one of them to be positive definite. We first analyze the convergence of the ADI iteration for solving such a class of Sylvester equations, then derive an upper bound for the contraction factor of this ADI iteration. To reduce its computational complexity, we further propose an inexact variant of the ADI iteration, which employs some Krylov subspace methods as its inner iteration processes at each step of the outer ADI iteration. The convergence is also analyzed in detail. The numerical experiments are given to illustrate the effectiveness of both ADI and inexact ADI iterations.

On preconditioned normal and skew-Hermitian splitting iteration method for continuous Sylvester equations AX + XB = C*

In this paper, we present a preconditioned normal and skew-Hermitian splitting (PNSS) iteration method for continuous Sylvester equations AX + XB = C with positive definite/semi-definite matrices. Theoretical analysis shows that the PNSS methods will converge unconditionally to the exact solution of the continuous Sylvester equations. An inexact variant of the PNSS iteration method(IPNSS) and the analysis of its convergence property in detail have been established. Numerical experiments further show that this new method is more efficient and robust than the existing ones.

Preconditioned Positive-Definite and Skew-Hermitian Splitting Iteration Methods for Continuous Sylvester Equations AX + XB = C

AbstractIn this paper, we present a preconditioned positive-definite and skew-Hermitian splitting (PPSS) iteration method for continuous Sylvester equations AX + XB = C with positive definite/semi-definite matrices. The analysis shows that the PPSS iteration method will converge under certain assumptions. An inexact variant of the PPSS iteration method (IPPSS) has been presented and the analysis of its convergence property in detail has been discussed. Numerical results show that this new method is more efficient and robust than the existing ones.

A NEW VERSION OF THE SMITH METHOD FOR SOLVING SYLVESTER EQUATION AND DISCRETE-TIME SYLVESTER EQUATION

Recently, Xue etc. (37) discussed the Smith method for solving Sylvester equation AX + XB = C, where one of the matrices A and B is at least a nonsingular M-matrix and the other is an (singular or nonsingular) M-matrix. Furthermore, in order to nd the minimal non-negative solution of a certain class of non-symmetric algebraic Riccati equations, Gao and Bai (17) considered a doubling iteration scheme to inexactly solve the Sylvester equations. This paper discusses the iterative error of the standard Smith method used in (17) and presents the prior estimations of the accurate solution X for the Sylvester equation. Furthermore, we give a new version of the Smith method for solving discrete-time Sylvester equation or Stein equation AXB + X = C, while the new version of the Smith method can also be used to solve Sylvester equation AX + XB = C, where both A and B are positive denite. We also study the convergence rate of the new Smith method. At last, numerical examples are given to illustrate the eectiveness of our methods.

A Parallel Preconditioned Modified Conjugate Gradient Method for Large Sylvester Matrix Equation

Computational effort of solving large-scale Sylvester equationsAX+XB+F=Ois frequently hindered in dealing with many complex control problems. In this work, a parallel preconditioned algorithm for solving it is proposed based on combination of a parameter iterative preconditioned method and modified form of conjugate gradient (MCG) method. Furthermore, Schur’s inequality and modified conjugate gradient method are employed to overcome the involved difficulties such as determination of parameter and calculation of inverse matrix. Several numerical results finally show that high performance of proposed parallel algorithm is obtained both in convergent rate and in parallel efficiency.

Least Squares Based Iterative Algorithm for the Coupled Sylvester Matrix Equations

By analyzing the eigenvalues of the related matrices, the convergence analysis of the least squares based iteration is given for solving the coupled Sylvester equationsAX+YB=CandDX+YE=Fin this paper. The analysis shows that the optimal convergence factor of this iterative algorithm is 1. In addition, the proposed iterative algorithm can solve the generalized Sylvester equationAXB+CXD=F. The analysis demonstrates that if the matrix equation has a unique solution then the least squares based iterative solution converges to the exact solution for any initial values. A numerical example illustrates the effectiveness of the proposed algorithm.

On the ★-Sylvester equation AX ± X★B★ = C

We consider the solution of the ★-Sylvester equations AX±X★B★=C, for ★=T, H and A,B,∈Cn×n, and the related linear matrix equations AXB★±X★=C, AXB★±CX★D★=E and AX±X★A★=C. Solvability conditions and numerical methods are considered, in terms of the (generalized and periodic) Schur and QR decompositions. We emphasize the square cases where m=n but the rectangular cases will be considered.

Accurate solutions of M-matrix Sylvester equations

This paper is concerned with a relative perturbation theory and its entrywise relatively accurate numerical solutions of an M-matrix Sylvester equation AX + XB = C by which we mean both A and B have positive diagonal entries and nonpositive off-diagonal entries and $${P=I_m \otimes A+B^{\rm T} \otimes I_n}$$ is a nonsingular M-matrix, and C is entrywise nonnegative. It is proved that small relative perturbations to the entries of A, B, and C introduce small relative errors to the entries of the solution X. Thus the smaller entries of X do not suffer bigger relative errors than its larger entries, unlikely the existing perturbation theory for (general) Sylvester equations. We then discuss some minor but crucial implementation changes to three existing numerical methods so that they can be used to compute X as accurately as the input data deserve.

Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle

In this paper, by extending the well-known Jacobi and Gauss–Seidel iterations for Ax = b , we study iterative solutions of matrix equations AXB = F and generalized Sylvester matrix equations AXB + CXD = F (including the Sylvester equation AX + XB = F as a special case), and present a gradient based and a least-squares based iterative algorithms for the solution. It is proved that the iterative solution always converges to the exact solution for any initial values. The basic idea is to regard the unknown matrix X to be solved as the parameters of a system to be identified, and to obtain the iterative solutions by applying the hierarchical identification principle. Finally, we test the algorithms and show their effectiveness using a numerical example.

A new version of successive approximations method for solving Sylvester matrix equations

This paper presents a new version of the successive approximations method for solving Sylvester equations AX − XB = C, where A and B are symmetric negative and positive definite matrices, respectively. This method is based on the block GMRES-Sylvester method. We also discuss the convergence of the new method. Some numerical experiments for obtaining the numerical solution of Sylvester equations are given. Numerical experiments show that the solution of Sylvester equations can be obtained with high accuracy and the new algorithm is a robust technique for solving Sylvester equations.

Preconditioned Galerkin and minimal residual methods for solving Sylvester equations

This paper presents preconditioned Galerkin and minimal residual algorithms for the solution of Sylvester equations AX − XB = C. Given two good preconditioner matrices M and N for matrices A and B, respectively, we solve the Sylvester equations MAXN − MXBN = MCN. The algorithms use the Arnoldi process to generate orthonormal bases of certain Krylov subspaces and simultaneously reduce the order of Sylvester equations. Numerical experiments show that the solution of Sylvester equations can be obtained with high accuracy by using the preconditioned versions of Galerkin and minimal residual algorithms and this versions are more robust and more efficient than those without preconditioning.

Approximate inverse preconditioner by computing approximate solution of Sylvester equation

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 = 2 I . 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.

Existence of a low rank or ℋ︁‐matrix approximant to the solution of a Sylvester equation

AbstractWe consider the Sylvester equation AX−XB+C=0 where the matrix C∈ℂn×m is of low rank and the spectra of A∈ℂn×n and B∈ℂm×m are separated by a line. We prove that the singular values of the solution X decay exponentially, that means for any ε∈(0,1) there exists a matrix X̃ of rank k=O(log(1/ε)) such that ∥X−X̃∥2⩽ε∥X∥2. As a generalization we prove that if A,B,C are hierarchical matrices then the solution X can be approximated by the hierarchical matrix format described in Hackbusch (Computing 2000; 62: 89–108). The blockwise rank of the approximation is again proportional to log(1/ε). Copyright © 2004 John Wiley & Sons, Ltd.

Computing tall skinny solutions of AX− XB= C

We will concentrate on the numerical computation of so-called tall and skinny solutions X of the Sylvester equations AX− XB= C. By this we mean that A is an n× n matrix with cheaply applicable action (A is for example sparse), and B a k× k matrix, with k≪ n. This type of Sylvester equation plays an important role in the computation of invariant subspaces when block Rayleigh quotient or block Jacobi–Davidson methods are used. There exist essentially two types of subspace projection methods for such equations. One in which the tall skinny matrices are treated as rigid objects, and one in which they are treated as a basis of a subspace. In this paper we review the two different types, design new methods, and aim to identify which application should make use of which solution method.

Fast algorithms for the Sylvester equation AX - XBT = C

For given matrices A Fm m, B Fn n, and C Fm n over an arbitrary field F, the matrix equation AX XBT=C has a unique solution X Fm n if and only if A and B have disjoint spectra. We describe an algor...