Krylov subspace methods for the generalized Sylvester equation

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Abstract

In the paper we propose Galerkin and minimal residual methods for iteratively solving generalized Sylvester equations of the form AXB  X = C. The algorithms use Krylov subspace for which orthogonal basis are generated by the Arnoldi process and reduce the storage space required by using the structure of the matrix. We give some convergence results and present numerical experiments for large problems to show that our methods are efficient.

Introduction

In this paper we present Krylov subspace methods for solving the generalized Sylvester matrix equationAXB-X=C,where ARN×N,BRM×M,andCRN×M are given matrices, and XRN×M is the solution matrix sought.

The matrix equation (1.1) plays an important role in control and communications theory; see [1] and the references therein. The analytical solution of the matrix equation (1.1) has been considered by many authors; see [3]. Direct methods for solving the matrix equation (1.1) such as those proposed in [3], [4] are attractive if the matrices are of small size. These methods are based on the Schur decomposition, by which the original equation is transformed into a form that is easily solved by a forward substitution.

Let X = [x1, x2,  , xM], where xi is the ith column of X. We define a linear operator vec:RN×MRMN,vec(X)=[x1T,x2T,,xMT]T.Hence the generalized Sylvester equation (1.1) can be written as systems of linear equationsAvec(X)=vec(C),where A=BTAf-IMNRMN×MN, IMN denotes the MN × MN identity matrix and ⊗ denotes the Kronecker product, see [2]. Let λ(A) be the spectrum of A, and λ(B) the spectrum of B. We haveλ(A)={λiλj-1:λiλ(A),λjλ(B),i=1,2,N;j=1,2,,M},which shows that the generalized Sylvester (1.1) has a unique solution if and only if λiλj  1, for λi  λ(A), λj  λ(B), i = 1, 2,  , N; j = 1, 2,  , M. In this paper we assume that this condition is satisfied. For two matrices X,YRm×n, we define the following inner product 〈 · , · F = tr(XTY), where tr(·) denotes the trace of the matrix XTY. The associated norm is the Frobenius norm denoted by ∥ · F. en(n) denotes the nth coordinate vector of Rn.

In the paper we propose the Galerkin and minimal residual algorithms for iteratively solving the generalized Sylvester matrix equation (1.1). The methods are based on the Arnoldi process [2] which allows us to construct an orthogonal basis of certain Krylov subspace and simultaneously reduce the order of the generalized Sylvester equation (1.1). The small generalized Sylvester equation obtained can be solved by direct methods or iterative methods. In our presentation and analysis of the algorithms, we use the form (1.2) of (1.1).

The remainder of the paper is organized as follows. In Section 2, we propose a Galerkin algorithm and a minimal residual algorithm for the solution of generalized Sylvester equation (1.1). Section 3 we discuss the result on the convergence of the algorithms introduced in Section 2. In Section 4, we present some numerical tests. Finally, we draw conclusions.

Section snippets

Galerkin method and minimal residual method for the generalized Sylvester equation

The full orthogonalization method, which is a Galerkin method introduced by Saad [8], and the GMRES method, which is a minimal residual method introduced by Saad and Schultz [9]. Both methods use the Arnoldi process to compute an orthonormal basis of certain Krylov subspaceKk(A,v)span{v,Av,,Ak-1v}.

Analysis of the convergence of the minimal residual method

We consider the convergence of the Algorithm 2.3 of Sections 2 in the special case when the residual vector satisfiesr0=gf.We derive an upper bound for the residual error (2.1). This bound shows how the rate of convergence depending on λ(A) and λ(B). Our results are most easily obtained by expressing the residual error (2.1) as a matrix.

Define the matrix R0RN×M, which satisfiesr0=vec(R0).It follows that R0=f·gT=f2g2v1w1T. SoR1=Vn+1f2g2e1(n+1)(e1(m+1))T-H¯AYH¯BT+In+1,nYIm+1,mTWm+1T.We

Numerical experiments

In this section, we present some numerical examples to illustrate the effectiveness of Algorithm 2.3 for large and sparse generalized Sylvester equations. Let SYL-MR(n, m) denote Algorithm 2.3, where n, m denote the dimension of the Krylov subspaces for A and BT, respectively. GMRES(k) method is based on the systems of linear equations (BT  A  IMN)vec(X) = vec(C), where k denotes the dimension of the Krylov subspace for BT  A  IMN. In the following examples, we compare SYL-MR(n, m) with restart GMRES(k)

Conclusion

In this paper we propose two iterative methods for the solution of generalized Sylvester equation. These methods reduce the given generalized Sylvester equation to a generalized Sylvester equation of smaller size by applying the Arnoldi process. We present Galerkin and minimal residual methods and obtain results on the convergence of the two methods, which is more appropriate for large and sparse generalized Sylvester equations than the usual Krylov subspace methods. Numerical experiments

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This work is supported by NSFC project 10471027 and Shanghai Education Commission.

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