Krylov subspace methods for the generalized Sylvester equation☆
Introduction
In this paper we present Krylov subspace methods for solving the generalized Sylvester matrix equationwhere are given matrices, and 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 ,Hence the generalized Sylvester equation (1.1) can be written as systems of linear equationswhere , 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 havewhich 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 , 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. denotes the nth coordinate vector of .
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 subspace
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 satisfiesWe 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 , which satisfiesIt follows that . SoWe
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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Cited by (0)
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This work is supported by NSFC project 10471027 and Shanghai Education Commission.