On inexact alternating direction implicit iteration for continuous Sylvester equations

Zhongyun Liu,Yang Zhou,Yulin Zhang

doi:10.1002/nla.2320

Abstract

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.

Highlights

Consider the iterative solution to the continuous Sylvester equation AX + XB = C, (1.1)by the alternating direction implicit (ADI)-like iterations, where A ∈ Cm×m, B ∈ Cn×n, C ∈ Cm×n are large sparse matrices.For definiteness, throughout this paper, both A and B in (1.1) are assumed to be positive semi-definite† and at least one of them to be positive definite.It is known that the Sylvester equation (1.1) has a unique solution if and only if A and −B haven’t the common eigenvalues, see e.g., [19, 21]
We have revisited 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, and at least one of them to be positive definite. For such a class of Sylvester equations, we have analyzed the convergence of the ADI iteration and derived an upper bound of its contraction factor
To reduce the computational complexity, we have proposed the IADI iteration whose convergence has been analyzed in detail

Summary

Introduction

Before giving the convergence theorem of the ADI iteration, we recall the following known results. The iterative sequence converges to the exact solution X∗ of (1.1), provided that −λmin < ∆ < μmin, where λmin and μmin denote the lower bounds of the real parts of the eigenvalues of the matrices A and B, respectively. In order to analyze the convergence of the above IADI iteration, we need to recall the following convergence theorem of iterative solution to a general linear system Ax = b by an inexact two-step splitting iterative method.

Results

Conclusion