On inexact alternating direction implicit iteration for continuous Sylvester equations

In 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.The authors are grateful to the anonymous referees for their valuable comments and suggestions which improved the quality of this paper. Also, The authors would like to thank the supports of the National Natural Science Foundation of China under Grant No. 11371075, the Hunan Key Laboratory of mathematical modeling and analysis in engineering, the Portuguese Funds through FCT-Fundacao para a Ciencia, within the Project UIDB/00013/2020 and UIDP/00013/2020. This work does not have any conflicts of interest