Nested splitting CG-like iterative method for solving the continuous Sylvester equation and preconditioning

Abstract

Abstract We present a nested splitting conjugate gradient iteration method for solving<br><br>large sparse continuous Sylvester equation, in which both coefficient matrices are<br><br>(non-Hermitian) positive semi-definite, and at least one of them is positive definite.<br><br>This method is actually inner/outer iterations, which employs the Sylvester conjugate<br><br>gradient method as inner iteration to approximate each outer iterate, while each outer<br><br>iteration is induced by a convergent and Hermitian positive definite splitting of the<br><br>coefficient matrices. Convergence conditions of this method are studied and numerical<br><br>experiments show the efficiency of this method. In addition, we show that the<br><br>quasi-Hermitian splitting can induce accurate, robust and effective preconditioned<br><br>Krylov subspace methods