A preconditioned nested splitting conjugate gradient iterative method for the large sparse generalized Sylvester equation

A nested splitting conjugate gradient (NSCG) iterative method and a preconditioned NSCG (PNSCG) iterative method are presented for solving the generalized Sylvester equation with large sparse coefficient matrices, respectively. Both methods are actually inner/outer iterations, which employ the CG-like method as inner iteration to approximate each outer iteration, while each outer iteration is induced by a convergent and symmetric positive definite splitting of the coefficient matrices. Convergence conditions of both methods are studied in depth and numerical experiments demonstrate the efficiency of the proposed methods. Moreover, experimental results show that the PNSCG method is more accurate, robust and effective than the NSCG method.