A parallel preconditioned modified conjugate gradient method for large Sylvester matrix equation (Q1718751)

Summary: Computational effort of solving large-scale Sylvester equations \(\mathbf{A} \mathbf{X} + \mathbf{X} \mathbf{B} + \mathbf{F} = \mathbf{O}\) is frequently hindered in dealing with many complex control problems. In this work, a parallel preconditioned algorithm for solving it is proposed based on combination of a parameter iterative preconditioned method and modified form of conjugate gradient (MCG) method. Furthermore, Schur's inequality and modified conjugate gradient method are employed to overcome the involved difficulties such as determination of parameter and calculation of inverse matrix. Several numerical results finally show that high performance of proposed parallel algorithm is obtained both in convergent rate and in parallel efficiency.