Shift-Splitting Iteration Method and Its Variants for Solving Continuous Sylvester Equations

Abstract

A shift-splitting iteration method for solving large sparse continuous Sylvester equations is developed. This single-step iteration algorithm demonstrates a better computational efficiency than the previously used two-step iterative methods. We also propose two variants — viz. inexact and accelerated shift-splitting iteration methods. The convergence properties of all algorithms are studied and the quasi-optimal iteration parameter of shift-splitting is derived. Numerical examples demonstrate the efficiencies of the three methods, especially for equations with ill-conditioned coefficient matrices.