Restrictively preconditioned Chebyshev method for solving systems of linear equations

The restrictive preconditioning technique applied to Chebyshev semi-iterative methods for solving a nonsingular system of linear equations is studied. Motivated by the fact that linear systems involving a preconditioner are solved inexactly at each step, we present the inexact restrictively preconditioned Chebyshev method. The convergence behaviors of the new methods are analyzed, particularly for solving two-by-two block systems. Numerical experiments show that the new methods may be more efficient than the restrictively preconditioned conjugate gradient method in the presence of inexactness. Moreover, the inexact restrictively preconditioned Chebyshev method is a strong competitor to the block upper triangular preconditioned GMRES and flexible GMRES methods for solving generalized saddle point problems.