Restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems

The restrictively preconditioned conjugate gradient (RPCG) method for solving large sparse system of linear equations of a symmetric positive definite and block two-by-two coefficient matrix is further studied. In fact, this RPCG method is essentially the classical preconditioned conjugate gradient (PCG) method with a specially structured preconditioner. Within this setting, we present algorithmic descriptions of two restrictive preconditioners that, respectively, employ the block Jacobi and the block symmetric Gauss-Seidel matrix splitting matrices as approximations to certain matrices involved in them, and give convergence analyses of the correspondingly induced two PCG methods. Numerical results show that these restrictive preconditioners can lead to practical and effective PCG methods for solving large sparse systems of linear equations of symmetric positive definite and block two-by-two coefficient matrices.