Convergences of splitting iterative methods for symmetric indefinite linear systems

In this paper, we generalize the saddle point problem to general symmetric indefinite systems, we also present a kind of convergent splitting iterative methods for the symmetric indefinite systems. A special divergent splitting is introduced. The sufficient condition is discussed that the eigenvalues of the iteration matrix are real. The spectral radius of the iteration matrix is discussed in detail, the convergence theories of the splitting iterative methods for the symmetric indefinite systems are obtained. Finally, we present a preconditioner and discuss the eigenvalues of preconditioned matrix.