Constraint Preconditioners for Symmetric Indefinite Matrices

We study the eigenvalue bounds of block two-by-two nonsingular and symmetric indefinite matrices whose (1, 1) block is symmetric positive definite and Schur complement with respect to its (2, 2) block is symmetric indefinite. A constraint preconditioner for this matrix is constructed by simply replacing the (1, 1) block by a symmetric and positive definite approximation, and the spectral properties of the preconditioned matrix are discussed. Numerical results show that, for a suitably chosen (1, 1) block-matrix, this constraint preconditioner outperforms the block-diagonal and the block-tridiagonal ones in iteration step and computing time when they are used to accelerate the GMRES method for solving these block two-by-two symmetric positive indefinite linear systems. The new results extend the existing ones about block two-by-two matrices of symmetric negative semidefinite (2, 2) blocks to those of general symmetric (2, 2) blocks.