We present a modified version of the PRESB preconditioner for two-by-two block systems of linear equations with the coefficient matrix
where $F∈\mathbb{C}^{n×n}$ is Hermitian positive definite and $G ∈\mathbb{C}^{n×n}$ is positive semidefinite. Spectral analysis of the preconditioned matrix is analyzed. In each iteration
of a Krylov subspace method, like GMRES, for solving the preconditioned system in
conjunction with proposed preconditioner two subsystems with Hermitian positive
definite coefficient matrix should be solved which can be accomplished exactly using
the Cholesky factorization or inexactly utilizing the conjugate gradient method. Application of the proposed preconditioner to the systems arising from finite element
discretization of PDE-constrained optimization problems is presented. Numerical results are given to demonstrate the efficiency of the preconditioner. Our theoretical
and numerical results show that the proposed preconditioner is efficient when the
norm of the skew-Hermitian part of $G$ is small.