  We present a modified version of the PRESB preconditioner for two-by-two
block system of linear equations with the coefficient matrix
$$\textbf{A}=\left(\begin{array}{cc}
  F & -G^*
  G & F \end{array}\right),$$ where $F\in\mathbb{C}^{n\times n}$ is Hermitian
positive definite and $G\in\mathbb{C}^{n\times n}$ 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.
