The set of H-infinity controllers with closed-loop performance gamma can be implicitly parametrized by the solutions (R,S) of a system of linear matrix inequalities (LMI). The matrices R and S play a role analogous to that of the Riccati solutions X(infinity) and Y-infinity in classical Riccati-based H-infinity control. Useful applications include LMI-based H-infinity synthesis, mixed H-2/H-infinity design, and H-infinity design with a pole-placement constraint. This paper is concerned with the reliable computation of H-infinity controllers given a solution (R, S) of the characteristic system of LMIs. Explicit formulas are derived for both the regular and singular cases. Remarkably, these formulas are extensions of the usual ’central controller’ formulas where the LMI solutions R and S replace the Riccati solutions X(infinity) and Y-infinity. Simple and numerically appealing new formulas for discrete-time H-infinity controllers are also derived.