This note proposes an efficient computational procedure for the continuous time, input constrained, infinite horizon, linear quadratic regulator problem (CLQR). To ensure satisfaction of the constraints, the input is approximated as a piecewise linear function on a finite time discretization. The solution of this approximate problem is a standard quadratic program. A novel lower bound on the infinite dimensional CLQR problem is developed, and the discretization is adaptively refined until a user supplied error tolerance on the CLQR cost is achieved. The offline storage of the required quadrature matrices at several levels of discretization tailors the method for online use as required in model predictive control (MPC). The performance of the proposed algorithm is then compared with the standard discrete time MPC algorithms. The proposed method is shown to be significantly more efficient than standard discrete time MPC that uses a sample time short enough to generate a cost close to the CLQR solution.