In this paper we evaluate two solvers for the algebraic Riccati equation –a crucial operation in control theory– based on the Newton iterative scheme for the matrix sign function. When the target architecture is a multicore server with a moderate number of cores, our Riccati solvers exploit the task-parallelism that exist in the computational kernels that compose the Newton iteration. On the other hand, for hybrid platforms that include a graphics processor (GPU), we further exploit the data-parallelism of the major kernels and off-load these computations to the graphics device. Our experimental results on a platform equipped with state-of-the-art technology from AMD, Intel and NVIDIA reveals significant acceleration factors from the GPU.