## 2 Matrix Sign Function-Based Solvers

Table 2: Execution time (in seconds) for the solution of Lyapunov equations. is dedicated to the sign function method (_i.e._, the Lyapunov solver). The ROS1_gpu implementation attains the best execution time for problems STEEL\({}_{S}\) and STEEL\({}_{L}\), while it is slightly slower than ROS1_mgpu for the larger problem. In all the cases, ROS1_gpu clearly outperforms the multi-core variant, due to the large computational cost of the algorithm and its large parallelism. ## 5 Conclusions

We have addressed the solution of three types of Lyapunov matrix equations, algebraic Riccati equations and differential matrix equations, that arise in control theory applications. A collection of implementations were evaluated for each solver targeting three different target platforms: multicore CPU, a hybrid system equipped with a multicore CPU and a GPU and a platform composed of a multicore CPU connected to several GPUs. All the routines make intensive use of high performance kernels from linear algebra libraries, like Intel's MKL and NVIDIA's CUBLAS, and parallel interfaces, like OpenMP. Numerical results employing three benchmarks, extracted from the Oberwolfach Model Reduction Benchmark Collection, show the efficiency attained by the parallel solvers. The hybrid implementations that exploit the use of a single GPU report remarkable performance, being more than four-times faster than their corresponding multicore counterparts for large problems. However, their applicability is limited by the size of the GPU memory. This limitation is partially overcome in the multi-GPU implementations, where the amount of aggregated memory is larger. \begin{table}
\begin{tabular}{c r r r r r r} \hline  & \multicolumn{2}{c}{**ROS1\_mc**} & \multicolumn{2}{c}{**ROS1\_gpu**} & \multicolumn{2}{c}{**ROS1\_mgpu**} \\ \cline{2-7}  & **Fsign** & **Total** & **Fsign** & **Total** & **Fsign** & **Total** \\ \hline STEEL\({}_{S}\) & 5.23 & 5.28 & 3.93 & 3.98 & 2.53 & 2.70 \\ STEEL\({}_{L}\) & 189.98 & 190.66 & 45.05 & 45.73 & 21.19 & 23.01 \\ FLOW\_METER & 1952.50 & 1954.77 & 404.54 & 406.82 & 258.56 & 251.75 \\ \hline \end{tabular}
\end{table} 
Table 4: Execution time (in seconds) for the solution of differential Riccati equations. \begin{table}
\begin{tabular}{c r r r r r r r r} \hline  & \multicolumn{2}{c}{\(\boldsymbol{H_{0}}\)} & \multicolumn{2}{c}{**System**} & \multicolumn{2}{c}{**GECRSG\_cpu**} & \multicolumn{2}{c}{**GECRSG\_gpu**} & \multicolumn{2}{c}{**GECRSG\_mgpu**} \\ \cline{3-9}  & & & **Fsign** & **Total** & **Fsign** & **Total** & **Fsign** & **Total** \\ \hline STEEL\({}_{S}\) & 0.03 & 0.28 & 15.38 & 17.04 & 5.34 & 6.64 & 15.48 & 16.17 \\ STEEL\({}_{L}\) & 0.45 & 12.03 & 505.96 & 545.36 & 109.26 & 129.68 & 162.75 & 188.86 \\ FLOW\_METER & 1.37 & 72.79 & 2945.08 & 3173.54 & – & – & 533.25 & 689.26 \\ \hline \end{tabular}
\end{table} 
Table 3: Execution time (in seconds) for the solution of algebraic Riccati equations. ## Acknowledgments

The researcher at Universidad Jaume I was supported by project CICYT TIN2011-23283 and FEDER. ## Conflicts of Interest

The authors declare no conflict of interest. ## References

* [PERSON] et al. (2003) [PERSON]; [PERSON]; [PERSON]; [PERSON] _Matrix Riccati Equations in Control and Systems Theory_; Birkhauser: Basel, Switzerland, 2003. * [PERSON] (1980) [PERSON] Linear model reduction and solution of the algebraic Riccati equation by use of the sign function. _Int. J. Control_**1980**, _32_, 677-687. (Reprint of Technical Report No. TR-13, CUED/B-Control, Cambridge University, Engineering Department, 1971). * [PERSON] et al. (2003) [PERSON]; [PERSON]; [PERSON] State-space truncation methods for parallel model reduction of large-scale systems. _Parallel Comput._**2003**, _29_, 1701-1722. * [PERSON] et al. (2008) [PERSON]; [PERSON]; [PERSON] Solving linear-quadratic optimal control problems on parallel computers. _Optim. Methods Softw._**2008**, _23_, 879-909. * [PERSON] et al.