The solution of large-scale Lyapunov equations is an important tool for the solution of several engineering problems arising in optimal control and model order reduction. In this work, we investigate the case when the coefficient matrix of the equations presents a band structure. Exploiting the structure of this matrix, we can achive relevant reductions in the memory requirements and the number of floating-point operations. Additionally, the new solver efficiently leverages the parallelism of CPU–GPU platforms. Furthermore, it is integrated in the lyapack library to facilitate its use. The new codes are evaluated on the solution of several benchmarks, exposing significant runtime reductions with respect to the original CPU version in lyapack.
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For numerical reasons, pivoting is required in the factorization of \(A\). Although, for simplicity, it is not included in our discussion, the lapack implementations and all the implementations presented include pivoting.
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Ernesto Dufrechou and Pablo Ezzatti acknowledge the support from Programa de Desarrollo de las Ciencias Básicas, and Agencia Nacional de Investigación e Innovación of Uruguay. Enrique S. Quintana-Ortí was supported by project TIN2011-23283 of the Ministry of Science and Competitiveness (MINECO) and EU FEDER, and project P1-1B2013-20 of the Fundació Caixa Castelló-Bancaixa and UJI.
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Benner, P., Remón, A., Dufrechou, E. et al. Extending lyapack for the solution of band Lyapunov equations on hybrid CPU–GPU platforms. J Supercomput 71, 740–750 (2015). https://doi.org/10.1007/s11227-014-1322-7
- Banded matrix Lyapunov equations