Parallel solver for shifted systems in a hybrid CPU-GPU framework

This paper proposes a combination of a hybrid CPU--GPU and a pure GPU software implementation of a direct algorithm for solving shifted linear systems $(A - \sigma I)X = B$ with large number of complex shifts $\sigma$ and multiple right-hand sides. Such problems often appear e.g. in control theory when evaluating the transfer function, or as a part of an algorithm performing interpolatory model reduction, as well as when computing pseudospectra and structured pseudospectra, or solving large linear systems of ordinary differential equations. The proposed algorithm first jointly reduces the general full $n\times n$ matrix $A$ and the $n\times m$ full right-hand side matrix $B$ to the controller Hessenberg canonical form that facilitates efficient solution: $A$ is transformed to a so-called $m$-Hessenberg form and $B$ is made upper-triangular. This is implemented as blocked highly parallel CPU--GPU hybrid algorithm; individual blocks are reduced by the CPU, and the necessary updates of the rest of the matrix are split among the cores of the CPU and the GPU. To enhance parallelization, the reduction and the updates are overlapped. In the next phase, the reduced $m$-Hessenberg--triangular systems are solved entirely on the GPU, with shifts divided into batches. The benefits of such load distribution are demonstrated by numerical experiments. In particular, we show that our proposed implementation provides an excellent basis for efficient implementations of computational methods in systems and control theory, from evaluation of transfer function to the interpolatory model reduction.