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Density functional calculations are used to compute the properties of materials, which can be stored in a database for later use, or used on the fly in the screening phase of the material designing process. Since during the screening process, the properties of a large number of materials have to be computed, the time and the resources needed for all the simulations have to be minimized. The time to solution of density functional calculation is dominated by dense matrix operations, and in particular the generalized eigenvalue solver, for which new implementations have to be developed in order to exploit the performance of the emerging hybrid CPU+GPU supercomputers. In the first part we present the algorithms used by the generalized eigenvalue solver, as well as the one-stage and two-stage approach for the tridiagonalization step, and their implementations for single-node hybrid CPU+GPU. The performance measurements of the solver show a significant speed-up of the hybrid solver compared to the state-of-art CPU-only implementations. This further show that the two-stage approach is preferable, especially when only a fraction of the eigenspace has to be computed. The implementation is then extended to support the single node multi-GPU architecture, and its benchmark show a good performance and scalability. We then present the implementation for the distributed memory hybrid CPU+GPU architecture of the generalized eigensolver, as well as its performance measurements. In the second part of the thesis we develop a model for the time to solution and the energy to solution for distributed memory applications, which explains the affine relationship between the resource time and the energy to solution empirically observed previously for two stencil based algorithms. This model shows that the affine relationship holds for different classes of algorithms, among which linear algebra solvers are included. Furthermore we study quantitatively the measured performance of the eigensolver, using an error analysis based on Bayesian inference. The results of the analysis allow us to extract two parameters which depends on the algorithm and the underlying architecture used to perform the calculation. The first parameter is the dynamic energy, and its scaling corresponds to the complexity of the underlying algorithm. The second parameter represents the static power consumed by the system, which has a strong dependence on the architecture and the parallelization scheme.

GPU accelerated implementations of a generalized eigenvalue solver for Hermitian matrices with systematic energy and time to solution analysis