high performance computing on graphics processing units: hgpu.org

We consider the computation of a few eigenpairs of a generalized eigenvalue problem Ax = lambda Bx with block-tridiagonal matrices, not necessarily symmetric, in the context of Krylov methods. In this kind of computation, it is often necessary to solve a linear system of equations in each iteration of the eigensolver, for instance when B is not the identity matrix or when computing interior eigenvalues with the shift-and-invert spectral transformation. The linear solve can be done in a direct fashion by means of an LU factorization or in an iterative way with preconditioned Krylov methods (inexact shift-and-invert). In this work, we aim to compare different direct linear solvers that can exploit the block-tridiagonal structure. Block cyclic reduction and the SPIKE algorithm are considered. A parallel implementation based on MPI is developed in the context of the SLEPc library. The use of GPU devices to accelerate local computations shows to be competitive for large block sizes.