Toward a parallel solver for generalized complex symmetric eigenvalue problems

Abstract

Abstract Methods for numerically solving generalized complex symmetric (non-Hermitian) eigenvalue problems (EVPs) A x = λ B x serially and in parallel are investigated. This research is motivated by two observations: Firstly, the conventional approach for solving such problems serially, as implemented, e.g., in zggev (LAPACK), is to treat complex symmetric problems as general complex and therefore does not exploit the structural properties. Secondly, there is currently no parallel solver for dense (generalized or standard) non-Hermitian EVPs in ScaLAPACK. The approach presented in this paper especially aims at exploiting the structural properties present in complex symmetric EVPs and at investigating the potential trade-offs between performance improvements and loss of numerical accuracy due to instabilities. For the serial case, a complete reduction based solver for computing eigenvalues of the generalized complex symmetric EVP has been designed, implemented, and is evaluated in terms of numerical accuracy as well as in terms of runtime performance. It is shown that the serial solver achieves a speedup of up to 43 compared to zggev (LAPACK), although at the cost of a reduced accuracy. Furthermore, the major parts of this reduction based solver have been parallelized based on ScaLAPACK and MPI. Their scaling behavior is evaluated on a cluster utilizing up to 1024 cores. Moreover, the parallel codes developed achieve encouraging parallel speedups comparable to the ones of ScaLAPACK routines for the complex Hermitian EVP.