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Optimally hybrid numerical solvers were constructed for massively parallelgeneralized eigenvalue problem (GEP).The strong scaling benchmark was carriedout on the K computer and other supercomputers for electronic structurecalculation problems in the matrix sizes of M = 10^4-10^6 with upto 105 cores.The procedure of GEP is decomposed into the two subprocedures of the reducer tothe standard eigenvalue problem (SEP) and the solver of SEP. A hybrid solver isconstructed, when a routine is chosen for each subprocedure from the threeparallel solver libraries of ScaLAPACK, ELPA and EigenExa. The hybrid solverswith the two newer libraries, ELPA and EigenExa, give better benchmark resultsthan the conventional ScaLAPACK library. The detailed analysis on the resultsimplies that the reducer can be a bottleneck in next-generation (exa-scale)supercomputers, which indicates the guidance for future research. The code wasdeveloped as a middleware and a mini-application and will appear online.

Hybrid Numerical Solvers for Massively Parallel Eigenvalue Computation and Their Benchmark with Electronic Structure Calculations