  Optimally hybrid numerical solvers were constructed for massively parallel
generalized eigenvalue problem (GEP).The strong scaling benchmark was carried
out on the K computer and other supercomputers for electronic structure
calculation 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 to
the standard eigenvalue problem (SEP) and the solver of SEP. A hybrid solver is
constructed, when a routine is chosen for each subprocedure from the three
parallel solver libraries of ScaLAPACK, ELPA and EigenExa. The hybrid solvers
with the two newer libraries, ELPA and EigenExa, give better benchmark results
than the conventional ScaLAPACK library. The detailed analysis on the results
implies that the reducer can be a bottleneck in next-generation (exa-scale)
supercomputers, which indicates the guidance for future research. The code was
developed as a middleware and a mini-application and will appear online.
