  In this paper, a parallel symmetric eigensolver with very small matrices in
massively parallel processing is considered. We define very small matrices that
fit the sizes of caches per node in a supercomputer. We assume that the sizes
also fit the exa-scale computing requirements of current production runs of an
application. To minimize communication time, we added several communication
avoiding and communication reducing algorithms based on Message Passing
Interface (MPI) non-blocking implementations. A performance evaluation with up
to full nodes of the FX10 system indicates that (1) the MPI non-blocking
implementation is 3x as efficient as the baseline implementation, (2) the
hybrid MPI execution is 1.9x faster than the pure MPI execution, (3) our
proposed solver is 2.3x and 22x faster than a ScaLAPACK routine with optimized
blocking size and cyclic-cyclic distribution, respectively.
