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We have developed a parallel eigensolver for very small-size matrices. Unlike conventional solvers, our design policy focusses on nature of non-blocking computations and reduced communications. A communication-avoiding approach for Householder pivot vectors is used to implement part of Householder inverse transformation. In addition to that, we implement some techniques for reducing communications by using non-blocking communications in tridiagonalization part. Performance of the solver with full nodes in the Fujitsu FX10 (76,800 cores) is also presented.