A parallel structured banded DC algorithm for symmetric eigenvalue problems

In this paper, a novel parallel structured divide-and-conquer (DC) algorithm is proposed for symmetric banded eigenvalue problems, denoted by PBSDC, which modifies the classical parallel banded DC (PBDC) algorithm by reducing its computational cost. The main tool that PBSDC uses is a parallel structured matrix multiplication algorithm (PSMMA), which can be much faster than the general dense matrix multiplication ScaLAPACK routine PDGEMM. Numerous experiments have been performed on Tianhe-2 supercomputer to compare PBSDC with PBDC and ELPA. For matrices with few deflations, PBSDC can be much faster than PBDC since computations are saved. For matrices with many deflations and/or small bandwidths, PBSDC can be faster than the tridiagonalization-based DC implemented in LAPACK and ELPA. However, PBSDC would become slower than ELPA for matrices with relatively large bandwidths.