High performance parallel approximate eigensolver for real symmetric matrices

摘要: In the first-principles calculation of electronic structures, one most timeconsuming tasks is that computing eigensystem a large symmetric nonlinear eigenvalue problem. The standard approach to use an iterative scheme involving solution linear problem in each iteration. early and intermediate iterations, significant gains efficiency may result from solving reduced accuracy. As iteration nears convergence, can be computed required The main contribution this dissertation efficient parallel approximate eigensolver computes eigenpairs real matrix This consists three major parts: (1) a block divide-and-conquer algorithm tridiagonal prescribed accuracy; (2) a tridiagonalization constructs sparse or "effectively" matrix---matrix with many small elements regarded as zeros without affecting accuracy eigenvalues; (3) a orthogonal reduction reduces dense form using similarity transformations high ratio level 3 BLAS operations. chooses proper combination algorithms depending on structure input all Numerical results show accurate tolerance. time for decreases significantly tolerance becomes larger.