An Accelerated Conjugate Gradient Algorithm to Compute Low-Lying Eigenvalues of Sparse Hermitian Matrices

摘要

The low-lying eigenvalues of a (sparse) hermitian matrix can be computed with controlled numerical errors by a conjugate gradient (CG) method. The algorithm presented here is accelerated by a factor 4-8 by alternating CG searches with exact diagonalizations in the subspace spanned by the numerically compted eigenvectors. The algorithm is numerically very stable and can be parallelized in an efficient way.