A Projected Preconditioned Conjugate Gradient Method For The Linear Response Eigenvalue Problem

The linear response eigenvalue problem aims at computing a few smallest positive eigenvalues together with the associated eigenvectors of a special Hamiltonian matrix and plays an important role for estimating the excited states of physical systems. A subspace version of the Thouless minimization principle was established by Bai and Li (SIAM J. Matrix Anal. Appl., 33:1075-1100, 2012) which characterizes the desired eigenpairs as its solution. In this paper, we propose a Projected Preconditioned Conjugate Gradient (PPCG-lrep) method to solve this subspace version of Thouless's minimization directly. We show that PPCG-lrep is an efficient implementation of the inverse power iteration and can be performed in parallel. It also enjoys several properties including the monotonicity and constraint preservation in the Thouless minimization principle. Convergence of both eigenvalues and eigenvectors are established and numerical experiences on various problems are reported.