Linear Response Eigenvalue Problem Solved by Extended Locally Optimal Preconditioned Conjugate Gradient Methods

The locally optimal block preconditioned 4-d conjugate gradient method (LOBP4dCG) for the linear response eigenvalue problem was proposed in [SIAM J. Matrix Anal. Appl., 34(2):392–416, 2013] and later was extended to the generalized linear response eigenvalue problem in [BIT Numer. Math., 54(1):31–54, 2014]. In this paper, we put forward two improvements to the method: a shifting deflation technique and an idea of extending the search subspace. The deflation technique is able to deflate away converged eigenpairs from future computation, and the idea of extending the search subspace increases convergence rate per iterative step. The resulting algorithm is called the extended LOBP4dCG (ELOBP4dCG). Numerical results of the ELOBP4dCG strongly demonstrate the capability of deflation technique and effectiveness the search space extension for solving linear response eigenvalue problems arising from linear response analysis of two molecule systems.