A CHEBYSHEV LOCALLY OPTIMAL BLOCK PRECONDITIONED CONJUGATE GRADIENT METHOD FOR PRODUCT AND STANDARD SYMMETRIC EIGENVALUE PROBLEMS

The discretized Bethe-Salpeter eigenvalue (BSE) problem arises in many-body physics and quantum chemistry. Discretization leads to an algebraic eigenvalue problem involving a matrix H in BbbC2ntimes2n with a Hamiltonian-like structure. With proper transformations, the real BSE eigenproblem of form I and the complex BSE eigenproblem of form II can be transformed into real product eigenvalue problems of order n and 2n, respectively. We propose a new variant of the locally optimal block preconditioned conjugate gradient (LOBPCG) enhanced with polynomial filters to improve the robustness and effectiveness of a few well-known algorithms for computing the lowest eigenvalues of the product eigenproblems. Furthermore, our proposed method can be easily employed to solve large sparse standard symmetric eigenvalue problems. We show that our ideal locally optimal algorithm delivers Rayleigh quotient approximation to the desired lowest eigenvalue that satisfies a global quasi-optimality, which is similar to the global optimality of the preconditioned conjugate gradient method for the iterative solution of a symmetric positive definite linear system. The robustness and efficiency of our proposed method is illustrated by numerical experiments. © 2024 Society for Industrial and Applied Mathematics Publications. All rights reserved.