Rayleigh--Ritz Approximation For the Linear Response Eigenvalue Problem

Abstract

Large scale eigenvalue computation is about approximating certain invariant subspaces associated with the interested part of the spectrum, and the interested eigenvalues are then extracted from projecting the problem by approximate invariant subspaces into a much smaller eigenvalue problem. In the case of the linear response eigenvalue problem (aka the random phase eigenvalue problem), it is the pair of deflating subspaces associated with the first few smallest positive eigenvalues that needs to be computed. This paper is concerned with approximation accuracy relationships between a pair of approximate deflating subspaces and approximate eigenvalues extracted by the pair. Lower and upper bounds on eigenvalue approximation errors are obtained in terms of canonical angles between exact and computed pair of deflating subspaces. These bounds can also be interpreted as lower/upper bounds on the canonical angles in terms of eigenvalue approximation errors. They are useful in analyzing numerical solutions to linear response eigenvalue problems.

Backward perturbation analysis and residual-based error bounds for the linear response eigenvalue problem

“…The goal of this paper is to perform a backward perturbation analysis and to establish error bounds on the accuracy (in eigenvalue/eigenspace approximations) using proper residuals associated with any given approximate deflating subspace pair. A related study is presented in [46]. The main difference among the results in [46] and those in the present paper is that the error bounds on eigenvalue/eigenspace approximations in [46] are characterized by the canonical angles between the approximate deflating subspace pair and the exact pair, whereas the error bounds in this paper use certain computable residuals.…”

“…A related study is presented in [46]. The main difference among the results in [46] and those in the present paper is that the error bounds on eigenvalue/eigenspace approximations in [46] are characterized by the canonical angles between the approximate deflating subspace pair and the exact pair, whereas the error bounds in this paper use certain computable residuals. These two types of error bounds are well-established in the standard eigenvalue problem (see, e.g., [30,33]), and both types are useful in analyzing the convergence and designing stopping criteria for iterative algorithms.…”

Weighted Block Golub-Kahan-Lanczos Algorithms for Linear Response Eigenvalue Problem

“…By using the approximate eigenvectors of H for thick restart, we post-multiply Ψ k and Φ k to the Equation (15), respectively, and get…”

“…It also known as the Bethe-Salpeter (BS) eigenvalue-problem [4] or the random phase approximation (RPA) eigenvalue problem [5]. There has immense past and recent work in developing efficient numerical algorithms and attractive theories for LREP [6][7][8][9][10][11][12][13][14][15].…”

Thick restarting the weighted harmonic Golub-Kahan-Lanczos algorithm for the linear response eigenvalue problem

“…The LREP has important applications in the area of computational quantum chemistry and physics such as silicon nanoparticles and nanoscale materials and the analysis of interstellar clouds and polarizabilities [3,17]. All the eigenvalues of H are real and appear in pairs {λ, −λ} [3,27], and we denote the ordered eigenvalues of H by…”