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

Abstract

The numerical solution of a large scale linear response eigenvalue problem is often accomplished by computing a pair of deflating subspaces associated with the interesting part of the spectrum. This paper is concerned with the backward perturbation analysis for a given pair of approximate deflating subspaces or an approximate eigenquadruple. Various optimal backward perturbation bounds are obtained, as well as bounds for approximate eigenvalues computed through the pair of approximate deflating subspaces or approximate eigenquadruple. These results are reminiscent of many existing classical ones for the standard eigenvalue problem.

Backward error analysis for linearizations in heavily damped quadratic eigenvalue problem

“…The perturbation bounds for a linear eigenvalue problem was widely discussed in other works. [9][10][11][12][13] Extending these works, we will establish the error bounds for heavily damped QEP. For a given approximate eigenpair of heavily damped QEP, it is important to know how accurate an approximate eigenpair of Q( ) via linearizations will be obtained.…”

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

“…We take Z = Y 1 (Ξ T (:,1:3) KY 1 ) −1 , then Z satisfies (6). We execute the weighted block Golub-Kahan-Lanczos method with full re-orthogonalization for LREP in MATLAB, and check the bounds in (7), (8), (13), and (14). Since the approximate eigenvalues are {σ 1 , σ 2 , σ 3 } and {σ nn b −2 , σ nn b −1 , σ nn b }, thus π i,k, = π i,k =πˆi ,k,ˆ = πˆi ,ˆ = 1, c =ĉ = 1, and we measure the following two groups of errors:…”

“…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

“…All of the above methods generate approximate deflating subspaces and then approximate eigenpairs of H from a far smaller matrix H S = 0 K S M S 0 , where K S and M S are both symmetric and positive definite. Some other effective methods and a rigorous theoretical analysis can be found in the references [9,26].…”