Structure-Preserving {\Gamma}QR and {\Gamma}-Lanczos Algorithms for Bethe-Salpeter Eigenvalue Problems

TL;DR

This paper introduces structure-preserving {3}QR and {33}-Lanczos algorithms tailored for Bethe-Salpeter eigenvalue problems, ensuring eigenvalues and eigenvectors retain key properties of the original matrix.

Contribution

The paper proposes novel {33}QR and {33}-Lanczos algorithms that preserve the structure of Bethe-Salpeter matrices, with theoretical validation and numerical demonstrations of their effectiveness.

Findings

Algorithms preserve eigenvalue properties.

Numerical results show superior performance.

Theoretical proofs confirm validity.

Abstract

To solve the Bethe-Salpeter eigenvalue problem with distinct sizes, two efficient methods, called {\Gamma}QR algorithm and {\Gamma}-Lanczos algorithm, are proposed in this paper. Both algorithms preserve the special structure of the initial matrix $H=\begin{bmatrix}A & B-\overline{B} & -\overline{A}\end{bmatrix}$, resulting the computed eigenvalues and the associated eigenvectors still hold the properties similar to those of $H$. Theorems are given to demonstrate the validity of the proposed two algorithms in theory. Numerical results are presented to illustrate the superiorities of our methods.