A symmetric structure-preserving ΓQR algorithm for linear response eigenvalue problems

In this paper, we present an efficient ΓQR algorithm for solving the linear response eigenvalue problem Hx=λx, where H is Π⁻-symmetric with respect to Γ₀=diag(I_n,−I_n). Based on newly introduced Γ-orthogonal transformations, the ΓQR algorithm preserves the Π⁻-symmetric structure of H throughout the whole process, and thus guarantees the computed eigenvalues to appear pairwise (λ,−λ) as they should. With the help of a newly established implicit Γ-orthogonality theorem, we incorporate the implicit multi-shift technique to accelerate the convergence of the ΓQR algorithm. Numerical experiments are given to show the effectiveness of the algorithm.