Efficient and accurate algorithms for solving the Bethe–Salpeter eigenvalue problem for crystalline systems

A Structure-Preserving Divide-and-Conquer Method for Pseudosymmetric Matrices

“…As a first result, we present the following theorem, clarifying the spectral structure of definite pseudosymmetric matrices. It is an extension of Theorem 5 in [16], additionally clarifying the structure of the eigenvectors, and a more general variant of Theorem 3 in [46]. Our version is independent of the additional structure of Bethe-Salpeter matrices given in (9).…”

“…A rational function g(x) = xh(x 2 ) which maps them close to 1, i.e. approximates the scalar sign function on the interval (ℓ, 1], can be used in an iteration (16). We see from (17) that the result will be an approximation to the polar factor W , which in our setting coincides with the matrix sign function.…”

“…In [47], the indefinite QR decomposition is presented, which improves stability by allowing 2 × 2 blocks on the diagonal of R and additional pivoting. This variant can also be computed via the (pivoted) LDL T decomposition of A T ΣA; a link which was exploited in [14] and [16]. There, the stability is improved by applying this method twice.…”

“…The setup (11) is essentially a complex version of (10). A more detailed analysis of the special structure in ( 9) and ( 11) is given in [16].…”

“…Consequently, employing Lemma 11 twice on a matrix A = W S with g(x) = Ẑ2r+1 (x; ℓ), we see that the eigenvalues of g(g(S)) will be in the interval [1 − 10 −15 , 1], under the condition that all eigenvalues of S are in [ℓ, 1] with ℓ ≥ 10 −16 . In this sense, G(G(A)) ≈ W has converged to the polar factor W , after two iterations of Iteration (16). Choosing a higher r, algorithms can be devised that converge in just one step.…”