Generalized Preconditioned Locally Harmonic Residual Method for Non-Hermitian Eigenproblems

Abstract

Abstract. We introduce the Generalized Preconditioned Locally Harmonic Residual (GPLHR) method for solving standard and generalized non-Hermitian eigenproblems. The method is particularly useful for computing a subset of eigenvalues, and their eigen-or Schur vectors, closest to a given shift. The proposed method is based on block iterations and can take advantage of a preconditioner if it is available. It does not need to perform exact shift-and-invert transformation. Standard and generalized eigenproblems are handled in a unified framework. Our numerical experiments demonstrate that GPLHR is generally more robust and efficient than existing methods, especially if the available memory is limited.

Calculation of Expectation Values of Operators in the Complex Scaling Method

“…However, since very soon the usefulness of the resonant states was realized for the theoretical description of time dependent processes (e.g. radioactive decays) and Open Quantum Systems (OQSs) [15], solid mathematical foundations were developed [16][17][18][19][20][21] in the framework of non-Hermitian QM and also advances in numerical techniques and non-Hermitian diagonalizers were called for [22][23][24][25].…”

iVI: An iterative vector interaction method for large eigenvalue problems

“…Although numerical results by the GPLHR method cannot be reported here, it can be said definitely that the present iVI‐Buf(0 + 3)/Buf(0 + 2)/Buf(0 + 2 + 3) will outperform GPLHR for interior roots because of the following reasons: (1) iVI‐Buf(2 + 3) shares the same search space as GPLHR when the so‐called space expansion parameter m is set to 1 in the latter; (2) iVI‐Buf(2 + 3) alongside the Rayleigh–Ritz procedure (32)/(44) for extracting the eigenvectors does not work for the energy windows considered here, whereas GPLHR ( m = 1) often fails to yield the desired interior roots even with the harmonic Rayleigh–Ritz procedure (33) for extracting the eigenvectors; (3) iVI‐Buf(0 + 3)/Buf(0 + 2)/Buf(0 + 2 + 3) is certainly more efficient than GPLHR ( m > 1) due to smaller search spaces. Instead, we here compare iVI directly with the recently proposed Chebyshev filter diagonalization (ChebFD) approach for the interior roots of the cc‐pV5Z and matrices.…”

“…It is just that the dimension of the reduced matrix (35) is in this case instead of . Similarly, the particular combination of Buf(2) with Buf(3) in the iVI method (denoted as iVI‐Buf(2 + 3)), which shares the same search space as the GPLHR method designed for interior eigenpairs, also does not increase the number of MVPs as compared with iVI‐Buf(2).…”

“…Although numerical results by the GPLHR method [23,24] cannot be reported here, it can be said definitely that the present iVI-Buf(0 1 3)/Buf(0 1 2)/Buf(0 1 2 1 3) will outperform GPLHR for interior roots because of the following reasons: (1) iVI-Buf(2 1 3) shares the same search space as GPLHR when the so-called space expansion parameter m is set to 1 in the latter;…”

TRPL+K: Thick-Restart Preconditioned Lanczos+K Method for Large Symmetric Eigenvalue Problems

“…For large problems, as exact matrix factorizations are prohibitive or infeasible, we focus on preconditioners such as incomplete ILU or LDL factorization [10]. Although GD (or JD) type methods use a preconditioner to build a general, non-Krylov subspace, a few methods have been proposed to exploit a preconditioned Krylov subspace [10,23,40]. This paper further explores this line of research.…”