Thick-restart block Lanczos method for large-scale shell-model calculations

Abstract

We propose a thick-restart block Lanczos method, which is an extension of the thick-restart Lanczos method with the block algorithm, as an eigensolver of the large-scale shell-model calculations. This method has two advantages over the conventional Lanczos method: the precise computations of the near-degenerate eigenvalues, and the efficient computations for obtaining a large number of eigenvalues. These features are quite advantageous to compute highly excited states where the eigenvalue density is rather high. A shell-model code, named KSHELL, equipped with this method was developed for massively parallel computations, and it enables us to reveal nuclear statistical properties which are intensively investigated by recent experimental facilities. We describe the algorithm and performance of the KSHELL code and demonstrate that the present method outperforms the conventional Lanczos method.

A Guided Tour of ab initio Nuclear Many-Body Theory

“…This is achieved by partitioning the single-particle basis into core, valence, and beyond-valence states, normal ordering all operators with respect to a Slater determinant describing the closed-shell core, and extending the definition of the off-diagonal Hamiltonian (23) to include all terms that couple valence and non-valence states. The eigenvalue problem for the evolved Hamiltonian can then be solved in the valence space with widely available Shell model codes [89][90][91][92][93]. After a study of the oxygen isotopic chain revealed an increasing overbinding away from the chosen core [26], we adopted a normal-ordering scheme that uses an ensemble of Slater determinants to account for partially filled shells in open-shell nuclei [27,54]…”

Ab Initio Limits of Atomic Nuclei

“…We discuss exceptions to this below. Finally the resulting valence-space Hamiltonians are diagonalized with the NuShellX@MSU shell-model code [46] (with the exception of a few of the heaviest Ca, Sc and Ti isotopes, which were computed with the m-scheme code K-shell [47]).…”

A thick-restart Lanczos type method for Hermitian J-symmetric eigenvalue problems

“…Together with Lemma 1 and the premise, we find Multiplying w H k+1 and v H k+1 to (17) and (18), respectively, yields…”

“…However, with finite precision arithmetic, the single-vector Lanczos process would generate a small overlap to W k via round-off errors, so that even after the convergence of an eigenvector, a late convergence to the other paired eigenvector would be possible. Because this behavior is rather accidental, a block-type Lanczos algorithm has to be applied to accelerate the convergence for degenerate eigenvectors [1,2,8,18,22].…”

“…If the matrix A has a dense cluster of eigenvalues or multiple eigenvalues other than those with the J-symmetry, we need to incorporate the block type Lanczos iteration in the algorithm for efficiency. We can extend the proposed algorithm to the blocked version in the same manner as it was conducted for the standard thick-restart Lanczos algorithm [18,22]. The study on the block version will be addressed in future studies and we have only shown the single vector version to demonstrate the idea for simplicity.…”