Volume 46, pp. 505-523, 2017.

A block Lanczos method for the linear response eigenvalue problem

Zhongming Teng and Lei-Hong Zhang


In the linear response eigenvalue problem arising from computational quantum chemistry and physics one needs to compute a small portion of eigenvalues around zero together with the associated eigenvectors. Lanczos-type methods are particularly suitable for such a task. However, single-vector Lanczos methods can only find one copy of any multiple eigenvalue and can be very slow when the desired eigenvalues form a cluster. In this paper, we propose a block Lanczos-type implementation for the linear response eigenvalue problem, which is able to compute a cluster of eigenvalues much faster and more efficiently than the single-vector version. Convergence results are established and reveal the accuracy of the approximations of eigenvalues in a cluster and of the eigenspace. A practical thick-restart procedure is introduced to alleviate the increasing numerical difficulties of the block Lanczos method in computational costs, memory demands, and numerical stability. Numerical examples are presented to support the effectiveness of the thick-restart block Lanczos method.

Full Text (PDF) [347 KB]

Key words

linear response eigenvalue problems, block Lanczos methods, convergence analysis, thick-restart

AMS subject classifications

65F15, 15A18

Links to the cited ETNA articles

[20]Vol. 44 (2015), pp. 624-638 Zhongming Teng, Linzhang Lu, and Ren-Cang Li: Perturbation of partitioned linear response eigenvalue problems

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