K ω  Opensource library for the shifted Krylov subspace method of the form (zI  H) x = b
Abstract
We develop K ω, an opensource linear algebra library for the shifted Krylov subspace methods. The methods solve a set of shifted linear equations (z_{k} I  H) ^{x (k)} = b(k = 0 , 1 , 2 , …) for a given matrix H and a vector b, simultaneously. The leading order of the operational cost is the same as that for a single equation. The shift invariance of the Krylov subspace is the mathematical foundation of the shifted Krylov subspace methods. Applications in materials science are presented to demonstrate the advantages of the algorithm over the standard Krylov subspace methods such as the Lanczos method. We introduce benchmark calculations of (i) an excited (optical) spectrum and (ii) intermediate eigenvalues by the contour integral on the complex plane. In combination with the quantum lattice solver HΦ , K ω can realize parallel computation of excitation spectra and intermediate eigenvalues for various quantum lattice models.
 Publication:

Computer Physics Communications
 Pub Date:
 January 2021
 DOI:
 10.1016/j.cpc.2020.107536
 arXiv:
 arXiv:2001.08707
 Bibcode:
 2021CoPhC.25807536H
 Keywords:

 Numerical linear algebra;
 Shifted linear equations;
 Krylov subspace method;
 Quantum lattice models;
 Mathematics  Numerical Analysis;
 Condensed Matter  Strongly Correlated Electrons;
 Physics  Computational Physics
 EPrint:
 12 pages, 5 figures