### Abstract

We develop Kω, an open-source 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, Kω can realize parallel computation of excitation spectra and intermediate eigenvalues for various quantum lattice models. Program summary: Program Title: Kω [kéi-óumig@] CPC Library link to program files: https://dx.doi.org/10.17632/mt928nz5r3.1 Developer's repository link: https://github.com/issp-center-dev/Komega Licensing provisions: GNU Lesser General Public License Version 3. Programming language: Fortran 90 External routines/libraries: BLAS library, LAPACK library (Used in the sample program), MPI library (Optional). Nature of problem: Efficient algorithms, called shifted Krylov subspace algorithms, designed to solve the shifted linear equations. Solution method: Shifted conjugate gradient method, Shifted conjugate orthogonal conjugate gradient method, shifted bi-conjugate gradient method. Additional comments: The present paper is accompanied by a frozen copy of Kω release 2.0.0 that is made publicly available on GitHub (repository https://github.com/issp-center-dev/Komega, commit hash fd5455328b102ec4fa13432496e41c404a0f5a9d).

Original language | English |
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Article number | 107536 |

Journal | Computer Physics Communications |

Volume | 258 |

DOIs | |

Publication status | Published - 2021 Jan |

### Keywords

- Krylov subspace method
- Numerical linear algebra
- Quantum lattice models
- Shifted linear equations

### ASJC Scopus subject areas

- Hardware and Architecture
- Physics and Astronomy(all)

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## Cite this

*Computer Physics Communications*,

*258*, [107536]. https://doi.org/10.1016/j.cpc.2020.107536