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We develop K$\omega$, an open-source linear algebra library for the shiftedKrylov 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 thatfor a single equation. The shift invariance of the Krylov subspace is themathematical foundation of the shifted Krylov subspace methods. Applications inmaterials science are presented to demonstrate the advantages of the algorithmover the standard Krylov subspace methods such as the Lanczos method. Weintroduce benchmark calculations of (i) an excited (optical) spectrum and (ii)intermediate eigenvalues by the contour integral on the complex plane. Incombination with the quantum lattice solver $\mathcal{H} \Phi$, K$\omega$ canrealize parallel computation of excitation spectra and intermediate eigenvaluesfor various quantum lattice models.

K$\omega$ -- Open-source library for the shifted Krylov subspace method of the form $(zI-H)x=b$