*2.2. Lanczos Method*

The Lanczos method [35] is one of the simplest Krylov subspace methods, in which the low-lying exact eigenvalues are approximated by the Ritz values of the Krylov subspace. The Krylov subspace is spanned by an initial vector, *v*0, and its vector multiplied by the shell-model Hamiltonian matrix as

$$\mathcal{K}\_n(H, v\_0) \quad = \quad \text{span}\{v\_0, Hv\_0, H^2v\_0, \dots, \quad H^{n-1}v\_0\}.\tag{5}$$

The exact eigenvalues are approximated by the eigenvalues of the Hamiltonian matrix in the subspace K*n*(*H*, *v*0). In the Lanczos method one performs the orthonormalization of the vector at every *n*, which makes the method numerically stable. While in the limit of *n* → *D* the approximated eigenvalues agree with the exact ones, a few of the smallest eigenvalues converge with small *n*, typically *n* 300 to obtain the 10 lowest eigenvalues [36]. In practical codes, its extension, the thick-restart Lanczos method, is widely used [37].
