The number of low-energy states N ( Λ Λ ) tends to grow rapidly with the UV cutoff, and it is common to perform Hamiltonian truncation computations taking into account several thousand states. Nonetheless, one is often only interested in the first O(10) energies and eigenstates of H. Instead of diagonalizing the full Hamiltonian explicitly (which is numerically very costly), approximation methods like Arnoldi iteration and the Lanczos algorithm are commonly used.