Simple Davidson iteration scheme

The preconditioned residual vector is calculated for each band resulting in a $2*N_{bands}$ basis-set

$$b_{i,i=1,2*{ N_{\rm bands}}} = \{ \phi_{n} / p_{n} \vert n=1,N_{\rm bands} \}.$$

Within this subspace the NBANDS lowest eigenfunctions are calculated solving the eigenvalue problem

$$\langle b_i \vert {\bf H} - \epsilon_j {\bf S} \vert b_j \rangle = 0.$$

The NBANDS lowest eigenfunctions are used in the next step.