Lanczos algorithm equations

However, there is a price to pay in a spectral transform Lanczos algorithm At each recursion step, the action of the filter operator onto the Lanczos vectors has to be evaluated. In the original version, Ericsson and Ruhe update the Lanczos vectors by solving the following linear equation ... 

Interestingly, the spectral transform Lanczos algorithm can be made more efficient if the filtering is not executed to the fullest extent. This can be achieved by truncating the Chebyshev expansion of the filter,76,81 or by terminating the recursive linear equation solver prematurely.82 In doing so, the number of vector-matrix multiplications can be reduced substantially. 

Summarizing, the calculation of the spectrum (equation 34) with the aid of the Lanczos algorithm proceeds in three steps. 

The non-adiabatic quantum simulation procedures we employ have been well described previously in the literature, so we describe them only briefly here. The model system consists of 200 classical SPC flexible water molecules," and one quantum mechanical electron interacting with the water molecules via a pseudopotential. 2 The equations of motion were integrated using the Verlet algorithm with a 1 fs time step in the microcanonical ensemble, and the adiabatic eigenstates at each time step were calculated with an iterative and block Lanczos scheme. Periodic boundary conditions were employed using a cubic simulation box of side 18.17A (water density 0.997 g/ml). 

---