Effect of spectral range on convergence in Lanczos algorithm, a numerical study

We investigate the numerical performance of the Lanczos algorithm applied to large eigenproblems in chemical physics. Particular attention is paid to the effect of the spectral range of the Hamiltonian (DeltaH) on the convergence of Lanczos eigenvalues in finite-precision arithmetic. A simple approximate scaling law is found in numerical tests involving one-, three-, and six-dimensional systems. The number of converged eigenlevels (n(conv)) increases linearly with the scaled length of the Lanczos recursion (K-norm), which is inversely proportional to the square root of the spectral range (K-norm = K/rootDeltaH). Discussions on controlling the spectral range and its effect on the performance of Lanczos algorithm are presented. (C) 2003 Elsevier Science B.V. All rights reserved.