PHYSICAL AND NUMERICAL ASPECTS IN LANCZOS AND MODIFIED LANCZOS CALCULATIONS

The use of recurrence relations in Lanczos procedure is often limited in practice by loss of numerical stability. In many applications this fact severely limits the numerical reliability of the results and hence the interpretation of the physical models proposed. In this paper we focus on the effects of finite precision arithmetic in the Lanczos algorithm and examine some methods for assessing and testing the reliability of the tridiagonal matrix generated. In addition to the residual vector test we propose an initial vector test, and an orthogonalization test to assess the quality of the eigenvalues. We also implement the use of multiple precision to obtain a significant improvement in the tridiagonal chain transformation. As a final resort, we consider the modified Lanczos procedure to obtain one by one, energies and wavefunctions of a system, within any desired energy range and with any desired accuracy.