ABSTRACT
Since the constants are of paramount importance for spectral analysis, it is imperative to search for more stable algorithms than the Lanczos recursion for state vectors. There are two recursive algorithms: Rutishauser’s quotient-difference algorithm and Gordon’s product-difference (PD) algorithm. This chapter describes the PD algorithm for obtaining Lanczos coupling parameters. The PD algorithm is error-free for signal points that are integers. Such integer data matrices are measured experimentally throughout magnetic resonance phenomena. The same infinite-order precision (no round-off errors) is achievable within the PD algorithm for auto-correlation functions or power moments given as rational numbers. In many cases of physical interest (e.g. systems exposed to external fields), the role of signal points is played by expansion coefficients that are obtained exactly as rational numbers from the quantum-mechanical perturbation theory.
