ABSTRACT
This chapter describes Gram–Schmidt versus the Lanczos orthogonalization with respect to the novel application of quantum mechanical methods to signal processing across a range of interdisciplinary research fields. It shows that the general Gram–Schmidt state is computed recursively. In other words, the whole string of the computed Gram–Schmidt states have to be kept in the computer memory throughout the recursion. Therefore, a more economical recursion is sought. One of the ways to obtain the simultaneous appearance of a few active orbitals at a time and still hold all the distant orbitals through a generating orbital is to use the evolution operator from the definition of the Schrödinger states. The non-standard derivation of the Lanczos recurive algorithm directly from the Gram–Schmidt orthogonalization is pedagogically instructive, since the former method is usually not taught in regular physics courses.
