Talks

Ein klassischer Satz von Friedrichs besagt, dass Schrödingeroperatoren kompakte Resolvente besitzen, wenn das zugrundeliegende Potential bei Unendlich gegen Unendlich geht. In diesem Vortrag werden wir einen einfachen Beweis einer Verallgemeinerung präsentieren, basierend auf einer gemeinsamen Arbeit mit D. Lenz (Jena) und D. Wingert.

In their seminal 1952 paper on the conjugate gradient (CG) method Hestenes and Stiefel pointed out that their method, which is applicable to linear systems of equations with symmetric positive definite matrix only, is closely related to certain orthogonal polynomials, the corresponding Gauss quadrature formulas, certain continued fractions, and their convergents (or `partial sums'). The latter can be seen to be Padé approximants of a function that involves the resolvent of the matrix.

Around the same time, in 1950 and 1952, Cornelius Lanczos published two related articles, of which the second one introduced a precursor of the biconjugate gradient (BCG or BiCG) method, which generalizes CG to the case of a nonsymmetric system. Here, the residual polynomials are formal orthogonal polynomials only, but the connections to continued fractions and Padé approximants persist. Moreover, there is a relation to the qd algorithm of Rutishauser (1954). The understanding of all these connections became probably the key to Rutishauser's discovery of the LR algorithm (1955, 1958), which was later enhanced by John G.F. Francis to the ubiquitous QR algorithm (1961/62).