Order Stars on Riemann Surfaces

Abstract

We have seen in the foregoing sections that the highest possible order of A -stable linear multistep methods is two; furthermore, the second derivative Enright methods as well as the SDBDF methods were seen to be A -stable for p ≤ 4; the three-stage Radau multistep methods were A -stable for p ≤ 6. In this section we shall see that these observations are special cases of a general principle, the so-called “Daniel-Moore conjecture” which says that the order of an A -stable multistep method involving either s derivatives or s implicit stages satisfies p ≤ 2s. Before proceeding to its proof, we should become familiar with Riemann surfaces.

Riemann ist der Mann der glänzenden Intuition. Durch seine umfassende Genialität überragt er alle seine Zeitgenossen ... Im Auftreten schüchtern, ja ungeschickt, musste sich der junge Dozent, zu dem wir Nachgeborenen wie zu einem Heiligen aufblicken, mancherlei Neckereien von seinen Kollegen gefallen lassen.

(F. Klein, Entwicklung der Mathematik im 19. Jhd., p. 246, 247)