The need for knowledge and reliability in numeric computation: Case study of multivariate Padé approximation

In this paper we review and link the numeric research projects carried out at the Department of Mathematics and Computer Science of the University of Antwerp since 1978. Results have and are being obtained in various areas. A lot of effort has been put in the theoretical investigation of the multivariate Padé approximation problem using different definitions (see Sections 3 and 7). The numerical implementation raises two delicate issues. First, there is the need to see the wood for the trees again: switching from one to many variables greatly increases the number of choices to be made on the way (see Sections 1 and 5). Second, there is the typical problem of breakdown when computing ratios of determinants: the added value of interval arithmetic combined with defect correction turns out to be significant (see Sections 2 and 4). In Section 6 these two problems are thoroughly illustrated and the interested reader is taken by the hand and guided through a typical computation session. On the way some open problems are indicated which motivate us to continue our research mainly in the area of gathering and offering more knowledge about the problem domain on one hand, and improving the arithmetic tools and numerical routines for a reliable computation of the approximants on the other hand.