Matrices for the direct determination of the barycentric weights of rational interpolation

✍ Scribed by Jean-Paul Berrut; Hans D. Mittelmann

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 813 KB
• Volume: 78
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(96)00163-x

✦ Synopsis

Let x0 .... ,XN be N + 1 interpolation points (nodes) and f0,...,fN be N+ 1 interpolation data. Then every rational function r with numerator and denominator degrees ~<N interpolating these values can be written in its barycentric form

r(x) = x-x---SJkl x-xk,

which is completely determined by a vector u of N+ 1 barycentric weights u,. Finding u is therefore an alternative to the determination of the coefficients in the canonical form of r; it is advantageous inasmuch as u contains information about unattainable points and poles.

In classical rational interpolation the numerator and the denominator of r are made unique (up to a constant factor) by restricting their respective degrees. We determine here the corresponding vectors u by applying a stabilized elimination algorithm to a matrix whose kernel is the space spanned by the u's. The method is of complexity (9(n 3) in terms of the denominator degree n; it seems on the other hand to be among the most stable ones.