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On the numerical stability of linear barycentric rational interpolation

Authors :
Fuda, C
Campagna, R
Hormann, K
Fuda, C
Campagna, R
Hormann, K
Source :
Numerische Mathematik. 152:761-786
Publication Year :
2022
Publisher :
Springer Science and Business Media LLC, 2022.

Abstract

The barycentric forms of polynomial and rational interpolation have recently gained popularity, because they can be computed with simple, efficient, and numerically stable algorithms. In this paper, we show more generally that the evaluation of any function that can be expressed as$$r(x)=\sum _{i=0}^n a_i(x) f_i\big /\sum _{j=0}^m b_j(x)$$r(x)=∑i=0nai(x)fi/∑j=0mbj(x)in terms of data values$$f_i$$fiand some functions$$a_i$$aiand$$b_j$$bjfor$$i=0,\ldots ,n$$i=0,…,nand$$j=0,\dots ,m$$j=0,⋯,mwith a simple algorithm that first sums up the terms in the numerator and the denominator, followed by a final division, is forward and backward stable under certain assumptions. This result includes the two barycentric forms of rational interpolation as special cases. Our analysis further reveals that the stability of the second barycentric form depends on the Lebesgue constant associated with the interpolation nodes, which typically grows withn, whereas the stability of the first barycentric form depends on a similar, but different quantity, that can be bounded in terms of the mesh ratio, regardless ofn. We support our theoretical results with numerical experiments.

Details

ISSN :
09453245 and 0029599X
Volume :
152
Database :
OpenAIRE
Journal :
Numerische Mathematik
Accession number :
edsair.doi.dedup.....53d8731c1931ac082a8bc83e3dc2d716