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In this paper, we show more generally that the evaluation of any function that can be expressed as<jats:inline-formula><jats:alternatives><jats:tex-math>$$r(x)=\\sum _{i=0}^n a_i(x) f_i\\big \/\\sum _{j=0}^m b_j(x)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>r<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mo>=<\/mml:mo><mml:msubsup><mml:mo>\u2211<\/mml:mo><mml:mrow><mml:mi>i<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:mrow><mml:mi>n<\/mml:mi><\/mml:msubsup><mml:msub><mml:mi>a<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><mml:msub><mml:mi>f<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><mml:mrow><mml:mo>\/<\/mml:mo><\/mml:mrow><mml:msubsup><mml:mo>\u2211<\/mml:mo><mml:mrow><mml:mi>j<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:mrow><mml:mi>m<\/mml:mi><\/mml:msubsup><mml:msub><mml:mi>b<\/mml:mi><mml:mi>j<\/mml:mi><\/mml:msub><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>in terms of data values<jats:inline-formula><jats:alternatives><jats:tex-math>$$f_i$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>f<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and some functions<jats:inline-formula><jats:alternatives><jats:tex-math>$$a_i$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>a<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:tex-math>$$b_j$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>b<\/mml:mi><mml:mi>j<\/mml:mi><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>for<jats:inline-formula><jats:alternatives><jats:tex-math>$$i=0,\\ldots ,n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>i<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:tex-math>$$j=0,\\dots ,m$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>j<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mo>\u22ef<\/mml:mo><mml:mo>,<\/mml:mo><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>with 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 with<jats:italic>n<\/jats:italic>, 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 of<jats:italic>n<\/jats:italic>. We support our theoretical results with numerical experiments.<\/jats:p>","DOI":"10.1007\/s00211-022-01316-w","type":"journal-article","created":{"date-parts":[[2022,9,17]],"date-time":"2022-09-17T14:02:26Z","timestamp":1663423346000},"page":"761-786","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["On the numerical stability of linear barycentric rational interpolation"],"prefix":"10.1007","volume":"152","author":[{"given":"Chiara","family":"Fuda","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rosanna","family":"Campagna","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kai","family":"Hormann","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,9,17]]},"reference":[{"issue":"4","key":"1316_CR1","doi-asserted-by":"publisher","first-page":"547","DOI":"10.1093\/imanum\/24.4.547","volume":"24","author":"NJ Higham","year":"2004","unstructured":"Higham, N.J.: The numerical stability of barycentric Lagrange interpolation. 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