The numerical stability of barycentric Lagrange interpolation

Abstract

The Lagrange representation of the interpolating polynomial can be rewritten in two\nmore computationally attractive forms: a modified Lagrange form and a barycentric\nform. We give an error analysis of the evaluation of the interpolating polynomial using\nthese two forms. The modified Lagrange formula is shown to be backward stable. The\nbarycentric formula has a less favourable error analysis, but is forward stable for any set of\ninterpolating points with a small Lebesgue constant. Therefore the barycentric formula can\nbe significantly less accurate than the modified Lagrange formula only for a poor choice of\ninterpolating points. This analysis provides further weight to the argument of Berrut and\nTrefethen that barycentric Lagrange interpolation should be the polynomial interpolation\nmethod of choice.