The numerical stability of barycentric Lagrange interpolation

Abstract

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

Chebfun: A New Kind of Numerical Computing

“…The formula has been proved to be stable by Higham in [9] and requires O(MN) operations to evaluate a chebfun at M points. The plot command, for instance, relies on evaluations at thousands of points.…”

Barycentric rational interpolation with asymptotically monitored poles

“…This formula is the barycentric form of the interpolating polynomial and is known to be a very stable way of evaluating the interpolating polynomial, see [5] or [6] for more details. For "special" points, such as equidistant points or Chebyshev points of the first and the second kind (see for example [7,8] or [5]), formulas for the weights w j are known.…”

Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

“…• Barycentric Form: (See [4,13]) Let P(z) be the matrix polynomial taking on the values [P 0 , P 1 , . .…”

“…This form is numerically stable to evaluate (in [13] it is shown that the numerical evaluation of a scalar polynomial is stable in this form; the generalization to matrix polynomials is immediate). Note that if…”