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Polynomial and Rational Interpolation

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A Graduate Introduction to Numerical Methods

Abstract

This chapter gives a detailed discussion of barycentric Lagrange and Hermite interpolation and extends this to rational interpolation with a specified denominator. We discuss the conditioning of these interpolants. A numerically stable method to find roots of polynomials expressed in barycentric form via a generalized eigenvalue problem is given. We conclude with a section on piecewise polynomial interpolants. ⊲

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Notes

  1. 1.

    Note the different use of “Lagrange interpolating polynomial” and “Lagrange basis polynomials.”

  2. 2.

    A useful fact, from the numerical point of view, is that we may scale all \(\beta _{k}\) by any common factor. This can prevent unnecessary overflow or underflow.

  3. 3.

    See also the forthcoming paper Stability of rootfinding for barycentric Lagrange enterpolants by Lawrence et al. (2013).

  4. 4.

    This series needs to be convergent for these manipulations to be valid. It is, for z close enough to τ ℓ —in fact, for all z closer to τ ℓ than t is. Since t is different from each node by hypothesis, this is possible.

  5. 5.

    Or by more efficient ones such as are described in van Deun et al. (2008).

  6. 6.

    Also, as Gerhard Wanner tells us, Runge used contour integrals too in his celebrated 1901 paper for just this purpose.

  7. 7.

    Some people might confuse this problem with the famous Wilkinson polynomial \(W(z) = (z - 1)(z - 2)\cdots (z - 20)\). Don’t. This problem has nothing to do with that polynomial.

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Authors and Affiliations

  1. Applied Mathematics, University of Western Ontario, London, ON, Canada

    Robert M. Corless & Nicolas Fillion

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  1. Robert M. Corless
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  2. Nicolas Fillion
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Corless, R.M., Fillion, N. (2013). Polynomial and Rational Interpolation. In: A Graduate Introduction to Numerical Methods. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8453-0_8

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