# Fast and Stable Rational Interpolation in Roots of Unity and Chebyshev Points

@article{Pachn2012FastAS, title={Fast and Stable Rational Interpolation in Roots of Unity and Chebyshev Points}, author={Ricardo Pach{\'o}n and Pedro Gonnet and Joris Van Deun}, journal={SIAM J. Numer. Anal.}, year={2012}, volume={50}, pages={1713-1734} }

A new method for interpolation by rational functions of prescribed numerator and denominator degrees is presented. When the interpolation nodes are roots of unity or Chebyshev points, the algorithm is particularly simple and relies on discrete Fourier transform matrices, which results in a fast implementation using the fast Fourier transform. The method is generalized for arbitrary grids, which requires the construction of polynomials orthogonal on the set of interpolation nodes. The appearance…

## 29 Citations

Algorithms for polynomial and rational approximation

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- 2010

Robust algorithms for the approximation of functions are studied and developed in this thesis. Novel results and algorithms on piecewise polynomial interpolation, rational interpolation and best…

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This article presents a technique of this kind that is related to previous work published in Japanese by Murakami, and shows that using rational interpolation per se suffers from instability; however, related techniques involving real rational filters can be very effective.

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A multivariate, robust, rational interpolation method for propagating uncertainties in several dimensions is presented. The algorithm for selecting numerator and denominator polynomial orders is…

Stable multivariate rational interpolation for parameter-dependent aerospace models

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A multivariate, robust, rational interpolation method for propagating uncertainties in several dimensions is presented. The algorithm for selecting numerator and denominator polynomial orders is…

COMPUTING COMPLEX SINGULARITIES OF DIFFERENTIAL EQUATIONS WITH CHEBFUN

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- 2016

This thesis discusses several topics related to interpolation and how it is used in numerical analysis. It begins with an overview of the aspects of interpolation theory that are relevant to the…

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- Computer Science, MathematicsSIAM J. Sci. Comput.
- 2012

It is shown that for evaluation at points in the complex plane outside $[-1,1]$, the algorithm becomes unstable and should be replaced by the alternative modified Lagrange or “first barycentric” formula dating to Jacobi in 1825.

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- 2011

The barycentric interpolation formula defines a stable algorithm for evaluation at points in [−1, 1] of polynomial interpolants through data on Chebyshev grids. Here it is shown that for evaluation…

Numerical Algorithms Based on Analytic Function Values at Roots of Unity

- Computer Science, MathematicsSIAM J. Numer. Anal.
- 2014

The distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of Cauchy integrals are emphasized and it is shown how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension.

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