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2016

A Numerical Study of Newton Interpolation with Extremely High Degrees

Breuß, Michael, Kemm, Friedemann, Vogel, Oliver

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In current textbooks the use of Chebyshev nodes with Newton interpolation is advocated as the most efficient numerical interpolation method in terms of approximation accuracy and computational effort.

However, we show numerically that the approximation quality obtained by Newton interpolation with Fast Leja (FL) points is competitive to the use of Chebyshev nodes, even for extremely high degree interpolation.
This is an experimental account of the analytic result that the limit distribution of FL points and Chebyshev nodes is the same when letting the number of points go to infinity.
Since the FL construction is easy to perform and allows to add interpolation nodes on the fly in contrast to the use of Chebyshev nodes, our study suggests that Newton interpolation with FL points is currently the most efficient numerical technique for polynomial interpolation.

A Numerical Study of Newton Interpolation with Extremely High Degrees · Around

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Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten
C. Runge · 1901
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Konstruktive Funktionentheorie
I.P. Natanson · 1955
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An introduction to numerical analysis
Kendall E. Atkinson · 1989
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Stability analysis of algorithms for solving confluent Vandermonde-like systems
Nicholas J. Higham · 1990
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Newton interpolation at Leja points
Lothar Reichel · 1990
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