Commun. Comput. Phys., 5 (2009), pp. 484-497.


Exponentially-Convergent Strategies for Defeating the Runge Phenomenon for the Approximation of Non-Periodic Functions, Part I: Single-Interval Schemes

John P. Boyd 1*, Jun Rong Ong 2

1 Department of Atmospheric, Oceanic and Space Science, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI 48109, USA.
2 Department of Electrical Engineering and Computer Science, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI 48109, USA.

Received 21 August 2007; Accepted (in revised version) 15 November 2007
Available online 1 August 2008

Abstract

Approximating a function from its values $f(x_{i})$ at a set of evenly spaced points $x_{i}$ through $(N+1)$-point polynomial interpolation often fails because of divergence near the endpoints, the ``Runge Phenomenon''. Here we briefly describe seven strategies, each employing a single polynomial over the entire interval, to wholly or partially defeat the Runge Phenomenon such that the error decreases exponentially fast with $N$. Each is successful in obtaining high accuracy for Runge's original example. Unfortunately, each of these single-interval strategies also has liabilities including, depending on the method, various permutations of inefficiency, ill-conditioning and a lack of theory. Even so, the Fourier Extension and Gaussian RBF methods are worthy of further development.

AMS subject classifications: 42C10, 65D05

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Key words: Interpolation, Runge Phenomenon, Fourier extension, radial basis functions.

*Corresponding author.
Email: jpboyd@umich.edu (J. P. Boyd), junrong@umich.edu (J. R. Ong)
 

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