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Volume 5, Issue 2-4
Exponentially-Convergent Strategies for Defeating the Runge Phenomenon for the Approximation of Non-Periodic Functions, Part I: Single-Interval Schemes

John P. Boyd & Jun Rong Ong

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

Published online: 2009-02

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  • Abstract

Approximating a function from its values f(xi) at a set of evenly spaced points xi 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. 

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@Article{CiCP-5-484, author = {}, title = {Exponentially-Convergent Strategies for Defeating the Runge Phenomenon for the Approximation of Non-Periodic Functions, Part I: Single-Interval Schemes}, journal = {Communications in Computational Physics}, year = {2009}, volume = {5}, number = {2-4}, pages = {484--497}, abstract = {

Approximating a function from its values f(xi) at a set of evenly spaced points xi 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. 

}, issn = {1991-7120}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cicp/7745.html} }
TY - JOUR T1 - Exponentially-Convergent Strategies for Defeating the Runge Phenomenon for the Approximation of Non-Periodic Functions, Part I: Single-Interval Schemes JO - Communications in Computational Physics VL - 2-4 SP - 484 EP - 497 PY - 2009 DA - 2009/02 SN - 5 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cicp/7745.html KW - AB -

Approximating a function from its values f(xi) at a set of evenly spaced points xi 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. 

John P. Boyd & Jun Rong Ong. (2020). Exponentially-Convergent Strategies for Defeating the Runge Phenomenon for the Approximation of Non-Periodic Functions, Part I: Single-Interval Schemes. Communications in Computational Physics. 5 (2-4). 484-497. doi:
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