A Fast, Simple, and Stable Chebyshev--Legendre Transform Using an Asymptotic Formula

Handle URI:
http://hdl.handle.net/10754/597266
Title:
A Fast, Simple, and Stable Chebyshev--Legendre Transform Using an Asymptotic Formula
Authors:
Hale, Nicholas; Townsend, Alex
Abstract:
A fast, simple, and numerically stable transform for converting between Legendre and Chebyshev coefficients of a degree N polynomial in O(N(log N)2/ log log N) operations is derived. The fundamental idea of the algorithm is to rewrite a well-known asymptotic formula for Legendre polynomials of large degree as a weighted linear combination of Chebyshev polynomials, which can then be evaluated by using the discrete cosine transform. Numerical results are provided to demonstrate the efficiency and numerical stability. Since the algorithm evaluates a Legendre expansion at an N +1 Chebyshev grid as an intermediate step, it also provides a fast transform between Legendre coefficients and values on a Chebyshev grid. © 2014 Society for Industrial and Applied Mathematics.
Citation:
Hale N, Townsend A (2014) A Fast, Simple, and Stable Chebyshev--Legendre Transform Using an Asymptotic Formula. SIAM Journal on Scientific Computing 36: A148–A167. Available: http://dx.doi.org/10.1137/130932223.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Scientific Computing
KAUST Grant Number:
KUK-C1-013-04
Issue Date:
6-Feb-2014
DOI:
10.1137/130932223
Type:
Article
ISSN:
1064-8275; 1095-7197
Sponsors:
The first author's work was supported by The MathWorks, Inc., and King Abdullah University of Science and Technology (KAUST), award KUK-C1-013-04. The second author's work was supported by EPSRC grant EP/P505666/1 and by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 291068.
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorHale, Nicholasen
dc.contributor.authorTownsend, Alexen
dc.date.accessioned2016-02-25T12:29:23Zen
dc.date.available2016-02-25T12:29:23Zen
dc.date.issued2014-02-06en
dc.identifier.citationHale N, Townsend A (2014) A Fast, Simple, and Stable Chebyshev--Legendre Transform Using an Asymptotic Formula. SIAM Journal on Scientific Computing 36: A148–A167. Available: http://dx.doi.org/10.1137/130932223.en
dc.identifier.issn1064-8275en
dc.identifier.issn1095-7197en
dc.identifier.doi10.1137/130932223en
dc.identifier.urihttp://hdl.handle.net/10754/597266en
dc.description.abstractA fast, simple, and numerically stable transform for converting between Legendre and Chebyshev coefficients of a degree N polynomial in O(N(log N)2/ log log N) operations is derived. The fundamental idea of the algorithm is to rewrite a well-known asymptotic formula for Legendre polynomials of large degree as a weighted linear combination of Chebyshev polynomials, which can then be evaluated by using the discrete cosine transform. Numerical results are provided to demonstrate the efficiency and numerical stability. Since the algorithm evaluates a Legendre expansion at an N +1 Chebyshev grid as an intermediate step, it also provides a fast transform between Legendre coefficients and values on a Chebyshev grid. © 2014 Society for Industrial and Applied Mathematics.en
dc.description.sponsorshipThe first author's work was supported by The MathWorks, Inc., and King Abdullah University of Science and Technology (KAUST), award KUK-C1-013-04. The second author's work was supported by EPSRC grant EP/P505666/1 and by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 291068.en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectAsymptotic formulaen
dc.subjectChebysheven
dc.subjectDiscrete cosine transformen
dc.subjectLegendreen
dc.subjectTransformen
dc.titleA Fast, Simple, and Stable Chebyshev--Legendre Transform Using an Asymptotic Formulaen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Scientific Computingen
dc.contributor.institutionUniversity of Oxford, Oxford, United Kingdomen
kaust.grant.numberKUK-C1-013-04en
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