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

Nicholas Hale, Alex Townsend

Research output: Contribution to journalArticlepeer-review

31 Scopus citations

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.
Original languageEnglish (US)
Pages (from-to)A148-A167
Number of pages1
JournalSIAM Journal on Scientific Computing
Volume36
Issue number1
DOIs
StatePublished - Feb 6 2014
Externally publishedYes

Fingerprint

Dive into the research topics of 'A Fast, Simple, and Stable Chebyshev--Legendre Transform Using an Asymptotic Formula'. Together they form a unique fingerprint.

Cite this