The Mathematical Institute, University of Oxford, Eprints Archive

A fast, simple, and stable Chebyshev-Legendre transform using an asymptotic formula

Hale, Nicholas and Townsend, Alex (2013) A fast, simple, and stable Chebyshev-Legendre transform using an asymptotic formula. Technical Report. Unspecified. (Submitted)

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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 basis 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.

Item Type:Technical Report (Technical Report)
Subjects:A - C > Approximations and expansions
H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1735
Deposited By: Lotti Ekert
Deposited On:08 Aug 2013 07:41
Last Modified:29 May 2015 19:25

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