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

Nicholas Hale | Zendy; Alex Townsend | Zendy

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

Author(s) -

Nicholas Hale,

Alex Townsend

Publication year - 2014

Publication title -

siam journal on scientific computing

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.674

H-Index - 147

eISSN - 1095-7197

pISSN - 1064-8275

DOI - 10.1137/130932223

Subject(s) - mathematics , legendre polynomials , chebyshev polynomials , chebyshev filter , legendre function , chebyshev nodes , chebyshev pseudospectral method , associated legendre polynomials , mathematical analysis , legendre wavelet , chebyshev equation , classical orthogonal polynomials , gegenbauer polynomials , orthogonal polynomials , wavelet transform , discrete wavelet transform , artificial intelligence , computer science , wavelet

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

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