853 research outputs found
A fast, simple, and stable Chebyshev-Legendre transform using an asymptotic formula
A fast, simple, and numerically stable transform for converting between Legendre and Chebyshev coefficients of a degree polynomial in 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 Chebyshev grid as an intermediate step, it also provides a fast transform between Legendre coefficients and values on a Chebyshev grid
A fast analysis-based discrete Hankel transform using asymptotic expansions
A fast and numerically stable algorithm is described for computing the
discrete Hankel transform of order as well as evaluating Schl\"{o}milch and
Fourier--Bessel expansions in
operations. The algorithm is based on an asymptotic expansion for Bessel
functions of large arguments, the fast Fourier transform, and the Neumann
addition formula. All the algorithmic parameters are selected from error bounds
to achieve a near-optimal computational cost for any accuracy goal. Numerical
results demonstrate the efficiency of the resulting algorithm.Comment: 22 page
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