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1 | 1 | # FastTransforms.jl
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2 | 2 |
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| -[](https://travis-ci.org/MikaelSlevinsky/FastTransforms.jl) [](http://fasttransformsjl.readthedocs.org/en/latest/?badge=latest) |
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| -The aim of this package is to provide a new class of fast transforms |
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| -based on the use of asymptotic formulae to relate the transforms to a small |
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| -number of fast Fourier transforms. This new class of fast transforms does not |
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| -require large pre-computation for fast execution, and they are designed |
| 3 | +[](https://travis-ci.org/MikaelSlevinsky/FastTransforms.jl) [](https://MikaelSlevinsky.github.io/FastTransforms.jl/stable) [](https://MikaelSlevinsky.github.io/FastTransforms.jl/latest) |
| 4 | + |
| 5 | +The aim of this package is to provide new classes of fast transforms with low |
| 6 | +pre-computation. One approach is based on the use of asymptotic formulae to |
| 7 | +relate the transforms to a small number of fast Fourier transforms. Another |
| 8 | +approach is based on a Toeplitz-dot-Hankel decomposition of the matrix of |
| 9 | +connection coefficients. Both new classes of fast transforms do not |
| 10 | +require large pre-computation for fast execution and they are designed |
9 | 11 | to work on expansions of functions with any degree of regularity.
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10 | 12 |
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11 | 13 | The Chebyshev—Jacobi transform and its inverse are implemented. This
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@@ -41,6 +43,11 @@ is valid for the half-open square `(α,β) ∈ (-1/2,1/2]^2`. Therefore, the fas
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41 | 43 | when the parameters are inside. If the parameters `(α,β)` are not exceptionally beyond the square,
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42 | 44 | then increment/decrement operators are used with linear complexity (and linear conditioning) in the degree.
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43 | 45 |
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| 46 | +The Padua transform and its inverse are also implemented thanks to |
| 47 | +[Michael Clarke](https://github.com/MikeAClarke). These are optimized methods |
| 48 | +designed for computing the bivariate Chebyshev coefficients by interpolating a |
| 49 | +bivariate function at the Padua points on `[-1,1]^2`. |
| 50 | + |
44 | 51 | # References:
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45 | 52 |
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46 | 53 | 1. N. Hale and A. Townsend. <a href="http://dx.doi.org/10.1137/130932223">A fast, simple, and stable Chebyshev—Legendre transform using and asymptotic formula</a>, SIAM J. Sci. Comput., 36:A148—A167, 2014.
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