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110 lines
3.2 KiB
ReStructuredText
110 lines
3.2 KiB
ReStructuredText
.. _acb-dft:
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**acb_dft.h** -- Discrete Fourier Transform on finite abelian groups
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===================================================================================
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*Warning: the interfaces in this module are experimental and may change
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without notice.*
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Let *G* be a finite abelian group, and `\chi` a character of *G*.
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For any map `f:G\to\mathbb C`, the discrete fourier transform
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`\hat f:\hat G\to \mathbb C` is defined by
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.. math::
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\hat f(\chi) = \sum_{x\in G}\overline{\chi(x)}f(x)
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Fast Fourier Transform techniques allow to compute efficiently
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all values `\hat f(\chi)` by reusing common computations.
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Specifically, if `H\triangleleft G` is a subgroup of size `M` and index
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`[G:H]=m`, then writing `f_x(h)=f(xh)` the translate of `f` by representatives
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`x` of `G/H`, one has a decomposition
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.. math::
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\hat f(\chi) = \sum_{x\in G/H} \overline{\chi(x)} \hat{f_x}(\chi_{H})
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so that the DFT on `G` can be computed using `m` DFT on `H` (of
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appropriate translates of `f`), then `M` DFT on `G/H`, one for
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each restriction `\chi_{H}`.
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This decomposition can be done recursively.
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Note that by inversion formula
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.. math::
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\widehat{\hat f}(\chi) = \#G\times f(\chi^{-1})
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it is straightforward to recover `f` from its DFT `\hat f`.
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DFT on Z/nZ
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-------------------------------------------------------------------------------
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If `G=\mathbb Z/n\mathbb Z`, we compute the DFT according to the usual convention
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.. math::
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w_x = \sum_{y\bmod n} v_y e^{-\frac{2iπ}nxy}
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.. function:: void acb_dirichlet_dft_pol(acb_ptr w, acb_srcptr v, slong n, slong prec)
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.. function:: void acb_dirichlet_dft_crt(acb_ptr w, acb_srcptr v, slong n, slong prec)
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.. function:: void acb_dirichlet_dft_cyc(acb_ptr w, acb_srcptr v, slong n, slong prec)
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Set *w* to the DFT of *v* of length *len*.
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The first variant uses the naive `O(n^2)` algorithm.
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The second one uses CRT to express `Z/nZ` as a product of cyclic groups.
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The *cyc* version uses each prime factor of `m` of `n` to decompose with
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the subgroup `H=m\mathbb Z/n\mathbb Z`.
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DFT on products
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-------------------------------------------------------------------------------
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A finite abelian group is isomorphic to a product of cyclic components
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.. math::
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G = \bigoplus_{i=1}^r \mathbb Z/n_i\mathbb Z
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then a character is a product of characters of all components and the DFT reads
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.. math::
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\hat f(x_1,\dots x_r) = \sum_{y_1\dots y_r} f(y_1,\dots y_r)
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e^{-2iπ\sum\frac{x_i y_i}{n_i}}
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We assume that `f` is given by a vector of length `\prod n_i` corresponding
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to a lexicographic ordering of the values `y_1,\dots y_r`, and the computation
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returns the same indexing for values of `\hat f`.
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.. function:: void acb_dirichlet_dft_prod(acb_ptr w, acb_srcptr v, slong * cyc, slong num, slong prec);
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Computes the DFT on the group product of *num* cyclic components of sizes *cyc*.
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Precomputations
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-------------------------------------------------------------------------------
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Convolution
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-------------------------------------------------------------------------------
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For functions `f` and `g` on `G` we consider the convolution
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.. math::
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(f \star g)(x) = \sum_{y\in G} f(x-y)g(y)
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which satisfies
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.. math::
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\widehat{f \star g}(\chi) = \hat f(\chi)\hat g(\chi)
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