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This manual is for FFTW
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(version 3.1.2, 23 June 2006).
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Copyright (C) 2003 Matteo Frigo.
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Copyright (C) 2003 Massachusetts Institute of Technology.
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<a name="More-DFTs-of-Real-Data"></a>
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Previous: <a rel="previous" accesskey="p" href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data">Multi-Dimensional DFTs of Real Data</a>,
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Up: <a rel="up" accesskey="u" href="Tutorial.html#Tutorial">Tutorial</a>
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<h3 class="section">2.5 More DFTs of Real Data</h3>
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<li><a accesskey="1" href="The-Halfcomplex_002dformat-DFT.html#The-Halfcomplex_002dformat-DFT">The Halfcomplex-format DFT</a>
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<li><a accesskey="2" href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029">Real even/odd DFTs (cosine/sine transforms)</a>
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<li><a accesskey="3" href="The-Discrete-Hartley-Transform.html#The-Discrete-Hartley-Transform">The Discrete Hartley Transform</a>
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<p>FFTW supports several other transform types via a unified <dfn>r2r</dfn>
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(real-to-real) interface,
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<a name="index-r2r-64"></a>so called because it takes a real (<code>double</code>) array and outputs a
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real array of the same size. These r2r transforms currently fall into
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three categories: DFTs of real input and complex-Hermitian output in
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halfcomplex format, DFTs of real input with even/odd symmetry
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(a.k.a. discrete cosine/sine transforms, DCTs/DSTs), and discrete
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Hartley transforms (DHTs), all described in more detail by the
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<p>The r2r transforms follow the by now familiar interface of creating an
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<code>fftw_plan</code>, executing it with <code>fftw_execute(plan)</code>, and
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destroying it with <code>fftw_destroy_plan(plan)</code>. Furthermore, all
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r2r transforms share the same planner interface:
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<pre class="example"> fftw_plan fftw_plan_r2r_1d(int n, double *in, double *out,
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fftw_r2r_kind kind, unsigned flags);
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fftw_plan fftw_plan_r2r_2d(int nx, int ny, double *in, double *out,
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fftw_r2r_kind kindx, fftw_r2r_kind kindy,
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fftw_plan fftw_plan_r2r_3d(int nx, int ny, int nz,
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double *in, double *out,
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fftw_plan fftw_plan_r2r(int rank, const int *n, double *in, double *out,
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const fftw_r2r_kind *kind, unsigned flags);
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<p><a name="index-fftw_005fplan_005fr2r_005f1d-65"></a><a name="index-fftw_005fplan_005fr2r_005f2d-66"></a><a name="index-fftw_005fplan_005fr2r_005f3d-67"></a><a name="index-fftw_005fplan_005fr2r-68"></a>
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Just as for the complex DFT, these plan 1d/2d/3d/multi-dimensional
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transforms for contiguous arrays in row-major order, transforming (real)
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input to output of the same size, where <code>n</code> specifies the
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<em>physical</em> dimensions of the arrays. All positive <code>n</code> are
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supported (with the exception of <code>n=1</code> for the <code>FFTW_REDFT00</code>
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kind, noted in the real-even subsection below); products of small
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factors are most efficient (factorizing <code>n-1</code> and <code>n+1</code> for
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<code>FFTW_REDFT00</code> and <code>FFTW_RODFT00</code> kinds, described below), but
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an <i>O</i>(<i>n</i> log <i>n</i>) algorithm is used even for prime sizes.
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<p>Each dimension has a <dfn>kind</dfn> parameter, of type
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<code>fftw_r2r_kind</code>, specifying the kind of r2r transform to be used
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<a name="index-kind-_0028r2r_0029-69"></a><a name="index-fftw_005fr2r_005fkind-70"></a>(In the case of <code>fftw_plan_r2r</code>, this is an array <code>kind[rank]</code>
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where <code>kind[i]</code> is the transform kind for the dimension
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<code>n[i]</code>.) The kind can be one of a set of predefined constants,
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defined in the following subsections.
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<p>In other words, FFTW computes the separable product of the specified
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r2r transforms over each dimension, which can be used e.g. for partial
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differential equations with mixed boundary conditions. (For some r2r
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kinds, notably the halfcomplex DFT and the DHT, such a separable
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product is somewhat problematic in more than one dimension, however,
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as is described below.)
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<p>In the current version of FFTW, all r2r transforms except for the
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halfcomplex type are computed via pre- or post-processing of
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halfcomplex transforms, and they are therefore not as fast as they
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could be. Since most other general DCT/DST codes employ a similar
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algorithm, however, FFTW's implementation should provide at least
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competitive performance.
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