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<a name="Complex-Multi_002dDimensional-DFTs"></a>
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<p>
Next: <a href="One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data" accesskey="n" rel="next">One-Dimensional DFTs of Real Data</a>, Previous: <a href="Complex-One_002dDimensional-DFTs.html#Complex-One_002dDimensional-DFTs" accesskey="p" rel="prev">Complex One-Dimensional DFTs</a>, Up: <a href="Tutorial.html#Tutorial" accesskey="u" rel="up">Tutorial</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Complex-Multi_002dDimensional-DFTs-1"></a>
<h3 class="section">2.2 Complex Multi-Dimensional DFTs</h3>

<p>Multi-dimensional transforms work much the same way as one-dimensional
transforms: you allocate arrays of <code>fftw_complex</code> (preferably
using <code>fftw_malloc</code>), create an <code>fftw_plan</code>, execute it as
many times as you want with <code>fftw_execute(plan)</code>, and clean up
with <code>fftw_destroy_plan(plan)</code> (and <code>fftw_free</code>).  
</p>
<p>FFTW provides two routines for creating plans for 2d and 3d transforms,
and one routine for creating plans of arbitrary dimensionality.
The 2d and 3d routines have the following signature:
</p><div class="example">
<pre class="example">fftw_plan fftw_plan_dft_2d(int n0, int n1,
                           fftw_complex *in, fftw_complex *out,
                           int sign, unsigned flags);
fftw_plan fftw_plan_dft_3d(int n0, int n1, int n2,
                           fftw_complex *in, fftw_complex *out,
                           int sign, unsigned flags);
</pre></div>
<a name="index-fftw_005fplan_005fdft_005f2d"></a>
<a name="index-fftw_005fplan_005fdft_005f3d"></a>

<p>These routines create plans for <code>n0</code> by <code>n1</code> two-dimensional
(2d) transforms and <code>n0</code> by <code>n1</code> by <code>n2</code> 3d transforms,
respectively.  All of these transforms operate on contiguous arrays in
the C-standard <em>row-major</em> order, so that the last dimension has the
fastest-varying index in the array.  This layout is described further in
<a href="Multi_002ddimensional-Array-Format.html#Multi_002ddimensional-Array-Format">Multi-dimensional Array Format</a>.
</p>
<p>FFTW can also compute transforms of higher dimensionality.  In order to
avoid confusion between the various meanings of the the word
&ldquo;dimension&rdquo;, we use the term <em>rank</em>
<a name="index-rank"></a>
to denote the number of independent indices in an array.<a name="DOCF2" href="#FOOT2"><sup>2</sup></a>  For
example, we say that a 2d transform has rank&nbsp;2, a 3d transform has
rank&nbsp;3, and so on.  You can plan transforms of arbitrary rank by
means of the following function:
</p>
<div class="example">
<pre class="example">fftw_plan fftw_plan_dft(int rank, const int *n,
                        fftw_complex *in, fftw_complex *out,
                        int sign, unsigned flags);
</pre></div>
<a name="index-fftw_005fplan_005fdft"></a>

<p>Here, <code>n</code> is a pointer to an array <code>n[rank]</code> denoting an
<code>n[0]</code> by <code>n[1]</code> by &hellip; by <code>n[rank-1]</code> transform.
Thus, for example, the call
</p><div class="example">
<pre class="example">fftw_plan_dft_2d(n0, n1, in, out, sign, flags);
</pre></div>
<p>is equivalent to the following code fragment:
</p><div class="example">
<pre class="example">int n[2];
n[0] = n0;
n[1] = n1;
fftw_plan_dft(2, n, in, out, sign, flags);
</pre></div>
<p><code>fftw_plan_dft</code> is not restricted to 2d and 3d transforms,
however, but it can plan transforms of arbitrary rank.
</p>
<p>You may have noticed that all the planner routines described so far
have overlapping functionality.  For example, you can plan a 1d or 2d
transform by using <code>fftw_plan_dft</code> with a <code>rank</code> of <code>1</code>
or <code>2</code>, or even by calling <code>fftw_plan_dft_3d</code> with <code>n0</code>
and/or <code>n1</code> equal to <code>1</code> (with no loss in efficiency).  This
pattern continues, and FFTW&rsquo;s planning routines in general form a
&ldquo;partial order,&rdquo; sequences of
<a name="index-partial-order"></a>
interfaces with strictly increasing generality but correspondingly
greater complexity.
</p>
<p><code>fftw_plan_dft</code> is the most general complex-DFT routine that we
describe in this tutorial, but there are also the advanced and guru interfaces,
<a name="index-advanced-interface-1"></a>
<a name="index-guru-interface-1"></a>
which allow one to efficiently combine multiple/strided transforms
into a single FFTW plan, transform a subset of a larger
multi-dimensional array, and/or to handle more general complex-number
formats.  For more information, see <a href="FFTW-Reference.html#FFTW-Reference">FFTW Reference</a>.
</p>
<div class="footnote">
<hr>
<h4 class="footnotes-heading">Footnotes</h4>

<h3><a name="FOOT2" href="#DOCF2">(2)</a></h3>
<p>The
term &ldquo;rank&rdquo; is commonly used in the APL, FORTRAN, and Common Lisp
traditions, although it is not so common in the C&nbsp;world.</p>
</div>
<hr>
<div class="header">
<p>
Next: <a href="One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data" accesskey="n" rel="next">One-Dimensional DFTs of Real Data</a>, Previous: <a href="Complex-One_002dDimensional-DFTs.html#Complex-One_002dDimensional-DFTs" accesskey="p" rel="prev">Complex One-Dimensional DFTs</a>, Up: <a href="Tutorial.html#Tutorial" accesskey="u" rel="up">Tutorial</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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