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<h3 class="section">7.2 Reversing array dimensions</h3>
<p><a name="index-row_002dmajor-517"></a><a name="index-column_002dmajor-518"></a>A minor annoyance in calling FFTW from Fortran is that FFTW's array
dimensions are defined in the C convention (row-major order), while
Fortran's array dimensions are the opposite convention (column-major
order). See <a href="Multi_002ddimensional-Array-Format.html#Multi_002ddimensional-Array-Format">Multi-dimensional Array Format</a>. This is just a
bookkeeping difference, with no effect on performance. The only
consequence of this is that, whenever you create an FFTW plan for a
multi-dimensional transform, you must always <em>reverse the
ordering of the dimensions</em>.
<p>For example, consider the three-dimensional (L × M × N) arrays:
<pre class="example"> complex(C_DOUBLE_COMPLEX), dimension(L,M,N) :: in, out
</pre>
<p>To plan a DFT for these arrays using <code>fftw_plan_dft_3d</code>, you could do:
<p><a name="index-fftw_005fplan_005fdft_005f3d-519"></a>
<pre class="example"> plan = fftw_plan_dft_3d(N,M,L, in,out, FFTW_FORWARD,FFTW_ESTIMATE)
</pre>
<p>That is, from FFTW's perspective this is a N × M × L array.
<em>No data transposition need occur</em>, as this is <em>only
notation</em>. Similarly, to use the more generic routine
<code>fftw_plan_dft</code> with the same arrays, you could do:
<pre class="example"> integer(C_INT), dimension(3) :: n = [N,M,L]
plan = fftw_plan_dft_3d(3, n, in,out, FFTW_FORWARD,FFTW_ESTIMATE)
</pre>
<p>Note, by the way, that this is different from the legacy Fortran
interface (see <a href="Fortran_002dinterface-routines.html#Fortran_002dinterface-routines">Fortran-interface routines</a>), which automatically
reverses the order of the array dimension for you. Here, you are
calling the C interface directly, so there is no “translation” layer.
<p><a name="index-r2c_002fc2r-multi_002ddimensional-array-format-520"></a>An important thing to keep in mind is the implication of this for
multidimensional real-to-complex transforms (see <a href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data">Multi-Dimensional DFTs of Real Data</a>). In C, a multidimensional real-to-complex DFT
chops the last dimension roughly in half (N × M × L real input
goes to N × M × L/2+1 complex output). In Fortran, because
the array dimension notation is reversed, the <em>first</em> dimension of
the complex data is chopped roughly in half. For example consider the
‘<samp><span class="samp">r2c</span></samp>’ transform of L × M × N real input in Fortran:
<p><a name="index-fftw_005fplan_005fdft_005fr2c_005f3d-521"></a><a name="index-fftw_005fexecute_005fdft_005fr2c-522"></a>
<pre class="example"> type(C_PTR) :: plan
real(C_DOUBLE), dimension(L,M,N) :: in
complex(C_DOUBLE_COMPLEX), dimension(L/2+1,M,N) :: out
plan = fftw_plan_dft_r2c_3d(N,M,L, in,out, FFTW_ESTIMATE)
...
call fftw_execute_dft_r2c(plan, in, out)
</pre>
<p><a name="index-in_002dplace-523"></a><a name="index-padding-524"></a>Alternatively, for an in-place r2c transform, as described in the C
documentation we must <em>pad</em> the <em>first</em> dimension of the
real input with an extra two entries (which are ignored by FFTW) so as
to leave enough space for the complex output. The input is
<em>allocated</em> as a 2[L/2+1] × M × N array, even though only
L × M × N of it is actually used. In this example, we will
allocate the array as a pointer type, using ‘<samp><span class="samp">fftw_alloc</span></samp>’ to
ensure aligned memory for maximum performance (see <a href="Allocating-aligned-memory-in-Fortran.html#Allocating-aligned-memory-in-Fortran">Allocating aligned memory in Fortran</a>); this also makes it easy to reference the
same memory as both a real array and a complex array.
<p><a name="index-fftw_005falloc_005fcomplex-525"></a><a name="index-c_005ff_005fpointer-526"></a>
<pre class="example"> real(C_DOUBLE), pointer :: in(:,:,:)
complex(C_DOUBLE_COMPLEX), pointer :: out(:,:,:)
type(C_PTR) :: plan, data
data = fftw_alloc_complex(int((L/2+1) * M * N, C_SIZE_T))
call c_f_pointer(data, in, [2*(L/2+1),M,N])
call c_f_pointer(data, out, [L/2+1,M,N])
plan = fftw_plan_dft_r2c_3d(N,M,L, in,out, FFTW_ESTIMATE)
...
call fftw_execute_dft_r2c(plan, in, out)
...
call fftw_destroy_plan(plan)
call fftw_free(data)
</pre>
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