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<h4 class="subsection">6.4.3 Transposed distributions</h4>
<p>Internally, FFTW's MPI transform algorithms work by first computing
transforms of the data local to each process, then by globally
<em>transposing</em> the data in some fashion to redistribute the data
among the processes, transforming the new data local to each process,
and transposing back. For example, a two-dimensional <code>n0</code> by
<code>n1</code> array, distributed across the <code>n0</code> dimension, is
transformd by: (i) transforming the <code>n1</code> dimension, which are
local to each process; (ii) transposing to an <code>n1</code> by <code>n0</code>
array, distributed across the <code>n1</code> dimension; (iii) transforming
the <code>n0</code> dimension, which is now local to each process; (iv)
transposing back.
<a name="index-transpose-379"></a>
<p>However, in many applications it is acceptable to compute a
multidimensional DFT whose results are produced in transposed order
(e.g., <code>n1</code> by <code>n0</code> in two dimensions). This provides a
significant performance advantage, because it means that the final
transposition step can be omitted. FFTW supports this optimization,
which you specify by passing the flag <code>FFTW_MPI_TRANSPOSED_OUT</code>
to the planner routines. To compute the inverse transform of
transposed output, you specify <code>FFTW_MPI_TRANSPOSED_IN</code> to tell
it that the input is transposed. In this section, we explain how to
interpret the output format of such a transform.
<a name="index-FFTW_005fMPI_005fTRANSPOSED_005fOUT-380"></a><a name="index-FFTW_005fMPI_005fTRANSPOSED_005fIN-381"></a>
<p>Suppose you have are transforming multi-dimensional data with (at
least two) dimensions n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub>. As always, it is distributed along
the first dimension n<sub>0</sub>. Now, if we compute its DFT with the
<code>FFTW_MPI_TRANSPOSED_OUT</code> flag, the resulting output data are stored
with the first <em>two</em> dimensions transposed: n<sub>1</sub> × n<sub>0</sub> × n<sub>2</sub> ×…× n<sub>d-1</sub>,
distributed along the n<sub>1</sub> dimension. Conversely, if we take the
n<sub>1</sub> × n<sub>0</sub> × n<sub>2</sub> ×…× n<sub>d-1</sub> data and transform it with the
<code>FFTW_MPI_TRANSPOSED_IN</code> flag, then the format goes back to the
original n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub> array.
<p>There are two ways to find the portion of the transposed array that
resides on the current process. First, you can simply call the
appropriate ‘<samp><span class="samp">local_size</span></samp>’ function, passing n<sub>1</sub> × n<sub>0</sub> × n<sub>2</sub> ×…× n<sub>d-1</sub> (the
transposed dimensions). This would mean calling the ‘<samp><span class="samp">local_size</span></samp>’
function twice, once for the transposed and once for the
non-transposed dimensions. Alternatively, you can call one of the
‘<samp><span class="samp">local_size_transposed</span></samp>’ functions, which returns both the
non-transposed and transposed data distribution from a single call.
For example, for a 3d transform with transposed output (or input), you
might call:
<pre class="example"> ptrdiff_t fftw_mpi_local_size_3d_transposed(
ptrdiff_t n0, ptrdiff_t n1, ptrdiff_t n2, MPI_Comm comm,
ptrdiff_t *local_n0, ptrdiff_t *local_0_start,
ptrdiff_t *local_n1, ptrdiff_t *local_1_start);
</pre>
<p><a name="index-fftw_005fmpi_005flocal_005fsize_005f3d_005ftransposed-382"></a>
Here, <code>local_n0</code> and <code>local_0_start</code> give the size and
starting index of the <code>n0</code> dimension for the
<em>non</em>-transposed data, as in the previous sections. For
<em>transposed</em> data (e.g. the output for
<code>FFTW_MPI_TRANSPOSED_OUT</code>), <code>local_n1</code> and
<code>local_1_start</code> give the size and starting index of the <code>n1</code>
dimension, which is the first dimension of the transposed data
(<code>n1</code> by <code>n0</code> by <code>n2</code>).
<p>(Note that <code>FFTW_MPI_TRANSPOSED_IN</code> is completely equivalent to
performing <code>FFTW_MPI_TRANSPOSED_OUT</code> and passing the first two
dimensions to the planner in reverse order, or vice versa. If you
pass <em>both</em> the <code>FFTW_MPI_TRANSPOSED_IN</code> and
<code>FFTW_MPI_TRANSPOSED_OUT</code> flags, it is equivalent to swapping the
first two dimensions passed to the planner and passing <em>neither</em>
flag.)
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