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As with the planner interface, the ‘`fftw_mpi_local_size`’
distribution interface is broken into basic and advanced
(‘`_many`’) interfaces, where the latter allows you to specify the
block size manually and also to request block sizes when computing
multiple transforms simultaneously. These functions are documented
more exhaustively by the FFTW MPI Reference, but we summarize the
basic ideas here using a couple of two-dimensional examples.

For the 100 × 200 complex-DFT example, above, we would find the distribution by calling the following function in the basic interface:

ptrdiff_t fftw_mpi_local_size_2d(ptrdiff_t n0, ptrdiff_t n1, MPI_Comm comm, ptrdiff_t *local_n0, ptrdiff_t *local_0_start);

Given the total size of the data to be transformed (here, ```
n0 =
100
```

and `n1 = 200`

) and an MPI communicator (`comm`

), this
function provides three numbers.

First, it describes the shape of the local data: the current process
should store a `local_n0`

by `n1`

slice of the overall
dataset, in row-major order (`n1`

dimension contiguous), starting
at index `local_0_start`

. That is, if the total dataset is
viewed as a `n0`

by `n1`

matrix, the current process should
store the rows `local_0_start`

to
`local_0_start+local_n0-1`

. Obviously, if you are running with
only a single MPI process, that process will store the entire array:
`local_0_start`

will be zero and `local_n0`

will be
`n0`

. See Row-major Format.

Second, the return value is the total number of data elements (e.g.,
complex numbers for a complex DFT) that should be allocated for the
input and output arrays on the current process (ideally with
`fftw_malloc`

or an ‘`fftw_alloc`’ function, to ensure optimal
alignment). It might seem that this should always be equal to
`local_n0 * n1`

, but this is *not* the case. FFTW’s
distributed FFT algorithms require data redistributions at
intermediate stages of the transform, and in some circumstances this
may require slightly larger local storage. This is discussed in more
detail below, under Load balancing.

The advanced-interface ‘`local_size`’ function for multidimensional
transforms returns the same three things (`local_n0`

,
`local_0_start`

, and the total number of elements to allocate),
but takes more inputs:

ptrdiff_t fftw_mpi_local_size_many(int rnk, const ptrdiff_t *n, ptrdiff_t howmany, ptrdiff_t block0, MPI_Comm comm, ptrdiff_t *local_n0, ptrdiff_t *local_0_start);

The two-dimensional case above corresponds to `rnk = 2`

and an
array `n`

of length 2 with `n[0] = n0`

and `n[1] = n1`

.
This routine is for any `rnk > 1`

; one-dimensional transforms
have their own interface because they work slightly differently, as
discussed below.

First, the advanced interface allows you to perform multiple
transforms at once, of interleaved data, as specified by the
`howmany`

parameter. (`hoamany`

is 1 for a single
transform.)

Second, here you can specify your desired block size in the `n0`

dimension, `block0`

. To use FFTW’s default block size, pass
`FFTW_MPI_DEFAULT_BLOCK`

(0) for `block0`

. Otherwise, on
`P`

processes, FFTW will return `local_n0`

equal to
`block0`

on the first `P / block0`

processes (rounded down),
return `local_n0`

equal to `n0 - block0 * (P / block0)`

on
the next process, and `local_n0`

equal to zero on any remaining
processes. In general, we recommend using the default block size
(which corresponds to `n0 / P`

, rounded up).

For example, suppose you have `P = 4`

processes and ```
n0 =
21
```

. The default will be a block size of `6`

, which will give
`local_n0 = 6`

on the first three processes and ```
local_n0 =
3
```

on the last process. Instead, however, you could specify
`block0 = 5`

if you wanted, which would give `local_n0 = 5`

on processes 0 to 2, `local_n0 = 6`

on process 3. (This choice,
while it may look superficially more “balanced,” has the same
critical path as FFTW’s default but requires more communications.)

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