For one-dimensional distributed DFTs using FFTW, matters are slightly more complicated because the data distribution is more closely tied to how the algorithm works. In particular, you can no longer pass an arbitrary block size and must accept FFTW's default; also, the block sizes may be different for input and output. Also, the data distribution depends on the flags and transform direction, in order for forward and backward transforms to work correctly.

ptrdiff_t fftw_mpi_local_size_1d(ptrdiff_t n0, MPI_Comm comm, int sign, unsigned flags, ptrdiff_t *local_ni, ptrdiff_t *local_i_start, ptrdiff_t *local_no, ptrdiff_t *local_o_start);

This function computes the data distribution for a 1d transform of
size `n0`

with the given transform `sign`

and `flags`

.
Both input and output data use block distributions. The input on the
current process will consist of `local_ni`

numbers starting at
index `local_i_start`

; e.g. if only a single process is used,
then `local_ni`

will be `n0`

and `local_i_start`

will
be `0`

. Similarly for the output, with `local_no`

numbers
starting at index `local_o_start`

. The return value of
`fftw_mpi_local_size_1d`

will be the total number of elements to
allocate on the current process (which might be slightly larger than
the local size due to intermediate steps in the algorithm).

As mentioned above (see Load balancing), the data will be divided
equally among the processes if `n0`

is divisible by the
*square* of the number of processes. In this case,
`local_ni`

will equal `local_no`

. Otherwise, they may be
different.

For some applications, such as convolutions, the order of the output
data is irrelevant. In this case, performance can be improved by
specifying that the output data be stored in an FFTW-defined
“scrambled” format. (In particular, this is the analogue of
transposed output in the multidimensional case: scrambled output saves
a communications step.) If you pass `FFTW_MPI_SCRAMBLED_OUT`

in
the flags, then the output is stored in this (undocumented) scrambled
order. Conversely, to perform the inverse transform of data in
scrambled order, pass the `FFTW_MPI_SCRAMBLED_IN`

flag.

In MPI FFTW, only composite sizes `n0`

can be parallelized; we
have not yet implemented a parallel algorithm for large prime sizes.