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# Parallel FFTW

In this chapter we discuss the use of FFTW in a parallel environment, documenting the different parallel libraries that we have provided. (Users calling FFTW from a multi-threaded program should also consult Section Thread safety.) The FFTW package currently contains three parallel transform implementations that leverage the uniprocessor FFTW code:

- The first set of routines utilizes shared-memory threads for parallel one- and multi-dimensional transforms of both real and complex data. Any program using FFTW can be trivially modified to use the multi-threaded routines. This code can use any common threads implementation, including POSIX threads. (POSIX threads are available on most Unix variants, including Linux.) These routines are located in the `threads` directory, and are documented in Section Multi-threaded FFTW.
- The `mpi` directory contains multi-dimensional transforms of real and complex data for parallel machines supporting MPI. It also includes parallel one-dimensional transforms for complex data. The main feature of this code is that it supports distributed-memory transforms, so it runs on everything from workstation clusters to massively-parallel supercomputers. More information on MPI can be found at the MPI home page. The FFTW MPI routines are documented in Section MPI FFTW.
- We also have an experimental parallel implementation written in Cilk, a C-like parallel language developed at MIT and currently available for several SMP platforms. For more information on Cilk see the Cilk home page. The FFTW Cilk code can be found in the `cilk` directory, with parallelized one- and multi-dimensional transforms of complex data. The Cilk FFTW routines are documented in `cilk/README`.

## Multi-threaded FFTW

In this section we document the parallel FFTW routines for shared-memory threads on SMP hardware. These routines, which support parallel one- and multi-dimensional transforms of both real and complex data, are the easiest way to take advantage of multiple processors with FFTW. They work just like the corresponding uniprocessor transform routines, except that they take the number of parallel threads to use as an extra parameter. Any program that uses the uniprocessor FFTW can be trivially modified to use the multi-threaded FFTW.

### Installation and Supported Hardware/Software

All of the FFTW threads code is located in the `threads` subdirectory of the FFTW package. On Unix systems, the FFTW threads libraries and header files can be automatically configured, compiled, and installed along with the uniprocessor FFTW libraries simply by including `--enable-threads` in the flags to the `configure` script (see Section Installation on Unix). (Note also that the threads routines, when enabled, are automatically tested by the ```make check'`` self-tests.)

The threads routines require your operating system to have some sort of shared-memory threads support. Specifically, the FFTW threads package works with POSIX threads (available on most Unix variants, including Linux), Solaris threads, BeOS threads (tested on BeOS DR8.2), Mach C threads (reported to work by users), and Win32 threads (reported to work by users). (There is also untested code to use MacOS MP threads.) If you have a shared-memory machine that uses a different threads API, it should be a simple matter of programming to include support for it; see the file `fftw_threads-int.h` for more detail.

SMP hardware is not required, although of course you need multiple processors to get any benefit from the multithreaded transforms.

### Usage of Multi-threaded FFTW

Here, it is assumed that the reader is already familiar with the usage of the uniprocessor FFTW routines, described elsewhere in this manual. We only describe what one has to change in order to use the multi-threaded routines.

First, instead of including `<fftw.h>` or `<rfftw.h>`, you should include the files `<fftw_threads.h>` or `<rfftw_threads.h>`, respectively.

Second, before calling any FFTW routines, you should call the function:

    int fftw_threads_init(void);

This function, which should only be called once (probably in your `main()` function), performs any one-time initialization required to use threads on your system. It returns zero if successful, and a non-zero value if there was an error (in which case, something is seriously wrong and you should probably exit the program).

Third, when you want to actually compute the transform, you should use one of the following transform routines instead of the ordinary FFTW functions:

    fftw_threads(nthreads, plan, howmany, in, istride,
                 idist, out, ostride, odist);

    fftw_threads_one(nthreads, plan, in, out);

    fftwnd_threads(nthreads, plan, howmany, in, istride,
                   idist, out, ostride, odist);

    fftwnd_threads_one(nthreads, plan, in, out);

    rfftw_threads(nthreads, plan, howmany, in, istride,
                  idist, out, ostride, odist);

    rfftw_threads_one(nthreads, plan, in, out);

    rfftwnd_threads_real_to_complex(nthreads, plan, howmany, in,
                                    istride, idist, out, ostride, odist);

    rfftwnd_threads_one_real_to_complex(nthreads, plan, in, out);

    rfftwnd_threads_complex_to_real(nthreads, plan, howmany, in,
                                    istride, idist, out, ostride, odist);

    rfftwnd_threads_one_real_to_complex(nthreads, plan, in, out);

    rfftwnd_threads_one_complex_to_real(nthreads, plan, in, out);

All of these routines take exactly the same arguments and have exactly the same effects as their uniprocessor counterparts (i.e. without the ```_threads`'`) except that they take one extra parameter, `nthreads` (of type `int`), before the normal parameters.(5) The `nthreads` parameter specifies the number of threads of execution to use when performing the transform (actually, the maximum number of threads).

For example, to parallelize a single one-dimensional transform of complex data, instead of calling the uniprocessor `fftw_one(plan, in, out)`, you would call `fftw_threads_one(nthreads, plan, in, out)`. Passing an `nthreads` of `1` means to use only one thread (the main thread), and is equivalent to calling the uniprocessor routine. Passing an `nthreads` of `2` means that the transform is potentially parallelized over two threads (and two processors, if you have them), and so on.

These are the only changes you need to make to your source code. Calls to all other FFTW routines (plan creation, destruction, wisdom, etcetera) are not parallelized and remain the same. (The same plans and wisdom are used by both uniprocessor and multi-threaded transforms.) Your arrays are allocated and formatted in the same way, and so on.

Programs using the parallel complex transforms should be linked with `-lfftw_threads -lfftw -lm` on Unix. Programs using the parallel real transforms should be linked with `-lrfftw_threads -lfftw_threads -lrfftw -lfftw -lm`. You will also need to link with whatever library is responsible for threads on your system (e.g. `-lpthread` on Linux).

### How Many Threads to Use?

There is a fair amount of overhead involved in spawning and synchronizing threads, so the optimal number of threads to use depends upon the size of the transform as well as on the number of processors you have.

As a general rule, you don't want to use more threads than you have processors. (Using more threads will work, but there will be extra overhead with no benefit.) In fact, if the problem size is too small, you may want to use fewer threads than you have processors.

You will have to experiment with your system to see what level of parallelization is best for your problem size. Useful tools to help you do this are the test programs that are automatically compiled along with the threads libraries, `fftw_threads_test` and `rfftw_threads_test` (in the `threads` subdirectory). These take the same arguments as the other FFTW test programs (see `tests/README`), except that they also take the number of threads to use as a first argument, and report the parallel speedup in speed tests. For example,

    fftw_threads_test 2 -s 128x128

will benchmark complex 128x128 transforms using two threads and report the speedup relative to the uniprocessor transform.

For instance, on a 4-processor 200MHz Pentium Pro system running Linux 2.2.0, we found that the "crossover" point at which 2 threads became beneficial for complex transforms was about 4k points, while 4 threads became beneficial at 8k points.

### Using Multi-threaded FFTW in a Multi-threaded Program

It is perfectly possible to use the multi-threaded FFTW routines from a multi-threaded program (e.g. have multiple threads computing multi-threaded transforms simultaneously). If you have the processors, more power to you! However, the same restrictions apply as for the uniprocessor FFTW routines (see Section Thread safety). In particular, you should recall that you may not create or destroy plans in parallel.

### Tips for Optimal Threading

Not all transforms are equally well-parallelized by the multi-threaded FFTW routines. (This is merely a consequence of laziness on the part of the implementors, and is not inherent to the algorithms employed.) Mainly, the limitations are in the parallel one-dimensional transforms. The things to avoid if you want optimal parallelization are as follows:

### Parallelization deficiencies in one-dimensional transforms

- Large prime factors can sometimes parallelize poorly. Of course, you should avoid these anyway if you want high performance.
- Single in-place transforms don't parallelize completely. (Multiple in-place transforms, i.e. `howmany > 1`, are fine.) Again, you should avoid these in any case if you want high performance, as they require transforming to a scratch array and copying back.
- Single real-complex (`rfftw`) transforms don't parallelize completely. This is unfortunate, but parallelizing this correctly would have involved a lot of extra code (and a much larger library). You still get some benefit from additional processors, but if you have a very large number of processors you will probably be better off using the parallel complex (`fftw`) transforms. Note that multi-dimensional real transforms or multiple one-dimensional real transforms are fine.

## MPI FFTW

This section describes the MPI FFTW routines for distributed-memory (and shared-memory) machines supporting MPI (Message Passing Interface). The MPI routines are significantly different from the ordinary FFTW because the transform data here are distributed over multiple processes, so that each process gets only a portion of the array. Currently, multi-dimensional transforms of both real and complex data, as well as one-dimensional transforms of complex data, are supported.

### MPI FFTW Installation

The FFTW MPI library code is all located in the `mpi` subdirectoy of the FFTW package (along with source code for test programs). On Unix systems, the FFTW MPI libraries and header files can be automatically configured, compiled, and installed along with the uniprocessor FFTW libraries simply by including `--enable-mpi` in the flags to the `configure` script (see Section Installation on Unix).

The only requirement of the FFTW MPI code is that you have the standard MPI 1.1 (or later) libraries and header files installed on your system. A free implementation of MPI is available from the MPICH home page.

Previous versions of the FFTW MPI routines have had an unfortunate tendency to expose bugs in MPI implementations. The current version has been largely rewritten, and hopefully avoids some of the problems. If you run into difficulties, try passing the optional workspace to `(r)fftwnd_mpi` (see below), as this allows us to use the standard (and hopefully well-tested) `MPI_Alltoall` primitive for communications. Please let us know (fftw@fftw.org) how things work out.

Several test programs are included in the `mpi` directory. The ones most useful to you are probably the `fftw_mpi_test` and `rfftw_mpi_test` programs, which are run just like an ordinary MPI program and accept the same parameters as the other FFTW test programs (c.f. `tests/README`). For example, `mpirun ...params... fftw_mpi_test -r 0` will run non-terminating complex-transform correctness tests of random dimensions. They can also do performance benchmarks.

### Usage of MPI FFTW for Complex Multi-dimensional Transforms

Usage of the MPI FFTW routines is similar to that of the uniprocessor FFTW. We assume that the reader already understands the usage of the uniprocessor FFTW routines, described elsewhere in this manual. Some familiarity with MPI is also helpful.

A typical program performing a complex two-dimensional MPI transform might look something like:

    #include <fftw_mpi.h>

    int main(int argc, char **argv)
    {
          const int NX = ..., NY = ...;
          fftwnd_mpi_plan plan;
          fftw_complex *data;

          MPI_Init(&argc,&argv);

          plan = fftw2d_mpi_create_plan(MPI_COMM_WORLD,
                                        NX, NY,
                                        FFTW_FORWARD, FFTW_ESTIMATE);

          ...allocate and initialize data...

          fftwnd_mpi(p, 1, data, NULL, FFTW_NORMAL_ORDER);

          ...

          fftwnd_mpi_destroy_plan(plan);
          MPI_Finalize();
    }

The calls to `MPI_Init` and `MPI_Finalize` are required in all MPI programs; see the MPI home page for more information. Note that all of your processes run the program in parallel, as a group; there is no explicit launching of threads/processes in an MPI program.

As in the ordinary FFTW, the first thing we do is to create a plan (of type `fftwnd_mpi_plan`), using:

    fftwnd_mpi_plan fftw2d_mpi_create_plan(MPI_Comm comm,
                                           int nx, int ny,
                                           fftw_direction dir, int flags);

Except for the first argument, the parameters are identical to those of `fftw2d_create_plan`. (There are also analogous `fftwnd_mpi_create_plan` and `fftw3d_mpi_create_plan` functions. Transforms of any rank greater than one are supported.) The first argument is an MPI communicator, which specifies the group of processes that are to be involved in the transform; the standard constant `MPI_COMM_WORLD` indicates all available processes.

Next, one has to allocate and initialize the data. This is somewhat tricky, because the transform data is distributed across the processes involved in the transform. It is discussed in detail by the next section (see Section MPI Data Layout).

The actual computation of the transform is performed by the function `fftwnd_mpi`, which differs somewhat from its uniprocessor equivalent and is described by:

    void fftwnd_mpi(fftwnd_mpi_plan p,
                    int n_fields,
                    fftw_complex *local_data, fftw_complex *work,
                    fftwnd_mpi_output_order output_order);

There are several things to notice here:

- First of all, all `fftw_mpi` transforms are in-place: the output is in the `local_data` parameter, and there is no need to specify `FFTW_IN_PLACE` in the plan flags.
- The MPI transforms also only support a limited subset of the `howmany`/`stride`/`dist` functionality of the uniprocessor routines: the `n_fields` parameter is equivalent to `howmany=n_fields`, `stride=n_fields`, and `dist=1`. (Conceptually, the `n_fields` parameter allows you to transform an array of contiguous vectors, each with length `n_fields`.) `n_fields` is `1` if you are only transforming a single, ordinary array.
- The `work` parameter is an optional workspace. If it is not `NULL`, it should be exactly the same size as the `local_data` array. If it is provided, FFTW is able to use the built-in `MPI_Alltoall` primitive for (often) greater efficiency at the expense of extra storage space.
- Finally, the last parameter specifies whether the output data has the same ordering as the input data (`FFTW_NORMAL_ORDER`), or if it is transposed (`FFTW_TRANSPOSED_ORDER`). Leaving the data transposed results in significant performance improvements due to a saved communication step (needed to un-transpose the data). Specifically, the first two dimensions of the array are transposed, as is described in more detail by the next section.

The output of `fftwnd_mpi` is identical to that of the corresponding uniprocessor transform. In particular, you should recall our conventions for normalization and the sign of the transform exponent.

The same plan can be used to compute many transforms of the same size. After you are done with it, you should deallocate it by calling `fftwnd_mpi_destroy_plan`.

Important: The FFTW MPI routines must be called in the same order by all processes involved in the transform. You should assume that they all are blocking, as if each contained a call to `MPI_Barrier`.

Programs using the FFTW MPI routines should be linked with `-lfftw_mpi -lfftw -lm` on Unix, in addition to whatever libraries are required for MPI.

### MPI Data Layout

The transform data used by the MPI FFTW routines is distributed: a distinct portion of it resides with each process involved in the transform. This allows the transform to be parallelized, for example, over a cluster of workstations, each with its own separate memory, so that you can take advantage of the total memory of all the processors you are parallelizing over.

In particular, the array is divided according to the rows (first dimension) of the data: each process gets a subset of the rows of the data. (This is sometimes called a "slab decomposition.") One consequence of this is that you can't take advantage of more processors than you have rows (e.g. `64x64x64` matrix can at most use 64 processors). This isn't usually much of a limitation, however, as each processor needs a fair amount of data in order for the parallel-computation benefits to outweight the communications costs.

Below, the first dimension of the data will be referred to as ```x`'` and the second dimension as ```y`'`.

FFTW supplies a routine to tell you exactly how much data resides on the current process:

    void fftwnd_mpi_local_sizes(fftwnd_mpi_plan p,
                                int *local_nx,
                                int *local_x_start,
                                int *local_ny_after_transpose,
                                int *local_y_start_after_transpose,
                                int *total_local_size);

Given a plan `p`, the other parameters of this routine are set to values describing the required data layout, described below.

`total_local_size` is the number of `fftw_complex` elements that you must allocate for your local data (and workspace, if you choose). (This value should, of course, be multiplied by `n_fields` if that parameter to `fftwnd_mpi` is not `1`.)

The data on the current process has `local_nx` rows, starting at row `local_x_start`. If `fftwnd_mpi` is called with `FFTW_TRANSPOSED_ORDER` output, then `y` will be the first dimension of the output, and the local `y` extent will be given by `local_ny_after_transpose` and `local_y_start_after_transpose`. Otherwise, the output has the same dimensions and layout as the input.

For instance, suppose you want to transform three-dimensional data of size `nx x ny x nz`. Then, the current process will store a subset of this data, of size `local_nx x ny x nz`, where the `x` indices correspond to the range `local_x_start` to `local_x_start+local_nx-1` in the "real" (i.e. logical) array. If `fftwnd_mpi` is called with `FFTW_TRANSPOSED_ORDER` output, then the result will be a `ny x nx x nz` array, of which a `local_ny_after_transpose x nx x nz` subset is stored on the current process (corresponding to `y` values starting at `local_y_start_after_transpose`).

The following is an example of allocating such a three-dimensional array array (`local_data`) before the transform and initializing it to some function `f(x,y,z)`:

            fftwnd_mpi_local_sizes(plan, &local_nx, &local_x_start,
                                   &local_ny_after_transpose,
                                   &local_y_start_after_transpose,
                                   &total_local_size);

            local_data = (fftw_complex*) malloc(sizeof(fftw_complex) *
                                                total_local_size);

            for (x = 0; x < local_nx; ++x)
                    for (y = 0; y < ny; ++y)
                            for (z = 0; z < nz; ++z)
                                    local_data[(x*ny + y)*nz + z]
                                            = f(x + local_x_start, y, z);

Some important things to remember:

- Although the local data is of dimensions `local_nx x ny x nz` in the above example, do not allocate the array to be of size `local_nx*ny*nz`. Use `total_local_size` instead.
- The amount of data on each process will not necessarily be the same; in fact, `local_nx` may even be zero for some processes. (For example, suppose you are doing a `6x6` transform on four processors. There is no way to effectively use the fourth processor in a slab decomposition, so we leave it empty. Proof left as an exercise for the reader.)
- All arrays are, of course, in row-major order (see Section Multi-dimensional Array Format).
- If you want to compute the inverse transform of the output of `fftwnd_mpi`, the dimensions of the inverse transform are given by the dimensions of the output of the forward transform. For example, if you are using `FFTW_TRANSPOSED_ORDER` output in the above example, then the inverse plan should be created with dimensions `ny x nx x nz`.
- The data layout only depends upon the dimensions of the array, not on the plan, so you are guaranteed that different plans for the same size (or inverse plans) will use the same (consistent) data layouts.

### Usage of MPI FFTW for Real Multi-dimensional Transforms

MPI transforms specialized for real data are also available, similiar to the uniprocessor `rfftwnd` transforms. Just as in the uniprocessor case, the real-data MPI functions gain roughly a factor of two in speed (and save a factor of two in space) at the expense of more complicated data formats in the calling program. Before reading this section, you should definitely understand how to call the uniprocessor `rfftwnd` functions and also the complex MPI FFTW functions.

The following is an example of a program using `rfftwnd_mpi`. It computes the size `nx x ny x nz` transform of a real function `f(x,y,z)`, multiplies the imaginary part by `2` for fun, then computes the inverse transform. (We'll also use `FFTW_TRANSPOSED_ORDER` output for the transform, and additionally supply the optional workspace parameter to `rfftwnd_mpi`, just to add a little spice.)

    #include <rfftw_mpi.h>

    int main(int argc, char **argv)
    {
         const int nx = ..., ny = ..., nz = ...;
         int local_nx, local_x_start, local_ny_after_transpose,
             local_y_start_after_transpose, total_local_size;
         int x, y, z;
         rfftwnd_mpi_plan plan, iplan;
         fftw_real *data, *work;
         fftw_complex *cdata;

         MPI_Init(&argc,&argv);

         /* create the forward and backward plans: */
         plan = rfftw3d_mpi_create_plan(MPI_COMM_WORLD,
                                        nx, ny, nz,
                                        FFTW_REAL_TO_COMPLEX,
                                        FFTW_ESTIMATE);
         iplan = rfftw3d_mpi_create_plan(MPI_COMM_WORLD,
          /* dim.'s of REAL data --> */  nx, ny, nz,
                                         FFTW_COMPLEX_TO_REAL,
                                         FFTW_ESTIMATE);

         rfftwnd_mpi_local_sizes(plan, &local_nx, &local_x_start,
                                &local_ny_after_transpose,
                                &local_y_start_after_transpose,
                                &total_local_size);

         data = (fftw_real*) malloc(sizeof(fftw_real) * total_local_size);

         /* workspace is the same size as the data: */
         work = (fftw_real*) malloc(sizeof(fftw_real) * total_local_size);

         /* initialize data to f(x,y,z): */
         for (x = 0; x < local_nx; ++x)
                 for (y = 0; y < ny; ++y)
                         for (z = 0; z < nz; ++z)
                                 data[(x*ny + y) * (2*(nz/2+1)) + z]
                                         = f(x + local_x_start, y, z);

         /* Now, compute the forward transform: */
         rfftwnd_mpi(plan, 1, data, work, FFTW_TRANSPOSED_ORDER);

         /* the data is now complex, so typecast a pointer: */
         cdata = (fftw_complex*) data;

         /* multiply imaginary part by 2, for fun:
            (note that the data is transposed) */
         for (y = 0; y < local_ny_after_transpose; ++y)
                 for (x = 0; x < nx; ++x)
                         for (z = 0; z < (nz/2+1); ++z)
                                 cdata[(y*nx + x) * (nz/2+1) + z].im
                                         *= 2.0;

         /* Finally, compute the inverse transform; the result
            is transposed back to the original data layout: */
         rfftwnd_mpi(iplan, 1, data, work, FFTW_TRANSPOSED_ORDER);

         free(data);
         free(work);
         rfftwnd_mpi_destroy_plan(plan);
         rfftwnd_mpi_destroy_plan(iplan);
         MPI_Finalize();
    }

There's a lot of stuff in this example, but it's all just what you would have guessed, right? We replaced all the `fftwnd_mpi*` functions by `rfftwnd_mpi*`, but otherwise the parameters were pretty much the same. The data layout distributed among the processes just like for the complex transforms (see Section MPI Data Layout), but in addition the final dimension is padded just like it is for the uniprocessor in-place real transforms (see Section Array Dimensions for Real Multi-dimensional Transforms). In particular, the `z` dimension of the real input data is padded to a size `2*(nz/2+1)`, and after the transform it contains `nz/2+1` complex values.

Some other important things to know about the real MPI transforms:

- As for the uniprocessor `rfftwnd_create_plan`, the dimensions passed for the `FFTW_COMPLEX_TO_REAL` plan are those of the real data. In particular, even when `FFTW_TRANSPOSED_ORDER` is used as in this case, the dimensions are those of the (untransposed) real output, not the (transposed) complex input. (For the complex MPI transforms, on the other hand, the dimensions are always those of the input array.)
- The output ordering of the transform (`FFTW_TRANSPOSED_ORDER` or `FFTW_TRANSPOSED_ORDER`) must be the same for both forward and backward transforms. (This is not required in the complex case.)
- `total_local_size` is the required size in `fftw_real` values, not `fftw_complex` values as it is for the complex transforms.
- `local_ny_after_transpose` and `local_y_start_after_transpose` describe the portion of the array after the transform; that is, they are indices in the complex array for an `FFTW_REAL_TO_COMPLEX` transform and in the real array for an `FFTW_COMPLEX_TO_REAL` transform.
- `rfftwnd_mpi` always expects `fftw_real*` array arguments, but of course these pointers can refer to either real or complex arrays, depending upon which side of the transform you are on. Just as for in-place uniprocessor real transforms (and also in the example above), this is most easily handled by typecasting to a complex pointer when handling the complex data.
- As with the complex transforms, there are also `rfftwnd_create_plan` and `rfftw2d_create_plan` functions, and any rank greater than one is supported.

Programs using the MPI FFTW real transforms should link with `-lrfftw_mpi -lfftw_mpi -lrfftw -lfftw -lm` on Unix.

### Usage of MPI FFTW for Complex One-dimensional Transforms

The MPI FFTW also includes routines for parallel one-dimensional transforms of complex data (only). Although the speedup is generally worse than it is for the multi-dimensional routines,(6) these distributed-memory one-dimensional transforms are especially useful for performing one-dimensional transforms that don't fit into the memory of a single machine.

The usage of these routines is straightforward, and is similar to that of the multi-dimensional MPI transform functions. You first include the header `<fftw_mpi.h>` and then create a plan by calling:

    fftw_mpi_plan fftw_mpi_create_plan(MPI_Comm comm, int n,
                                       fftw_direction dir, int flags);

The last three arguments are the same as for `fftw_create_plan` (except that all MPI transforms are automatically `FFTW_IN_PLACE`). The first argument specifies the group of processes you are using, and is usually `MPI_COMM_WORLD` (all processes). A plan can be used for many transforms of the same size, and is destroyed when you are done with it by calling `fftw_mpi_destroy_plan(plan)`.

If you don't care about the ordering of the input or output data of the transform, you can include `FFTW_SCRAMBLED_INPUT` and/or `FFTW_SCRAMBLED_OUTPUT` in the `flags`. These save some communications at the expense of having the input and/or output reordered in an undocumented way. For example, if you are performing an FFT-based convolution, you might use `FFTW_SCRAMBLED_OUTPUT` for the forward transform and `FFTW_SCRAMBLED_INPUT` for the inverse transform.

The transform itself is computed by:

    void fftw_mpi(fftw_mpi_plan p, int n_fields,
                  fftw_complex *local_data, fftw_complex *work);

`n_fields`, as in `fftwnd_mpi`, is equivalent to `howmany=n_fields`, `stride=n_fields`, and `dist=1`, and should be `1` when you are computing the transform of a single array. `local_data` contains the portion of the array local to the current process, described below. `work` is either `NULL` or an array exactly the same size as `local_data`; in the latter case, FFTW can use the `MPI_Alltoall` communications primitive which is (usually) faster at the expense of extra storage. Upon return, `local_data` contains the portion of the output local to the current process (see below).

To find out what portion of the array is stored local to the current process, you call the following routine:

    void fftw_mpi_local_sizes(fftw_mpi_plan p,
                              int *local_n, int *local_start,
                              int *local_n_after_transform,
                              int *local_start_after_transform,
                              int *total_local_size);

`total_local_size` is the number of `fftw_complex` elements you should actually allocate for `local_data` (and `work`). `local_n` and `local_start` indicate that the current process stores `local_n` elements corresponding to the indices `local_start` to `local_start+local_n-1` in the "real" array. After the transform, the process may store a different portion of the array. The portion of the data stored on the process after the transform is given by `local_n_after_transform` and `local_start_after_transform`. This data is exactly the same as a contiguous segment of the corresponding uniprocessor transform output (i.e. an in-order sequence of sequential frequency bins).

Note that, if you compute both a forward and a backward transform of the same size, the local sizes are guaranteed to be consistent. That is, the local size after the forward transform will be the same as the local size before the backward transform, and vice versa.

Programs using the FFTW MPI routines should be linked with `-lfftw_mpi -lfftw -lm` on Unix, in addition to whatever libraries are required for MPI.

### MPI Tips

There are several things you should consider in order to get the best performance out of the MPI FFTW routines.

First, if possible, the first and second dimensions of your data should be divisible by the number of processes you are using. (If only one can be divisible, then you should choose the first dimension.) This allows the computational load to be spread evenly among the processes, and also reduces the communications complexity and overhead. In the one-dimensional transform case, the size of the transform should ideally be divisible by the square of the number of processors.

Second, you should consider using the `FFTW_TRANSPOSED_ORDER` output format if it is not too burdensome. The speed gains from communications savings are usually substantial.

Third, you should consider allocating a workspace for `(r)fftw(nd)_mpi`, as this can often (but not always) improve performance (at the cost of extra storage).

Fourth, you should experiment with the best number of processors to use for your problem. (There comes a point of diminishing returns, when the communications costs outweigh the computational benefits.(7)) The `fftw_mpi_test` program can output helpful performance benchmarks. It accepts the same parameters as the uniprocessor test programs (c.f. `tests/README`) and is run like an ordinary MPI program. For example, `mpirun -np 4 fftw_mpi_test -s 128x128x128` will benchmark a `128x128x128` transform on four processors, reporting timings and parallel speedups for all variants of `fftwnd_mpi` (transposed, with workspace, etcetera). (Note also that there is the `rfftw_mpi_test` program for the real transforms.)

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