This is Info file fftw.info, produced by Makeinfo version 1.68 from the input file fftw.texi.
This is the FFTW User's manual.
Copyright (C) 1997-1999 Massachusetts Institute of Technology
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File: fftw.info, Node: Array Dimensions for Real Multi-dimensional Transforms, Next: Strides in In-place RFFTWND, Prev: rfftwnd, Up: Real Multi-dimensional Transforms Reference
Array Dimensions for Real Multi-dimensional Transforms ------------------------------------------------------
The output of a multi-dimensional transform of real data contains symmetries that, in principle, make half of the outputs redundant (*note What RFFTWND Really Computes::.). In practice, it is not possible to entirely realize these savings in an efficient and understandable format. Instead, the output of the rfftwnd transforms is *slightly* over half of the output of the corresponding complex transform. We do not "pack" the data in any way, but store it as an ordinary array of `fftw_complex' values. In fact, this data is simply a subsection of what would be the array in the corresponding complex transform.
Specifically, for a real transform of dimensions n1 x n2 x ... x nd, the complex data is an n1 x n2 x ... x (nd/2+1) array of `fftw_complex' values in row-major order (with the division rounded down). That is, we only store the lower half (plus one element) of the last dimension of the data from the ordinary complex transform. (We could have instead taken half of any other dimension, but implementation turns out to be simpler if the last, contiguous, dimension is used.)
Since the complex data is slightly larger than the real data, some complications arise for in-place transforms. In this case, the final dimension of the real data must be padded with extra values to accommodate the size of the complex data--two extra if the last dimension is even and one if it is odd. That is, the last dimension of the real data must physically contain 2 * (nd/2+1) `fftw_real' values (exactly enough to hold the complex data). This physical array size does not, however, change the *logical* array size--only nd values are actually stored in the last dimension, and nd is the last dimension passed to `rfftwnd_create_plan'.
File: fftw.info, Node: Strides in In-place RFFTWND, Next: rfftwnd_destroy_plan, Prev: Array Dimensions for Real Multi-dimensional Transforms, Up: Real Multi-dimensional Transforms Reference
Strides in In-place RFFTWND ---------------------------
The fact that the input and output datatypes are different for rfftwnd complicates the meaning of the `stride' and `dist' parameters of in-place transforms--are they in units of `fftw_real' or `fftw_complex' elements? When reading the input, they are interpreted in units of the datatype of the input data. When writing the output, the `istride' and `idist' are translated to the output datatype's "units" in one of two ways, corresponding to the two most common situations in which `stride' and `dist' parameters are useful. Below, we refer to these "translated" parameters as `ostride_t' and `odist_t'. (Note that these are computed internally by rfftwnd; the actual `ostride' and `odist' parameters are ignored for in-place transforms.)
First, there is the case where you are transforming a number of contiguous arrays located one after another in memory. In this situation, `istride' is `1' and `idist' is the product of the physical dimensions of the array. `ostride_t' and `odist_t' are then chosen so that the output arrays are contiguous and lie on top of the input arrays. `ostride_t' is therefore `1'. For a real-to-complex transform, `odist_t' is `idist/2'; for a complex-to-real transform, `odist_t' is `idist*2'.
The second case is when you have an array in which each element has `nc' components (e.g. a structure with `nc' numeric fields), and you want to transform all of the components at once. Here, `istride' is `nc' and `idist' is `1'. For this case, it is natural to want the output to also have `nc' consecutive components, now of the output data type; this is exactly what rfftwnd does. Specifically, it uses an `ostride_t' equal to `istride', and an `odist_t' of `1'. (Astute readers will realize that some extra buffer space is required in order to perform such a transform; this is handled automatically by rfftwnd.)
The general rule is as follows. `ostride_t' equals `istride'. If `idist' is `1' and `idist' is less than `istride', then `odist_t' is `1'. Otherwise, for a real-to-complex transform `odist_t' is `idist/2' and for a complex-to-real transform `odist_t' is `idist*2'.
File: fftw.info, Node: rfftwnd_destroy_plan, Next: What RFFTWND Really Computes, Prev: Strides in In-place RFFTWND, Up: Real Multi-dimensional Transforms Reference
Destroying a Multi-dimensional Plan -----------------------------------
#include <rfftw.h> void rfftwnd_destroy_plan(rfftwnd_plan plan);
The function `rfftwnd_destroy_plan' frees the plan `plan' and releases all the memory associated with it. After destruction, a plan is no longer valid.
File: fftw.info, Node: What RFFTWND Really Computes, Prev: rfftwnd_destroy_plan, Up: Real Multi-dimensional Transforms Reference
What RFFTWND Really Computes ----------------------------
The conventions that we follow for the real multi-dimensional transform are analogous to those for the complex multi-dimensional transform. In particular, the forward transform has a negative sign in the exponent and neither the forward nor the backward transforms will perform any normalization. Computing the backward transform of the forward transform will multiply the array by the product of its dimensions (that is, the logical dimensions of the real data). The forward transform is real-to-complex and the backward transform is complex-to-real.
The TeX version of this manual contains the exact definition of the n-dimensional transform RFFTWND uses. It is not possible to display the definition on a ASCII terminal properly.
File: fftw.info, Node: Wisdom Reference, Next: Memory Allocator Reference, Prev: Real Multi-dimensional Transforms Reference, Up: FFTW Reference
Wisdom Reference ================
* Menu:
* fftw_export_wisdom:: * fftw_import_wisdom:: * fftw_forget_wisdom::
File: fftw.info, Node: fftw_export_wisdom, Next: fftw_import_wisdom, Prev: Wisdom Reference, Up: Wisdom Reference
Exporting Wisdom ----------------
#include <fftw.h> void fftw_export_wisdom(void (*emitter)(char c, void *), void *data); void fftw_export_wisdom_to_file(FILE *output_file); char *fftw_export_wisdom_to_string(void);
These functions allow you to export all currently accumulated `wisdom' in a form from which it can be later imported and restored, even during a separate run of the program. (*Note Words of Wisdom::.) The current store of `wisdom' is not affected by calling any of these routines.
`fftw_export_wisdom' exports the `wisdom' to any output medium, as specified by the callback function `emitter'. `emitter' is a `putc'-like function that writes the character `c' to some output; its second parameter is the `data' pointer passed to `fftw_export_wisdom'. For convenience, the following two "wrapper" routines are provided:
`fftw_export_wisdom_to_file' writes the `wisdom' to the current position in `output_file', which should be open with write permission. Upon exit, the file remains open and is positioned at the end of the `wisdom' data.
`fftw_export_wisdom_to_string' returns a pointer to a `NULL'-terminated string holding the `wisdom' data. This string is dynamically allocated, and it is the responsibility of the caller to deallocate it with `fftw_free' when it is no longer needed.
All of these routines export the wisdom in the same format, which we will not document here except to say that it is LISP-like ASCII text that is insensitive to white space.
File: fftw.info, Node: fftw_import_wisdom, Next: fftw_forget_wisdom, Prev: fftw_export_wisdom, Up: Wisdom Reference
Importing Wisdom ----------------
#include <fftw.h> fftw_status fftw_import_wisdom(int (*get_input)(void *), void *data); fftw_status fftw_import_wisdom_from_file(FILE *input_file); fftw_status fftw_import_wisdom_from_string(const char *input_string);
These functions import `wisdom' into a program from data stored by the `fftw_export_wisdom' functions above. (*Note Words of Wisdom::.) The imported `wisdom' supplements rather than replaces any `wisdom' already accumulated by the running program (except when there is conflicting `wisdom', in which case the existing wisdom is replaced).
`fftw_import_wisdom' imports `wisdom' from any input medium, as specified by the callback function `get_input'. `get_input' is a `getc'-like function that returns the next character in the input; its parameter is the `data' pointer passed to `fftw_import_wisdom'. If the end of the input data is reached (which should never happen for valid data), it may return either `NULL' (ASCII 0) or `EOF' (as defined in `<stdio.h>'). For convenience, the following two "wrapper" routines are provided:
`fftw_import_wisdom_from_file' reads `wisdom' from the current position in `input_file', which should be open with read permission. Upon exit, the file remains open and is positioned at the end of the `wisdom' data.
`fftw_import_wisdom_from_string' reads `wisdom' from the `NULL'-terminated string `input_string'.
The return value of these routines is `FFTW_SUCCESS' if the wisdom was read successfully, and `FFTW_FAILURE' otherwise. Note that, in all of these functions, any data in the input stream past the end of the `wisdom' data is simply ignored (it is not even read if the `wisdom' data is well-formed).
File: fftw.info, Node: fftw_forget_wisdom, Prev: fftw_import_wisdom, Up: Wisdom Reference
Forgetting Wisdom -----------------
#include <fftw.h> void fftw_forget_wisdom(void);
Calling `fftw_forget_wisdom' causes all accumulated `wisdom' to be discarded and its associated memory to be freed. (New `wisdom' can still be gathered subsequently, however.)
File: fftw.info, Node: Memory Allocator Reference, Next: Thread safety, Prev: Wisdom Reference, Up: FFTW Reference
Memory Allocator Reference ==========================
#include <fftw.h> void *(*fftw_malloc_hook) (size_t n); void (*fftw_free_hook) (void *p);
Whenever it has to allocate and release memory, FFTW ordinarily calls `malloc' and `free'. If `malloc' fails, FFTW prints an error message and exits. This behavior may be undesirable in some applications. Also, special memory-handling functions may be necessary in certain environments. Consequently, FFTW provides means by which you can install your own memory allocator and take whatever error-correcting action you find appropriate. The variables `fftw_malloc_hook' and `fftw_free_hook' are pointers to functions, and they are normally `NULL'. If you set those variables to point to other functions, then FFTW will use your routines instead of `malloc' and `free'. `fftw_malloc_hook' must point to a `malloc'-like function, and `fftw_free_hook' must point to a `free'-like function.
File: fftw.info, Node: Thread safety, Prev: Memory Allocator Reference, Up: FFTW Reference
Thread safety =============
Users writing multi-threaded programs must concern themselves with the "thread safety" of the libraries they use--that is, whether it is safe to call routines in parallel from multiple threads. FFTW can be used in such an environment, but some care must be taken because certain parts of FFTW use private global variables to share data between calls. In particular, the plan-creation functions share trigonometric tables and accumulated `wisdom'. (Users should note that these comments only apply to programs using shared-memory threads. Parallelism using MPI or forked processes involves a separate address-space and global variables for each process, and is not susceptible to problems of this sort.)
The central restriction of FFTW is that it is not safe to create multiple plans in parallel. You must either create all of your plans from a single thread, or instead use a semaphore, mutex, or other mechanism to ensure that different threads don't attempt to create plans at the same time. The same restriction also holds for destruction of plans and importing/forgetting `wisdom'. Once created, a plan may safely be used in any thread.
The actual transform routines in FFTW (`fftw_one', etcetera) are re-entrant and thread-safe, so it is fine to call them simultaneously from multiple threads. Another question arises, however--is it safe to use the *same plan* for multiple transforms in parallel? (It would be unsafe if, for example, the plan were modified in some way by the transform.) We address this question by defining an additional planner flag, `FFTW_THREADSAFE'. When included in the flags for any of the plan-creation routines, `FFTW_THREADSAFE' guarantees that the resulting plan will be read-only and safe to use in parallel by multiple threads.
File: fftw.info, Node: Parallel FFTW, Next: Calling FFTW from Fortran, Prev: FFTW Reference, Up: Top
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 *Note 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 *Note 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 (http://www.mcs.anl.gov/mpi). The FFTW MPI routines are documented in *Note 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 (http://supertech.lcs.mit.edu/cilk). 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'.
* Menu:
* Multi-threaded FFTW:: * MPI FFTW::
File: fftw.info, Node: Multi-threaded FFTW, Next: MPI FFTW, Prev: Parallel FFTW, Up: Parallel FFTW
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.
* Menu:
* Installation and Supported Hardware/Software:: * Usage of Multi-threaded FFTW:: * How Many Threads to Use?:: * Using Multi-threaded FFTW in a Multi-threaded Program:: * Tips for Optimal Threading::
File: fftw.info, Node: Installation and Supported Hardware/Software, Next: Usage of Multi-threaded FFTW, Prev: Multi-threaded FFTW, Up: 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 (*note 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 (http://www.be.com) 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.
File: fftw.info, Node: Usage of Multi-threaded FFTW, Next: How Many Threads to Use?, Prev: Installation and Supported Hardware/Software, Up: Multi-threaded FFTW
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.(1) 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).
---------- Footnotes ----------
(1) There is one exception: when performing one-dimensional in-place transforms, the `out' parameter is always ignored by the multi-threaded routines, instead of being used as a workspace if it is non-`NULL' as in the uniprocessor routines. The multi-threaded routines always allocate their own workspace (the size of which depends upon the number of threads).
File: fftw.info, Node: How Many Threads to Use?, Next: Using Multi-threaded FFTW in a Multi-threaded Program, Prev: Usage of Multi-threaded FFTW, Up: Multi-threaded FFTW
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.
File: fftw.info, Node: Using Multi-threaded FFTW in a Multi-threaded Program, Next: Tips for Optimal Threading, Prev: How Many Threads to Use?, Up: Multi-threaded FFTW
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 (*note Thread safety::.). In particular, you should recall that you may not create or destroy plans in parallel.
File: fftw.info, Node: Tips for Optimal Threading, Prev: Using Multi-threaded FFTW in a Multi-threaded Program, Up: Multi-threaded FFTW
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.
File: fftw.info, Node: MPI FFTW, Prev: Multi-threaded FFTW, Up: Parallel FFTW
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.
* Menu:
* MPI FFTW Installation:: * Usage of MPI FFTW for Complex Multi-dimensional Transforms:: * MPI Data Layout:: * Usage of MPI FFTW for Real Multi-dimensional Transforms:: * Usage of MPI FFTW for Complex One-dimensional Transforms:: * MPI Tips::
File: fftw.info, Node: MPI FFTW Installation, Next: Usage of MPI FFTW for Complex Multi-dimensional Transforms, Prev: MPI FFTW, Up: MPI FFTW
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 (*note 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 (http://www-unix.mcs.anl.gov/mpi/mpich/).
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.
File: fftw.info, Node: Usage of MPI FFTW for Complex Multi-dimensional Transforms, Next: MPI Data Layout, Prev: MPI FFTW Installation, Up: MPI FFTW
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 (http://www.mcs.anl.gov/mpi/) 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 (*note 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.
File: fftw.info, Node: MPI Data Layout, Next: Usage of MPI FFTW for Real Multi-dimensional Transforms, Prev: Usage of MPI FFTW for Complex Multi-dimensional Transforms, Up: MPI FFTW
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 (*note 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.
File: fftw.info, Node: Usage of MPI FFTW for Real Multi-dimensional Transforms, Next: Usage of MPI FFTW for Complex One-dimensional Transforms, Prev: MPI Data Layout, Up: MPI FFTW
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 (*note MPI Data Layout::.), but in addition the final dimension is padded just like it is for the uniprocessor in-place real transforms (*note 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.
File: fftw.info, Node: Usage of MPI FFTW for Complex One-dimensional Transforms, Next: MPI Tips, Prev: Usage of MPI FFTW for Real Multi-dimensional Transforms, Up: MPI FFTW
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,(1) 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.
---------- Footnotes ----------
(1) The 1D transforms require much more communication. All the communication in our FFT routines takes the form of an all-to-all communication: the multi-dimensional transforms require two all-to-all communications (or one, if you use `FFTW_TRANSPOSED_ORDER'), while the one-dimensional transforms require *three* (or two, if you use scrambled input or output).
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