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FFTW Reference

This chapter provides a complete reference for all sequential (i.e., one-processor) FFTW functions. We first define the data types upon which FFTW operates, that is, real, complex, and "halfcomplex" numbers (see Section Data Types). Then, in four sections, we explain the FFTW program interface for complex one-dimensional transforms (see Section One-dimensional Transforms Reference), complex multi-dimensional transforms (see Section Multi-dimensional Transforms Reference), and real one-dimensional transforms (see Section Real One-dimensional Transforms Reference), real multi-dimensional transforms (see Section Real Multi-dimensional Transforms Reference). Section Wisdom Reference describes the wisdom mechanism for exporting and importing plans. Finally, Section Memory Allocator Reference describes how to change FFTW's default memory allocator. For parallel transforms, See Section Parallel FFTW.

Data Types

The routines in the FFTW package use three main kinds of data types. Real and complex numbers should be already known to the reader. We also use the term halfcomplex to describe complex arrays in a special packed format used by the one-dimensional real transforms (taking advantage of the hermitian symmetry that arises in those cases).

By including <fftw.h> or <rfftw.h>, you will have access to the following definitions: 

typedef double fftw_real;

typedef struct {
     fftw_real re, im;
} fftw_complex;

#define c_re(c)  ((c).re)
#define c_im(c)  ((c).im)

All FFTW operations are performed on the fftw_real and fftw_complex data types. For fftw_complex numbers, the two macros c_re and c_im retrieve, respectively, the real and imaginary parts of the number.

A real array is an array of real numbers. A complex array is an array of complex numbers. A one-dimensional array X of n complex numbers is hermitian if the following property holds: for all 0 <= i < n, we have Xi = conj(Xn-i)}. Hermitian arrays are relevant to FFTW because the Fourier transform of a real array is hermitian.

Because of its symmetry, a hermitian array can be stored in half the space of a complex array of the same size. FFTW's one-dimensional real transforms store hermitian arrays as halfcomplex arrays. A halfcomplex array of size n is a one-dimensional array of n fftw_real numbers. A hermitian array X in stored into a halfcomplex array Y as follows. For all integers i such that 0 <= i <= n / 2, we have Yi = Re(Xi). For all integers i such that 0 < i < n / 2, we have Yn-i = Im(Xi).

We now illustrate halfcomplex storage for n = 4 and n = 5, since the scheme depends on the parity of n. Let n = 4. In this case, we have Y0 = Re(X0), Y1 = Re(X1), Y2 = Re(X2), and Y3 = Im(X1). Let now n = 5. In this case, we have Y0 = Re(X0), Y1 = Re(X1), Y2 = Re(X2), Y3 = Im(X2), and Y4 = Im(X1).

By default, the type fftw_real equals the C type double. To work in single precision rather than double precision, #define the symbol FFTW_ENABLE_FLOAT in fftw.h and then recompile the library. On Unix systems, you can instead use configure --enable-float at installation time (see Section Installation and Customization).

In version 1 of FFTW, the data types were called FFTW_REAL and FFTW_COMPLEX. We changed the capitalization for consistency with the rest of FFTW's conventions. The old names are still supported, but their use is deprecated.

One-dimensional Transforms Reference

The one-dimensional complex routines are generally prefixed with fftw_. Programs using FFTW should be linked with -lfftw -lm on Unix systems, or with the FFTW and standard math libraries in general.

Plan Creation for One-dimensional Transforms

#include <fftw.h>

fftw_plan fftw_create_plan(int n, fftw_direction dir,
                           int flags);

fftw_plan fftw_create_plan_specific(int n, fftw_direction dir,
                                    int flags,
                                    fftw_complex *in, int istride,
                                    fftw_complex *out, int ostride);

The function fftw_create_plan creates a plan, which is a data structure containing all the information that fftw needs in order to compute the 1D Fourier transform. You can create as many plans as you need, but only one plan for a given array size is required (a plan can be reused many times).

fftw_create_plan returns a valid plan, or NULL if, for some reason, the plan can't be created. In the default installation, this cannot happen, but it is possible to configure FFTW in such a way that some input sizes are forbidden, and FFTW cannot create a plan.

The fftw_create_plan_specific variant takes as additional arguments specific input/output arrays and their strides. For the last four arguments, you should pass the arrays and strides that you will eventually be passing to fftw. The resulting plans will be optimized for those arrays and strides, although they may be used on other arrays as well. Note: the contents of the in and out arrays are destroyed by the specific planner (the initial contents are ignored, so the arrays need not have been initialized).

Arguments

Discussion on Specific Plans

We recommend the use of the specific planners, even in cases where you will be transforming arrays different from those passed to the specific planners, as they confer the following advantages:

On the other hand, note that the specific planner destroys the contents of the in and out arrays.

Computing the One-dimensional Transform

#include <fftw.h>

void fftw(fftw_plan plan, int howmany,
          fftw_complex *in, int istride, int idist,
          fftw_complex *out, int ostride, int odist);

void fftw_one(fftw_plan plan, fftw_complex *in, 
          fftw_complex *out);

The function fftw computes the one-dimensional Fourier transform, using a plan created by fftw_create_plan (See Section Plan Creation for One-dimensional Transforms.) The function fftw_one provides a simplified interface for the common case of single input array of stride 1.

Arguments

The function fftw_one transforms a single, contiguous input array to a contiguous output array. By definition, the call 

fftw_one(plan, in, out)

is equivalent to 

fftw(plan, 1, in, 1, 0, out, 1, 0)

Destroying a One-dimensional Plan

#include <fftw.h>

void fftw_destroy_plan(fftw_plan plan);

The function fftw_destroy_plan frees the plan plan and releases all the memory associated with it. After destruction, a plan is no longer valid.

What FFTW Really Computes

In this section, we define precisely what FFTW computes. Please be warned that different authors and software packages might employ different conventions than FFTW does.

The forward transform of a complex array X of size n computes an array Y, where

The backward transform computes

FFTW computes an unnormalized transform, that is, the equation IFFT(FFT(X)) = n X holds. In other words, applying the forward and then the backward transform will multiply the input by n.

An FFTW_FORWARD transform corresponds to a sign of -1 in the exponent of the DFT. Note also that we use the standard "in-order" output ordering--the k-th output corresponds to the frequency k/n (or k/T, where T is your total sampling period). For those who like to think in terms of positive and negative frequencies, this means that the positive frequencies are stored in the first half of the output and the negative frequencies are stored in backwards order in the second half of the output. (The frequency -k/n is the same as the frequency (n-k)/n.)

Multi-dimensional Transforms Reference

The multi-dimensional complex routines are generally prefixed with fftwnd_. Programs using FFTWND should be linked with -lfftw -lm on Unix systems, or with the FFTW and standard math libraries in general.

Plan Creation for Multi-dimensional Transforms

#include <fftw.h>

fftwnd_plan fftwnd_create_plan(int rank, const int *n,
                               fftw_direction dir, int flags);

fftwnd_plan fftw2d_create_plan(int nx, int ny,
                               fftw_direction dir, int flags);

fftwnd_plan fftw3d_create_plan(int nx, int ny, int nz,
                               fftw_direction dir, int flags);

fftwnd_plan fftwnd_create_plan_specific(int rank, const int *n,
                                        fftw_direction dir,
                                        int flags,
                                        fftw_complex *in, int istride,
                                        fftw_complex *out, int ostride);

fftwnd_plan fftw2d_create_plan_specific(int nx, int ny,
                                        fftw_direction dir,
                                        int flags,
                                        fftw_complex *in, int istride,
                                        fftw_complex *out, int ostride);

fftwnd_plan fftw3d_create_plan_specific(int nx, int ny, int nz,
                                        fftw_direction dir, int flags,
                                        fftw_complex *in, int istride,
                                        fftw_complex *out, int ostride);

The function fftwnd_create_plan creates a plan, which is a data structure containing all the information that fftwnd needs in order to compute a multi-dimensional Fourier transform. You can create as many plans as you need, but only one plan for a given array size is required (a plan can be reused many times). The functions fftw2d_create_plan and fftw3d_create_plan are optional, alternative interfaces to fftwnd_create_plan for two and three dimensions, respectively.

fftwnd_create_plan returns a valid plan, or NULL if, for some reason, the plan can't be created. This can happen if memory runs out or if the arguments are invalid in some way (e.g. if rank < 0).

The create_plan_specific variants take as additional arguments specific input/output arrays and their strides. For the last four arguments, you should pass the arrays and strides that you will eventually be passing to fftwnd. The resulting plans will be optimized for those arrays and strides, although they may be used on other arrays as well. Note: the contents of the in and out arrays are destroyed by the specific planner (the initial contents are ignored, so the arrays need not have been initialized). See Section Discussion on Specific Plans, for a discussion on specific plans.

Arguments

Computing the Multi-dimensional Transform

#include <fftw.h>

void fftwnd(fftwnd_plan plan, int howmany,
            fftw_complex *in, int istride, int idist,
            fftw_complex *out, int ostride, int odist);

void fftwnd_one(fftwnd_plan p, fftw_complex *in, 
                fftw_complex *out);

The function fftwnd computes one or more multi-dimensional Fourier Transforms, using a plan created by fftwnd_create_plan (see Section Plan Creation for Multi-dimensional Transforms). (Note that the plan determines the rank and dimensions of the array to be transformed.) The function fftwnd_one provides a simplified interface for the common case of single input array of stride 1.

Arguments

The function fftwnd_one transforms a single, contiguous input array to a contiguous output array. By definition, the call 

fftwnd_one(plan, in, out)

is equivalent to 

fftwnd(plan, 1, in, 1, 0, out, 1, 0)

Destroying a Multi-dimensional Plan

#include <fftw.h>

void fftwnd_destroy_plan(fftwnd_plan plan);

The function fftwnd_destroy_plan frees the plan plan and releases all the memory associated with it. After destruction, a plan is no longer valid.

What FFTWND Really Computes

The conventions that we follow for the multi-dimensional transform are analogous to those for the one-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. The output is in-order, and the zeroth element of the output is the amplitude of the zero frequency component. The Gods forbade using HTML to display mathematical formulas. Please see the TeX or Postscript version of this manual for the proper definition of the n-dimensional Fourier transform that FFTW uses. For completeness, we include a bitmap of the TeX output below:

Real One-dimensional Transforms Reference

The one-dimensional real routines are generally prefixed with rfftw_. (4) Programs using RFFTW should be linked with -lrfftw -lfftw -lm on Unix systems, or with the RFFTW, the FFTW, and the standard math libraries in general.

Plan Creation for Real One-dimensional Transforms

#include <rfftw.h>

rfftw_plan rfftw_create_plan(int n, fftw_direction dir, int flags);

rfftw_plan rfftw_create_plan_specific(int n, fftw_direction dir,
	    int flags, fftw_real *in, int istride,
	    fftw_real *out, int ostride);

The function rfftw_create_plan creates a plan, which is a data structure containing all the information that rfftw needs in order to compute the 1D real Fourier transform. You can create as many plans as you need, but only one plan for a given array size is required (a plan can be reused many times).

rfftw_create_plan returns a valid plan, or NULL if, for some reason, the plan can't be created. In the default installation, this cannot happen, but it is possible to configure RFFTW in such a way that some input sizes are forbidden, and RFFTW cannot create a plan.

The rfftw_create_plan_specific variant takes as additional arguments specific input/output arrays and their strides. For the last four arguments, you should pass the arrays and strides that you will eventually be passing to rfftw. The resulting plans will be optimized for those arrays and strides, although they may be used on other arrays as well. Note: the contents of the in and out arrays are destroyed by the specific planner (the initial contents are ignored, so the arrays need not have been initialized). See Section Discussion on Specific Plans, for a discussion on specific plans.

Arguments

Computing the Real One-dimensional Transform

#include <rfftw.h>

void rfftw(rfftw_plan plan, int howmany, 
           fftw_real *in, int istride, int idist, 
           fftw_real *out, int ostride, int odist);

void rfftw_one(rfftw_plan plan, fftw_real *in, fftw_real *out);

The function rfftw computes the Real One-dimensional Fourier Transform, using a plan created by rfftw_create_plan (see Section Plan Creation for Real One-dimensional Transforms). The function rfftw_one provides a simplified interface for the common case of single input array of stride 1.

Important: When invoked for an out-of-place, FFTW_COMPLEX_TO_REAL transform, the input array is overwritten with scratch values by these routines. The input array is not modified for FFTW_REAL_TO_COMPLEX transforms.

Arguments

The function rfftw_one transforms a single, contiguous input array to a contiguous output array. By definition, the call 

rfftw_one(plan, in, out)

is equivalent to 

rfftw(plan, 1, in, 1, 0, out, 1, 0)

Destroying a Real One-dimensional Plan

#include <rfftw.h>

void rfftw_destroy_plan(rfftw_plan plan);

The function rfftw_destroy_plan frees the plan plan and releases all the memory associated with it. After destruction, a plan is no longer valid.

What RFFTW Really Computes

In this section, we define precisely what RFFTW computes.

The real to complex (FFTW_REAL_TO_COMPLEX) transform of a real array X of size n computes an hermitian array Y, where

(That Y is a hermitian array is not intended to be obvious, although the proof is easy.) The hermitian array Y is stored in halfcomplex order (see Section Data Types). Currently, RFFTW provides no way to compute a real to complex transform with a positive sign in the exponent.

The complex to real (FFTW_COMPLEX_TO_REAL) transform of a hermitian array X of size n computes a real array Y, where

(That Y is a real array is not intended to be obvious, although the proof is easy.) The hermitian input array X is stored in halfcomplex order (see Section Data Types). Currently, RFFTW provides no way to compute a complex to real transform with a negative sign in the exponent.

Like FFTW, RFFTW computes an unnormalized transform. In other words, applying the real to complex (forward) and then the complex to real (backward) transform will multiply the input by n.

Real Multi-dimensional Transforms Reference

The multi-dimensional real routines are generally prefixed with rfftwnd_. Programs using RFFTWND should be linked with -lrfftw -lfftw -lm on Unix systems, or with the FFTW, RFFTW, and standard math libraries in general.

Plan Creation for Real Multi-dimensional Transforms

#include <rfftw.h>

rfftwnd_plan rfftwnd_create_plan(int rank, const int *n,
                                 fftw_direction dir, int flags);

rfftwnd_plan rfftw2d_create_plan(int nx, int ny,
                                 fftw_direction dir, int flags);

rfftwnd_plan rfftw3d_create_plan(int nx, int ny, int nz,
                                 fftw_direction dir, int flags);

The function rfftwnd_create_plan creates a plan, which is a data structure containing all the information that rfftwnd needs in order to compute a multi-dimensional real Fourier transform. You can create as many plans as you need, but only one plan for a given array size is required (a plan can be reused many times). The functions rfftw2d_create_plan and rfftw3d_create_plan are optional, alternative interfaces to rfftwnd_create_plan for two and three dimensions, respectively.

rfftwnd_create_plan returns a valid plan, or NULL if, for some reason, the plan can't be created. This can happen if the arguments are invalid in some way (e.g. if rank < 0).

Arguments

Computing the Real Multi-dimensional Transform

#include <rfftw.h>

void rfftwnd_real_to_complex(rfftwnd_plan plan, int howmany,
                             fftw_real *in, int istride, int idist,
                             fftw_complex *out, int ostride, int odist);
void rfftwnd_complex_to_real(rfftwnd_plan plan, int howmany,
                             fftw_complex *in, int istride, int idist,
                             fftw_real *out, int ostride, int odist);

void rfftwnd_one_real_to_complex(rfftwnd_plan p, fftw_real *in,
                                 fftw_complex *out);
void rfftwnd_one_complex_to_real(rfftwnd_plan p, fftw_complex *in,
                                 fftw_real *out);

These functions compute the real multi-dimensional Fourier Transform, using a plan created by rfftwnd_create_plan (see Section Plan Creation for Real Multi-dimensional Transforms). (Note that the plan determines the rank and dimensions of the array to be transformed.) The `rfftwnd_one_' functions provide a simplified interface for the common case of single input array of stride 1. Unlike other transform routines in FFTW, we here use separate functions for the two directions of the transform in order to correctly express the datatypes of the parameters.

Important: When invoked for an out-of-place, FFTW_COMPLEX_TO_REAL transform with rank > 1, the input array is overwritten with scratch values by these routines. The input array is not modified for FFTW_REAL_TO_COMPLEX transforms or for FFTW_COMPLEX_TO_REAL with rank == 1.

Arguments

The function rfftwnd_one transforms a single, contiguous input array to a contiguous output array. By definition, the call 

rfftwnd_one_...(plan, in, out)

is equivalent to 

rfftwnd_...(plan, 1, in, 1, 0, out, 1, 0)

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 (see Section 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.

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.

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.

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 Gods forbade using HTML to display mathematical formulas. Please see the TeX or Postscript version of this manual for the proper definition of the n-dimensional real Fourier transform that RFFTW uses. For completeness, we include a bitmap of the TeX output below:

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. (See Section 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.

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. (See Section 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).

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.)

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.

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.

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