# 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

• n is the size of the transform. It can be any positive integer.
• FFTW is best at handling sizes of the form 2a 3b 5c 7d 11e 13f, where e+f is either 0 or 1, and the other exponents are arbitrary. Other sizes are computed by means of a slow, general-purpose routine (which nevertheless retains O(n lg n) performance, even for prime sizes). (It is possible to customize FFTW for different array sizes. See Section Installation and Customization, for more information.) Transforms whose sizes are powers of 2 are especially fast.
• dir is the sign of the exponent in the formula that defines the Fourier transform. It can be -1 or +1. The aliases FFTW_FORWARD and FFTW_BACKWARD are provided, where FFTW_FORWARD stands for -1.
• flags is a boolean OR (`|') of zero or more of the following:
• FFTW_MEASURE: this flag tells FFTW to find the optimal plan by actually computing several FFTs and measuring their execution time. Depending on the installation, this can take some time. (2)
• FFTW_ESTIMATE: do not run any FFT and provide a "reasonable" plan (for a RISC processor with many registers). If neither FFTW_ESTIMATE nor FFTW_MEASURE is provided, the default is FFTW_ESTIMATE.
• FFTW_OUT_OF_PLACE: produce a plan assuming that the input and output arrays will be distinct (this is the default).
• FFTW_IN_PLACE: produce a plan assuming that you want the output in the input array. The algorithm used is not necessarily in place: FFTW is able to compute true in-place transforms only for small values of n. If FFTW is not able to compute the transform in-place, it will allocate a temporary array (unless you provide one yourself), compute the transform out of place, and copy the result back. Warning: This option changes the meaning of some parameters of fftw (see Section Computing the One-dimensional Transform). The in-place option is mainly provided for people who want to write their own in-place multi-dimensional Fourier transform, using FFTW as a base. For example, consider a three-dimensional n * n * n transform. An out-of-place algorithm will need another array (which may be huge). However, FFTW can compute the in-place transform along each dimension using only a temporary array of size n. Moreover, if FFTW happens to be able to compute the transform truly in-place, no temporary array and no copying are needed. As distributed, FFTW `knows' how to compute in-place transforms of size 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 32 and 64. The default mode of operation is FFTW_OUT_OF_PLACE.
• FFTW_USE_WISDOM: use any wisdom that is available to help in the creation of the plan. (See Section Words of Wisdom.) This can greatly speed the creation of plans, especially with the FFTW_MEASURE option. FFTW_ESTIMATE plans can also take advantage of wisdom to produce a more optimal plan (based on past measurements) than the estimation heuristic would normally generate. When the FFTW_MEASURE option is used, new wisdom will also be generated if the current transform size is not completely understood by existing wisdom.
• in, out, istride, ostride (only for fftw_create_plan_specific): see corresponding arguments in the description of fftw. (See Section Computing the One-dimensional Transform.) In particular, the out and ostride parameters have the same special meaning for FFTW_IN_PLACE transforms as they have for fftw.

### 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:

• The resulting plans will be optimized for your specific arrays and strides. This may or may not make a significant difference, but it certainly doesn't hurt. (The ordinary planner does its planning based upon a stride-one temporary array that it allocates.)
• Less intermediate storage is required during the planning process. (The ordinary planner uses O(N) temporary storage, where N is the maximum dimension, while it is creating the plan.)
• For multi-dimensional transforms, new parameters become accessible for optimization by the planner. (Since multi-dimensional arrays can be very large, we don't dare to allocate one in the ordinary planner for experimentation. This prevents us from doing certain optimizations that can yield dramatic improvements in some cases.)

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

• plan is the plan created by fftw_create_plan (see Section Plan Creation for One-dimensional Transforms).
• howmany is the number of transforms fftw will compute. It is faster to tell FFTW to compute many transforms, instead of simply calling fftw many times.
• in, istride and idist describe the input array(s). There are howmany input arrays; the first one is pointed to by in, the second one is pointed to by in + idist, and so on, up to in + (howmany - 1) * idist. Each input array consists of complex numbers (see Section Data Types), which are not necessarily contiguous in memory. Specifically, in[0] is the first element of the first array, in[istride] is the second element of the first array, and so on. In general, the i-th element of the j-th input array will be in position in[i * istride + j * idist].
• out, ostride and odist describe the output array(s). The format is the same as for the input array.
• In-place transforms: If the plan specifies an in-place transform, ostride and odist are always ignored. If out is NULL, out is ignored, too. Otherwise, out is interpreted as a pointer to an array of n complex numbers, that FFTW will use as temporary space to perform the in-place computation. out is used as scratch space and its contents destroyed. In this case, out must be an ordinary array whose elements are contiguous in memory (no striding).

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

• rank is the dimensionality of the arrays to be transformed. It can be any non-negative integer.
• n is a pointer to an array of rank integers, giving the size of each dimension of the arrays to be transformed. These sizes, which must be positive integers, correspond to the dimensions of row-major arrays--i.e. n[0] is the size of the dimension whose indices vary most slowly, and so on. (See Section Multi-dimensional Array Format, for more information on row-major storage.) See Section Plan Creation for One-dimensional Transforms, for more information regarding optimal array sizes.
• nx and ny in fftw2d_create_plan are positive integers specifying the dimensions of the rank 2 array to be transformed. i.e. they specify that the transform will operate on nx x ny arrays in row-major order, where nx is the number of rows and ny is the number of columns.
• nx, ny and nz in fftw3d_create_plan are positive integers specifying the dimensions of the rank 3 array to be transformed. i.e. they specify that the transform will operate on nx x ny x nz arrays in row-major order.
• dir is the sign of the exponent in the formula that defines the Fourier transform. It can be -1 or +1. The aliases FFTW_FORWARD and FFTW_BACKWARD are provided, where FFTW_FORWARD stands for -1.
• flags is a boolean OR (`|') of zero or more of the following:
• FFTW_MEASURE: this flag tells FFTW to find the optimal plan by actually computing several FFTs and measuring their execution time.
• FFTW_ESTIMATE: do not run any FFT and provide a "reasonable" plan (for a RISC processor with many registers). If neither FFTW_ESTIMATE nor FFTW_MEASURE is provided, the default is FFTW_ESTIMATE.
• FFTW_OUT_OF_PLACE: produce a plan assuming that the input and output arrays will be distinct (this is the default).
• FFTW_IN_PLACE: produce a plan assuming that you want to perform the transform in-place. (Unlike the one-dimensional transform, this "really" (3) performs the transform in-place.) Note that, if you want to perform in-place transforms, you must use a plan created with this option. The default mode of operation is FFTW_OUT_OF_PLACE.
• FFTW_USE_WISDOM: use any wisdom that is available to help in the creation of the plan. (See Section Words of Wisdom.) This can greatly speed the creation of plans, especially with the FFTW_MEASURE option. FFTW_ESTIMATE plans can also take advantage of wisdom to produce a more optimal plan (based on past measurements) than the estimation heuristic would normally generate. When the FFTW_MEASURE option is used, new wisdom will also be generated if the current transform size is not completely understood by existing wisdom. Note that the same wisdom is shared between one-dimensional and multi-dimensional transforms.
• in, out, istride, ostride (only for the _create_plan_specific variants): see corresponding arguments in the description of fftwnd. (See Section Computing the Multi-dimensional Transform.)

### 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

• plan is the plan created by fftwnd_create_plan. (see Section Plan Creation for Multi-dimensional Transforms). In the case of two and three-dimensional transforms, it could also have been created by fftw2d_create_plan or fftw3d_create_plan, respectively.
• howmany is the number of multi-dimensional transforms fftwnd will compute.
• in, istride and idist describe the input array(s). There are howmany multi-dimensional input arrays; the first one is pointed to by in, the second one is pointed to by in + idist, and so on, up to in + (howmany - 1) * idist. Each multi-dimensional input array consists of complex numbers (see Section Data Types), stored in row-major format (see Section Multi-dimensional Array Format), which are not necessarily contiguous in memory. Specifically, in[0] is the first element of the first array, in[istride] is the second element of the first array, and so on. In general, the i-th element of the j-th input array will be in position in[i * istride + j * idist]. Note that, here, i refers to an index into the row-major format for the multi-dimensional array, rather than an index in any particular dimension.
• In-place transforms: For plans created with the FFTW_IN_PLACE option, the transform is computed in-place--the output is returned in the in array, using the same strides, etcetera, as were used in the input.
• out, ostride and odist describe the output array(s). The format is the same as for the input array.
• In-place transforms: These parameters are ignored for plans created with the FFTW_IN_PLACE option.

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

• n is the size of the transform. It can be any positive integer.
• RFFTW is best at handling sizes of the form 2a 3b 5c 7d 11e 13f, where e+f is either 0 or 1, and the other exponents are arbitrary. Other sizes are computed by means of a slow, general-purpose routine (reducing to O(n2) performance for prime sizes). (It is possible to customize RFFTW for different array sizes. See Section Installation and Customization, for more information.) Transforms whose sizes are powers of 2 are especially fast. If you have large prime factors, it may be faster to switch over to the complex FFTW routines, which have O(n lg n) performance even for prime sizes (we don't know of a similar algorithm specialized for real data, unfortunately).
• dir is the direction of the desired transform, either FFTW_REAL_TO_COMPLEX or FFTW_COMPLEX_TO_REAL, corresponding to FFTW_FORWARD or FFTW_BACKWARD, respectively.
• flags is a boolean OR (`|') of zero or more of the following:
• FFTW_MEASURE: this flag tells RFFTW to find the optimal plan by actually computing several FFTs and measuring their execution time. Depending on the installation, this can take some time.
• FFTW_ESTIMATE: do not run any FFT and provide a "reasonable" plan (for a RISC processor with many registers). If neither FFTW_ESTIMATE nor FFTW_MEASURE is provided, the default is FFTW_ESTIMATE.
• FFTW_OUT_OF_PLACE: produce a plan assuming that the input and output arrays will be distinct (this is the default).
• FFTW_IN_PLACE: produce a plan assuming that you want the output in the input array. The algorithm used is not necessarily in place: RFFTW is able to compute true in-place transforms only for small values of n. If RFFTW is not able to compute the transform in-place, it will allocate a temporary array (unless you provide one yourself), compute the transform out of place, and copy the result back. Warning: This option changes the meaning of some parameters of rfftw (see Section Computing the Real One-dimensional Transform). The default mode of operation is FFTW_OUT_OF_PLACE.
• FFTW_USE_WISDOM: use any wisdom that is available to help in the creation of the plan. (See Section Words of Wisdom.) This can greatly speed the creation of plans, especially with the FFTW_MEASURE option. FFTW_ESTIMATE plans can also take advantage of wisdom to produce a more optimal plan (based on past measurements) than the estimation heuristic would normally generate. When the FFTW_MEASURE option is used, new wisdom will also be generated if the current transform size is not completely understood by existing wisdom.
• in, out, istride, ostride (only for rfftw_create_plan_specific): see corresponding arguments in the description of rfftw. (See Section Computing the Real One-dimensional Transform.) In particular, the out and ostride parameters have the same special meaning for FFTW_IN_PLACE transforms as they have for rfftw.

### 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

• plan is the plan created by rfftw_create_plan (see Section Plan Creation for Real One-dimensional Transforms).
• howmany is the number of transforms rfftw will compute. It is faster to tell RFFTW to compute many transforms, instead of simply calling rfftw many times.
• in, istride and idist describe the input array(s). There are two cases. If the plan defines a FFTW_REAL_TO_COMPLEX transform, in is a real array. Otherwise, for FFTW_COMPLEX_TO_REAL transforms, in is a halfcomplex array whose contents will be destroyed.
• out, ostride and odist describe the output array(s), and have the same meaning as the corresponding parameters for the input array.
• In-place transforms: If the plan specifies an in-place transform, ostride and odist are always ignored. If out is NULL, out is ignored, too. Otherwise, out is interpreted as a pointer to an array of n complex numbers, that FFTW will use as temporary space to perform the in-place computation. out is used as scratch space and its contents destroyed. In this case, out must be an ordinary array whose elements are contiguous in memory (no striding).

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

• rank is the dimensionality of the arrays to be transformed. It can be any non-negative integer.
• n is a pointer to an array of rank integers, giving the size of each dimension of the arrays to be transformed. Note that these are always the dimensions of the real arrays; the complex arrays have different dimensions (see Section Array Dimensions for Real Multi-dimensional Transforms). These sizes, which must be positive integers, correspond to the dimensions of row-major arrays--i.e. n[0] is the size of the dimension whose indices vary most slowly, and so on. (See Section Multi-dimensional Array Format, for more information.)
• nx and ny in rfftw2d_create_plan are positive integers specifying the dimensions of the rank 2 array to be transformed. i.e. they specify that the transform will operate on nx x ny arrays in row-major order, where nx is the number of rows and ny is the number of columns.
• nx, ny and nz in rfftw3d_create_plan are positive integers specifying the dimensions of the rank 3 array to be transformed. i.e. they specify that the transform will operate on nx x ny x nz arrays in row-major order.
• dir is the direction of the desired transform, either FFTW_REAL_TO_COMPLEX or FFTW_COMPLEX_TO_REAL, corresponding to FFTW_FORWARD or FFTW_BACKWARD, respectively.
• flags is a boolean OR (`|') of zero or more of the following:
• FFTW_MEASURE: this flag tells FFTW to find the optimal plan by actually computing several FFTs and measuring their execution time.
• FFTW_ESTIMATE: do not run any FFT and provide a "reasonable" plan (for a RISC processor with many registers). If neither FFTW_ESTIMATE nor FFTW_MEASURE is provided, the default is FFTW_ESTIMATE.
• FFTW_OUT_OF_PLACE: produce a plan assuming that the input and output arrays will be distinct (this is the default).
• FFTW_IN_PLACE: produce a plan assuming that you want to perform the transform in-place. (Unlike the one-dimensional transform, this "really" performs the transform in-place.) Note that, if you want to perform in-place transforms, you must use a plan created with this option. The use of this option has important implications for the size of the input/output array (see Section Computing the Real Multi-dimensional Transform). The default mode of operation is FFTW_OUT_OF_PLACE.
• FFTW_USE_WISDOM: use any wisdom that is available to help in the creation of the plan. (See Section Words of Wisdom.) This can greatly speed the creation of plans, especially with the FFTW_MEASURE option. FFTW_ESTIMATE plans can also take advantage of wisdom to produce a more optimal plan (based on past measurements) than the estimation heuristic would normally generate. When the FFTW_MEASURE option is used, new wisdom will also be generated if the current transform size is not completely understood by existing wisdom. Note that the same wisdom is shared between one-dimensional and multi-dimensional transforms.

### 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

• plan is the plan created by rfftwnd_create_plan. (see Section Plan Creation for Real Multi-dimensional Transforms). In the case of two and three-dimensional transforms, it could also have been created by rfftw2d_create_plan or rfftw3d_create_plan, respectively. FFTW_REAL_TO_COMPLEX plans must be used with the `real_to_complex' functions, and FFTW_COMPLEX_TO_REAL plans must be used with the `complex_to_real' functions. It is an error to mismatch the plan direction and the transform function.
• howmany is the number of transforms to be computed.
• in, istride and idist describe the input array(s). There are howmany input arrays; the first one is pointed to by in, the second one is pointed to by in + idist, and so on, up to in + (howmany - 1) * idist. Each input array is stored in row-major format (see Section Multi-dimensional Array Format), and is not necessarily contiguous in memory. Specifically, in[0] is the first element of the first array, in[istride] is the second element of the first array, and so on. In general, the i-th element of the j-th input array will be in position in[i * istride + j * idist]. Note that, here, i refers to an index into the row-major format for the multi-dimensional array, rather than an index in any particular dimension. The dimensions of the arrays are different for real and complex data, and are discussed in more detail below (see Section Array Dimensions for Real Multi-dimensional Transforms).
• In-place transforms: For plans created with the FFTW_IN_PLACE option, the transform is computed in-place--the output is returned in the in array. The meaning of the stride and dist parameters in this case is subtle and is discussed below (see Section Strides in In-place RFFTWND).
• out, ostride and odist describe the output array(s). The format is the same as that for the input array. See below for a discussion of the dimensions of the output array for real and complex data.
• In-place transforms: These parameters are ignored for plans created with the FFTW_IN_PLACE option.

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.