     [file version.texi not included]

This is the FFTW User's manual.

Copyright © 1997--1999 Massachusetts Institute of Technology

Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies.

Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one.

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FFTW User Manual
****************

Welcome to FFTW, the Fastest Fourier Transform in the West.  FFTW is a collection of fast C routines to compute the discrete Fourier transform. This manual documents FFTW version VERSION.

Introduction
************

This manual documents version VERSION of FFTW, the Fastest
Fourier Transform in the West.  FFTW is a comprehensive collection of fast C routines for computing the discrete Fourier transform (DFT) in one or more dimensions, of both real and complex data, and of arbitrary input size.  FFTW also includes parallel transforms for both shared- and distributed-memory systems.  We assume herein that the reader is already familiar with the properties and uses of the DFT that are relevant to her application.  Otherwise, see e.g. The Fast Fourier Transform by E. O. Brigham (Prentice-Hall, Englewood Cliffs, NJ, 1974). Our web page (http://www.fftw.org) also has links to FFT-related information online.

FFTW is usually faster (and sometimes much faster) than all other freely-available Fourier transform programs found on the Net.  For transforms whose size is a power of two, it compares favorably with the FFT codes in Sun's Performance Library and IBM's ESSL library, which are targeted at specific machines.  Moreover, FFTW's performance is portable.  Indeed, FFTW is unique in that it automatically adapts itself to your machine, your cache, the size of your memory, the number of registers, and all the other factors that normally make it impossible to optimize a program for more than one machine.  An extensive comparison of FFTW's performance with that of other Fourier transform codes has been made. The results are available on the Web at the benchFFT home page (http://theory.lcs.mit.edu/~benchfft).

In order to use FFTW effectively, you need to understand one basic concept of FFTW's internal structure.  FFTW does not used a fixed algorithm for computing the transform, but it can adapt the DFT algorithm to details of the underlying hardware in order to achieve best performance.  Hence, the computation of the transform is split into two phases.  First, FFTW's planner is called, which ``learns'' the fastest way to compute the transform on your machine.  The planner produces a data structure called a plan that contains this information.  Subsequently, the plan is passed to FFTW's executor, along with an array of input data.  The executor computes the actual transform, as dictated by the plan.  The plan can be reused as many times as needed.  In typical high-performance applications, many transforms of the same size are computed, and consequently a relatively-expensive initialization of this sort is acceptable.  On the other hand, if you need a single transform of a given size, the one-time cost of the planner becomes significant.  For this case, FFTW provides fast planners based on heuristics or on previously computed plans.

The pattern of planning/execution applies to all four operation modes of FFTW, that is, I) one-dimensional complex transforms (FFTW), II) multi-dimensional complex transforms (FFTWND), III) one-dimensional transforms of real data (RFFTW), IV) multi-dimensional transforms of real data (RFFTWND).  Each mode comes with its own planner and executor.

Besides the automatic performance adaptation performed by the planner, it is also possible for advanced users to customize FFTW for their special needs.  As distributed, FFTW works most efficiently for arrays whose size can be factored into small primes (2, 3, 5, and 7), and uses a slower general-purpose routine for other factors.  FFTW, however, comes with a code generator that can produce fast C programs for any particular array size you may care about. For example, if you need transforms of size 513 = 19 x 3^3, you can customize FFTW to support the factor 19 efficiently.

FFTW can exploit multiple processors if you have them.  FFTW comes with a shared-memory implementation on top of POSIX (and similar) threads, as well as a distributed-memory implementation based on MPI. We also provide an experimental parallel implementation written in Cilk, the superior programming tool of choice for discriminating hackers (Olin Shivers).  (See the Cilk home page (http://supertech.lcs.mit.edu/cilk).)

For more information regarding FFTW, see the paper, ``The Fastest Fourier Transform in the West,'' by M. Frigo and S. G. Johnson, which is the technical report MIT-LCS-TR-728 (Sep. '97).  See also, ``FFTW: An Adaptive Software Architecture for the FFT,'' by M. Frigo and S. G. Johnson, which appeared in the 23rd International Conference on Acoustics, Speech, and Signal Processing (Proc. ICASSP 1998 3, p. 1381).  The code generator is described in the paper ``A Fast Fourier Transform Compiler'', by M. Frigo, to appear in the Proceedings of the 1999 ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI), Atlanta, Georgia, May 1999.  These papers, along with the latest version of FFTW, the FAQ, benchmarks, and other links, are available at the FFTW home page (http://www.fftw.org).  The current version of FFTW incorporates many good ideas from the past thirty years of FFT literature.  In one way or another, FFTW uses the Cooley-Tukey algorithm, the Prime Factor algorithm, Rader's algorithm for prime sizes, and the split-radix algorithm (with a variation due to Dan Bernstein).  Our code generator also produces new algorithms that we do not yet completely understand. The reader is referred to the cited papers for the appropriate references.

The rest of this manual is organized as follows.  We first discuss the sequential (one-processor) implementation.  We start by describing the basic features of FFTW in Tutorial.  This discussion includes the storage scheme of multi-dimensional arrays (Multi-dimensional Array Format) and FFTW's mechanisms for storing plans on disk (Words of Wisdom).  Next, FFTW Reference provides comprehensive documentation of all FFTW's features.  Parallel transforms are discussed in their own chapter Parallel FFTW.  Fortran programmers can also use FFTW, as described in Calling FFTW from Fortran. Installation and Customization explains how to install FFTW in your computer system and how to adapt FFTW to your needs.  License and copyright information is given in License and Copyright.  Finally, we thank all the people who helped us in Acknowledgments.

Tutorial
********

This chapter describes the basic usage of FFTW, i.e., how to compute the Fourier transform of a single array.  This chapter tells the truth, but not the whole truth. Specifically, FFTW implements additional routines and flags, providing extra functionality, that are not documented here.  See FFTW Reference, for more complete information. (Note that you need to compile and install FFTW before you can use it in a program.  See Installation and Customization, for the details of the installation.)

Here, we assume a default installation of FFTW.  In some installations (particulary from binary packages), the FFTW header files and libraries are prefixed with d or s to indicate versions in double or single precision, respectively.  The usage of FFTW in that case is the same, except that #include directives and link commands must use the appropriate prefix.  See Installing FFTW in both single and double precision, for more information.

This tutorial chapter is structured as follows.  Complex One-dimensional Transforms Tutorial describes the basic usage of the one-dimensional transform of complex data.  Complex Multi-dimensional Transforms Tutorial describes the basic usage of the multi-dimensional transform of complex data.  Real One-dimensional Transforms Tutorial describes the one-dimensional transform of real data and its inverse.  Finally, Real Multi-dimensional Transforms Tutorial describes the multi-dimensional transform of real data and its inverse.  We recommend that you read these sections in the order that they are presented.  We then discuss two topics in detail.  In Multi-dimensional Array Format, we discuss the various alternatives for storing multi-dimensional arrays in memory.  Words of Wisdom shows how you can save FFTW's plans for future use.

Complex One-dimensional Transforms Tutorial
===========================================

The basic usage of FFTW is simple.  A typical call to FFTW looks like:

     #include <fftw.h>
     ...
     {
          fftw_complex in[N], out[N];
          fftw_plan p;
          ...
          p = fftw_create_plan(N, FFTW_FORWARD, FFTW_ESTIMATE);
          ...
          fftw_one(p, in, out);
          ...
          fftw_destroy_plan(p);
     }

The first thing we do is to create a plan, which is an object that contains all the data that FFTW needs to compute the FFT, using the following function:

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

The first argument, n, is the size of the transform you are trying to compute.  The size n can be any positive integer, but sizes that are products of small factors are transformed most efficiently.  The second argument, dir, can be either FFTW_FORWARD or FFTW_BACKWARD, and indicates the direction of the transform you are interested in.  Alternatively, you can use the sign of the exponent in the transform, -1 or +1, which corresponds to FFTW_FORWARD or FFTW_BACKWARD respectively.  The flags argument is either FFTW_MEASURE or FFTW_ESTIMATE.  FFTW_MEASURE means that FFTW actually runs and measures the execution time of several FFTs in order to find the best way to compute the transform of size n.  This may take some time, depending on your installation and on the precision of the timer in your machine.  FFTW_ESTIMATE, on the contrary, does not run any computation, and just builds a reasonable plan, which may be sub-optimal.  In other words, if your program performs many transforms of the same size and initialization time is not important, use FFTW_MEASURE; otherwise use the estimate.  (A compromise between these two extremes exists.  See Words of Wisdom.)

Once the plan has been created, you can use it as many times as you like for transforms on arrays of the same size.  When you are done with the plan, you deallocate it by calling fftw_destroy_plan(plan).

The transform itself is computed by passing the plan along with the input and output arrays to fftw_one:

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

Note that the transform is out of place: in and out must point to distinct arrays. It operates on data of type fftw_complex, a data structure with real (in[i].re) and imaginary (in[i].im) floating-point components.  The in and out arrays should have the length specified when the plan was created.  An alternative function, fftw, allows you to efficiently perform multiple and/or strided transforms (see FFTW Reference).

The DFT results are stored in-order in the array out, with the zero-frequency (DC) component in out[0]. The array in is not modified.  Users should note that FFTW computes an unnormalized DFT, the sign of whose exponent is given by the dir parameter of fftw_create_plan.  Thus, computing a forward followed by a backward transform (or vice versa) results in the original array scaled by n.  See What FFTW Really Computes, for the definition of DFT.

A program using FFTW should be linked with -lfftw -lm on Unix systems, or with the FFTW and standard math libraries in general.

Complex Multi-dimensional Transforms Tutorial
=============================================

FFTW can also compute transforms of any number of dimensions (rank).  The syntax is similar to that for the one-dimensional transforms, with fftw_ replaced by fftwnd_ (which stands for ``fftw in N dimensions'').

As before, we #include <fftw.h> and create a plan for the transforms, this time of type fftwnd_plan:

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

rank is the dimensionality of the array, and can be any non-negative integer.  The next argument, n, is a pointer to an integer array of length rank containing the (positive) sizes of each dimension of the array.  (Note that the array will be stored in row-major order. See Multi-dimensional Array Format, for information on row-major order.)  The last two parameters are the same as in fftw_create_plan.  We now, however, have an additional possible flag, FFTW_IN_PLACE, since fftwnd supports true in-place transforms.  Multiple flags are combined using a bitwise or (|).  (An in-place transform is one in which the output data overwrite the input data.  It thus requires half as much memory as---and is often faster than---its opposite, an out-of-place transform.)

For two- and three-dimensional transforms, FFTWND provides alternative routines that accept the sizes of each dimension directly, rather than indirectly through a rank and an array of sizes.  These are otherwise identical to fftwnd_create_plan, and are sometimes more convenient:

     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);

Once the plan has been created, you can use it any number of times for transforms of the same size.  When you do not need a plan anymore, you can deallocate the plan by calling fftwnd_destroy_plan(plan).

Given a plan, you can compute the transform of an array of data by calling:

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

Here, in and out point to multi-dimensional arrays in row-major order, of the size specified when the plan was created.  In the case of an in-place transform, the out parameter is ignored and the output data are stored in the in array.  The results are stored in-order, unnormalized, with the zero-frequency component in out[0]. A forward followed by a backward transform (or vice-versa) yields the original data multiplied by the size of the array (i.e. the product of the dimensions).  See What FFTWND Really Computes, for a discussion of what FFTWND computes.

For example, code to perform an in-place FFT of a three-dimensional array might look like:

     #include <fftw.h>
     ...
     {
          fftw_complex in[L][M][N];
          fftwnd_plan p;
          ...
          p = fftw3d_create_plan(L, M, N, FFTW_FORWARD,
                                 FFTW_MEASURE | FFTW_IN_PLACE);
          ...
          fftwnd_one(p, &in[0][0][0], NULL);
          ...
          fftwnd_destroy_plan(p);
     }

Note that in is a statically-declared array, which is automatically in row-major order, but we must take the address of the first element in order to fit the type expected by fftwnd_one. (see Multi-dimensional Array Format.)

Real One-dimensional Transforms Tutorial
========================================

If the input data are purely real, you can save roughly a factor of two in both time and storage by using the rfftw transforms, which are FFTs specialized for real data.  The output of a such a transform is a halfcomplex array, which consists of only half of the complex DFT amplitudes (since the negative-frequency amplitudes for real data are the complex conjugate of the positive-frequency amplitudes).

In exchange for these speed and space advantages, the user sacrifices some of the simplicity of FFTW's complex transforms.  First of all, to allow maximum performance, the output format of the one-dimensional real transforms is different from that used by the multi-dimensional transforms.  Second, the inverse transform (halfcomplex to real) has the side-effect of destroying its input array.  Neither of these inconveniences should pose a serious problem for users, but it is important to be aware of them.  (Both the inconvenient output format and the side-effect of the inverse transform can be ameliorated for one-dimensional transforms, at the expense of some performance, by using instead the multi-dimensional transform routines with a rank of one.)

The computation of the plan is similar to that for the complex transforms.  First, you #include <rfftw.h>.  Then, you create a plan (of type rfftw_plan) by calling:

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

n is the length of the real array in the transform (even for halfcomplex-to-real transforms), and can be any positive integer (although sizes with small factors are transformed more efficiently). dir is either FFTW_REAL_TO_COMPLEX or FFTW_COMPLEX_TO_REAL. The flags parameter is the same as in fftw_create_plan.

Once created, a plan can be used for any number of transforms, and is deallocated when you are done with it by calling rfftw_destroy_plan(plan).

Given a plan, a real-to-complex or complex-to-real transform is computed by calling:

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

(Note that fftw_real is an alias for the floating-point type for which FFTW was compiled.)  Depending upon the direction of the plan, either the input or the output array is halfcomplex, and is stored in the following format:

r0, r1, r2, r(n/2), i((n+1)/2-1), ..., i2, i1

Here, rk is the real part of the kth output, and ik is the imaginary part.  (We follow here the C convention that integer division is rounded down, e.g. 7 / 2 = 3.) For a halfcomplex array hc[], the kth component has its real part in hc[k] and its imaginary part in hc[n-k], with the exception of k == 0 or n/2 (the latter only if n is even)---in these two cases, the imaginary part is zero due to symmetries of the real-complex transform, and is not stored.  Thus, the transform of n real values is a halfcomplex array of length n, and vice versa.   (The output for the multi-dimensional rfftw is a more-conventional array of fftw_complex values, but the format here permitted us greater efficiency in one dimension.)  This is actually only half of the DFT spectrum of the data.  Although the other half can be obtained by complex conjugation, it is not required by many applications such as convolution and filtering.

Like the complex transforms, the RFFTW transforms are unnormalized, so a forward followed by a backward transform (or vice-versa) yields the original data scaled by the length of the array, n.

Let us reiterate here our warning that an FFTW_COMPLEX_TO_REAL transform has the side-effect of destroying its (halfcomplex) input. The FFTW_REAL_TO_COMPLEX transform, however, leaves its (real) input untouched, just as you would hope.

As an example, here is an outline of how you might use RFFTW to compute the power spectrum of a real array (i.e. the squares of the absolute values of the DFT amplitudes):

     #include <rfftw.h>
     ...
     {
          fftw_real in[N], out[N], power_spectrum[N/2+1];
          rfftw_plan p;
          int k;
          ...
          p = rfftw_create_plan(N, FFTW_REAL_TO_COMPLEX, FFTW_ESTIMATE);
          ...
          rfftw_one(p, in, out);
          power_spectrum[0] = out[0]*out[0];  /* DC component */
          for (k = 1; k < (N+1)/2; ++k)  /* (k < N/2 rounded up) */
               power_spectrum[k] = out[k]*out[k] + out[N-k]*out[N-k];
          if (N % 2 == 0) /* N is even */
               power_spectrum[N/2] = out[N/2]*out[N/2];  /* Nyquist freq. */
          ...
          rfftw_destroy_plan(p);
     }

Programs using RFFTW should link with -lrfftw -lfftw -lm on Unix, or with the FFTW, RFFTW, and math libraries in general.

Real Multi-dimensional Transforms Tutorial
==========================================

FFTW includes multi-dimensional transforms for real data of any rank. As with the one-dimensional real transforms, they save roughly a factor of two in time and storage over complex transforms of the same size. Also as in one dimension, these gains come at the expense of some increase in complexity---the output format is different from the one-dimensional RFFTW (and is more similar to that of the complex FFTW) and the inverse (complex to real) transforms have the side-effect of overwriting their input data.

To use the real multi-dimensional transforms, you first #include <rfftw.h> and then create a plan for the size and direction of transform that you are interested in:

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

The first two parameters describe the size of the real data (not the halfcomplex data, which will have different dimensions).  The last two parameters are the same as those for rfftw_create_plan.  Just as for fftwnd, there are two alternate versions of this routine, rfftw2d_create_plan and rfftw3d_create_plan, that are sometimes more convenient for two- and three-dimensional transforms. Also as in fftwnd, rfftwnd supports true in-place transforms, specified by including FFTW_IN_PLACE in the flags.

Once created, a plan can be used for any number of transforms, and is deallocated by calling rfftwnd_destroy_plan(plan).

Given a plan, the transform is computed by calling one of the following two routines:

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

As is clear from their names and parameter types, the former function is for FFTW_REAL_TO_COMPLEX transforms and the latter is for FFTW_COMPLEX_TO_REAL transforms.  (We could have used only a single routine, since the direction of the transform is encoded in the plan, but we wanted to correctly express the datatypes of the parameters.)  The latter routine, as we discuss elsewhere, has the side-effect of overwriting its input (except when the rank of the array is one).  In both cases, the out parameter is ignored for in-place transforms.

The format of the complex arrays deserves careful attention. Suppose that the real data has dimensions n1 x n2 x ... x nd (in row-major order).  Then, after a real-to-complex transform, the output is an n1 x n2 x ... x (nd/2+1) array of fftw_complex values in row-major order, corresponding to slightly over half of the output of the corresponding complex transform. (Note that the division is rounded down.)  The ordering of the data is otherwise exactly the same as in the complex case.  (In principle, the output could be exactly half the size of the complex transform output, but in more than one dimension this requires too complicated a format to be practical.)  Note that, unlike the one-dimensional RFFTW, the real and imaginary parts of the DFT amplitudes are here stored together in the natural way.

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.

For example, consider the transform of a two-dimensional real array of size nx by ny.  The output of the rfftwnd transform is a two-dimensional complex array of size nx by ny/2+1, where the y dimension has been cut nearly in half because of redundancies in the output.  Because fftw_complex is twice the size of fftw_real, the output array is slightly bigger than the input array.  Thus, if we want to compute the transform in place, we must pad the input array so that it is of size nx by 2*(ny/2+1).  If ny is even, then there are two padding elements at the end of each row (which need not be initialized, as they are only used for output).

The RFFTWND transforms are unnormalized, so a forward followed by a backward transform will result in the original data scaled by the number of real data elements---that is, the product of the (logical) dimensions of the real data.

Below, we illustrate the use of RFFTWND by showing how you might use it to compute the (cyclic) convolution of two-dimensional real arrays a and b (using the identity that a convolution corresponds to a pointwise product of the Fourier transforms).  For variety, in-place transforms are used for the forward FFTs and an out-of-place transform is used for the inverse transform.

     #include <rfftw.h>
     ...
     {
          fftw_real a[M][2*(N/2+1)], b[M][2*(N/2+1)], c[M][N];
          fftw_complex *A, *B, C[M][N/2+1];
          rfftwnd_plan p, pinv;
          fftw_real scale = 1.0 / (M * N);
          int i, j;
          ...
          p    = rfftw2d_create_plan(M, N, FFTW_REAL_TO_COMPLEX,
                                     FFTW_ESTIMATE | FFTW_IN_PLACE);
          pinv = rfftw2d_create_plan(M, N, FFTW_COMPLEX_TO_REAL,
                                     FFTW_ESTIMATE);

          /* aliases for accessing complex transform outputs: */
          A = (fftw_complex*) &a[0][0];
          B = (fftw_complex*) &b[0][0];
          ...
          for (i = 0; i < M; ++i)
               for (j = 0; j < N; ++j) {
                    a[i][j] = ... ;
                    b[i][j] = ... ;
               }
          ...
          rfftwnd_one_real_to_complex(p, &a[0][0], NULL);
          rfftwnd_one_real_to_complex(p, &b[0][0], NULL);

          for (i = 0; i < M; ++i)
               for (j = 0; j < N/2+1; ++j) {
                    int ij = i*(N/2+1) + j;
                    C[i][j].re = (A[ij].re * B[ij].re
                                  - A[ij].im * B[ij].im) * scale;
                    C[i][j].im = (A[ij].re * B[ij].im
                                  + A[ij].im * B[ij].re) * scale;
               }

          /* inverse transform to get c, the convolution of a and b;
             this has the side effect of overwriting C */
          rfftwnd_one_complex_to_real(pinv, &C[0][0], &c[0][0]);
          ...
          rfftwnd_destroy_plan(p);
          rfftwnd_destroy_plan(pinv);
     }

We access the complex outputs of the in-place transforms by casting each real array to a fftw_complex pointer.  Because this is a ``flat'' pointer, we have to compute the row-major index ij explicitly in the convolution product loop. In order to normalize the convolution, we must multiply by a scale factor---we can do so either before or after the inverse transform, and choose the former because it obviates the necessity of an additional loop. Notice the limits of the loops and the dimensions of the various arrays.

As with the one-dimensional RFFTW, an out-of-place FFTW_COMPLEX_TO_REAL transform has the side-effect of overwriting its input array.  (The real-to-complex transform, on the other hand, leaves its input array untouched.)  If you use RFFTWND for a rank-one transform, however, this side-effect does not occur.  Because of this fact (and the simpler output format), users may find the RFFTWND interface more convenient than RFFTW for one-dimensional transforms. However, RFFTWND in one dimension is slightly slower than RFFTW because RFFTWND uses an extra buffer array internally.

Multi-dimensional Array Format
==============================

This section describes the format in which multi-dimensional arrays are stored.  We felt that a detailed discussion of this topic was necessary, since it is often a source of confusion among users and several different formats are common.  Although the comments below refer to fftwnd, they are also applicable to the rfftwnd routines.

Row-major Format
----------------

The multi-dimensional arrays passed to fftwnd are expected to be stored as a single contiguous block in row-major order (sometimes called ``C order'').  Basically, this means that as you step through adjacent memory locations, the first dimension's index varies most slowly and the last dimension's index varies most quickly.

To be more explicit, let us consider an array of rank d whose dimensions are n1 x n2 x n3 x ... x nd. Now, we specify a location in the array by a sequence of (zero-based) indices, one for each dimension: (i1, i2, ..., id). If the array is stored in row-major order, then this element is located at the position id + nd * (id-1 + nd-1 * (... + n2 * i1)).

Note that each element of the array must be of type fftw_complex; i.e. a (real, imaginary) pair of (double-precision) numbers. Note also that, in fftwnd, the expression above is multiplied by the stride to get the actual array index---this is useful in situations where each element of the multi-dimensional array is actually a data structure or another array, and you just want to transform a single field. In most cases, however, you use a stride of 1.

Column-major Format
-------------------

Readers from the Fortran world are used to arrays stored in column-major order (sometimes called ``Fortran order'').  This is essentially the exact opposite of row-major order in that, here, the first dimension's index varies most quickly.

If you have an array stored in column-major order and wish to transform it using fftwnd, it is quite easy to do.  When creating the plan, simply pass the dimensions of the array to fftwnd_create_plan in reverse order.  For example, if your array is a rank three N x M x L matrix in column-major order, you should pass the dimensions of the array as if it were an L x M x N matrix (which it is, from the perspective of fftwnd).  This is done for you automatically by the FFTW Fortran wrapper routines (see Calling FFTW from Fortran).

Static Arrays in C
------------------

Multi-dimensional arrays declared statically (that is, at compile time, not necessarily with the static keyword) in C are already in row-major order.  You don't have to do anything special to transform them.  (see Complex Multi-dimensional Transforms Tutorial, for an example of this sort of code.)

Dynamic Arrays in C
-------------------

Often, especially for large arrays, it is desirable to allocate the arrays dynamically, at runtime.  This isn't too hard to do, although it is not as straightforward for multi-dimensional arrays as it is for one-dimensional arrays.

Creating the array is simple: using a dynamic-allocation routine like malloc, allocate an array big enough to store N fftw_complex values, where N is the product of the sizes of the array dimensions (i.e. the total number of complex values in the array).  For example, here is code to allocate a 5x12x27 rank 3 array:

     fftw_complex *an_array;

     an_array = (fftw_complex *) malloc(5 * 12 * 27 * sizeof(fftw_complex));

Accessing the array elements, however, is more tricky---you can't simply use multiple applications of the [] operator like you could for static arrays.  Instead, you have to explicitly compute the offset into the array using the formula given earlier for row-major arrays.  For example, to reference the (i,j,k)-th element of the array allocated above, you would use the expression an_array[k + 27 * (j + 12 * i)].

This pain can be alleviated somewhat by defining appropriate macros, or, in C++, creating a class and overloading the () operator.

Dynamic Arrays in C---The Wrong Way
-----------------------------------

A different method for allocating multi-dimensional arrays in C is often suggested that is incompatible with fftwnd: using it will cause FFTW to die a painful death.  We discuss the technique here, however, because it is so commonly known and used.  This method is to create arrays of pointers of arrays of pointers of ...etcetera.  For example, the analogue in this method to the example above is:

     int i,j;
     fftw_complex ***a_bad_array;  /* another way to make a 5x12x27 array */

     a_bad_array = (fftw_complex ***) malloc(5 * sizeof(fftw_complex **));
     for (i = 0; i < 5; ++i) {
          a_bad_array[i] =
             (fftw_complex **) malloc(12 * sizeof(fftw_complex *));
          for (j = 0; j < 12; ++j)
               a_bad_array[i][j] =
                     (fftw_complex *) malloc(27 * sizeof(fftw_complex));
     }

As you can see, this sort of array is inconvenient to allocate (and deallocate).  On the other hand, it has the advantage that the (i,j,k)-th element can be referenced simply by a_bad_array[i][j][k].

If you like this technique and want to maximize convenience in accessing the array, but still want to pass the array to FFTW, you can use a hybrid method.  Allocate the array as one contiguous block, but also declare an array of arrays of pointers that point to appropriate places in the block.  That sort of trick is beyond the scope of this documentation; for more information on multi-dimensional arrays in C, see the comp.lang.c FAQ (http://www.eskimo.com/~scs/C-faq/s6.html).

Words of Wisdom
===============

FFTW implements a method for saving plans to disk and restoring them. In fact, what FFTW does is more general than just saving and loading plans.  The mechanism is called wisdom.  Here, we describe this feature at a high level. See FFTW Reference, for a less casual (but more complete) discussion of how to use wisdom in FFTW.

Plans created with the FFTW_MEASURE option produce near-optimal FFT performance, but it can take a long time to compute a plan because FFTW must actually measure the runtime of many possible plans and select the best one.  This is designed for the situations where so many transforms of the same size must be computed that the start-up time is irrelevant.  For short initialization times but slightly slower transforms, we have provided FFTW_ESTIMATE.  The wisdom mechanism is a way to get the best of both worlds.  There are, however, certain caveats that the user must be aware of in using wisdom. For this reason, wisdom is an optional feature which is not enabled by default.

At its simplest, wisdom provides a way of saving plans to disk so that they can be reused in other program runs.  You create a plan with the flags FFTW_MEASURE and FFTW_USE_WISDOM, and then save the wisdom using fftw_export_wisdom:

          plan = fftw_create_plan(..., ... | FFTW_MEASURE | FFTW_USE_WISDOM);
          fftw_export_wisdom(...);

The next time you run the program, you can restore the wisdom with fftw_import_wisdom, and then recreate the plan using the same flags as before.  This time, however, the same optimal plan will be created very quickly without measurements. (FFTW still needs some time to compute trigonometric tables, however.)  The basic outline is:

          fftw_import_wisdom(...);
          plan = fftw_create_plan(..., ... | FFTW_USE_WISDOM);

Wisdom is more than mere rote memorization, however.  FFTW's wisdom encompasses all of the knowledge and measurements that were used to create the plan for a given size.  Therefore, existing wisdom is also applied to the creation of other plans of different sizes.

Whenever a plan is created with the FFTW_MEASURE and FFTW_USE_WISDOM flags, wisdom is generated.  Thereafter, plans for any transform with a similar factorization will be computed more quickly, so long as they use the FFTW_USE_WISDOM flag.  In fact, for transforms with the same factors and of equal or lesser size, no measurements at all need to be made and an optimal plan can be created with negligible delay!

For example, suppose that you create a plan for N = 2^16. Then, for any equal or smaller power of two, FFTW can create a plan (with the same direction and flags) quickly, using the precomputed wisdom. Even for larger powers of two, or sizes that are a power of two times some other prime factors, plans will be computed more quickly than they would otherwise (although some measurements still have to be made).

The wisdom is cumulative, and is stored in a global, private data structure managed internally by FFTW.  The storage space required is minimal, proportional to the logarithm of the sizes the wisdom was generated from.  The wisdom can be forgotten (and its associated memory freed) by a call to fftw_forget_wisdom(); otherwise, it is remembered until the program terminates.  It can also be exported to a file, a string, or any other medium using fftw_export_wisdom and restored during a subsequent execution of the program (or a different program) using fftw_import_wisdom (these functions are described below).

Because wisdom is incorporated into FFTW at a very low level, the same wisdom can be used for one-dimensional transforms, multi-dimensional transforms, and even the parallel extensions to FFTW. Just include FFTW_USE_WISDOM in the flags for whatever plans you create (i.e., always plan wisely).

Plans created with the FFTW_ESTIMATE plan can use wisdom, but cannot generate it;  only FFTW_MEASURE plans actually produce wisdom.  Also, plans can only use wisdom generated from plans created with the same direction and flags.  For example, a size 42 FFTW_BACKWARD transform will not use wisdom produced by a size 42 FFTW_FORWARD transform.  The only exception to this rule is that FFTW_ESTIMATE plans can use wisdom from FFTW_MEASURE plans.

Caveats in Using Wisdom
-----------------------

     For in much wisdom is much grief, and he that increaseth knowledge increaseth sorrow. [Ecclesiastes 1:18]

There are pitfalls to using wisdom, in that it can negate FFTW's ability to adapt to changing hardware and other conditions. For example, it would be perfectly possible to export wisdom from a program running on one processor and import it into a program running on another processor.  Doing so, however, would mean that the second program would use plans optimized for the first processor, instead of the one it is running on.

It should be safe to reuse wisdom as long as the hardware and program binaries remain unchanged. (Actually, the optimal plan may change even between runs of the same binary on identical hardware, due to differences in the virtual memory environment, etcetera.  Users seriously interested in performance should worry about this problem, too.)  It is likely that, if the same wisdom is used for two different program binaries, even running on the same machine, the plans may be sub-optimal because of differing code alignments.  It is therefore wise to recreate wisdom every time an application is recompiled.  The more the underlying hardware and software changes between the creation of wisdom and its use, the greater grows the risk of sub-optimal plans.

Importing and Exporting Wisdom
------------------------------

     void fftw_export_wisdom_to_file(FILE *output_file);
     fftw_status fftw_import_wisdom_from_file(FILE *input_file);

fftw_export_wisdom_to_file writes the wisdom to output_file, which must be a file open for writing. fftw_import_wisdom_from_file reads the wisdom from input_file, which must be a file open for reading, and returns FFTW_SUCCESS if successful and FFTW_FAILURE otherwise. In both cases, the file is left open and must be closed by the caller.  It is perfectly fine if other data lie before or after the wisdom in the file, as long as the file is positioned at the beginning of the wisdom data before import.

     char *fftw_export_wisdom_to_string(void);
     fftw_status fftw_import_wisdom_from_string(const char *input_string)

fftw_export_wisdom_to_string allocates a string, exports the wisdom to it in NULL-terminated format, and returns a pointer to the string.  If there is an error in allocating or writing the data, it returns NULL.  The caller is responsible for deallocating the string (with fftw_free) when she is done with it. fftw_import_wisdom_from_string imports the wisdom from input_string, returning FFTW_SUCCESS if successful and FFTW_FAILURE otherwise.

Exporting wisdom does not affect the store of wisdom. Imported wisdom supplements the current store rather than replacing it (except when there is conflicting wisdom, in which case the older wisdom is discarded). The format of the exported wisdom is ``nerd-readable'' LISP-like ASCII text; we will not document it here except to note that it is insensitive to white space (interested users can contact us for more details).

See FFTW Reference, for more information, and for a description of how you can implement wisdom import/export for other media besides files and strings.

The following is a brief example in which the wisdom is read from a file, a plan is created (possibly generating more wisdom), and then the wisdom is exported to a string and printed to stdout.

     {
          fftw_plan plan;
          char *wisdom_string;
          FILE *input_file;

          /* open file to read wisdom from */
          input_file = fopen("sample.wisdom", "r");
          if (FFTW_FAILURE == fftw_import_wisdom_from_file(input_file))
               printf("Error reading wisdom!\n");
          fclose(input_file); /* be sure to close the file! */

          /* create a plan for N=64, possibly creating and/or using wisdom */
          plan = fftw_create_plan(64,FFTW_FORWARD,
                                  FFTW_MEASURE | FFTW_USE_WISDOM);

          /* ... do some computations with the plan ... */

          /* always destroy plans when you are done */
          fftw_destroy_plan(plan);

          /* write the wisdom to a string */
          wisdom_string = fftw_export_wisdom_to_string();
          if (wisdom_string != NULL) {
               printf("Accumulated wisdom: %s\n",wisdom_string);

               /* Just for fun, destroy and restore the wisdom */
               fftw_forget_wisdom(); /* all gone! */
               fftw_import_wisdom_from_string(wisdom_string);
               /* wisdom is back! */

               fftw_free(wisdom_string); /* deallocate it since we're done */
          }
     }

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 Data Types).  Then, in four sections, we explain the FFTW program interface for complex one-dimensional transforms (see One-dimensional Transforms Reference), complex multi-dimensional transforms (see Multi-dimensional Transforms Reference), and real one-dimensional transforms (see Real One-dimensional Transforms Reference), real multi-dimensional transforms (see Real Multi-dimensional Transforms Reference). Wisdom Reference describes the wisdom mechanism for exporting and importing plans.  Finally, Memory Allocator Reference describes how to change FFTW's default memory allocator. For parallel transforms, see 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 X[i] = conj(X[n-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 Y[i] = Re(X[i]).  For all integers i such that 0 < i < n / 2, we have Y[n-i] = Im(X[i]).

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 Y[0] = Re(X[0]), Y[1] = Re(X[1]), Y[2] = Re(X[2]), and  Y[3] = Im(X[1]). Let now n = 5.  In this case, we have Y[0] = Re(X[0]), Y[1] = Re(X[1]), Y[2] = Re(X[2]), Y[3] = Im(X[2]), and Y[4] = Im(X[1]).

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 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 2^a 3^b 5^c 7^d 11^e 13^f, 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 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.  (The basic problem is the resolution of the clock: FFTW needs to run for a certain time for the clock to be reliable.)

       - 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 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 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 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 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 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 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
Y[i] = sum for j = 0 to (n - 1) of X[j] * exp(-2 pi i j sqrt(-1)/n) .

The backward transform computes
Y[i] = sum for j = 0 to (n - 1) of X[j] * exp(2 pi i j sqrt(-1)/n) .

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 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 Multi-dimensional Array Format, for more information on row-major storage.) See 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''  (fftwnd actually may use some temporary storage (hidden in the plan), but this storage space is only the size of the largest dimension of the array, rather than being as big as the entire array.  (Unless you use fftwnd to perform one-dimensional transforms, in which case the temporary storage required for in-place transforms is as big as the entire array.)) 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 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 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 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 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 Data Types), stored in row-major format (see 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 TeX version of this manual contains the exact definition of the n-dimensional transform FFTW uses.  It is not possible to display the definition on a ASCII terminal properly.

Real One-dimensional Transforms Reference
=========================================

The one-dimensional real routines are generally prefixed with rfftw_.  (The etymologically-correct spelling would be frftw_, but it is hard to remember.)  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 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 2^a 3^b 5^c 7^d 11^e 13^f, 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(n^2) performance for prime sizes).  (It is possible to customize RFFTW for different array sizes.  See 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 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 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 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 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 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
Y[i] = sum for j = 0 to (n - 1) of X[j] * exp(-2 pi i j sqrt(-1)/n)
(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 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
Y[i] = sum for j = 0 to (n - 1) of X[j] * exp(2 pi i j sqrt(-1)/n)
(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 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 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 Multi-dimensional Array Format, for more information.)

       - See Plan Creation for Real One-dimensional Transforms, for more information regarding optimal array sizes.

  * 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 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 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 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 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 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 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 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 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 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.

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 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 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.

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 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 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 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.

Multi-threaded FFTW
===================

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

Installation and Supported Hardware/Software
--------------------------------------------

All of the FFTW threads code is located in the threads subdirectory of the FFTW package.  On Unix systems, the FFTW threads libraries and header files can be automatically configured, compiled, and installed along with the uniprocessor FFTW libraries simply by including --enable-threads in the flags to the configure script (see 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.)  We also support using OpenMP (http://www.openmp.org) or SGI MP compiler directives to launch threads, enabled by using --with-openmp or --with-sgimp in addition to --enable-threads.  This is especially useful if you are employing that sort of directive in your own code, in order to minimize conflicts.  If you have a shared-memory machine that uses a different threads API, it should be a simple matter of programming to include support for it; see the file fftw_threads-int.h for more detail.

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

Usage of Multi-threaded FFTW
----------------------------

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

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

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

     int fftw_threads_init(void);

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

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

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

     fftw_threads_one(nthreads, plan, in, out);

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

     fftwnd_threads_one(nthreads, plan, in, out);

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

     rfftw_threads_one(nthreads, plan, in, out);

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

     rfftwnd_threads_one_real_to_complex(nthreads, plan, in, out);

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

     rfftwnd_threads_one_real_to_complex(nthreads, plan, in, out);

     rfftwnd_threads_one_complex_to_real(nthreads, plan, in, out);

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

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

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

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

How Many Threads to Use?
------------------------

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

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

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

     fftw_threads_test 2 -s 128x128

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

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

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

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

Tips for Optimal Threading
--------------------------

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

Parallelization deficiencies in one-dimensional transforms
----------------------------------------------------------

  * Large prime factors can sometimes parallelize poorly.  Of course, you should avoid these anyway if you want high performance.

  * Single in-place transforms don't parallelize completely.  (Multiple in-place transforms, i.e. howmany > 1, are fine.)  Again, you should avoid these in any case if you want high performance, as they require transforming to a scratch array and copying back.

  * Single real-complex (rfftw) transforms don't parallelize completely.  This is unfortunate, but parallelizing this correctly would have involved a lot of extra code (and a much larger library).  You still get some benefit from additional processors, but if you have a very large number of processors you will probably be better off using the parallel complex (fftw) transforms.  Note that multi-dimensional real transforms or multiple one-dimensional real transforms are fine.

MPI FFTW
========

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

MPI FFTW Installation
---------------------

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

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 (see MPI Data Layout).

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

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

There are several things to notice here:

  * First of all, all fftw_mpi transforms are in-place: the output is in the local_data parameter, and there is no need to specify FFTW_IN_PLACE in the plan flags.

  * The MPI transforms also only support a limited subset of the howmany/stride/dist functionality of the uniprocessor routines: the n_fields parameter is equivalent to howmany=n_fields, stride=n_fields, and dist=1. (Conceptually, the n_fields parameter allows you to transform an array of contiguous vectors, each with length n_fields.) n_fields is 1 if you are only transforming a single, ordinary array.

  * The work parameter is an optional workspace.  If it is not NULL, it should be exactly the same size as the local_data array.  If it is provided, FFTW is able to use the built-in MPI_Alltoall primitive for (often) greater efficiency at the expense of extra storage space.

  * Finally, the last parameter specifies whether the output data has the same ordering as the input data (FFTW_NORMAL_ORDER), or if it is transposed (FFTW_TRANSPOSED_ORDER).  Leaving the data transposed results in significant performance improvements due to a saved communication step (needed to un-transpose the data).  Specifically, the first two dimensions of the array are transposed, as is described in more detail by the next section.

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

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

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

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

MPI Data Layout
---------------

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

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

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

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

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

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

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

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

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

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

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

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

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

Some important things to remember:

  * Although the local data is of dimensions local_nx x ny x nz in the above example, do not allocate the array to be of size local_nx*ny*nz.  Use total_local_size instead.

  * The amount of data on each process will not necessarily be the same; in fact, local_nx may even be zero for some processes.  (For example, suppose you are doing a 6x6 transform on four processors.  There is no way to effectively use the fourth processor in a slab decomposition, so we leave it empty.  Proof left as an exercise for the reader.)

  * All arrays are, of course, in row-major order (see Multi-dimensional Array Format).

  * If you want to compute the inverse transform of the output of fftwnd_mpi, the dimensions of the inverse transform are given by the dimensions of the output of the forward transform.  For example, if you are using FFTW_TRANSPOSED_ORDER output in the above example, then the inverse plan should be created with dimensions ny x nx x nz.

  * The data layout only depends upon the dimensions of the array, not on the plan, so you are guaranteed that different plans for the same size (or inverse plans) will use the same (consistent) data layouts.

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

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

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

     #include <rfftw_mpi.h>

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

          MPI_Init(&argc,&argv);

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

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

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

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

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

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

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

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

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

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

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

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

  * As for the uniprocessor rfftwnd_create_plan, the dimensions passed for the FFTW_COMPLEX_TO_REAL plan are those of the real data.  In particular, even when FFTW_TRANSPOSED_ORDER is used as in this case, the dimensions are those of the (untransposed) real output, not the (transposed) complex input.  (For the complex MPI transforms, on the other hand, the dimensions are always those of the input array.)

  * The output ordering of the transform (FFTW_TRANSPOSED_ORDER or FFTW_TRANSPOSED_ORDER) must be the same for both forward and backward transforms.  (This is not required in the complex case.)

  * total_local_size is the required size in fftw_real values, not fftw_complex values as it is for the complex transforms.

  * local_ny_after_transpose and local_y_start_after_transpose describe the portion of the array after the transform; that is, they are indices in the complex array for an FFTW_REAL_TO_COMPLEX transform and in the real array for an FFTW_COMPLEX_TO_REAL transform.

  * rfftwnd_mpi always expects fftw_real* array arguments, but of course these pointers can refer to either real or complex arrays, depending upon which side of the transform you are on.  Just as for in-place uniprocessor real transforms (and also in the example above), this is most easily handled by typecasting to a complex pointer when handling the complex data.

  * As with the complex transforms, there are also rfftwnd_create_plan and rfftw2d_create_plan functions, and any rank greater than one is supported.

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

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

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

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

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

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

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

The transform itself is computed by:

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

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

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

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

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

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

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

MPI Tips
--------

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

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

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

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

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

Calling FFTW from Fortran
*************************

The standard FFTW libraries include special wrapper functions that allow Fortran programs to call FFTW subroutines.  This chapter describes how those functions may be employed to use FFTW from Fortran.  We assume here that the reader is already familiar with the usage of FFTW in C, as described elsewhere in this manual.

In general, it is not possible to call C functions directly from Fortran, due to Fortran's inability to pass arguments by value and also because Fortran compilers typically expect identifiers to be mangled somehow for linking.  However, if C functions are written in a special way, they are callable from Fortran, and we have employed this technique to create Fortran-callable ``wrapper'' functions around the main FFTW routines.  These wrapper functions are included in the FFTW libraries by default, unless a Fortran compiler isn't found on your system or --disable-fortran is included in the configure flags.

As a result, calling FFTW from Fortran requires little more than appending _f77 to the function names and then linking normally with the FFTW libraries.  There are a few wrinkles, however, as we shall discuss below.

Wrapper Routines
================

All of the uniprocessor and multi-threaded transform routines have Fortran-callable wrappers, except for the wisdom import/export functions (since it is not possible to exchange string and file arguments portably with Fortran) and the specific planner routines (see Discussion on Specific Plans).  The name of the wrapper routine is the same as that of the corresponding C routine, but with fftw/fftwnd/rfftw/rfftwnd replaced by fftw_f77/fftwnd_f77/rfftw_f77/rfftwnd_f77.  For example, in Fortran, instead of calling fftw_one you would call fftw_f77_one. (Technically, Fortran 77 identifiers are not allowed to have more than 6 characters, nor may they contain underscores.  Any compiler that enforces this limitation doesn't deserve to link to FFTW.) For the most part, all of the arguments to the functions are the same, with the following exceptions:

  * plan variables (what would be of type fftw_plan, rfftwnd_plan, etcetera, in C), must be declared as a type that is the same size as a pointer (address) on your machine.  (Fortran has no generic pointer type.)  The Fortran integer type is usually the same size as a pointer, but you need to be wary (especially on 64-bit machines).  (You could also use integer*4 on a 32-bit machine and integer*8 on a 64-bit machine.)  Ugh.  (g77 has a special type, integer(kind=7), that is defined to be the same size as a pointer.)

  * Any function that returns a value (e.g. fftw_create_plan) is converted into a subroutine.  The return value is converted into an additional (first) parameter of the wrapper subroutine.  (The reason for this is that some Fortran implementations seem to have trouble with C function return values.)

  * When performing one-dimensional FFTW_IN_PLACE transforms, you don't have the option of passing NULL for the out argument (since there is no way to pass NULL from Fortran).  Therefore, when performing such transforms, you must allocate and pass a contiguous scratch array of the same size as the transform.  Note that for in-place multi-dimensional ((r)fftwnd) transforms, the out argument is ignored, so you can pass anything for that parameter.

  * The wrapper routines expect multi-dimensional arrays to be in column-major order, which is the ordinary format of Fortran arrays. They do this transparently and costlessly simply by reversing the order of the dimensions passed to FFTW, but this has one important consequence for multi-dimensional real-complex transforms, discussed below.

In general, you should take care to use Fortran data types that correspond to (i.e. are the same size as) the C types used by FFTW.  If your C and Fortran compilers are made by the same vendor, the correspondence is usually straightforward (i.e. integer corresponds to int, real corresponds to float, etcetera).  Such simple correspondences are assumed in the examples below.  The examples also assume that FFTW was compiled in double precision (the default).

FFTW Constants in Fortran
=========================

When creating plans in FFTW, a number of constants are used to specify options, such as FFTW_FORWARD or FFTW_USE_WISDOM.  The same constants must be used with the wrapper routines, but of course the C header files where the constants are defined can't be incorporated directly into Fortran code.

Instead, we have placed Fortran equivalents of the FFTW constant definitions in the file fortran/fftw_f77.i of the FFTW package. If your Fortran compiler supports a preprocessor, you can use that to incorporate this file into your code whenever you need to call FFTW. Otherwise, you will have to paste the constant definitions in directly. They are:

           integer FFTW_FORWARD,FFTW_BACKWARD
           parameter (FFTW_FORWARD=-1,FFTW_BACKWARD=1)

           integer FFTW_REAL_TO_COMPLEX,FFTW_COMPLEX_TO_REAL
           parameter (FFTW_REAL_TO_COMPLEX=-1,FFTW_COMPLEX_TO_REAL=1)

           integer FFTW_ESTIMATE,FFTW_MEASURE
           parameter (FFTW_ESTIMATE=0,FFTW_MEASURE=1)

           integer FFTW_OUT_OF_PLACE,FFTW_IN_PLACE,FFTW_USE_WISDOM
           parameter (FFTW_OUT_OF_PLACE=0)
           parameter (FFTW_IN_PLACE=8,FFTW_USE_WISDOM=16)

           integer FFTW_THREADSAFE
           parameter (FFTW_THREADSAFE=128)

In C, you combine different flags (like FFTW_USE_WISDOM and FFTW_MEASURE) using the | operator; in Fortran you should just use +.

Fortran Examples
================

In C you might have something like the following to transform a one-dimensional complex array:

             fftw_complex in[N], *out[N];
             fftw_plan plan;

             plan = fftw_create_plan(N,FFTW_FORWARD,FFTW_ESTIMATE);
             fftw_one(plan,in,out);
             fftw_destroy_plan(plan);

In Fortran, you use the following to accomplish the same thing:

             double complex in, out
             dimension in(N), out(N)
             integer plan

             call fftw_f77_create_plan(plan,N,FFTW_FORWARD,FFTW_ESTIMATE)
             call fftw_f77_one(plan,in,out)
             call fftw_f77_destroy_plan(plan)

Notice how all routines are called as Fortran subroutines, and the plan is returned via the first argument to fftw_f77_create_plan. Important: these examples assume that integer is the same size as a pointer, and may need modification on a 64-bit machine. See Wrapper Routines, above.  To do the same thing, but using 8 threads in parallel (see Multi-threaded FFTW), you would simply replace the call to fftw_f77_one with:

             call fftw_f77_threads_one(8,plan,in,out)

To transform a three-dimensional array in-place with C, you might do:

             fftw_complex arr[L][M][N];
             fftwnd_plan plan;
             int n[3] = {L,M,N};

             plan = fftwnd_create_plan(3,n,FFTW_FORWARD,
                                       FFTW_ESTIMATE | FFTW_IN_PLACE);
             fftwnd_one(plan, arr, 0);
             fftwnd_destroy_plan(plan);

In Fortran, you would use this instead:

             double complex arr
             dimension arr(L,M,N)
             integer n
             dimension n(3)
             integer plan

             n(1) = L
             n(2) = M
             n(3) = N
             call fftwnd_f77_create_plan(plan,3,n,FFTW_FORWARD,
            +                            FFTW_ESTIMATE + FFTW_IN_PLACE)
             call fftwnd_f77_one(plan, arr, 0)
             call fftwnd_f77_destroy_plan(plan)

Instead of calling fftwnd_f77_create_plan(plan,3,n,...), we could also have called fftw3d_f77_create_plan(plan,L,M,N,...).

Note that we pass the array dimensions in the "natural" order; also note that the last argument to fftwnd_f77 is ignored since the transform is FFTW_IN_PLACE.

To transform a one-dimensional real array in Fortran, you might do:

             double precision in, out
             dimension in(N), out(N)
             integer plan

             call rfftw_f77_create_plan(plan,N,FFTW_REAL_TO_COMPLEX,
            +                           FFTW_ESTIMATE)
             call rfftw_f77_one(plan,in,out)
             call rfftw_f77_destroy_plan(plan)

To transform a two-dimensional real array, out of place, you might use the following:

             double precision in
             double complex out
             dimension in(M,N), out(M/2 + 1, N)
             integer plan

             call rfftw2d_f77_create_plan(plan,M,N,FFTW_REAL_TO_COMPLEX,
            +                             FFTW_ESTIMATE)
             call rfftwnd_f77_one_real_to_complex(plan, in, out)
             call rfftwnd_f77_destroy_plan(plan)

Important: Notice that it is the first dimension of the complex output array that is cut in half in Fortran, rather than the last dimension as in C.  This is a consequence of the wrapper routines reversing the order of the array dimensions passed to FFTW so that the Fortran program can use its ordinary column-major order.

Installation and Customization
******************************

This chapter describes the installation and customization of FFTW, the latest version of which may be downloaded from the FFTW home page (http://www.fftw.org).

As distributed, FFTW makes very few assumptions about your system.  All you need is an ANSI C compiler (gcc is fine, although vendor-provided compilers often produce faster code). However, installation of FFTW is somewhat simpler if you have a Unix or a GNU system, such as Linux.  In this chapter, we first describe the installation of FFTW on Unix and non-Unix systems.  We then describe how you can customize FFTW to achieve better performance.  Specifically, you can I) enable gcc/x86-specific hacks that improve performance on Pentia and PentiumPro's; II) adapt FFTW to use the high-resolution clock of your machine, if any; III) produce code (codelets) to support fast transforms of sizes that are not supported efficiently by the standard FFTW distribution.

Installation on Unix
====================

FFTW comes with a configure program in the GNU style. Installation can be as simple as:

     ./configure
     make
     make install

This will build the uniprocessor complex and real transform libraries along with the test programs.  We strongly recommend that you use GNU make if it is available; on some systems it is called gmake.  The ``make install'' command installs the fftw and rfftw libraries in standard places, and typically requires root privileges (unless you specify a different install directory with the --prefix flag to configure).  You can also type ``make check'' to put the FFTW test programs through their paces. If you have problems during configuration or compilation, you may want to run ``make distclean'' before trying again; this ensures that you don't have any stale files left over from previous compilation attempts.

The configure script knows good CFLAGS (C compiler flags) for a few systems.  If your system is not known, the configure script will print out a warning.   (Each version of cc seems to have its own magic incantation to get the fastest code most of the time---you'd think that people would have agreed upon some convention, e.g. "-Omax", by now.)  In this case, you can compile FFTW with the command

     make CFLAGS="<write your CFLAGS here>"

If you do find an optimal set of CFLAGS for your system, please let us know what they are (along with the output of config.guess) so that we can include them in future releases.

The configure program supports all the standard flags defined by the GNU Coding Standards; see the INSTALL file in FFTW or the GNU web page (http://www.gnu.org/prep/standards_toc.html). Note especially --help to list all flags and --enable-shared to create shared, rather than static, libraries. configure also accepts a few FFTW-specific flags, particularly:

  * --enable-float Produces a single-precision version of FFTW (float) instead of the default double-precision (double). See Installing FFTW in both single and double precision.

  * --enable-type-prefix Adds a d or s prefix to all installed libraries and header files to indicate the floating-point precision.  See Installing FFTW in both single and double precision.  (--enable-type-prefix=<prefix> lets you add an arbitrary prefix.)  By default, no prefix is used.

  * --enable-threads Enables compilation and installation of the FFTW threads library (see Multi-threaded FFTW), which provides a simple interface to parallel transforms for SMP systems.  (By default, the threads routines are not compiled.)

  * --with-openmp, --with-sgimp In conjunction with --enable-threads, causes the multi-threaded FFTW library to use OpenMP or SGI MP compiler directives in order to induce parallelism, rather than spawning its own threads directly. (Useful especially for programs already employing such directives, in order to minimize conflicts between different parallelization mechanisms.)

  * --enable-mpi Enables compilation and installation of the FFTW MPI library (see MPI FFTW), which provides parallel transforms for distributed-memory systems with MPI.  (By default, the MPI routines are not compiled.)

  * --disable-fortran Disables inclusion of Fortran-callable wrapper routines (see Calling FFTW from Fortran) in the standard FFTW libraries.  These wrapper routines increase the library size by only a negligible amount, so they are included by default as long as the configure script finds a Fortran compiler on your system.

  * --with-gcc Enables the use of gcc.  By default, FFTW uses the vendor-supplied cc compiler if present.  Unfortunately, gcc produces slower code than cc on many systems.

  * --enable-i386-hacks  see gcc and Pentium hacks, below.

  * --enable-pentium-timer  see gcc and Pentium hacks, below.

To force configure to use a particular C compiler (instead of the default, usually cc), set the environment variable CC to the name of the desired compiler before running configure; you may also need to set the flags via the variable CFLAGS.

Installation on non-Unix Systems
================================

It is quite straightforward to install FFTW even on non-Unix systems lacking the niceties of the configure script.  The FFTW Home Page may include some FFTW packages preconfigured for particular systems/compilers, and also contains installation notes sent in by users.  All you really need to do, though, is to compile all of the .c files in the appropriate directories of the FFTW package. (You needn't worry about the many extraneous files lying around.)

For the complex transforms, compile all of the .c files in the fftw directory and link them into a library.  Similarly, for the real transforms, compile all of the .c files in the rfftw directory into a library.  Note that these sources #include various files in the fftw and rfftw directories, so you may need to set up the #include paths for your compiler appropriately.  Be sure to enable the highest-possible level of optimization in your compiler.

By default, FFTW is compiled for double-precision transforms.  To work in single precision rather than double precision, #define the symbol FFTW_ENABLE_FLOAT in fftw.h (in the fftw directory) and (re)compile FFTW.

These libraries should be linked with any program that uses the corresponding transforms.  The required header files, fftw.h and rfftw.h, are located in the fftw and rfftw directories respectively; you may want to put them with the libraries, or wherever header files normally go on your system.

FFTW includes test programs, fftw_test and rfftw_test, in the tests directory.  These are compiled and linked like any program using FFTW, except that they use additional header files located in the fftw and rfftw directories, so you will need to set your compiler #include paths appropriately.  fftw_test is compiled from fftw_test.c and test_main.c, while rfftw_test is compiled from rfftw_test.c and test_main.c.  When you run these programs, you will be prompted interactively for various possible tests to perform; see also tests/README for more information.

Installing FFTW in both single and double precision
===================================================

It is often useful to install both single- and double-precision versions of the FFTW libraries on the same machine, and we provide a convenient mechanism for achieving this on Unix systems.

When the --enable-type-prefix option of configure is used, the FFTW libraries and header files are installed with a prefix of d or s, depending upon whether you compiled in double or single precision.  Then, instead of linking your program with -lrfftw -lfftw, for example, you would link with -ldrfftw -ldfftw to use the double-precision version or with -lsrfftw -lsfftw to use the single-precision version.  Also, you would #include <drfftw.h> or <srfftw.h> instead of <rfftw.h>, and so on.

The names of FFTW functions, data types, and constants remain unchanged!  You still call, for instance, fftw_one and not dfftw_one.  Only the names of header files and libraries are modified.  One consequence of this is that you cannot use both the single- and double-precision FFTW libraries in the same program, simultaneously, as the function names would conflict.

So, to install both the single- and double-precision libraries on the same machine, you would do:

     ./configure --enable-type-prefix [ other options ]
     make
     make install
     make clean
     ./configure --enable-float --enable-type-prefix [ other options ]
     make
     make install

gcc and Pentium hacks
=====================

The configure option --enable-i386-hacks enables specific optimizations for the Pentium and later x86 CPUs under gcc, which can significantly improve performance of double-precision transforms. Specifically, we have tested these hacks on Linux with gcc 2.[789] and versions of egcs since 1.0.3.  These optimizations affect only the performance and not the correctness of FFTW (i.e. it is always safe to try them out).

These hacks provide a workaround to the incorrect alignment of local double variables in gcc.  The compiler aligns these variables to multiples of 4 bytes, but execution is much faster (on Pentium and PentiumPro) if doubles are aligned to a multiple of 8 bytes.  By carefully counting the number of variables allocated by the compiler in performance-critical regions of the code, we have been able to introduce dummy allocations (using alloca) that align the stack properly.  The hack depends crucially on the compiler flags that are used.  For example, it won't work without -fomit-frame-pointer.

In principle, these hacks are no longer required under gcc versions 2.95 and later, which automatically align the stack correctly (see -mpreferred-stack-boundary in the gcc manual). However, we have encountered a bug (http://egcs.cygnus.com/ml/gcc-bugs/1999-11/msg00259.html) in the stack alignment of versions 2.95.[012] that causes FFTW's stack to be misaligned under some circumstances.  The configure script automatically detects this bug and disables gcc's stack alignment in favor of our own hacks when --enable-i386-hacks is used.

The fftw_test program outputs speed measurements that you can use to see if these hacks are beneficial.

The configure option --enable-pentium-timer enables the use of the Pentium and PentiumPro cycle counter for timing purposes.  In order to get correct results, you must define FFTW_CYCLES_PER_SEC in fftw/config.h to be the clock speed of your processor; the resulting FFTW library will be nonportable.  The use of this option is deprecated.  On serious operating systems (such as Linux), FFTW uses gettimeofday(), which has enough resolution and is portable. (Note that Win32 has its own high-resolution timing routines as well. FFTW contains unsupported code to use these routines.)

Customizing the timer
=====================

FFTW needs a reasonably-precise clock in order to find the optimal way to compute a transform.  On Unix systems, configure looks for gettimeofday and other system-specific timers.  If it does not find any high resolution clock, it defaults to using the clock() function, which is very portable, but forces FFTW to run for a long time in order to get reliable measurements.

If your machine supports a high-resolution clock not recognized by FFTW, it is therefore advisable to use it.  You must edit fftw/fftw-int.h.  There are a few macros you must redefine.  The code is documented and should be self-explanatory.  (By the way, fftw-int stands for fftw-internal, but for some inexplicable reason people are still using primitive systems with 8.3 filenames.)

Even if you don't install high-resolution timing code, we still recommend that you look at the FFTW_TIME_MIN constant in fftw/fftw-int.h. This constant holds the minimum time interval (in seconds) required to get accurate timing measurements, and should be (at least) several hundred times the resolution of your clock.  The default constants are on the conservative side, and may cause FFTW to take longer than necessary when you create a plan. Set FFTW_TIME_MIN to whatever is appropriate on your system (be sure to set the right FFTW_TIME_MIN...there are several definitions in fftw-int.h, corresponding to different platforms and timers).

As an aid in checking the resolution of your clock, you can use the tests/fftw_test program with the -t option (c.f. tests/README). Remember, the mere fact that your clock reports times in, say, picoseconds, does not mean that it is actually accurate to that resolution.

Generating your own code
========================

If you know that you will only use transforms of a certain size (say, powers of 2) and want to reduce the size of the library, you can reconfigure FFTW to support only those sizes you are interested in.  You may even generate code to enable efficient transforms of a size not supported by the default distribution.  The default distribution supports transforms of any size, but not all sizes are equally fast. The default installation of FFTW is best at handling sizes of the form 2^a 3^b 5^c 7^d 11^e 13^f, 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.  However, if you have an application that requires fast transforms of size, say, 17, there is a way to generate specialized code to handle that.

The directory gensrc contains all the programs and scripts that were used to generate FFTW.  In particular, the program gensrc/genfft.ml was used to generate the code that FFTW uses to compute the transforms.  We do not expect casual users to use it. genfft is a rather sophisticated program that generates directed acyclic graphs of FFT algorithms and performs algebraic simplifications on them.  genfft is written in Objective Caml, a dialect of ML. Objective Caml is described at http://pauillac.inria.fr/ocaml/ and can be downloaded from from ftp://ftp.inria.fr/lang/caml-light.

If you have Objective Caml installed, you can type sh bootstrap.sh in the top-level directory to re-generate the files.  If you change the gensrc/config file, you can optimize FFTW for sizes that are not currently supported efficiently (say, 17 or 19).

We do not provide more details about the code-generation process, since we do not expect that users will need to generate their own code. However, feel free to contact us at fftw@fftw.org if you are interested in the subject.

You might find it interesting to learn Caml and/or some modern programming techniques that we used in the generator (including monadic programming), especially if you heard the rumor that Java and object-oriented programming are the latest advancement in the field. The internal operation of the codelet generator is described in the paper, ``A Fast Fourier Transform Compiler,'' by M. Frigo, which is available from the FFTW home page (http://www.fftw.org) and will appear in the Proceedings of the 1999 ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI).

Acknowledgments
***************

Matteo Frigo was supported in part by the Defense Advanced Research Projects Agency (DARPA) under Grants N00014-94-1-0985 and F30602-97-1-0270, and by a Digital Equipment Corporation Fellowship. Steven G. Johnson was supported in part by a DoD NDSEG Fellowship, an MIT Karl Taylor Compton Fellowship, and by the Materials Research Science and Engineering Center program of the National Science Foundation under award DMR-9400334.

Both authors were also supported in part by their respective girlfriends, by the letters ``Q'' and ``R'', and by the number 12.

We are grateful to SUN Microsystems Inc. for its donation of a cluster of 9 8-processor Ultra HPC 5000 SMPs (24 Gflops peak). These machines served as the primary platform for the development of earlier versions of FFTW.

We thank Intel Corporation for donating a four-processor Pentium Pro machine.  We thank the Linux community for giving us a decent OS to run on that machine.

The genfft program was written using Objective Caml, a dialect of ML.  Objective Caml is a small and elegant language developed by Xavier Leroy.  The implementation is available from ftp.inria.fr in the directory lang/caml-light.  We used versions 1.07 and 2.00 of the software.  In previous releases of FFTW, genfft was written in Caml Light, by the same authors.  An even earlier implementation of genfft was written in Scheme, but Caml is definitely better for this kind of application.

FFTW uses many tools from the GNU project, including automake, texinfo, and libtool.

Prof. Charles E. Leiserson of MIT provided continuous support and encouragement.  This program would not exist without him.  Charles also proposed the name ``codelets'' for the basic FFT blocks.

Prof. John D. Joannopoulos of MIT demonstrated continuing tolerance of Steven's ``extra-curricular'' computer-science activities.  Steven's chances at a physics degree would not exist without him.

Andrew Sterian contributed the Windows timing code.

Didier Miras reported a bug in the test procedure used in FFTW 1.2.  We now use a completely different test algorithm by Funda Ergun that does not require a separate FFT program to compare against.

Wolfgang Reimer contributed the Pentium cycle counter and a few fixes that help portability.

Ming-Chang Liu uncovered a well-hidden bug in the complex transforms of FFTW 2.0 and supplied a patch to correct it.

The FFTW FAQ was written in bfnn (Bizarre Format With No Name) and formatted using the tools developed by Ian Jackson for the Linux FAQ.

We are especially thankful to all of our users for their continuing support, feedback, and interest during our development of FFTW.

License and Copyright
*********************

FFTW is copyright © 1997--1999 Massachusetts Institute of Technology.

FFTW is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA.  You can also find the GPL on the GNU web site (http://www.gnu.org/copyleft/gpl.html).

In addition, we kindly ask you to acknowledge FFTW and its authors in any program or publication in which you use FFTW.  (You are not required to do so; it is up to your common sense to decide whether you want to comply with this request or not.)

Non-free versions of FFTW are available under terms different than the General Public License. (e.g. they do not require you to accompany any object code using FFTW with the corresponding source code.)  For these alternate terms you must purchase a license from MIT's Technology Licensing Office.  Users interested in such a license should contact us (fftw@fftw.org) for more information.