@node FFTW Reference, Multi-threaded FFTW, Other Important Topics, Top
@chapter FFTW Reference

This chapter provides a complete reference for all sequential (i.e.,
one-processor) FFTW functions.  Parallel transforms are described in
later chapters.

@menu
* Data Types and Files::
* Using Plans::
* Basic Interface::
* Advanced Interface::
* Guru Interface::
* New-array Execute Functions::
* Wisdom::
* What FFTW Really Computes::
@end menu

@c ------------------------------------------------------------
@node Data Types and Files, Using Plans, FFTW Reference, FFTW Reference
@section Data Types and Files

All programs using FFTW should include its header file:

@example
#include <fftw3.h>
@end example

You must also link to the FFTW library.  On Unix, this
means adding @code{-lfftw3 -lm} at the @emph{end} of the link command.

@menu
* Complex numbers::
* Precision::
* Memory Allocation::
@end menu

@c =========>
@node Complex numbers, Precision, Data Types and Files, Data Types and Files
@subsection Complex numbers

The default FFTW interface uses @code{double} precision for all
floating-point numbers, and defines a @code{fftw_complex} type to hold
complex numbers as:

@example
typedef double fftw_complex[2];
@end example
@tindex fftw_complex

Here, the @code{[0]} element holds the real part and the @code{[1]}
element holds the imaginary part.

Alternatively, if you have a C compiler (such as @code{gcc}) that
supports the C99 revision of the ANSI C standard, you can use C's new
native complex type (which is binary-compatible with the typedef above).
In particular, if you @code{#include <complex.h>} @emph{before}
@code{<fftw3.h>}, then @code{fftw_complex} is defined to be the native
complex type and you can manipulate it with ordinary arithmetic
(e.g. @code{x = y * (3+4*I)}, where @code{x} and @code{y} are
@code{fftw_complex} and @code{I} is the standard symbol for the
imaginary unit);
@cindex C99


C++ has its own @code{complex<T>} template class, defined in the
standard @code{<complex>} header file.  Reportedly, the C++ standards
committee has recently agreed to mandate that the storage format used
for this type be binary-compatible with the C99 type, i.e. an array
@code{T[2]} with consecutive real @code{[0]} and imaginary @code{[1]}
parts.  (See report
@uref{http://www.open-std.org/jtc1/sc22/WG21/docs/papers/2002/n1388.pdf
WG21/N1388}.)  Although not part of the official standard as of this
writing, the proposal stated that: ``This solution has been tested with
all current major implementations of the standard library and shown to
be working.''  To the extent that this is true, if you have a variable
@code{complex<double> *x}, you can pass it directly to FFTW via
@code{reinterpret_cast<fftw_complex*>(x)}.
@cindex C++
@cindex portability

@c =========>
@node Precision, Memory Allocation, Complex numbers, Data Types and Files
@subsection Precision
@cindex precision

You can install single and long-double precision versions of FFTW,
which replace @code{double} with @code{float} and @code{long double},
respectively (@pxref{Installation and Customization}).  To use these
interfaces, you:

@itemize @bullet

@item
Link to the single/long-double libraries; on Unix, @code{-lfftw3f} or
@code{-lfftw3l} instead of (or in addition to) @code{-lfftw3}.  (You
can link to the different-precision libraries simultaneously.)

@item
Include the @emph{same} @code{<fftw3.h>} header file.

@item
Replace all lowercase instances of @samp{fftw_} with @samp{fftwf_} or
@samp{fftwl_} for single or long-double precision, respectively.
(@code{fftw_complex} becomes @code{fftwf_complex}, @code{fftw_execute}
becomes @code{fftwf_execute}, etcetera.)

@item
Uppercase names, i.e. names beginning with @samp{FFTW_}, remain the
same.

@item
Replace @code{double} with @code{float} or @code{long double} for
subroutine parameters.

@end itemize

Depending upon your compiler and/or hardware, @code{long double} may not
be any more precise than @code{double} (or may not be supported at all,
although it is standard in C99).
@cindex C99


We also support using the nonstandard @code{__float128}
quadruple-precision type provided by recent versions of @code{gcc} on
32- and 64-bit x86 hardware (@pxref{Installation and Customization}).
To use this type, link with @code{-lfftw3q -lquadmath -lm} (the
@code{libquadmath} library provided by @code{gcc} is needed for
quadruple-precision trigonometric functions) and use @samp{fftwq_}
identifiers.

@c =========>
@node Memory Allocation,  , Precision, Data Types and Files
@subsection Memory Allocation

@example
void *fftw_malloc(size_t n);
void fftw_free(void *p);
@end example
@findex fftw_malloc
@findex fftw_free

These are functions that behave identically to @code{malloc} and
@code{free}, except that they guarantee that the returned pointer obeys
any special alignment restrictions imposed by any algorithm in FFTW
(e.g. for SIMD acceleration).  @xref{SIMD alignment and fftw_malloc}.
@cindex alignment


Data allocated by @code{fftw_malloc} @emph{must} be deallocated by
@code{fftw_free} and not by the ordinary @code{free}.

These routines simply call through to your operating system's
@code{malloc} or, if necessary, its aligned equivalent
(e.g. @code{memalign}), so you normally need not worry about any
significant time or space overhead.  You are @emph{not required} to use
them to allocate your data, but we strongly recommend it.

Note: in C++, just as with ordinary @code{malloc}, you must typecast
the output of @code{fftw_malloc} to whatever pointer type you are
allocating.
@cindex C++


We also provide the following two convenience functions to allocate
real and complex arrays with @code{n} elements, which are equivalent
to @code{(double *) fftw_malloc(sizeof(double) * n)} and
@code{(fftw_complex *) fftw_malloc(sizeof(fftw_complex) * n)},
respectively:

@example
double *fftw_alloc_real(size_t n);
fftw_complex *fftw_alloc_complex(size_t n);
@end example
@findex fftw_alloc_real
@findex fftw_alloc_complex

The equivalent functions in other precisions allocate arrays of @code{n}
elements in that precision.  e.g. @code{fftwf_alloc_real(n)} is
equivalent to @code{(float *) fftwf_malloc(sizeof(float) * n)}.
@cindex precision

@c ------------------------------------------------------------
@node Using Plans, Basic Interface, Data Types and Files, FFTW Reference
@section Using Plans

Plans for all transform types in FFTW are stored as type
@code{fftw_plan} (an opaque pointer type), and are created by one of the
various planning routines described in the following sections.
@tindex fftw_plan
An @code{fftw_plan} contains all information necessary to compute the
transform, including the pointers to the input and output arrays.

@example
void fftw_execute(const fftw_plan plan);
@end example
@findex fftw_execute

This executes the @code{plan}, to compute the corresponding transform on
the arrays for which it was planned (which must still exist).  The plan
is not modified, and @code{fftw_execute} can be called as many times as
desired.

To apply a given plan to a different array, you can use the new-array execute
interface.  @xref{New-array Execute Functions}.

@code{fftw_execute} (and equivalents) is the only function in FFTW
guaranteed to be thread-safe; see @ref{Thread safety}.

This function:
@example
void fftw_destroy_plan(fftw_plan plan);
@end example
@findex fftw_destroy_plan
deallocates the @code{plan} and all its associated data.

FFTW's planner saves some other persistent data, such as the
accumulated wisdom and a list of algorithms available in the current
configuration.  If you want to deallocate all of that and reset FFTW
to the pristine state it was in when you started your program, you can
call:

@example
void fftw_cleanup(void);
@end example
@findex fftw_cleanup

After calling @code{fftw_cleanup}, all existing plans become undefined,
and you should not attempt to execute them nor to destroy them.  You can
however create and execute/destroy new plans, in which case FFTW starts
accumulating wisdom information again.

@code{fftw_cleanup} does not deallocate your plans, however.  To prevent
memory leaks, you must still call @code{fftw_destroy_plan} before
executing @code{fftw_cleanup}.

Occasionally, it may useful to know FFTW's internal ``cost'' metric
that it uses to compare plans to one another; this cost is
proportional to an execution time of the plan, in undocumented units,
if the plan was created with the @code{FFTW_MEASURE} or other
timing-based options, or alternatively is a heuristic cost function
for @code{FFTW_ESTIMATE} plans.  (The cost values of measured and
estimated plans are not comparable, being in different units.  Also,
costs from different FFTW versions or the same version compiled
differently may not be in the same units.  Plans created from wisdom
have a cost of 0 since no timing measurement is performed for them.
Finally, certain problems for which only one top-level algorithm was
possible may have required no measurements of the cost of the whole
plan, in which case @code{fftw_cost} will also return 0.)  The cost
metric for a given plan is returned by:

@example
double fftw_cost(const fftw_plan plan);
@end example
@findex fftw_cost

The following two routines are provided purely for academic purposes
(that is, for entertainment).

@example
void fftw_flops(const fftw_plan plan, 
                double *add, double *mul, double *fma);
@end example
@findex fftw_flops

Given a @code{plan}, set @code{add}, @code{mul}, and @code{fma} to an
exact count of the number of floating-point additions, multiplications,
and fused multiply-add operations involved in the plan's execution.  The
total number of floating-point operations (flops) is @code{add + mul +
2*fma}, or @code{add + mul + fma} if the hardware supports fused
multiply-add instructions (although the number of FMA operations is only
approximate because of compiler voodoo).  (The number of operations
should be an integer, but we use @code{double} to avoid overflowing
@code{int} for large transforms; the arguments are of type @code{double}
even for single and long-double precision versions of FFTW.)

@example
void fftw_fprint_plan(const fftw_plan plan, FILE *output_file);
void fftw_print_plan(const fftw_plan plan);
char *fftw_sprint_plan(const fftw_plan plan);
@end example
@findex fftw_fprint_plan
@findex fftw_print_plan

This outputs a ``nerd-readable'' representation of the @code{plan} to
the given file, to @code{stdout}, or two a newly allocated
NUL-terminated string (which the caller is responsible for deallocating
with @code{free}), respectively.

@c ------------------------------------------------------------
@node Basic Interface, Advanced Interface, Using Plans, FFTW Reference
@section Basic Interface
@cindex basic interface

Recall that the FFTW API is divided into three parts@footnote{@i{Gallia est
omnis divisa in partes tres} (Julius Caesar).}: the @dfn{basic interface}
computes a single transform of contiguous data, the @dfn{advanced
interface} computes transforms of multiple or strided arrays, and the
@dfn{guru interface} supports the most general data layouts,
multiplicities, and strides.  This section describes the the basic
interface, which we expect to satisfy the needs of most users.

@menu
* Complex DFTs::
* Planner Flags::
* Real-data DFTs::
* Real-data DFT Array Format::
* Real-to-Real Transforms::
* Real-to-Real Transform Kinds::
@end menu

@c =========>
@node Complex DFTs, Planner Flags, Basic Interface, Basic Interface
@subsection Complex DFTs

@example
fftw_plan fftw_plan_dft_1d(int n0,
                           fftw_complex *in, fftw_complex *out,
                           int sign, unsigned flags);
fftw_plan fftw_plan_dft_2d(int n0, int n1,
                           fftw_complex *in, fftw_complex *out,
                           int sign, unsigned flags);
fftw_plan fftw_plan_dft_3d(int n0, int n1, int n2,
                           fftw_complex *in, fftw_complex *out,
                           int sign, unsigned flags);
fftw_plan fftw_plan_dft(int rank, const int *n,
                        fftw_complex *in, fftw_complex *out,
                        int sign, unsigned flags);
@end example
@findex fftw_plan_dft_1d
@findex fftw_plan_dft_2d
@findex fftw_plan_dft_3d
@findex fftw_plan_dft

Plan a complex input/output discrete Fourier transform (DFT) in zero or
more dimensions, returning an @code{fftw_plan} (@pxref{Using Plans}).

Once you have created a plan for a certain transform type and
parameters, then creating another plan of the same type and parameters,
but for different arrays, is fast and shares constant data with the
first plan (if it still exists).

The planner returns @code{NULL} if the plan cannot be created.  In the
standard FFTW distribution, the basic interface is guaranteed to return
a non-@code{NULL} plan.  A plan may be @code{NULL}, however, if you are
using a customized FFTW configuration supporting a restricted set of
transforms.

@subsubheading Arguments
@itemize @bullet

@item
@code{rank} is the rank of the transform (it should be the size of the
array @code{*n}), and can be any non-negative integer.  (@xref{Complex
Multi-Dimensional DFTs}, for the definition of ``rank''.)  The
@samp{_1d}, @samp{_2d}, and @samp{_3d} planners correspond to a
@code{rank} of @code{1}, @code{2}, and @code{3}, respectively.  The rank
may be zero, which is equivalent to a rank-1 transform of size 1, i.e. a
copy of one number from input to output.

@item
@code{n0}, @code{n1}, @code{n2}, or @code{n[0..rank-1]} (as appropriate
for each routine) specify the size of the transform dimensions.  They
can be any positive integer.
 
@itemize @minus
@item
@cindex row-major
Multi-dimensional arrays are stored in row-major order with dimensions:
@code{n0} x @code{n1}; or @code{n0} x @code{n1} x @code{n2}; or
@code{n[0]} x @code{n[1]} x ... x @code{n[rank-1]}.
@xref{Multi-dimensional Array Format}.
@item
FFTW is best at handling sizes of the form
@ifinfo
@math{2^a 3^b 5^c 7^d 11^e 13^f},
@end ifinfo
@tex
$2^a 3^b 5^c 7^d 11^e 13^f$,
@end tex
@html
2<sup>a</sup> 3<sup>b</sup> 5<sup>c</sup> 7<sup>d</sup>
        11<sup>e</sup> 13<sup>f</sup>,
@end html
where @math{e+f} is either @math{0} or @math{1}, and the other exponents
are arbitrary.  Other sizes are computed by means of a slow,
general-purpose algorithm (which nevertheless retains @Onlogn{} performance even for prime sizes).  It is possible to customize FFTW
for different array sizes; see @ref{Installation and Customization}.
Transforms whose sizes are powers of @math{2} are especially fast.
@end itemize

@item
@code{in} and @code{out} point to the input and output arrays of the
transform, which may be the same (yielding an in-place transform).
@cindex in-place
These arrays are overwritten during planning, unless
@code{FFTW_ESTIMATE} is used in the flags.  (The arrays need not be
initialized, but they must be allocated.)

If @code{in == out}, the transform is @dfn{in-place} and the input
array is overwritten. If @code{in != out}, the two arrays must
not overlap (but FFTW does not check for this condition).

@item
@ctindex FFTW_FORWARD
@ctindex FFTW_BACKWARD
@code{sign} is the sign of the exponent in the formula that defines the
Fourier transform.  It can be @math{-1} (= @code{FFTW_FORWARD}) or
@math{+1} (= @code{FFTW_BACKWARD}).

@item
@cindex flags
@code{flags} is a bitwise OR (@samp{|}) of zero or more planner flags,
as defined in @ref{Planner Flags}.

@end itemize

FFTW computes an unnormalized transform: computing a forward followed by
a backward transform (or vice versa) will result in the original data
multiplied by the size of the transform (the product of the dimensions).
@cindex normalization
For more information, see @ref{What FFTW Really Computes}.

@c =========>
@node Planner Flags, Real-data DFTs, Complex DFTs, Basic Interface
@subsection Planner Flags

All of the planner routines in FFTW accept an integer @code{flags}
argument, which is a bitwise OR (@samp{|}) of zero or more of the flag
constants defined below.  These flags control the rigor (and time) of
the planning process, and can also impose (or lift) restrictions on the
type of transform algorithm that is employed.

@emph{Important:} the planner overwrites the input array during
planning unless a saved plan (@pxref{Wisdom}) is available for that
problem, so you should initialize your input data after creating the
plan.  The only exceptions to this are the @code{FFTW_ESTIMATE} and
@code{FFTW_WISDOM_ONLY} flags, as mentioned below.

In all  cases, if  wisdom is  available for the  given problem  that was
created  with equal-or-greater  planning rigor,  then the  more rigorous
wisdom is used.  For example, in @code{FFTW_ESTIMATE} mode any available
wisdom is used, whereas  in @code{FFTW_PATIENT} mode only wisdom created
in patient or exhaustive mode can be used.  @xref{Words of Wisdom-Saving
Plans}.

@subsubheading Planning-rigor flags
@itemize @bullet

@item
@ctindex FFTW_ESTIMATE
@code{FFTW_ESTIMATE} specifies that, instead of actual measurements of
different algorithms, a simple heuristic is used to pick a (probably
sub-optimal) plan quickly.  With this flag, the input/output arrays are
not overwritten during planning.

@item
@ctindex FFTW_MEASURE
@code{FFTW_MEASURE} tells FFTW to find an optimized plan by actually
@emph{computing} several FFTs and measuring their execution time.
Depending on your machine, this can take some time (often a few
seconds).  @code{FFTW_MEASURE} is the default planning option.

@item
@ctindex FFTW_PATIENT
@code{FFTW_PATIENT} is like @code{FFTW_MEASURE}, but considers a wider
range of algorithms and often produces a ``more optimal'' plan
(especially for large transforms), but at the expense of several times
longer planning time (especially for large transforms).

@item
@ctindex FFTW_EXHAUSTIVE
@code{FFTW_EXHAUSTIVE} is like @code{FFTW_PATIENT}, but considers an
even wider range of algorithms, including many that we think are
unlikely to be fast, to produce the most optimal plan but with a
substantially increased planning time.

@item
@ctindex FFTW_WISDOM_ONLY
@code{FFTW_WISDOM_ONLY} is a special planning mode in which the plan
is only created if wisdom is available for the given problem, and
otherwise a @code{NULL} plan is returned.  This can be combined with
other flags, e.g. @samp{FFTW_WISDOM_ONLY | FFTW_PATIENT} creates a
plan only if wisdom is available that was created in
@code{FFTW_PATIENT} or @code{FFTW_EXHAUSTIVE} mode.  The
@code{FFTW_WISDOM_ONLY} flag is intended for users who need to detect
whether wisdom is available; for example, if wisdom is not available
one may wish to allocate new arrays for planning so that user data is
not overwritten.

@end itemize

@subsubheading Algorithm-restriction flags
@itemize @bullet

@item
@ctindex FFTW_DESTROY_INPUT
@code{FFTW_DESTROY_INPUT} specifies that an out-of-place transform is
allowed to @emph{overwrite its input} array with arbitrary data; this
can sometimes allow more efficient algorithms to be employed.
@cindex out-of-place

@item
@ctindex FFTW_PRESERVE_INPUT
@code{FFTW_PRESERVE_INPUT} specifies that an out-of-place transform must
@emph{not change its input} array.  This is ordinarily the
@emph{default}, except for c2r and hc2r (i.e. complex-to-real)
transforms for which @code{FFTW_DESTROY_INPUT} is the default.  In the
latter cases, passing @code{FFTW_PRESERVE_INPUT} will attempt to use
algorithms that do not destroy the input, at the expense of worse
performance; for multi-dimensional c2r transforms, however, no
input-preserving algorithms are implemented and the planner will return
@code{NULL} if one is requested.
@cindex c2r
@cindex hc2r

@item
@ctindex FFTW_UNALIGNED
@cindex alignment
@findex fftw_malloc
@findex fftw_alignment_of
@code{FFTW_UNALIGNED} specifies that the algorithm may not impose any
unusual alignment requirements on the input/output arrays (i.e. no
SIMD may be used).  This flag is normally @emph{not necessary}, since
the planner automatically detects misaligned arrays.  The only use for
this flag is if you want to use the new-array execute interface to
execute a given plan on a different array that may not be aligned like
the original.  (Using @code{fftw_malloc} makes this flag unnecessary
even then.  You can also use @code{fftw_alignment_of} to detect
whether two arrays are equivalently aligned.)

@end itemize

@subsubheading Limiting planning time

@example
extern void fftw_set_timelimit(double seconds);
@end example
@findex fftw_set_timelimit

This function instructs FFTW to spend at most @code{seconds} seconds
(approximately) in the planner.  If @code{seconds ==
FFTW_NO_TIMELIMIT} (the default value, which is negative), then
planning time is unbounded.  Otherwise, FFTW plans with a
progressively wider range of algorithms until the the given time limit
is reached or the given range of algorithms is explored, returning the
best available plan.
@ctindex FFTW_NO_TIMELIMIT


For example, specifying @code{FFTW_PATIENT} first plans in
@code{FFTW_ESTIMATE} mode, then in @code{FFTW_MEASURE} mode, then
finally (time permitting) in @code{FFTW_PATIENT}.  If
@code{FFTW_EXHAUSTIVE} is specified instead, the planner will further
progress to @code{FFTW_EXHAUSTIVE} mode.

Note that the @code{seconds} argument specifies only a rough limit; in
practice, the planner may use somewhat more time if the time limit is
reached when the planner is in the middle of an operation that cannot
be interrupted.  At the very least, the planner will complete planning
in @code{FFTW_ESTIMATE} mode (which is thus equivalent to a time limit
of 0).


@c =========>
@node Real-data DFTs, Real-data DFT Array Format, Planner Flags, Basic Interface
@subsection Real-data DFTs

@example
fftw_plan fftw_plan_dft_r2c_1d(int n0,
                               double *in, fftw_complex *out,
                               unsigned flags);
fftw_plan fftw_plan_dft_r2c_2d(int n0, int n1,
                               double *in, fftw_complex *out,
                               unsigned flags);
fftw_plan fftw_plan_dft_r2c_3d(int n0, int n1, int n2,
                               double *in, fftw_complex *out,
                               unsigned flags);
fftw_plan fftw_plan_dft_r2c(int rank, const int *n,
                            double *in, fftw_complex *out,
                            unsigned flags);
@end example
@findex fftw_plan_dft_r2c_1d
@findex fftw_plan_dft_r2c_2d
@findex fftw_plan_dft_r2c_3d
@findex fftw_plan_dft_r2c
@cindex r2c

Plan a real-input/complex-output discrete Fourier transform (DFT) in
zero or more dimensions, returning an @code{fftw_plan} (@pxref{Using
Plans}).

Once you have created a plan for a certain transform type and
parameters, then creating another plan of the same type and parameters,
but for different arrays, is fast and shares constant data with the
first plan (if it still exists).

The planner returns @code{NULL} if the plan cannot be created.  A
non-@code{NULL} plan is always returned by the basic interface unless
you are using a customized FFTW configuration supporting a restricted
set of transforms, or if you use the @code{FFTW_PRESERVE_INPUT} flag
with a multi-dimensional out-of-place c2r transform (see below).

@subsubheading Arguments
@itemize @bullet

@item
@code{rank} is the rank of the transform (it should be the size of the
array @code{*n}), and can be any non-negative integer.  (@xref{Complex
Multi-Dimensional DFTs}, for the definition of ``rank''.)  The
@samp{_1d}, @samp{_2d}, and @samp{_3d} planners correspond to a
@code{rank} of @code{1}, @code{2}, and @code{3}, respectively.  The rank
may be zero, which is equivalent to a rank-1 transform of size 1, i.e. a
copy of one real number (with zero imaginary part) from input to output.

@item
@code{n0}, @code{n1}, @code{n2}, or @code{n[0..rank-1]}, (as appropriate
for each routine) specify the size of the transform dimensions.  They
can be any positive integer.  This is different in general from the
@emph{physical} array dimensions, which are described in @ref{Real-data
DFT Array Format}.
 
@itemize @minus
@item
FFTW is best at handling sizes of the form
@ifinfo
@math{2^a 3^b 5^c 7^d 11^e 13^f},
@end ifinfo
@tex
$2^a 3^b 5^c 7^d 11^e 13^f$,
@end tex
@html
2<sup>a</sup> 3<sup>b</sup> 5<sup>c</sup> 7<sup>d</sup>
        11<sup>e</sup> 13<sup>f</sup>,
@end html
where @math{e+f} is either @math{0} or @math{1}, and the other exponents
are arbitrary.  Other sizes are computed by means of a slow,
general-purpose algorithm (which nevertheless retains @Onlogn{} performance even for prime sizes).  (It is possible to customize FFTW
for different array sizes; see @ref{Installation and Customization}.)
Transforms whose sizes are powers of @math{2} are especially fast, and
it is generally beneficial for the @emph{last} dimension of an r2c/c2r
transform to be @emph{even}.
@end itemize

@item
@code{in} and @code{out} point to the input and output arrays of the
transform, which may be the same (yielding an in-place transform).
@cindex in-place
These arrays are overwritten during planning, unless
@code{FFTW_ESTIMATE} is used in the flags.  (The arrays need not be
initialized, but they must be allocated.)  For an in-place transform, it
is important to remember that the real array will require padding,
described in @ref{Real-data DFT Array Format}.
@cindex padding

@item
@cindex flags
@code{flags} is a bitwise OR (@samp{|}) of zero or more planner flags,
as defined in @ref{Planner Flags}.

@end itemize

The inverse transforms, taking complex input (storing the non-redundant
half of a logically Hermitian array) to real output, are given by:

@example
fftw_plan fftw_plan_dft_c2r_1d(int n0,
                               fftw_complex *in, double *out,
                               unsigned flags);
fftw_plan fftw_plan_dft_c2r_2d(int n0, int n1,
                               fftw_complex *in, double *out,
                               unsigned flags);
fftw_plan fftw_plan_dft_c2r_3d(int n0, int n1, int n2,
                               fftw_complex *in, double *out,
                               unsigned flags);
fftw_plan fftw_plan_dft_c2r(int rank, const int *n,
                            fftw_complex *in, double *out,
                            unsigned flags);
@end example
@findex fftw_plan_dft_c2r_1d
@findex fftw_plan_dft_c2r_2d
@findex fftw_plan_dft_c2r_3d
@findex fftw_plan_dft_c2r
@cindex c2r

The arguments are the same as for the r2c transforms, except that the
input and output data formats are reversed.

FFTW computes an unnormalized transform: computing an r2c followed by a
c2r transform (or vice versa) will result in the original data
multiplied by the size of the transform (the product of the logical
dimensions).
@cindex normalization
An r2c transform produces the same output as a @code{FFTW_FORWARD}
complex DFT of the same input, and a c2r transform is correspondingly
equivalent to @code{FFTW_BACKWARD}.  For more information, see @ref{What
FFTW Really Computes}.

@c =========>
@node Real-data DFT Array Format, Real-to-Real Transforms, Real-data DFTs, Basic Interface
@subsection Real-data DFT Array Format
@cindex r2c/c2r multi-dimensional array format

The output of a DFT of real data (r2c) contains symmetries that, in
principle, make half of the outputs redundant (@pxref{What FFTW Really
Computes}).  (Similarly for the input of an inverse c2r transform.)  In
practice, it is not possible to entirely realize these savings in an
efficient and understandable format that generalizes to
multi-dimensional transforms.  Instead, the output of the r2c
transforms is @emph{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 @code{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 @math{d} (= @code{rank})
dimensions @ndims{}, the complex data is an @ndimshalf array of
@code{fftw_complex} values in row-major order (with the division rounded
down).  That is, we only store the @emph{lower} half (non-negative
frequencies), 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.)

@cindex out-of-place
For an out-of-place transform, the real data is simply an array with
physical dimensions @ndims in row-major order.

@cindex in-place
@cindex padding
For an in-place transform, some complications arise since the complex data
is slightly larger than the real data.  In this case, the final
dimension of the real data must be @emph{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
@tex
$2 (n_{d-1}/2+1)$
@end tex
@ifinfo
2 * (n[d-1]/2+1)
@end ifinfo
@html
2 * (n<sub>d-1</sub>/2+1)
@end html
@code{double} values (exactly enough to hold the complex data).  This
physical array size does not, however, change the @emph{logical} array
size---only
@tex
$n_{d-1}$
@end tex
@ifinfo
n[d-1]
@end ifinfo
@html
n<sub>d-1</sub>
@end html
values are actually stored in the last dimension, and
@tex
$n_{d-1}$
@end tex
@ifinfo
n[d-1]
@end ifinfo
@html
n<sub>d-1</sub>
@end html
is the last dimension passed to the planner.

@c =========>
@node Real-to-Real Transforms, Real-to-Real Transform Kinds, Real-data DFT Array Format, Basic Interface
@subsection Real-to-Real Transforms
@cindex r2r

@example
fftw_plan fftw_plan_r2r_1d(int n, double *in, double *out,
                           fftw_r2r_kind kind, unsigned flags);
fftw_plan fftw_plan_r2r_2d(int n0, int n1, double *in, double *out,
                           fftw_r2r_kind kind0, fftw_r2r_kind kind1,
                           unsigned flags);
fftw_plan fftw_plan_r2r_3d(int n0, int n1, int n2,
                           double *in, double *out,
                           fftw_r2r_kind kind0,
                           fftw_r2r_kind kind1,
                           fftw_r2r_kind kind2,
                           unsigned flags);
fftw_plan fftw_plan_r2r(int rank, const int *n, double *in, double *out,
                        const fftw_r2r_kind *kind, unsigned flags);
@end example
@findex fftw_plan_r2r_1d
@findex fftw_plan_r2r_2d
@findex fftw_plan_r2r_3d
@findex fftw_plan_r2r

Plan a real input/output (r2r) transform of various kinds in zero or
more dimensions, returning an @code{fftw_plan} (@pxref{Using Plans}).

Once you have created a plan for a certain transform type and
parameters, then creating another plan of the same type and parameters,
but for different arrays, is fast and shares constant data with the
first plan (if it still exists).

The planner returns @code{NULL} if the plan cannot be created.  A
non-@code{NULL} plan is always returned by the basic interface unless
you are using a customized FFTW configuration supporting a restricted
set of transforms, or for size-1 @code{FFTW_REDFT00} kinds (which are
not defined).
@ctindex FFTW_REDFT00

@subsubheading Arguments
@itemize @bullet

@item
@code{rank} is the dimensionality of the transform (it should be the
size of the arrays @code{*n} and @code{*kind}), and can be any
non-negative integer.  The @samp{_1d}, @samp{_2d}, and @samp{_3d}
planners correspond to a @code{rank} of @code{1}, @code{2}, and
@code{3}, respectively.  A @code{rank} of zero is equivalent to a copy
of one number from input to output.

@item
@code{n}, or @code{n0}/@code{n1}/@code{n2}, or @code{n[rank]},
respectively, gives the (physical) size of the transform dimensions.
They can be any positive integer.
 
@itemize @minus
@item
@cindex row-major
Multi-dimensional arrays are stored in row-major order with dimensions:
@code{n0} x @code{n1}; or @code{n0} x @code{n1} x @code{n2}; or
@code{n[0]} x @code{n[1]} x ... x @code{n[rank-1]}.
@xref{Multi-dimensional Array Format}.
@item
FFTW is generally best at handling sizes of the form
@ifinfo
@math{2^a 3^b 5^c 7^d 11^e 13^f},
@end ifinfo
@tex
$2^a 3^b 5^c 7^d 11^e 13^f$,
@end tex
@html
2<sup>a</sup> 3<sup>b</sup> 5<sup>c</sup> 7<sup>d</sup>
        11<sup>e</sup> 13<sup>f</sup>,
@end html
where @math{e+f} is either @math{0} or @math{1}, and the other exponents
are arbitrary.  Other sizes are computed by means of a slow,
general-purpose algorithm (which nevertheless retains @Onlogn{} performance even for prime sizes).  (It is possible to customize FFTW
for different array sizes; see @ref{Installation and Customization}.)
Transforms whose sizes are powers of @math{2} are especially fast.
@item
For a @code{REDFT00} or @code{RODFT00} transform kind in a dimension of
size @math{n}, it is @math{n-1} or @math{n+1}, respectively, that
should be factorizable in the above form.
@end itemize

@item
@code{in} and @code{out} point to the input and output arrays of the
transform, which may be the same (yielding an in-place transform).
@cindex in-place
These arrays are overwritten during planning, unless
@code{FFTW_ESTIMATE} is used in the flags.  (The arrays need not be
initialized, but they must be allocated.)

@item
@code{kind}, or @code{kind0}/@code{kind1}/@code{kind2}, or
@code{kind[rank]}, is the kind of r2r transform used for the
corresponding dimension.  The valid kind constants are described in
@ref{Real-to-Real Transform Kinds}.  In a multi-dimensional transform,
what is computed is the separable product formed by taking each
transform kind along the corresponding dimension, one dimension after
another.

@item
@cindex flags
@code{flags} is a bitwise OR (@samp{|}) of zero or more planner flags,
as defined in @ref{Planner Flags}.

@end itemize

@c =========>
@node Real-to-Real Transform Kinds,  , Real-to-Real Transforms, Basic Interface
@subsection Real-to-Real Transform Kinds
@cindex kind (r2r)

FFTW currently supports 11 different r2r transform kinds, specified by
one of the constants below.  For the precise definitions of these
transforms, see @ref{What FFTW Really Computes}.  For a more colloquial
introduction to these transform kinds, see @ref{More DFTs of Real Data}.

For dimension of size @code{n}, there is a corresponding ``logical''
dimension @code{N} that determines the normalization (and the optimal
factorization); the formula for @code{N} is given for each kind below.
Also, with each transform kind is listed its corrsponding inverse
transform.  FFTW computes unnormalized transforms: a transform followed
by its inverse will result in the original data multiplied by @code{N}
(or the product of the @code{N}'s for each dimension, in
multi-dimensions).
@cindex normalization

@itemize @bullet

@item
@ctindex FFTW_R2HC
@code{FFTW_R2HC} computes a real-input DFT with output in
``halfcomplex'' format, i.e. real and imaginary parts for a transform of
size @code{n} stored as:
@tex
$$
r_0, r_1, r_2, \ldots, r_{n/2}, i_{(n+1)/2-1}, \ldots, i_2, i_1
$$
@end tex
@ifinfo
r0, r1, r2, r(n/2), i((n+1)/2-1), ..., i2, i1
@end ifinfo
@html
<p align=center>
r<sub>0</sub>, r<sub>1</sub>, r<sub>2</sub>, ..., r<sub>n/2</sub>, i<sub>(n+1)/2-1</sub>, ..., i<sub>2</sub>, i<sub>1</sub>
</p>
@end html
(Logical @code{N=n}, inverse is @code{FFTW_HC2R}.)

@item
@ctindex FFTW_HC2R
@code{FFTW_HC2R} computes the reverse of @code{FFTW_R2HC}, above.
(Logical @code{N=n}, inverse is @code{FFTW_R2HC}.)

@item
@ctindex FFTW_DHT
@code{FFTW_DHT} computes a discrete Hartley transform.
(Logical @code{N=n}, inverse is @code{FFTW_DHT}.)
@cindex discrete Hartley transform

@item
@ctindex FFTW_REDFT00
@code{FFTW_REDFT00} computes an REDFT00 transform, i.e. a DCT-I.
(Logical @code{N=2*(n-1)}, inverse is @code{FFTW_REDFT00}.)
@cindex discrete cosine transform
@cindex DCT

@item
@ctindex FFTW_REDFT10
@code{FFTW_REDFT10} computes an REDFT10 transform, i.e. a DCT-II (sometimes called ``the'' DCT).
(Logical @code{N=2*n}, inverse is @code{FFTW_REDFT01}.)

@item
@ctindex FFTW_REDFT01
@code{FFTW_REDFT01} computes an REDFT01 transform, i.e. a DCT-III (sometimes called ``the'' IDCT, being the inverse of DCT-II).
(Logical @code{N=2*n}, inverse is @code{FFTW_REDFT=10}.)
@cindex IDCT

@item
@ctindex FFTW_REDFT11
@code{FFTW_REDFT11} computes an REDFT11 transform, i.e. a DCT-IV.
(Logical @code{N=2*n}, inverse is @code{FFTW_REDFT11}.)

@item
@ctindex FFTW_RODFT00
@code{FFTW_RODFT00} computes an RODFT00 transform, i.e. a DST-I.
(Logical @code{N=2*(n+1)}, inverse is @code{FFTW_RODFT00}.)
@cindex discrete sine transform
@cindex DST

@item
@ctindex FFTW_RODFT10
@code{FFTW_RODFT10} computes an RODFT10 transform, i.e. a DST-II.
(Logical @code{N=2*n}, inverse is @code{FFTW_RODFT01}.)

@item
@ctindex FFTW_RODFT01
@code{FFTW_RODFT01} computes an RODFT01 transform, i.e. a DST-III.
(Logical @code{N=2*n}, inverse is @code{FFTW_RODFT=10}.)

@item
@ctindex FFTW_RODFT11
@code{FFTW_RODFT11} computes an RODFT11 transform, i.e. a DST-IV.
(Logical @code{N=2*n}, inverse is @code{FFTW_RODFT11}.)

@end itemize

@c ------------------------------------------------------------
@node Advanced Interface, Guru Interface, Basic Interface, FFTW Reference
@section Advanced Interface
@cindex advanced interface

FFTW's ``advanced'' interface supplements the basic interface with four
new planner routines, providing a new level of flexibility: you can plan
a transform of multiple arrays simultaneously, operate on non-contiguous
(strided) data, and transform a subset of a larger multi-dimensional
array.  Other than these additional features, the planner operates in
the same fashion as in the basic interface, and the resulting
@code{fftw_plan} is used in the same way (@pxref{Using Plans}).

@menu
* Advanced Complex DFTs::
* Advanced Real-data DFTs::
* Advanced Real-to-real Transforms::
@end menu

@c =========>
@node Advanced Complex DFTs, Advanced Real-data DFTs, Advanced Interface, Advanced Interface
@subsection Advanced Complex DFTs

@example
fftw_plan fftw_plan_many_dft(int rank, const int *n, int howmany,
                             fftw_complex *in, const int *inembed,
                             int istride, int idist,
                             fftw_complex *out, const int *onembed,
                             int ostride, int odist,
                             int sign, unsigned flags);
@end example
@findex fftw_plan_many_dft

This routine plans multiple multidimensional complex DFTs, and it
extends the @code{fftw_plan_dft} routine (@pxref{Complex DFTs}) to
compute @code{howmany} transforms, each having rank @code{rank} and size
@code{n}.  In addition, the transform data need not be contiguous, but
it may be laid out in memory with an arbitrary stride.  To account for
these possibilities, @code{fftw_plan_many_dft} adds the new parameters
@code{howmany}, @{@code{i},@code{o}@}@code{nembed},
@{@code{i},@code{o}@}@code{stride}, and
@{@code{i},@code{o}@}@code{dist}.  The FFTW basic interface
(@pxref{Complex DFTs}) provides routines specialized for ranks 1, 2,
and@tie{}3, but the advanced interface handles only the general-rank
case.

@code{howmany} is the number of transforms to compute.  The resulting
plan computes @code{howmany} transforms, where the input of the
@code{k}-th transform is at location @code{in+k*idist} (in C pointer
arithmetic), and its output is at location @code{out+k*odist}.  Plans
obtained in this way can often be faster than calling FFTW multiple
times for the individual transforms.  The basic @code{fftw_plan_dft}
interface corresponds to @code{howmany=1} (in which case the @code{dist}
parameters are ignored).
@cindex howmany parameter
@cindex dist


Each of the @code{howmany} transforms has rank @code{rank} and size
@code{n}, as in the basic interface.  In addition, the advanced
interface allows the input and output arrays of each transform to be
row-major subarrays of larger rank-@code{rank} arrays, described by
@code{inembed} and @code{onembed} parameters, respectively.
@{@code{i},@code{o}@}@code{nembed} must be arrays of length @code{rank},
and @code{n} should be elementwise less than or equal to
@{@code{i},@code{o}@}@code{nembed}.  Passing @code{NULL} for an
@code{nembed} parameter is equivalent to passing @code{n} (i.e. same
physical and logical dimensions, as in the basic interface.)

The @code{stride} parameters indicate that the @code{j}-th element of
the input or output arrays is located at @code{j*istride} or
@code{j*ostride}, respectively.  (For a multi-dimensional array,
@code{j} is the ordinary row-major index.)  When combined with the
@code{k}-th transform in a @code{howmany} loop, from above, this means
that the (@code{j},@code{k})-th element is at @code{j*stride+k*dist}.
(The basic @code{fftw_plan_dft} interface corresponds to a stride of 1.)
@cindex stride


For in-place transforms, the input and output @code{stride} and
@code{dist} parameters should be the same; otherwise, the planner may
return @code{NULL}.

Arrays @code{n}, @code{inembed}, and @code{onembed} are not used after
this function returns.  You can safely free or reuse them.

@strong{Examples}:
One transform of one 5 by 6 array contiguous in memory:
@example
   int rank = 2;
   int n[] = @{5, 6@};
   int howmany = 1;
   int idist = odist = 0; /* unused because howmany = 1 */
   int istride = ostride = 1; /* array is contiguous in memory */
   int *inembed = n, *onembed = n;
@end example

Transform of three 5 by 6 arrays, each contiguous in memory,
stored in memory one after another:
@example
   int rank = 2;
   int n[] = @{5, 6@};
   int howmany = 3;
   int idist = odist = n[0]*n[1]; /* = 30, the distance in memory
                                     between the first element
                                     of the first array and the
                                     first element of the second array */
   int istride = ostride = 1; /* array is contiguous in memory */
   int *inembed = n, *onembed = n;
@end example

Transform each column of a 2d array with 10 rows and 3 columns:
@example
   int rank = 1; /* not 2: we are computing 1d transforms */
   int n[] = @{10@}; /* 1d transforms of length 10 */
   int howmany = 3;
   int idist = odist = 1;
   int istride = ostride = 3; /* distance between two elements in 
                                 the same column */
   int *inembed = n, *onembed = n;
@end example

@c =========>
@node Advanced Real-data DFTs, Advanced Real-to-real Transforms, Advanced Complex DFTs, Advanced Interface
@subsection Advanced Real-data DFTs

@example
fftw_plan fftw_plan_many_dft_r2c(int rank, const int *n, int howmany,
                                 double *in, const int *inembed,
                                 int istride, int idist,
                                 fftw_complex *out, const int *onembed,
                                 int ostride, int odist,
                                 unsigned flags);
fftw_plan fftw_plan_many_dft_c2r(int rank, const int *n, int howmany,
                                 fftw_complex *in, const int *inembed,
                                 int istride, int idist,
                                 double *out, const int *onembed,
                                 int ostride, int odist,
                                 unsigned flags);
@end example
@findex fftw_plan_many_dft_r2c
@findex fftw_plan_many_dft_c2r

Like @code{fftw_plan_many_dft}, these two functions add @code{howmany},
@code{nembed}, @code{stride}, and @code{dist} parameters to the
@code{fftw_plan_dft_r2c} and @code{fftw_plan_dft_c2r} functions, but
otherwise behave the same as the basic interface.

The interpretation of @code{howmany}, @code{stride}, and @code{dist} are
the same as for @code{fftw_plan_many_dft}, above.  Note that the
@code{stride} and @code{dist} for the real array are in units of
@code{double}, and for the complex array are in units of
@code{fftw_complex}.

If an @code{nembed} parameter is @code{NULL}, it is interpreted as what
it would be in the basic interface, as described in @ref{Real-data DFT
Array Format}.  That is, for the complex array the size is assumed to be
the same as @code{n}, but with the last dimension cut roughly in half.
For the real array, the size is assumed to be @code{n} if the transform
is out-of-place, or @code{n} with the last dimension ``padded'' if the
transform is in-place.

If an @code{nembed} parameter is non-@code{NULL}, it is interpreted as
the physical size of the corresponding array, in row-major order, just
as for @code{fftw_plan_many_dft}.  In this case, each dimension of
@code{nembed} should be @code{>=} what it would be in the basic
interface (e.g. the halved or padded @code{n}).

Arrays @code{n}, @code{inembed}, and @code{onembed} are not used after
this function returns.  You can safely free or reuse them.

@c =========>
@node Advanced Real-to-real Transforms,  , Advanced Real-data DFTs, Advanced Interface
@subsection Advanced Real-to-real Transforms

@example
fftw_plan fftw_plan_many_r2r(int rank, const int *n, int howmany,
                             double *in, const int *inembed,
                             int istride, int idist,
                             double *out, const int *onembed,
                             int ostride, int odist,
                             const fftw_r2r_kind *kind, unsigned flags);
@end example
@findex fftw_plan_many_r2r

Like @code{fftw_plan_many_dft}, this functions adds @code{howmany},
@code{nembed}, @code{stride}, and @code{dist} parameters to the
@code{fftw_plan_r2r} function, but otherwise behave the same as the
basic interface.  The interpretation of those additional parameters are
the same as for @code{fftw_plan_many_dft}.  (Of course, the
@code{stride} and @code{dist} parameters are now in units of
@code{double}, not @code{fftw_complex}.)

Arrays @code{n}, @code{inembed}, @code{onembed}, and @code{kind} are not
used after this function returns.  You can safely free or reuse them.

@c ------------------------------------------------------------
@node Guru Interface, New-array Execute Functions, Advanced Interface, FFTW Reference
@section Guru Interface
@cindex guru interface

The ``guru'' interface to FFTW is intended to expose as much as possible
of the flexibility in the underlying FFTW architecture.  It allows one
to compute multi-dimensional ``vectors'' (loops) of multi-dimensional
transforms, where each vector/transform dimension has an independent
size and stride.
@cindex vector
One can also use more general complex-number formats, e.g. separate real
and imaginary arrays.

For those users who require the flexibility of the guru interface, it is
important that they pay special attention to the documentation lest they
shoot themselves in the foot.

@menu
* Interleaved and split arrays::
* Guru vector and transform sizes::
* Guru Complex DFTs::
* Guru Real-data DFTs::
* Guru Real-to-real Transforms::
* 64-bit Guru Interface::
@end menu

@c =========>
@node  Interleaved and split arrays, Guru vector and transform sizes, Guru Interface, Guru Interface
@subsection Interleaved and split arrays

The guru interface supports two representations of complex numbers,
which we call the interleaved and the split format.

The @dfn{interleaved} format is the same one used by the basic and
advanced interfaces, and it is documented in @ref{Complex numbers}.
In the interleaved format, you provide pointers to the real part of a
complex number, and the imaginary part understood to be stored in the
next memory location.
@cindex interleaved format


The @dfn{split} format allows separate pointers to the real and
imaginary parts of a complex array.
@cindex split format


Technically, the interleaved format is redundant, because you can
always express an interleaved array in terms of a split array with
appropriate pointers and strides.  On the other hand, the interleaved
format is simpler to use, and it is common in practice.  Hence, FFTW
supports it as a special case.

@c =========>
@node Guru vector and transform sizes, Guru Complex DFTs, Interleaved and split arrays, Guru Interface
@subsection Guru vector and transform sizes

The guru interface introduces one basic new data structure,
@code{fftw_iodim}, that is used to specify sizes and strides for
multi-dimensional transforms and vectors:

@example
typedef struct @{
     int n;
     int is;
     int os;
@} fftw_iodim;
@end example
@tindex fftw_iodim

Here, @code{n} is the size of the dimension, and @code{is} and @code{os}
are the strides of that dimension for the input and output arrays.  (The
stride is the separation of consecutive elements along this dimension.)

The meaning of the stride parameter depends on the type of the array
that the stride refers to.  @emph{If the array is interleaved complex,
strides are expressed in units of complex numbers
(@code{fftw_complex}).  If the array is split complex or real, strides
are expressed in units of real numbers (@code{double}).}  This
convention is consistent with the usual pointer arithmetic in the C
language.  An interleaved array is denoted by a pointer @code{p} to
@code{fftw_complex}, so that @code{p+1} points to the next complex
number.  Split arrays are denoted by pointers to @code{double}, in
which case pointer arithmetic operates in units of
@code{sizeof(double)}.
@cindex stride


The guru planner interfaces all take a (@code{rank}, @code{dims[rank]})
pair describing the transform size, and a (@code{howmany_rank},
@code{howmany_dims[howmany_rank]}) pair describing the ``vector'' size (a
multi-dimensional loop of transforms to perform), where @code{dims} and
@code{howmany_dims} are arrays of @code{fftw_iodim}.

For example, the @code{howmany} parameter in the advanced complex-DFT
interface corresponds to @code{howmany_rank} = 1,
@code{howmany_dims[0].n} = @code{howmany}, @code{howmany_dims[0].is} =
@code{idist}, and @code{howmany_dims[0].os} = @code{odist}.
@cindex howmany loop
@cindex dist
(To compute a single transform, you can just use @code{howmany_rank} = 0.)


A row-major multidimensional array with dimensions @code{n[rank]}
(@pxref{Row-major Format}) corresponds to @code{dims[i].n} =
@code{n[i]} and the recurrence @code{dims[i].is} = @code{n[i+1] *
dims[i+1].is} (similarly for @code{os}).  The stride of the last
(@code{i=rank-1}) dimension is the overall stride of the array.
e.g. to be equivalent to the advanced complex-DFT interface, you would
have @code{dims[rank-1].is} = @code{istride} and
@code{dims[rank-1].os} = @code{ostride}.
@cindex row-major


In general, we only guarantee FFTW to return a non-@code{NULL} plan if
the vector and transform dimensions correspond to a set of distinct
indices, and for in-place transforms the input/output strides should
be the same.

@c =========>
@node Guru Complex DFTs, Guru Real-data DFTs, Guru vector and transform sizes, Guru Interface
@subsection Guru Complex DFTs

@example
fftw_plan fftw_plan_guru_dft(
     int rank, const fftw_iodim *dims,
     int howmany_rank, const fftw_iodim *howmany_dims,
     fftw_complex *in, fftw_complex *out,
     int sign, unsigned flags);

fftw_plan fftw_plan_guru_split_dft(
     int rank, const fftw_iodim *dims,
     int howmany_rank, const fftw_iodim *howmany_dims,
     double *ri, double *ii, double *ro, double *io,
     unsigned flags);
@end example
@findex fftw_plan_guru_dft
@findex fftw_plan_guru_split_dft

These two functions plan a complex-data, multi-dimensional DFT
for the interleaved and split format, respectively.
Transform dimensions are given by (@code{rank}, @code{dims}) over a
multi-dimensional vector (loop) of dimensions (@code{howmany_rank},
@code{howmany_dims}).  @code{dims} and @code{howmany_dims} should point
to @code{fftw_iodim} arrays of length @code{rank} and
@code{howmany_rank}, respectively.

@cindex flags
@code{flags} is a bitwise OR (@samp{|}) of zero or more planner flags,
as defined in @ref{Planner Flags}.

In the @code{fftw_plan_guru_dft} function, the pointers @code{in} and
@code{out} point to the interleaved input and output arrays,
respectively.  The sign can be either @math{-1} (=
@code{FFTW_FORWARD}) or @math{+1} (= @code{FFTW_BACKWARD}).  If the
pointers are equal, the transform is in-place.

In the @code{fftw_plan_guru_split_dft} function,
@code{ri} and @code{ii} point to the real and imaginary input arrays,
and @code{ro} and @code{io} point to the real and imaginary output
arrays.  The input and output pointers may be the same, indicating an
in-place transform.  For example, for @code{fftw_complex} pointers
@code{in} and @code{out}, the corresponding parameters are:

@example
ri = (double *) in;
ii = (double *) in + 1;
ro = (double *) out;
io = (double *) out + 1;
@end example

Because @code{fftw_plan_guru_split_dft} accepts split arrays, strides
are expressed in units of @code{double}.  For a contiguous
@code{fftw_complex} array, the overall stride of the transform should
be 2, the distance between consecutive real parts or between
consecutive imaginary parts; see @ref{Guru vector and transform
sizes}.  Note that the dimension strides are applied equally to the
real and imaginary parts; real and imaginary arrays with different
strides are not supported.

There is no @code{sign} parameter in @code{fftw_plan_guru_split_dft}.
This function always plans for an @code{FFTW_FORWARD} transform.  To
plan for an @code{FFTW_BACKWARD} transform, you can exploit the
identity that the backwards DFT is equal to the forwards DFT with the
real and imaginary parts swapped.  For example, in the case of the
@code{fftw_complex} arrays above, the @code{FFTW_BACKWARD} transform
is computed by the parameters:

@example
ri = (double *) in + 1;
ii = (double *) in;
ro = (double *) out + 1;
io = (double *) out;
@end example

@c =========>
@node Guru Real-data DFTs, Guru Real-to-real Transforms, Guru Complex DFTs, Guru Interface
@subsection Guru Real-data DFTs

@example
fftw_plan fftw_plan_guru_dft_r2c(
     int rank, const fftw_iodim *dims,
     int howmany_rank, const fftw_iodim *howmany_dims,
     double *in, fftw_complex *out,
     unsigned flags);

fftw_plan fftw_plan_guru_split_dft_r2c(
     int rank, const fftw_iodim *dims,
     int howmany_rank, const fftw_iodim *howmany_dims,
     double *in, double *ro, double *io,
     unsigned flags);

fftw_plan fftw_plan_guru_dft_c2r(
     int rank, const fftw_iodim *dims,
     int howmany_rank, const fftw_iodim *howmany_dims,
     fftw_complex *in, double *out,
     unsigned flags);

fftw_plan fftw_plan_guru_split_dft_c2r(
     int rank, const fftw_iodim *dims,
     int howmany_rank, const fftw_iodim *howmany_dims,
     double *ri, double *ii, double *out,
     unsigned flags);
@end example
@findex fftw_plan_guru_dft_r2c
@findex fftw_plan_guru_split_dft_r2c
@findex fftw_plan_guru_dft_c2r
@findex fftw_plan_guru_split_dft_c2r

Plan a real-input (r2c) or real-output (c2r), multi-dimensional DFT with
transform dimensions given by (@code{rank}, @code{dims}) over a
multi-dimensional vector (loop) of dimensions (@code{howmany_rank},
@code{howmany_dims}).  @code{dims} and @code{howmany_dims} should point
to @code{fftw_iodim} arrays of length @code{rank} and
@code{howmany_rank}, respectively.  As for the basic and advanced
interfaces, an r2c transform is @code{FFTW_FORWARD} and a c2r transform
is @code{FFTW_BACKWARD}.

The @emph{last} dimension of @code{dims} is interpreted specially:
that dimension of the real array has size @code{dims[rank-1].n}, but
that dimension of the complex array has size @code{dims[rank-1].n/2+1}
(division rounded down).  The strides, on the other hand, are taken to
be exactly as specified.  It is up to the user to specify the strides
appropriately for the peculiar dimensions of the data, and we do not
guarantee that the planner will succeed (return non-@code{NULL}) for
any dimensions other than those described in @ref{Real-data DFT Array
Format} and generalized in @ref{Advanced Real-data DFTs}.  (That is,
for an in-place transform, each individual dimension should be able to
operate in place.)
@cindex in-place


@code{in} and @code{out} point to the input and output arrays for r2c
and c2r transforms, respectively.  For split arrays, @code{ri} and
@code{ii} point to the real and imaginary input arrays for a c2r
transform, and @code{ro} and @code{io} point to the real and imaginary
output arrays for an r2c transform.  @code{in} and @code{ro} or
@code{ri} and @code{out} may be the same, indicating an in-place
transform.   (In-place transforms where @code{in} and @code{io} or
@code{ii} and @code{out} are the same are not currently supported.)

@cindex flags
@code{flags} is a bitwise OR (@samp{|}) of zero or more planner flags,
as defined in @ref{Planner Flags}.

In-place transforms of rank greater than 1 are currently only
supported for interleaved arrays.  For split arrays, the planner will
return @code{NULL}.
@cindex in-place

@c =========>
@node Guru Real-to-real Transforms, 64-bit Guru Interface, Guru Real-data DFTs, Guru Interface
@subsection Guru Real-to-real Transforms

@example
fftw_plan fftw_plan_guru_r2r(int rank, const fftw_iodim *dims,
                             int howmany_rank,
                             const fftw_iodim *howmany_dims,
                             double *in, double *out,
                             const fftw_r2r_kind *kind,
                             unsigned flags);
@end example
@findex fftw_plan_guru_r2r

Plan a real-to-real (r2r) multi-dimensional @code{FFTW_FORWARD}
transform with transform dimensions given by (@code{rank}, @code{dims})
over a multi-dimensional vector (loop) of dimensions
(@code{howmany_rank}, @code{howmany_dims}).  @code{dims} and
@code{howmany_dims} should point to @code{fftw_iodim} arrays of length
@code{rank} and @code{howmany_rank}, respectively.

The transform kind of each dimension is given by the @code{kind}
parameter, which should point to an array of length @code{rank}.  Valid
@code{fftw_r2r_kind} constants are given in @ref{Real-to-Real Transform
Kinds}.

@code{in} and @code{out} point to the real input and output arrays; they
may be the same, indicating an in-place transform.

@cindex flags
@code{flags} is a bitwise OR (@samp{|}) of zero or more planner flags,
as defined in @ref{Planner Flags}.

@c =========>
@node 64-bit Guru Interface,  , Guru Real-to-real Transforms, Guru Interface
@subsection 64-bit Guru Interface
@cindex 64-bit architecture

When compiled in 64-bit mode on a 64-bit architecture (where addresses
are 64 bits wide), FFTW uses 64-bit quantities internally for all
transform sizes, strides, and so on---you don't have to do anything
special to exploit this.  However, in the ordinary FFTW interfaces,
you specify the transform size by an @code{int} quantity, which is
normally only 32 bits wide.  This means that, even though FFTW is
using 64-bit sizes internally, you cannot specify a single transform
dimension larger than
@ifinfo
2^31-1
@end ifinfo
@html
2<sup><small>31</small></sup>&minus;1
@end html
@tex
$2^{31}-1$
@end tex
numbers.

We expect that few users will require transforms larger than this, but,
for those who do, we provide a 64-bit version of the guru interface in
which all sizes are specified as integers of type @code{ptrdiff_t}
instead of @code{int}.  (@code{ptrdiff_t} is a signed integer type
defined by the C standard to be wide enough to represent address
differences, and thus must be at least 64 bits wide on a 64-bit
machine.)  We stress that there is @emph{no performance advantage} to
using this interface---the same internal FFTW code is employed
regardless---and it is only necessary if you want to specify very
large transform sizes.
@tindex ptrdiff_t


In particular, the 64-bit guru interface is a set of planner routines
that are exactly the same as the guru planner routines, except that
they are named with @samp{guru64} instead of @samp{guru} and they take
arguments of type @code{fftw_iodim64} instead of @code{fftw_iodim}.
For example, instead of @code{fftw_plan_guru_dft}, we have
@code{fftw_plan_guru64_dft}.

@example
fftw_plan fftw_plan_guru64_dft(
     int rank, const fftw_iodim64 *dims,
     int howmany_rank, const fftw_iodim64 *howmany_dims,
     fftw_complex *in, fftw_complex *out,
     int sign, unsigned flags);
@end example
@findex fftw_plan_guru64_dft

The @code{fftw_iodim64} type is similar to @code{fftw_iodim}, with the
same interpretation, except that it uses type @code{ptrdiff_t} instead
of type @code{int}.

@example
typedef struct @{
     ptrdiff_t n;
     ptrdiff_t is;
     ptrdiff_t os;
@} fftw_iodim64;
@end example
@tindex fftw_iodim64

Every other @samp{fftw_plan_guru} function also has a
@samp{fftw_plan_guru64} equivalent, but we do not repeat their
documentation here since they are identical to the 32-bit versions
except as noted above.

@c -----------------------------------------------------------
@node New-array Execute Functions, Wisdom, Guru Interface, FFTW Reference
@section New-array Execute Functions
@cindex execute
@cindex new-array execution

Normally, one executes a plan for the arrays with which the plan was
created, by calling @code{fftw_execute(plan)} as described in @ref{Using
Plans}.
@findex fftw_execute
However, it is possible for sophisticated users to apply a given plan
to a @emph{different} array using the ``new-array execute'' functions
detailed below, provided that the following conditions are met:

@itemize @bullet

@item
The array size, strides, etcetera are the same (since those are set by
the plan).

@item
The input and output arrays are the same (in-place) or different
(out-of-place) if the plan was originally created to be in-place or
out-of-place, respectively.

@item
For split arrays, the separations between the real and imaginary
parts, @code{ii-ri} and @code{io-ro}, are the same as they were for
the input and output arrays when the plan was created.  (This
condition is automatically satisfied for interleaved arrays.)

@item
The @dfn{alignment} of the new input/output arrays is the same as that
of the input/output arrays when the plan was created, unless the plan
was created with the @code{FFTW_UNALIGNED} flag.
@ctindex FFTW_UNALIGNED
Here, the alignment is a platform-dependent quantity (for example, it is
the address modulo 16 if SSE SIMD instructions are used, but the address
modulo 4 for non-SIMD single-precision FFTW on the same machine).  In
general, only arrays allocated with @code{fftw_malloc} are guaranteed to
be equally aligned (@pxref{SIMD alignment and fftw_malloc}).

@end itemize

@cindex alignment
The alignment issue is especially critical, because if you don't use
@code{fftw_malloc} then you may have little control over the alignment
of arrays in memory.  For example, neither the C++ @code{new} function
nor the Fortran @code{allocate} statement provide strong enough
guarantees about data alignment.  If you don't use @code{fftw_malloc},
therefore, you probably have to use @code{FFTW_UNALIGNED} (which
disables most SIMD support).  If possible, it is probably better for
you to simply create multiple plans (creating a new plan is quick once
one exists for a given size), or better yet re-use the same array for
your transforms.

@findex fftw_alignment_of
For rare circumstances in which you cannot control the alignment of
allocated memory, but wish to determine where a given array is
aligned like the original array for which a plan was created, you can
use the @code{fftw_alignment_of} function:
@example
int fftw_alignment_of(double *p);
@end example
Two arrays have equivalent alignment (for the purposes of applying a
plan) if and only if @code{fftw_alignment_of} returns the same value
for the corresponding pointers to their data (typecast to @code{double*} 
if necessary).

If you are tempted to use the new-array execute interface because you
want to transform a known bunch of arrays of the same size, you should
probably go use the advanced interface instead (@pxref{Advanced
Interface})).

The new-array execute functions are:

@example
void fftw_execute_dft(
     const fftw_plan p, 
     fftw_complex *in, fftw_complex *out);

void fftw_execute_split_dft(
     const fftw_plan p, 
     double *ri, double *ii, double *ro, double *io);

void fftw_execute_dft_r2c(
     const fftw_plan p,
     double *in, fftw_complex *out);

void fftw_execute_split_dft_r2c(
     const fftw_plan p,
     double *in, double *ro, double *io);

void fftw_execute_dft_c2r(
     const fftw_plan p,
     fftw_complex *in, double *out);

void fftw_execute_split_dft_c2r(
     const fftw_plan p,
     double *ri, double *ii, double *out);

void fftw_execute_r2r(
     const fftw_plan p, 
     double *in, double *out);
@end example
@findex fftw_execute_dft
@findex fftw_execute_split_dft
@findex fftw_execute_dft_r2c
@findex fftw_execute_split_dft_r2c
@findex fftw_execute_dft_c2r
@findex fftw_execute_split_dft_c2r
@findex fftw_execute_r2r

These execute the @code{plan} to compute the corresponding transform on
the input/output arrays specified by the subsequent arguments.  The
input/output array arguments have the same meanings as the ones passed
to the guru planner routines in the preceding sections.  The @code{plan}
is not modified, and these routines can be called as many times as
desired, or intermixed with calls to the ordinary @code{fftw_execute}.

The @code{plan} @emph{must} have been created for the transform type
corresponding to the execute function, e.g. it must be a complex-DFT
plan for @code{fftw_execute_dft}.  Any of the planner routines for that
transform type, from the basic to the guru interface, could have been
used to create the plan, however.

@c ------------------------------------------------------------
@node Wisdom, What FFTW Really Computes, New-array Execute Functions, FFTW Reference
@section Wisdom
@cindex wisdom
@cindex saving plans to disk

This section documents the FFTW mechanism for saving and restoring
plans from disk.  This mechanism is called @dfn{wisdom}.

@menu
* Wisdom Export::
* Wisdom Import::
* Forgetting Wisdom::
* Wisdom Utilities::
@end menu

@c =========>
@node Wisdom Export, Wisdom Import, Wisdom, Wisdom
@subsection Wisdom Export

@example
int fftw_export_wisdom_to_filename(const char *filename);
void fftw_export_wisdom_to_file(FILE *output_file);
char *fftw_export_wisdom_to_string(void);
void fftw_export_wisdom(void (*write_char)(char c, void *), void *data);
@end example
@findex fftw_export_wisdom
@findex fftw_export_wisdom_to_filename
@findex fftw_export_wisdom_to_file
@findex fftw_export_wisdom_to_string

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. (@xref{Words of Wisdom-Saving
Plans}.)  The current store of wisdom is not affected by calling any
of these routines.

@code{fftw_export_wisdom} exports the wisdom to any output
medium, as specified by the callback function
@code{write_char}. @code{write_char} is a @code{putc}-like function that
writes the character @code{c} to some output; its second parameter is
the @code{data} pointer passed to @code{fftw_export_wisdom}.  For
convenience, the following three ``wrapper'' routines are provided:

@code{fftw_export_wisdom_to_filename} writes wisdom to a file named
@code{filename} (which is created or overwritten), returning @code{1}
on success and @code{0} on failure.  A lower-level function, which
requires you to open and close the file yourself (e.g. if you want to
write wisdom to a portion of a larger file) is
@code{fftw_export_wisdom_to_file}.  This writes the wisdom to the
current position in @code{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.

@code{fftw_export_wisdom_to_string} returns a pointer to a
@code{NULL}-terminated string holding the wisdom data. This string is
dynamically allocated, and it is the responsibility of the caller to
deallocate it with @code{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.

@c =========>
@node Wisdom Import, Forgetting Wisdom, Wisdom Export, Wisdom
@subsection Wisdom Import

@example
int fftw_import_system_wisdom(void);
int fftw_import_wisdom_from_filename(const char *filename);
int fftw_import_wisdom_from_string(const char *input_string);
int fftw_import_wisdom(int (*read_char)(void *), void *data);
@end example
@findex fftw_import_wisdom
@findex fftw_import_system_wisdom
@findex fftw_import_wisdom_from_filename
@findex fftw_import_wisdom_from_file
@findex fftw_import_wisdom_from_string

These functions import wisdom into a program from data stored by the
@code{fftw_export_wisdom} functions above. (@xref{Words of
Wisdom-Saving Plans}.)  The imported wisdom replaces any wisdom
already accumulated by the running program.

@code{fftw_import_wisdom} imports wisdom from any input medium, as
specified by the callback function @code{read_char}. @code{read_char} is
a @code{getc}-like function that returns the next character in the
input; its parameter is the @code{data} pointer passed to
@code{fftw_import_wisdom}. If the end of the input data is reached
(which should never happen for valid data), @code{read_char} should
return @code{EOF} (as defined in @code{<stdio.h>}).  For convenience,
the following three ``wrapper'' routines are provided:

@code{fftw_import_wisdom_from_filename} reads wisdom from a file named
@code{filename}.  A lower-level function, which requires you to open
and close the file yourself (e.g. if you want to read wisdom from a
portion of a larger file) is @code{fftw_import_wisdom_from_file}. This
reads wisdom from the current position in @code{input_file} (which
should be open with read permission); upon exit, the file remains
open, but the position of the read pointer is unspecified.

@code{fftw_import_wisdom_from_string} reads wisdom from the
@code{NULL}-terminated string @code{input_string}.

@code{fftw_import_system_wisdom} reads wisdom from an
implementation-defined standard file (@code{/etc/fftw/wisdom} on Unix
and GNU systems).
@cindex wisdom, system-wide


The return value of these import routines is @code{1} if the wisdom was
read successfully and @code{0} otherwise. Note that, in all of these
functions, any data in the input stream past the end of the wisdom data
is simply ignored.

@c =========>
@node Forgetting Wisdom, Wisdom Utilities, Wisdom Import, Wisdom
@subsection Forgetting Wisdom

@example
void fftw_forget_wisdom(void);
@end example
@findex fftw_forget_wisdom

Calling @code{fftw_forget_wisdom} causes all accumulated @code{wisdom}
to be discarded and its associated memory to be freed. (New
@code{wisdom} can still be gathered subsequently, however.)

@c =========>
@node Wisdom Utilities,  , Forgetting Wisdom, Wisdom
@subsection Wisdom Utilities

FFTW includes two standalone utility programs that deal with wisdom.  We
merely summarize them here, since they come with their own @code{man}
pages for Unix and GNU systems (with HTML versions on our web site).

The first program is @code{fftw-wisdom} (or @code{fftwf-wisdom} in
single precision, etcetera), which can be used to create a wisdom file
containing plans for any of the transform sizes and types supported by
FFTW.  It is preferable to create wisdom directly from your executable
(@pxref{Caveats in Using Wisdom}), but this program is useful for
creating global wisdom files for @code{fftw_import_system_wisdom}.
@cindex fftw-wisdom utility


The second program is @code{fftw-wisdom-to-conf}, which takes a wisdom
file as input and produces a @dfn{configuration routine} as output.  The
latter is a C subroutine that you can compile and link into your
program, replacing a routine of the same name in the FFTW library, that
determines which parts of FFTW are callable by your program.
@code{fftw-wisdom-to-conf} produces a configuration routine that links
to only those parts of FFTW needed by the saved plans in the wisdom,
greatly reducing the size of statically linked executables (which should
only attempt to create plans corresponding to those in the wisdom,
however).
@cindex fftw-wisdom-to-conf utility
@cindex configuration routines

@c ------------------------------------------------------------
@node What FFTW Really Computes,  , Wisdom, FFTW Reference
@section What FFTW Really Computes

In this section, we provide precise mathematical definitions for the
transforms that FFTW computes.  These transform definitions are fairly
standard, but some authors follow slightly different conventions for the
normalization of the transform (the constant factor in front) and the
sign of the complex exponent.  We begin by presenting the
one-dimensional (1d) transform definitions, and then give the
straightforward extension to multi-dimensional transforms.

@menu
* The 1d Discrete Fourier Transform (DFT)::
* The 1d Real-data DFT::
* 1d Real-even DFTs (DCTs)::
* 1d Real-odd DFTs (DSTs)::
* 1d Discrete Hartley Transforms (DHTs)::
* Multi-dimensional Transforms::
@end menu

@c =========>
@node The 1d Discrete Fourier Transform (DFT), The 1d Real-data DFT, What FFTW Really Computes, What FFTW Really Computes
@subsection The 1d Discrete Fourier Transform (DFT)

@cindex discrete Fourier transform
@cindex DFT
The forward (@code{FFTW_FORWARD}) discrete Fourier transform (DFT) of a
1d complex array @math{X} of size @math{n} computes an array @math{Y},
where:
@tex
$$
Y_k = \sum_{j = 0}^{n - 1} X_j e^{-2\pi j k \sqrt{-1}/n} \ .
$$
@end tex
@ifinfo
@center Y[k] = sum for j = 0 to (n - 1) of X[j] * exp(-2 pi j k sqrt(-1)/n) .
@end ifinfo
@html
<center><img src="equation-dft.png" align="top">.</center>
@end html
The backward (@code{FFTW_BACKWARD}) DFT computes:
@tex
$$
Y_k = \sum_{j = 0}^{n - 1} X_j e^{2\pi j k \sqrt{-1}/n} \ .
$$
@end tex
@ifinfo
@center Y[k] = sum for j = 0 to (n - 1) of X[j] * exp(2 pi j k sqrt(-1)/n) .
@end ifinfo
@html
<center><img src="equation-idft.png" align="top">.</center>
@end html

@cindex normalization
FFTW computes an unnormalized transform, in that there is no coefficient
in front of the summation in the DFT.  In other words, applying the
forward and then the backward transform will multiply the input by
@math{n}.

@cindex frequency
From above, an @code{FFTW_FORWARD} transform corresponds to a sign of
@math{-1} in the exponent of the DFT.  Note also that we use the
standard ``in-order'' output ordering---the @math{k}-th output
corresponds to the frequency @math{k/n} (or @math{k/T}, where @math{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 @math{-k/n} is the same as the frequency
@math{(n-k)/n}.)

@c =========>
@node The 1d Real-data DFT, 1d Real-even DFTs (DCTs), The 1d Discrete Fourier Transform (DFT), What FFTW Really Computes
@subsection The 1d Real-data DFT

The real-input (r2c) DFT in FFTW computes the @emph{forward} transform
@math{Y} of the size @code{n} real array @math{X}, exactly as defined
above, i.e.
@tex
$$
Y_k = \sum_{j = 0}^{n - 1} X_j e^{-2\pi j k \sqrt{-1}/n} \ .
$$
@end tex
@ifinfo
@center Y[k] = sum for j = 0 to (n - 1) of X[j] * exp(-2 pi j k sqrt(-1)/n) .
@end ifinfo
@html
<center><img src="equation-dft.png" align="top">.</center>
@end html
This output array @math{Y} can easily be shown to possess the
``Hermitian'' symmetry
@cindex Hermitian
@tex
$Y_k = Y_{n-k}^*$,
@end tex
@ifinfo
Y[k] = Y[n-k]*,
@end ifinfo
@html
<i>Y<sub>k</sub> = Y<sub>n-k</sub></i><sup>*</sup>,
@end html
where we take @math{Y} to be periodic so that
@tex
$Y_n = Y_0$.
@end tex
@ifinfo
Y[n] = Y[0].
@end ifinfo
@html
<i>Y<sub>n</sub> = Y</i><sub>0</sub>.
@end html

As a result of this symmetry, half of the output @math{Y} is redundant
(being the complex conjugate of the other half), and so the 1d r2c
transforms only output elements @math{0}@dots{}@math{n/2} of @math{Y}
(@math{n/2+1} complex numbers), where the division by @math{2} is
rounded down. 

Moreover, the Hermitian symmetry implies that
@tex
$Y_0$
@end tex
@ifinfo
Y[0]
@end ifinfo
@html
<i>Y</i><sub>0</sub>
@end html
and, if @math{n} is even, the
@tex
$Y_{n/2}$
@end tex
@ifinfo
Y[n/2]
@end ifinfo
@html
<i>Y</i><sub><i>n</i>/2</sub>
@end html
element, are purely real.  So, for the @code{R2HC} r2r transform, the
halfcomplex format does not store the imaginary parts of these elements.
@cindex r2r
@ctindex R2HC
@cindex halfcomplex format


The c2r and @code{H2RC} r2r transforms compute the backward DFT of the
@emph{complex} array @math{X} with Hermitian symmetry, stored in the
r2c/@code{R2HC} output formats, respectively, where the backward
transform is defined exactly as for the complex case:
@tex
$$
Y_k = \sum_{j = 0}^{n - 1} X_j e^{2\pi j k \sqrt{-1}/n} \ .
$$
@end tex
@ifinfo
@center Y[k] = sum for j = 0 to (n - 1) of X[j] * exp(2 pi j k sqrt(-1)/n) .
@end ifinfo
@html
<center><img src="equation-idft.png" align="top">.</center>
@end html
The outputs @code{Y} of this transform can easily be seen to be purely
real, and are stored as an array of real numbers.

@cindex normalization
Like FFTW's complex DFT, these transforms are unnormalized.  In other
words, applying the real-to-complex (forward) and then the
complex-to-real (backward) transform will multiply the input by
@math{n}.

@c =========>
@node 1d Real-even DFTs (DCTs), 1d Real-odd DFTs (DSTs), The 1d Real-data DFT, What FFTW Really Computes
@subsection 1d Real-even DFTs (DCTs)

The Real-even symmetry DFTs in FFTW are exactly equivalent to the unnormalized
forward (and backward) DFTs as defined above, where the input array
@math{X} of length @math{N} is purely real and is also @dfn{even} symmetry.  In
this case, the output array is likewise real and even symmetry.
@cindex real-even DFT
@cindex REDFT


@ctindex REDFT00
For the case of @code{REDFT00}, this even symmetry means that
@tex
$X_j = X_{N-j}$,
@end tex
@ifinfo
X[j] = X[N-j],
@end ifinfo
@html
<i>X<sub>j</sub> = X<sub>N-j</sub></i>,
@end html
where we take @math{X} to be periodic so that
@tex
$X_N = X_0$.
@end tex
@ifinfo
X[N] = X[0].
@end ifinfo
@html
<i>X<sub>N</sub> = X</i><sub>0</sub>.
@end html
Because of this redundancy, only the first @math{n} real numbers are
actually stored, where @math{N = 2(n-1)}.

The proper definition of even symmetry for @code{REDFT10},
@code{REDFT01}, and @code{REDFT11} transforms is somewhat more intricate
because of the shifts by @math{1/2} of the input and/or output, although
the corresponding boundary conditions are given in @ref{Real even/odd
DFTs (cosine/sine transforms)}.  Because of the even symmetry, however,
the sine terms in the DFT all cancel and the remaining cosine terms are
written explicitly below.  This formulation often leads people to call
such a transform a @dfn{discrete cosine transform} (DCT), although it is
really just a special case of the DFT.
@cindex discrete cosine transform
@cindex DCT


In each of the definitions below, we transform a real array @math{X} of
length @math{n} to a real array @math{Y} of length @math{n}:

@subsubheading REDFT00 (DCT-I)
@ctindex REDFT00
An @code{REDFT00} transform (type-I DCT) in FFTW is defined by:
@tex
$$
Y_k = X_0 + (-1)^k X_{n-1}
       + 2 \sum_{j=1}^{n-2} X_j \cos [ \pi j k / (n-1)].
$$
@end tex
@ifinfo
Y[k] = X[0] + (-1)^k X[n-1] + 2 (sum for j = 1 to n-2 of X[j] cos(pi jk /(n-1))).
@end ifinfo
@html
<center><img src="equation-redft00.png" align="top">.</center>
@end html
Note that this transform is not defined for @math{n=1}.  For @math{n=2},
the summation term above is dropped as you might expect.

@subsubheading REDFT10 (DCT-II)
@ctindex REDFT10
An @code{REDFT10} transform (type-II DCT, sometimes called ``the'' DCT) in FFTW is defined by:
@tex
$$
Y_k = 2 \sum_{j=0}^{n-1} X_j \cos [\pi (j+1/2) k / n].
$$
@end tex
@ifinfo
Y[k] = 2 (sum for j = 0 to n-1 of X[j] cos(pi (j+1/2) k / n)).
@end ifinfo
@html
<center><img src="equation-redft10.png" align="top">.</center>
@end html

@subsubheading REDFT01 (DCT-III)
@ctindex REDFT01
An @code{REDFT01} transform (type-III DCT) in FFTW is defined by:
@tex
$$
Y_k = X_0 + 2 \sum_{j=1}^{n-1} X_j \cos [\pi j (k+1/2) / n].
$$
@end tex
@ifinfo
Y[k] = X[0] + 2 (sum for j = 1 to n-1 of X[j] cos(pi j (k+1/2) / n)).
@end ifinfo
@html
<center><img src="equation-redft01.png" align="top">.</center>
@end html
In the case of @math{n=1}, this reduces to
@tex
$Y_0 = X_0$.
@end tex
@ifinfo
Y[0] = X[0].
@end ifinfo
@html
<i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.
@end html
Up to a scale factor (see below), this is the inverse of @code{REDFT10} (``the'' DCT), and so the @code{REDFT01} (DCT-III) is sometimes called the ``IDCT''.
@cindex IDCT

@subsubheading REDFT11 (DCT-IV)
@ctindex REDFT11
An @code{REDFT11} transform (type-IV DCT) in FFTW is defined by:
@tex
$$
Y_k = 2 \sum_{j=0}^{n-1} X_j \cos [\pi (j+1/2) (k+1/2) / n].
$$
@end tex
@ifinfo
Y[k] = 2 (sum for j = 0 to n-1 of X[j] cos(pi (j+1/2) (k+1/2) / n)).
@end ifinfo
@html
<center><img src="equation-redft11.png" align="top">.</center>
@end html

@subsubheading Inverses and Normalization

These definitions correspond directly to the unnormalized DFTs used
elsewhere in FFTW (hence the factors of @math{2} in front of the
summations).  The unnormalized inverse of @code{REDFT00} is
@code{REDFT00}, of @code{REDFT10} is @code{REDFT01} and vice versa, and
of @code{REDFT11} is @code{REDFT11}.  Each unnormalized inverse results
in the original array multiplied by @math{N}, where @math{N} is the
@emph{logical} DFT size.  For @code{REDFT00}, @math{N=2(n-1)} (note that
@math{n=1} is not defined); otherwise, @math{N=2n}.
@cindex normalization


In defining the discrete cosine transform, some authors also include
additional factors of
@ifinfo
sqrt(2)
@end ifinfo
@html
&radic;2
@end html
@tex
$\sqrt{2}$
@end tex
(or its inverse) multiplying selected inputs and/or outputs.  This is a
mostly cosmetic change that makes the transform orthogonal, but
sacrifices the direct equivalence to a symmetric DFT.

@c =========>
@node 1d Real-odd DFTs (DSTs), 1d Discrete Hartley Transforms (DHTs), 1d Real-even DFTs (DCTs), What FFTW Really Computes
@subsection 1d Real-odd DFTs (DSTs)

The Real-odd symmetry DFTs in FFTW are exactly equivalent to the unnormalized
forward (and backward) DFTs as defined above, where the input array
@math{X} of length @math{N} is purely real and is also @dfn{odd} symmetry.  In
this case, the output is odd symmetry and purely imaginary.
@cindex real-odd DFT
@cindex RODFT


@ctindex RODFT00
For the case of @code{RODFT00}, this odd symmetry means that
@tex
$X_j = -X_{N-j}$,
@end tex
@ifinfo
X[j] = -X[N-j],
@end ifinfo
@html
<i>X<sub>j</sub> = -X<sub>N-j</sub></i>,
@end html
where we take @math{X} to be periodic so that
@tex
$X_N = X_0$.
@end tex
@ifinfo
X[N] = X[0].
@end ifinfo
@html
<i>X<sub>N</sub> = X</i><sub>0</sub>.
@end html
Because of this redundancy, only the first @math{n} real numbers
starting at @math{j=1} are actually stored (the @math{j=0} element is
zero), where @math{N = 2(n+1)}.

The proper definition of odd symmetry for @code{RODFT10},
@code{RODFT01}, and @code{RODFT11} transforms is somewhat more intricate
because of the shifts by @math{1/2} of the input and/or output, although
the corresponding boundary conditions are given in @ref{Real even/odd
DFTs (cosine/sine transforms)}.  Because of the odd symmetry, however,
the cosine terms in the DFT all cancel and the remaining sine terms are
written explicitly below.  This formulation often leads people to call
such a transform a @dfn{discrete sine transform} (DST), although it is
really just a special case of the DFT.
@cindex discrete sine transform
@cindex DST


In each of the definitions below, we transform a real array @math{X} of
length @math{n} to a real array @math{Y} of length @math{n}:

@subsubheading RODFT00 (DST-I)
@ctindex RODFT00
An @code{RODFT00} transform (type-I DST) in FFTW is defined by:
@tex
$$
Y_k = 2 \sum_{j=0}^{n-1} X_j \sin [ \pi (j+1) (k+1) / (n+1)].
$$
@end tex
@ifinfo
Y[k] = 2 (sum for j = 0 to n-1 of X[j] sin(pi (j+1)(k+1) / (n+1))).
@end ifinfo
@html
<center><img src="equation-rodft00.png" align="top">.</center>
@end html

@subsubheading RODFT10 (DST-II)
@ctindex RODFT10
An @code{RODFT10} transform (type-II DST) in FFTW is defined by:
@tex
$$
Y_k = 2 \sum_{j=0}^{n-1} X_j \sin [\pi (j+1/2) (k+1) / n].
$$
@end tex
@ifinfo
Y[k] = 2 (sum for j = 0 to n-1 of X[j] sin(pi (j+1/2) (k+1) / n)).
@end ifinfo
@html
<center><img src="equation-rodft10.png" align="top">.</center>
@end html

@subsubheading RODFT01 (DST-III)
@ctindex RODFT01
An @code{RODFT01} transform (type-III DST) in FFTW is defined by:
@tex
$$
Y_k = (-1)^k X_{n-1} + 2 \sum_{j=0}^{n-2} X_j \sin [\pi (j+1) (k+1/2) / n].
$$
@end tex
@ifinfo
Y[k] = (-1)^k X[n-1] + 2 (sum for j = 0 to n-2 of X[j] sin(pi (j+1) (k+1/2) / n)).
@end ifinfo
@html
<center><img src="equation-rodft01.png" align="top">.</center>
@end html
In the case of @math{n=1}, this reduces to
@tex
$Y_0 = X_0$.
@end tex
@ifinfo
Y[0] = X[0].
@end ifinfo
@html
<i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.
@end html

@subsubheading RODFT11 (DST-IV)
@ctindex RODFT11
An @code{RODFT11} transform (type-IV DST) in FFTW is defined by:
@tex
$$
Y_k = 2 \sum_{j=0}^{n-1} X_j \sin [\pi (j+1/2) (k+1/2) / n].
$$
@end tex
@ifinfo
Y[k] = 2 (sum for j = 0 to n-1 of X[j] sin(pi (j+1/2) (k+1/2) / n)).
@end ifinfo
@html
<center><img src="equation-rodft11.png" align="top">.</center>
@end html

@subsubheading Inverses and Normalization

These definitions correspond directly to the unnormalized DFTs used
elsewhere in FFTW (hence the factors of @math{2} in front of the
summations).  The unnormalized inverse of @code{RODFT00} is
@code{RODFT00}, of @code{RODFT10} is @code{RODFT01} and vice versa, and
of @code{RODFT11} is @code{RODFT11}.  Each unnormalized inverse results
in the original array multiplied by @math{N}, where @math{N} is the
@emph{logical} DFT size.  For @code{RODFT00}, @math{N=2(n+1)};
otherwise, @math{N=2n}.
@cindex normalization


In defining the discrete sine transform, some authors also include
additional factors of
@ifinfo
sqrt(2)
@end ifinfo
@html
&radic;2
@end html
@tex
$\sqrt{2}$
@end tex
(or its inverse) multiplying selected inputs and/or outputs.  This is a
mostly cosmetic change that makes the transform orthogonal, but
sacrifices the direct equivalence to an antisymmetric DFT.

@c =========>
@node 1d Discrete Hartley Transforms (DHTs), Multi-dimensional Transforms, 1d Real-odd DFTs (DSTs), What FFTW Really Computes
@subsection 1d Discrete Hartley Transforms (DHTs)

@cindex discrete Hartley transform
@cindex DHT
The discrete Hartley transform (DHT) of a 1d real array @math{X} of size
@math{n} computes a real array @math{Y} of the same size, where:
@tex
$$
Y_k = \sum_{j = 0}^{n - 1} X_j [ \cos(2\pi j k / n) + \sin(2\pi j k / n)].
$$
@end tex
@ifinfo
@center Y[k] = sum for j = 0 to (n - 1) of X[j] * [cos(2 pi j k / n) + sin(2 pi j k / n)].
@end ifinfo
@html
<center><img src="equation-dht.png" align="top">.</center>
@end html

@cindex normalization
FFTW computes an unnormalized transform, in that there is no coefficient
in front of the summation in the DHT.  In other words, applying the
transform twice (the DHT is its own inverse) will multiply the input by
@math{n}.

@c =========>
@node Multi-dimensional Transforms,  , 1d Discrete Hartley Transforms (DHTs), What FFTW Really Computes
@subsection Multi-dimensional Transforms

The multi-dimensional transforms of FFTW, in general, compute simply the
separable product of the given 1d transform along each dimension of the
array.  Since each of these transforms is unnormalized, computing the
forward followed by the backward/inverse multi-dimensional transform
will result in the original array scaled by the product of the
normalization factors for each dimension (e.g. the product of the
dimension sizes, for a multi-dimensional DFT).

@tex
As an explicit example, consider the following exact mathematical
definition of our multi-dimensional DFT.  Let $X$ be a $d$-dimensional
complex array whose elements are $X[j_1, j_2, \ldots, j_d]$, where $0
\leq j_s < n_s$ for all~$s \in \{ 1, 2, \ldots, d \}$.  Let also
$\omega_s = e^{2\pi \sqrt{-1}/n_s}$, for all ~$s \in \{ 1, 2, \ldots, d
\}$.

The forward transform computes a complex array~$Y$, whose
structure is the same as that of~$X$, defined by

$$
Y[k_1, k_2, \ldots, k_d] =
    \sum_{j_1 = 0}^{n_1 - 1}
        \sum_{j_2 = 0}^{n_2 - 1}
           \cdots
              \sum_{j_d = 0}^{n_d - 1}
                  X[j_1, j_2, \ldots, j_d] 
                      \omega_1^{-j_1 k_1}
                      \omega_2^{-j_2 k_2}
                      \cdots
                      \omega_d^{-j_d k_d} \ .
$$

The backward transform computes
$$
Y[k_1, k_2, \ldots, k_d] =
    \sum_{j_1 = 0}^{n_1 - 1}
        \sum_{j_2 = 0}^{n_2 - 1}
           \cdots
              \sum_{j_d = 0}^{n_d - 1}
                  X[j_1, j_2, \ldots, j_d] 
                      \omega_1^{j_1 k_1}
                      \omega_2^{j_2 k_2}
                      \cdots
                      \omega_d^{j_d k_d} \ .
$$

Computing the forward transform followed by the backward transform
will multiply the array by $\prod_{s=1}^{d} n_d$.
@end tex

@cindex r2c
The definition of FFTW's multi-dimensional DFT of real data (r2c)
deserves special attention.  In this case, we logically compute the full
multi-dimensional DFT of the input data; since the input data are purely
real, the output data have the Hermitian symmetry and therefore only one
non-redundant half need be stored.  More specifically, for an @ndims multi-dimensional real-input DFT, the full (logical) complex output array
@tex
$Y[k_0, k_1, \ldots, k_{d-1}]$
@end tex
@html
<i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
<i>k</i><sub><i>d-1</i></sub>]
@end html
@ifinfo
Y[k[0], k[1], ..., k[d-1]]
@end ifinfo
has the symmetry:
@tex
$$
Y[k_0, k_1, \ldots, k_{d-1}] = Y[n_0 - k_0, n_1 - k_1, \ldots, n_{d-1} - k_{d-1}]^*
$$
@end tex
@html
<i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
<i>k</i><sub><i>d-1</i></sub>] = <i>Y</i>[<i>n</i><sub>0</sub> -
<i>k</i><sub>0</sub>, <i>n</i><sub>1</sub> - <i>k</i><sub>1</sub>, ...,
<i>n</i><sub><i>d-1</i></sub> - <i>k</i><sub><i>d-1</i></sub>]<sup>*</sup>
@end html
@ifinfo
Y[k[0], k[1], ..., k[d-1]] = Y[n[0] - k[0], n[1] - k[1], ..., n[d-1] - k[d-1]]*
@end ifinfo
(where each dimension is periodic).  Because of this symmetry, we only
store the
@tex
$k_{d-1} = 0 \cdots n_{d-1}/2$
@end tex
@html
<i>k</i><sub><i>d-1</i></sub> = 0...<i>n</i><sub><i>d-1</i></sub>/2+1
@end html
@ifinfo
k[d-1] = 0...n[d-1]/2
@end ifinfo
elements of the @emph{last} dimension (division by @math{2} is rounded
down).  (We could instead have cut any other dimension in half, but the
last dimension proved computationally convenient.)  This results in the
peculiar array format described in more detail by @ref{Real-data DFT
Array Format}.

The multi-dimensional c2r transform is simply the unnormalized inverse
of the r2c transform.  i.e. it is the same as FFTW's complex backward
multi-dimensional DFT, operating on a Hermitian input array in the
peculiar format mentioned above and outputting a real array (since the
DFT output is purely real).

We should remind the user that the separable product of 1d transforms
along each dimension, as computed by FFTW, is not always the same thing
as the usual multi-dimensional transform.  A multi-dimensional
@code{R2HC} (or @code{HC2R}) transform is not identical to the
multi-dimensional DFT, requiring some post-processing to combine the
requisite real and imaginary parts, as was described in @ref{The
Halfcomplex-format DFT}.  Likewise, FFTW's multidimensional
@code{FFTW_DHT} r2r transform is not the same thing as the logical
multi-dimensional discrete Hartley transform defined in the literature,
as discussed in @ref{The Discrete Hartley Transform}.

