Chris@10: @node Other Important Topics, FFTW Reference, Tutorial, Top
Chris@10: @chapter Other Important Topics
Chris@10: @menu
Chris@10: * SIMD alignment and fftw_malloc::  
Chris@10: * Multi-dimensional Array Format::  
Chris@10: * Words of Wisdom-Saving Plans::  
Chris@10: * Caveats in Using Wisdom::     
Chris@10: @end menu
Chris@10: 
Chris@10: @c ------------------------------------------------------------
Chris@10: @node SIMD alignment and fftw_malloc, Multi-dimensional Array Format, Other Important Topics, Other Important Topics
Chris@10: @section SIMD alignment and fftw_malloc
Chris@10: 
Chris@10: SIMD, which stands for ``Single Instruction Multiple Data,'' is a set of
Chris@10: special operations supported by some processors to perform a single
Chris@10: operation on several numbers (usually 2 or 4) simultaneously.  SIMD
Chris@10: floating-point instructions are available on several popular CPUs:
Chris@10: SSE/SSE2/AVX on recent x86/x86-64 processors, AltiVec (single precision)
Chris@10: on some PowerPCs (Apple G4 and higher), NEON on some ARM models, and MIPS Paired Single
Chris@10: (currently only in FFTW 3.2.x).  FFTW can be compiled to support the
Chris@10: SIMD instructions on any of these systems.
Chris@10: @cindex SIMD
Chris@10: @cindex SSE
Chris@10: @cindex SSE2
Chris@10: @cindex AVX
Chris@10: @cindex AltiVec
Chris@10: @cindex MIPS PS
Chris@10: @cindex precision
Chris@10: 
Chris@10: 
Chris@10: A program linking to an FFTW library compiled with SIMD support can
Chris@10: obtain a nonnegligible speedup for most complex and r2c/c2r
Chris@10: transforms.  In order to obtain this speedup, however, the arrays of
Chris@10: complex (or real) data passed to FFTW must be specially aligned in
Chris@10: memory (typically 16-byte aligned), and often this alignment is more
Chris@10: stringent than that provided by the usual @code{malloc} (etc.)
Chris@10: allocation routines.
Chris@10: 
Chris@10: @cindex portability
Chris@10: In order to guarantee proper alignment for SIMD, therefore, in case
Chris@10: your program is ever linked against a SIMD-using FFTW, we recommend
Chris@10: allocating your transform data with @code{fftw_malloc} and
Chris@10: de-allocating it with @code{fftw_free}.
Chris@10: @findex fftw_malloc
Chris@10: @findex fftw_free
Chris@10: These have exactly the same interface and behavior as
Chris@10: @code{malloc}/@code{free}, except that for a SIMD FFTW they ensure
Chris@10: that the returned pointer has the necessary alignment (by calling
Chris@10: @code{memalign} or its equivalent on your OS).
Chris@10: 
Chris@10: You are not @emph{required} to use @code{fftw_malloc}.  You can
Chris@10: allocate your data in any way that you like, from @code{malloc} to
Chris@10: @code{new} (in C++) to a fixed-size array declaration.  If the array
Chris@10: happens not to be properly aligned, FFTW will not use the SIMD
Chris@10: extensions.
Chris@10: @cindex C++
Chris@10: 
Chris@10: @findex fftw_alloc_real
Chris@10: @findex fftw_alloc_complex
Chris@10: Since @code{fftw_malloc} only ever needs to be used for real and
Chris@10: complex arrays, we provide two convenient wrapper routines
Chris@10: @code{fftw_alloc_real(N)} and @code{fftw_alloc_complex(N)} that are
Chris@10: equivalent to @code{(double*)fftw_malloc(sizeof(double) * N)} and
Chris@10: @code{(fftw_complex*)fftw_malloc(sizeof(fftw_complex) * N)},
Chris@10: respectively (or their equivalents in other precisions).
Chris@10: 
Chris@10: @c ------------------------------------------------------------
Chris@10: @node Multi-dimensional Array Format, Words of Wisdom-Saving Plans, SIMD alignment and fftw_malloc, Other Important Topics
Chris@10: @section Multi-dimensional Array Format
Chris@10: 
Chris@10: This section describes the format in which multi-dimensional arrays
Chris@10: are stored in FFTW.  We felt that a detailed discussion of this topic
Chris@10: was necessary.  Since several different formats are common, this topic
Chris@10: is often a source of confusion.
Chris@10: 
Chris@10: @menu
Chris@10: * Row-major Format::            
Chris@10: * Column-major Format::         
Chris@10: * Fixed-size Arrays in C::      
Chris@10: * Dynamic Arrays in C::         
Chris@10: * Dynamic Arrays in C-The Wrong Way::  
Chris@10: @end menu
Chris@10: 
Chris@10: @c =========>
Chris@10: @node Row-major Format, Column-major Format, Multi-dimensional Array Format, Multi-dimensional Array Format
Chris@10: @subsection Row-major Format
Chris@10: @cindex row-major
Chris@10: 
Chris@10: The multi-dimensional arrays passed to @code{fftw_plan_dft} etcetera
Chris@10: are expected to be stored as a single contiguous block in
Chris@10: @dfn{row-major} order (sometimes called ``C order'').  Basically, this
Chris@10: means that as you step through adjacent memory locations, the first
Chris@10: dimension's index varies most slowly and the last dimension's index
Chris@10: varies most quickly.
Chris@10: 
Chris@10: To be more explicit, let us consider an array of rank @math{d} whose
Chris@10: dimensions are @ndims{}. Now, we specify a location in the array by a
Chris@10: sequence of @math{d} (zero-based) indices, one for each dimension:
Chris@10: @tex
Chris@10: $(i_0, i_1, i_2, \ldots, i_{d-1})$.
Chris@10: @end tex
Chris@10: @ifinfo
Chris@10: (i[0], i[1], ..., i[d-1]).
Chris@10: @end ifinfo
Chris@10: @html
Chris@10: (i<sub>0</sub>, i<sub>1</sub>, i<sub>2</sub>,..., i<sub>d-1</sub>).
Chris@10: @end html
Chris@10: If the array is stored in row-major
Chris@10: order, then this element is located at the position
Chris@10: @tex
Chris@10: $i_{d-1} + n_{d-1} (i_{d-2} + n_{d-2} (\ldots + n_1 i_0))$.
Chris@10: @end tex
Chris@10: @ifinfo
Chris@10: i[d-1] + n[d-1] * (i[d-2] + n[d-2] * (... + n[1] * i[0])).
Chris@10: @end ifinfo
Chris@10: @html
Chris@10: i<sub>d-1</sub> + n<sub>d-1</sub> * (i<sub>d-2</sub> + n<sub>d-2</sub> * (... + n<sub>1</sub> * i<sub>0</sub>)).
Chris@10: @end html
Chris@10: 
Chris@10: Note that, for the ordinary complex DFT, each element of the array
Chris@10: must be of type @code{fftw_complex}; i.e. a (real, imaginary) pair of
Chris@10: (double-precision) numbers. 
Chris@10: 
Chris@10: In the advanced FFTW interface, the physical dimensions @math{n} from
Chris@10: which the indices are computed can be different from (larger than)
Chris@10: the logical dimensions of the transform to be computed, in order to
Chris@10: transform a subset of a larger array.
Chris@10: @cindex advanced interface
Chris@10: Note also that, in the advanced interface, the expression above is
Chris@10: multiplied by a @dfn{stride} to get the actual array index---this is
Chris@10: useful in situations where each element of the multi-dimensional array
Chris@10: is actually a data structure (or another array), and you just want to
Chris@10: transform a single field. In the basic interface, however, the stride
Chris@10: is 1.
Chris@10: @cindex stride
Chris@10: 
Chris@10: @c =========>
Chris@10: @node Column-major Format, Fixed-size Arrays in C, Row-major Format, Multi-dimensional Array Format
Chris@10: @subsection Column-major Format
Chris@10: @cindex column-major
Chris@10: 
Chris@10: Readers from the Fortran world are used to arrays stored in
Chris@10: @dfn{column-major} order (sometimes called ``Fortran order'').  This is
Chris@10: essentially the exact opposite of row-major order in that, here, the
Chris@10: @emph{first} dimension's index varies most quickly.
Chris@10: 
Chris@10: If you have an array stored in column-major order and wish to
Chris@10: transform it using FFTW, it is quite easy to do.  When creating the
Chris@10: plan, simply pass the dimensions of the array to the planner in
Chris@10: @emph{reverse order}.  For example, if your array is a rank three
Chris@10: @code{N x M x L} matrix in column-major order, you should pass the
Chris@10: dimensions of the array as if it were an @code{L x M x N} matrix
Chris@10: (which it is, from the perspective of FFTW).  This is done for you
Chris@10: @emph{automatically} by the FFTW legacy-Fortran interface
Chris@10: (@pxref{Calling FFTW from Legacy Fortran}), but you must do it
Chris@10: manually with the modern Fortran interface (@pxref{Reversing array
Chris@10: dimensions}).
Chris@10: @cindex Fortran interface
Chris@10: 
Chris@10: @c =========>
Chris@10: @node Fixed-size Arrays in C, Dynamic Arrays in C, Column-major Format, Multi-dimensional Array Format
Chris@10: @subsection Fixed-size Arrays in C
Chris@10: @cindex C multi-dimensional arrays
Chris@10: 
Chris@10: A multi-dimensional array whose size is declared at compile time in C
Chris@10: is @emph{already} in row-major order.  You don't have to do anything
Chris@10: special to transform it.  For example:
Chris@10: 
Chris@10: @example
Chris@10: @{
Chris@10:      fftw_complex data[N0][N1][N2];
Chris@10:      fftw_plan plan;
Chris@10:      ...
Chris@10:      plan = fftw_plan_dft_3d(N0, N1, N2, &data[0][0][0], &data[0][0][0],
Chris@10:                              FFTW_FORWARD, FFTW_ESTIMATE);
Chris@10:      ...
Chris@10: @}
Chris@10: @end example
Chris@10: 
Chris@10: This will plan a 3d in-place transform of size @code{N0 x N1 x N2}.
Chris@10: Notice how we took the address of the zero-th element to pass to the
Chris@10: planner (we could also have used a typecast).
Chris@10: 
Chris@10: However, we tend to @emph{discourage} users from declaring their
Chris@10: arrays in this way, for two reasons.  First, this allocates the array
Chris@10: on the stack (``automatic'' storage), which has a very limited size on
Chris@10: most operating systems (declaring an array with more than a few
Chris@10: thousand elements will often cause a crash).  (You can get around this
Chris@10: limitation on many systems by declaring the array as
Chris@10: @code{static} and/or global, but that has its own drawbacks.)
Chris@10: Second, it may not optimally align the array for use with a SIMD
Chris@10: FFTW (@pxref{SIMD alignment and fftw_malloc}).  Instead, we recommend
Chris@10: using @code{fftw_malloc}, as described below.
Chris@10: 
Chris@10: @c =========>
Chris@10: @node Dynamic Arrays in C, Dynamic Arrays in C-The Wrong Way, Fixed-size Arrays in C, Multi-dimensional Array Format
Chris@10: @subsection Dynamic Arrays in C
Chris@10: 
Chris@10: We recommend allocating most arrays dynamically, with
Chris@10: @code{fftw_malloc}.  This isn't too hard to do, although it is not as
Chris@10: straightforward for multi-dimensional arrays as it is for
Chris@10: one-dimensional arrays.
Chris@10: 
Chris@10: Creating the array is simple: using a dynamic-allocation routine like
Chris@10: @code{fftw_malloc}, allocate an array big enough to store N
Chris@10: @code{fftw_complex} values (for a complex DFT), where N is the product
Chris@10: of the sizes of the array dimensions (i.e. the total number of complex
Chris@10: values in the array).  For example, here is code to allocate a
Chris@10: @threedims{5,12,27} rank-3 array:
Chris@10: @findex fftw_malloc
Chris@10: 
Chris@10: @example
Chris@10: fftw_complex *an_array;
Chris@10: an_array = (fftw_complex*) fftw_malloc(5*12*27 * sizeof(fftw_complex));
Chris@10: @end example
Chris@10: 
Chris@10: Accessing the array elements, however, is more tricky---you can't
Chris@10: simply use multiple applications of the @samp{[]} operator like you
Chris@10: could for fixed-size arrays.  Instead, you have to explicitly compute
Chris@10: the offset into the array using the formula given earlier for
Chris@10: row-major arrays.  For example, to reference the @math{(i,j,k)}-th
Chris@10: element of the array allocated above, you would use the expression
Chris@10: @code{an_array[k + 27 * (j + 12 * i)]}.
Chris@10: 
Chris@10: This pain can be alleviated somewhat by defining appropriate macros,
Chris@10: or, in C++, creating a class and overloading the @samp{()} operator.
Chris@10: The recent C99 standard provides a way to reinterpret the dynamic
Chris@10: array as a ``variable-length'' multi-dimensional array amenable to
Chris@10: @samp{[]}, but this feature is not yet widely supported by compilers.
Chris@10: @cindex C99
Chris@10: @cindex C++
Chris@10: 
Chris@10: @c =========>
Chris@10: @node Dynamic Arrays in C-The Wrong Way,  , Dynamic Arrays in C, Multi-dimensional Array Format
Chris@10: @subsection Dynamic Arrays in C---The Wrong Way
Chris@10: 
Chris@10: A different method for allocating multi-dimensional arrays in C is
Chris@10: often suggested that is incompatible with FFTW: @emph{using it will
Chris@10: cause FFTW to die a painful death}.  We discuss the technique here,
Chris@10: however, because it is so commonly known and used.  This method is to
Chris@10: create arrays of pointers of arrays of pointers of @dots{}etcetera.
Chris@10: For example, the analogue in this method to the example above is:
Chris@10: 
Chris@10: @example
Chris@10: int i,j;
Chris@10: fftw_complex ***a_bad_array;  /* @r{another way to make a 5x12x27 array} */
Chris@10: 
Chris@10: a_bad_array = (fftw_complex ***) malloc(5 * sizeof(fftw_complex **));
Chris@10: for (i = 0; i < 5; ++i) @{
Chris@10:      a_bad_array[i] = 
Chris@10:         (fftw_complex **) malloc(12 * sizeof(fftw_complex *));
Chris@10:      for (j = 0; j < 12; ++j)
Chris@10:           a_bad_array[i][j] =
Chris@10:                 (fftw_complex *) malloc(27 * sizeof(fftw_complex));
Chris@10: @}
Chris@10: @end example
Chris@10: 
Chris@10: As you can see, this sort of array is inconvenient to allocate (and
Chris@10: deallocate).  On the other hand, it has the advantage that the
Chris@10: @math{(i,j,k)}-th element can be referenced simply by
Chris@10: @code{a_bad_array[i][j][k]}.
Chris@10: 
Chris@10: If you like this technique and want to maximize convenience in accessing
Chris@10: the array, but still want to pass the array to FFTW, you can use a
Chris@10: hybrid method.  Allocate the array as one contiguous block, but also
Chris@10: declare an array of arrays of pointers that point to appropriate places
Chris@10: in the block.  That sort of trick is beyond the scope of this
Chris@10: documentation; for more information on multi-dimensional arrays in C,
Chris@10: see the @code{comp.lang.c}
Chris@10: @uref{http://c-faq.com/aryptr/dynmuldimary.html, FAQ}.
Chris@10: 
Chris@10: @c ------------------------------------------------------------
Chris@10: @node Words of Wisdom-Saving Plans, Caveats in Using Wisdom, Multi-dimensional Array Format, Other Important Topics
Chris@10: @section Words of Wisdom---Saving Plans
Chris@10: @cindex wisdom
Chris@10: @cindex saving plans to disk
Chris@10: 
Chris@10: FFTW implements a method for saving plans to disk and restoring them.
Chris@10: In fact, what FFTW does is more general than just saving and loading
Chris@10: plans.  The mechanism is called @dfn{wisdom}.  Here, we describe
Chris@10: this feature at a high level. @xref{FFTW Reference}, for a less casual
Chris@10: but more complete discussion of how to use wisdom in FFTW.
Chris@10: 
Chris@10: Plans created with the @code{FFTW_MEASURE}, @code{FFTW_PATIENT}, or
Chris@10: @code{FFTW_EXHAUSTIVE} options produce near-optimal FFT performance,
Chris@10: but may require a long time to compute because FFTW must measure the
Chris@10: runtime of many possible plans and select the best one.  This setup is
Chris@10: designed for the situations where so many transforms of the same size
Chris@10: must be computed that the start-up time is irrelevant.  For short
Chris@10: initialization times, but slower transforms, we have provided
Chris@10: @code{FFTW_ESTIMATE}.  The @code{wisdom} mechanism is a way to get the
Chris@10: best of both worlds: you compute a good plan once, save it to
Chris@10: disk, and later reload it as many times as necessary.  The wisdom
Chris@10: mechanism can actually save and reload many plans at once, not just
Chris@10: one.
Chris@10: @ctindex FFTW_MEASURE
Chris@10: @ctindex FFTW_PATIENT
Chris@10: @ctindex FFTW_EXHAUSTIVE
Chris@10: @ctindex FFTW_ESTIMATE
Chris@10: 
Chris@10: 
Chris@10: Whenever you create a plan, the FFTW planner accumulates wisdom, which
Chris@10: is information sufficient to reconstruct the plan.  After planning,
Chris@10: you can save this information to disk by means of the function:
Chris@10: @example
Chris@10: int fftw_export_wisdom_to_filename(const char *filename);
Chris@10: @end example
Chris@10: @findex fftw_export_wisdom_to_filename
Chris@10: (This function returns non-zero on success.)
Chris@10: 
Chris@10: The next time you run the program, you can restore the wisdom with
Chris@10: @code{fftw_import_wisdom_from_filename} (which also returns non-zero on success),
Chris@10: and then recreate the plan using the same flags as before.
Chris@10: @example
Chris@10: int fftw_import_wisdom_from_filename(const char *filename);
Chris@10: @end example
Chris@10: @findex fftw_import_wisdom_from_filename
Chris@10: 
Chris@10: Wisdom is automatically used for any size to which it is applicable, as
Chris@10: long as the planner flags are not more ``patient'' than those with which
Chris@10: the wisdom was created.  For example, wisdom created with
Chris@10: @code{FFTW_MEASURE} can be used if you later plan with
Chris@10: @code{FFTW_ESTIMATE} or @code{FFTW_MEASURE}, but not with
Chris@10: @code{FFTW_PATIENT}.
Chris@10: 
Chris@10: The @code{wisdom} is cumulative, and is stored in a global, private
Chris@10: data structure managed internally by FFTW.  The storage space required
Chris@10: is minimal, proportional to the logarithm of the sizes the wisdom was
Chris@10: generated from.  If memory usage is a concern, however, the wisdom can
Chris@10: be forgotten and its associated memory freed by calling:
Chris@10: @example
Chris@10: void fftw_forget_wisdom(void);
Chris@10: @end example
Chris@10: @findex fftw_forget_wisdom
Chris@10: 
Chris@10: Wisdom can be exported to a file, a string, or any other medium.
Chris@10: For details, see @ref{Wisdom}.
Chris@10: 
Chris@10: @node Caveats in Using Wisdom,  , Words of Wisdom-Saving Plans, Other Important Topics
Chris@10: @section Caveats in Using Wisdom
Chris@10: @cindex wisdom, problems with
Chris@10: 
Chris@10: @quotation
Chris@10: @html
Chris@10: <i>
Chris@10: @end html
Chris@10: For in much wisdom is much grief, and he that increaseth knowledge
Chris@10: increaseth sorrow.
Chris@10: @html
Chris@10: </i>
Chris@10: @end html
Chris@10: [Ecclesiastes 1:18]
Chris@10: @cindex Ecclesiastes
Chris@10: @end quotation
Chris@10: @iftex
Chris@10: @medskip
Chris@10: @end iftex
Chris@10: 
Chris@10: @cindex portability
Chris@10: There are pitfalls to using wisdom, in that it can negate FFTW's
Chris@10: ability to adapt to changing hardware and other conditions. For
Chris@10: example, it would be perfectly possible to export wisdom from a
Chris@10: program running on one processor and import it into a program running
Chris@10: on another processor.  Doing so, however, would mean that the second
Chris@10: program would use plans optimized for the first processor, instead of
Chris@10: the one it is running on.
Chris@10: 
Chris@10: It should be safe to reuse wisdom as long as the hardware and program
Chris@10: binaries remain unchanged. (Actually, the optimal plan may change even
Chris@10: between runs of the same binary on identical hardware, due to
Chris@10: differences in the virtual memory environment, etcetera.  Users
Chris@10: seriously interested in performance should worry about this problem,
Chris@10: too.)  It is likely that, if the same wisdom is used for two
Chris@10: different program binaries, even running on the same machine, the
Chris@10: plans may be sub-optimal because of differing code alignments.  It is
Chris@10: therefore wise to recreate wisdom every time an application is
Chris@10: recompiled.  The more the underlying hardware and software changes
Chris@10: between the creation of wisdom and its use, the greater grows
Chris@10: the risk of sub-optimal plans.
Chris@10: 
Chris@10: Nevertheless, if the choice is between using @code{FFTW_ESTIMATE} or
Chris@10: using possibly-suboptimal wisdom (created on the same machine, but for a
Chris@10: different binary), the wisdom is likely to be better.  For this reason,
Chris@10: we provide a function to import wisdom from a standard system-wide
Chris@10: location (@code{/etc/fftw/wisdom} on Unix):
Chris@10: @cindex wisdom, system-wide
Chris@10: 
Chris@10: @example
Chris@10: int fftw_import_system_wisdom(void);
Chris@10: @end example
Chris@10: @findex fftw_import_system_wisdom
Chris@10: 
Chris@10: FFTW also provides a standalone program, @code{fftw-wisdom} (described
Chris@10: by its own @code{man} page on Unix) with which users can create wisdom,
Chris@10: e.g. for a canonical set of sizes to store in the system wisdom file.
Chris@10: @xref{Wisdom Utilities}.
Chris@10: @cindex fftw-wisdom utility
Chris@10: