Chris@42: @node Introduction, Tutorial, Top, Top Chris@42: @chapter Introduction Chris@42: This manual documents version @value{VERSION} of FFTW, the Chris@42: @emph{Fastest Fourier Transform in the West}. FFTW is a comprehensive Chris@42: collection of fast C routines for computing the discrete Fourier Chris@42: transform (DFT) and various special cases thereof. Chris@42: @cindex discrete Fourier transform Chris@42: @cindex DFT Chris@42: @itemize @bullet Chris@42: @item FFTW computes the DFT of complex data, real data, even- Chris@42: or odd-symmetric real data (these symmetric transforms are usually Chris@42: known as the discrete cosine or sine transform, respectively), and the Chris@42: discrete Hartley transform (DHT) of real data. Chris@42: Chris@42: @item The input data can have arbitrary length. Chris@42: FFTW employs @Onlogn{} algorithms for all lengths, including Chris@42: prime numbers. Chris@42: Chris@42: @item FFTW supports arbitrary multi-dimensional data. Chris@42: Chris@42: @item FFTW supports the SSE, SSE2, AVX, AVX2, AVX512, KCVI, Altivec, VSX, and Chris@42: NEON vector instruction sets. Chris@42: Chris@42: @item FFTW includes parallel (multi-threaded) transforms Chris@42: for shared-memory systems. Chris@42: @item Starting with version 3.3, FFTW includes distributed-memory parallel Chris@42: transforms using MPI. Chris@42: @end itemize Chris@42: Chris@42: We assume herein that you are familiar with the properties and uses of Chris@42: the DFT that are relevant to your application. Otherwise, see Chris@42: e.g. @cite{The Fast Fourier Transform and Its Applications} by E. O. Brigham Chris@42: (Prentice-Hall, Englewood Cliffs, NJ, 1988). Chris@42: @uref{http://www.fftw.org, Our web page} also has links to FFT-related Chris@42: information online. Chris@42: @cindex FFTW Chris@42: Chris@42: @c TODO: revise. We don't need to brag any longer Chris@42: @c Chris@42: @c FFTW is usually faster (and sometimes much faster) than all other Chris@42: @c freely-available Fourier transform programs found on the Net. It is Chris@42: @c competitive with (and often faster than) the FFT codes in Sun's Chris@42: @c Performance Library, IBM's ESSL library, HP's CXML library, and Chris@42: @c Intel's MKL library, which are targeted at specific machines. Chris@42: @c Moreover, FFTW's performance is @emph{portable}. Indeed, FFTW is Chris@42: @c unique in that it automatically adapts itself to your machine, your Chris@42: @c cache, the size of your memory, your number of registers, and all the Chris@42: @c other factors that normally make it impossible to optimize a program Chris@42: @c for more than one machine. An extensive comparison of FFTW's Chris@42: @c performance with that of other Fourier transform codes has been made, Chris@42: @c and the results are available on the Web at Chris@42: @c @uref{http://fftw.org/benchfft, the benchFFT home page}. Chris@42: @c @cindex benchmark Chris@42: @c @fpindex benchfft Chris@42: Chris@42: In order to use FFTW effectively, you need to learn one basic concept Chris@42: of FFTW's internal structure: FFTW does not use a fixed algorithm for Chris@42: computing the transform, but instead it adapts the DFT algorithm to Chris@42: details of the underlying hardware in order to maximize performance. Chris@42: Hence, the computation of the transform is split into two phases. Chris@42: First, FFTW's @dfn{planner} ``learns'' the fastest way to compute the Chris@42: transform on your machine. The planner Chris@42: @cindex planner Chris@42: produces a data structure called a @dfn{plan} that contains this Chris@42: @cindex plan Chris@42: information. Subsequently, the plan is @dfn{executed} Chris@42: @cindex execute Chris@42: to transform the array of input data as dictated by the plan. The Chris@42: plan can be reused as many times as needed. In typical Chris@42: high-performance applications, many transforms of the same size are Chris@42: computed and, consequently, a relatively expensive initialization of Chris@42: this sort is acceptable. On the other hand, if you need a single Chris@42: transform of a given size, the one-time cost of the planner becomes Chris@42: significant. For this case, FFTW provides fast planners based on Chris@42: heuristics or on previously computed plans. Chris@42: Chris@42: FFTW supports transforms of data with arbitrary length, rank, Chris@42: multiplicity, and a general memory layout. In simple cases, however, Chris@42: this generality may be unnecessary and confusing. Consequently, we Chris@42: organized the interface to FFTW into three levels of increasing Chris@42: generality. Chris@42: @itemize @bullet Chris@42: @item The @dfn{basic interface} computes a single Chris@42: transform of contiguous data. Chris@42: @item The @dfn{advanced interface} computes transforms Chris@42: of multiple or strided arrays. Chris@42: @item The @dfn{guru interface} supports the most general data Chris@42: layouts, multiplicities, and strides. Chris@42: @end itemize Chris@42: We expect that most users will be best served by the basic interface, Chris@42: whereas the guru interface requires careful attention to the Chris@42: documentation to avoid problems. Chris@42: @cindex basic interface Chris@42: @cindex advanced interface Chris@42: @cindex guru interface Chris@42: Chris@42: Chris@42: Besides the automatic performance adaptation performed by the planner, Chris@42: it is also possible for advanced users to customize FFTW manually. For Chris@42: example, if code space is a concern, we provide a tool that links only Chris@42: the subset of FFTW needed by your application. Conversely, you may need Chris@42: to extend FFTW because the standard distribution is not sufficient for Chris@42: your needs. For example, the standard FFTW distribution works most Chris@42: efficiently for arrays whose size can be factored into small primes Chris@42: (@math{2}, @math{3}, @math{5}, and @math{7}), and otherwise it uses a Chris@42: slower general-purpose routine. If you need efficient transforms of Chris@42: other sizes, you can use FFTW's code generator, which produces fast C Chris@42: programs (``codelets'') for any particular array size you may care Chris@42: about. Chris@42: @cindex code generator Chris@42: @cindex codelet Chris@42: For example, if you need transforms of size Chris@42: @ifinfo Chris@42: @math{513 = 19 x 3^3}, Chris@42: @end ifinfo Chris@42: @tex Chris@42: $513 = 19 \cdot 3^3$, Chris@42: @end tex Chris@42: @html Chris@42: 513 = 19*33, Chris@42: @end html Chris@42: you can customize FFTW to support the factor @math{19} efficiently. Chris@42: Chris@42: For more information regarding FFTW, see the paper, ``The Design and Chris@42: Implementation of FFTW3,'' by M. Frigo and S. G. Johnson, which was an Chris@42: invited paper in @cite{Proc. IEEE} @b{93} (2), p. 216 (2005). The Chris@42: code generator is described in the paper ``A fast Fourier transform Chris@42: compiler'', Chris@42: @cindex compiler Chris@42: by M. Frigo, in the @cite{Proceedings of the 1999 ACM SIGPLAN Conference Chris@42: on Programming Language Design and Implementation (PLDI), Atlanta, Chris@42: Georgia, May 1999}. These papers, along with the latest version of Chris@42: FFTW, the FAQ, benchmarks, and other links, are available at Chris@42: @uref{http://www.fftw.org, the FFTW home page}. Chris@42: Chris@42: The current version of FFTW incorporates many good ideas from the past Chris@42: thirty years of FFT literature. In one way or another, FFTW uses the Chris@42: Cooley-Tukey algorithm, the prime factor algorithm, Rader's algorithm Chris@42: for prime sizes, and a split-radix algorithm (with a Chris@42: ``conjugate-pair'' variation pointed out to us by Dan Bernstein). Chris@42: FFTW's code generator also produces new algorithms that we do not Chris@42: completely understand. Chris@42: @cindex algorithm Chris@42: The reader is referred to the cited papers for the appropriate Chris@42: references. Chris@42: Chris@42: The rest of this manual is organized as follows. We first discuss the Chris@42: sequential (single-processor) implementation. We start by describing Chris@42: the basic interface/features of FFTW in @ref{Tutorial}. Chris@42: Next, @ref{Other Important Topics} discusses data alignment Chris@42: (@pxref{SIMD alignment and fftw_malloc}), Chris@42: the storage scheme of multi-dimensional arrays Chris@42: (@pxref{Multi-dimensional Array Format}), and FFTW's mechanism for Chris@42: storing plans on disk (@pxref{Words of Wisdom-Saving Plans}). Next, Chris@42: @ref{FFTW Reference} provides comprehensive documentation of all Chris@42: FFTW's features. Parallel transforms are discussed in their own Chris@42: chapters: @ref{Multi-threaded FFTW} and @ref{Distributed-memory FFTW Chris@42: with MPI}. Fortran programmers can also use FFTW, as described in Chris@42: @ref{Calling FFTW from Legacy Fortran} and @ref{Calling FFTW from Chris@42: Modern Fortran}. @ref{Installation and Customization} explains how to Chris@42: install FFTW in your computer system and how to adapt FFTW to your Chris@42: needs. License and copyright information is given in @ref{License and Chris@42: Copyright}. Finally, we thank all the people who helped us in Chris@42: @ref{Acknowledgments}. Chris@42: