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<!-- This manual is for FFTW
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(version 3.3.10, 10 December 2020).
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Copyright (C) 2003 Matteo Frigo.
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Copyright (C) 2003 Massachusetts Institute of Technology.
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Permission is granted to make and distribute verbatim copies of this
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manual provided the copyright notice and this permission notice are
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<head>
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<title>Introduction (FFTW 3.3.10)</title>
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<link href="Concept-Index.html" rel="index" title="Concept Index">
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<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
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<span id="Introduction"></span><div class="header">
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<p>
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Next: <a href="Tutorial.html" accesskey="n" rel="next">Tutorial</a>, Previous: <a href="index.html" accesskey="p" rel="prev">Top</a>, Up: <a href="index.html" accesskey="u" rel="up">Top</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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</div>
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<hr>
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<span id="Introduction-1"></span><h2 class="chapter">1 Introduction</h2>
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<p>This manual documents version 3.3.10 of FFTW, the
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<em>Fastest Fourier Transform in the West</em>. FFTW is a comprehensive
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collection of fast C routines for computing the discrete Fourier
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transform (DFT) and various special cases thereof.
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<span id="index-discrete-Fourier-transform"></span>
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<span id="index-DFT"></span>
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</p><ul>
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<li> FFTW computes the DFT of complex data, real data, even-
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or odd-symmetric real data (these symmetric transforms are usually
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known as the discrete cosine or sine transform, respectively), and the
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discrete Hartley transform (DHT) of real data.
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</li><li> The input data can have arbitrary length.
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FFTW employs <i>O</i>(<i>n</i> log <i>n</i>)
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algorithms for all lengths, including
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prime numbers.
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</li><li> FFTW supports arbitrary multi-dimensional data.
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</li><li> FFTW supports the SSE, SSE2, AVX, AVX2, AVX512, KCVI, Altivec, VSX, and
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NEON vector instruction sets.
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</li><li> FFTW includes parallel (multi-threaded) transforms
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for shared-memory systems.
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</li><li> Starting with version 3.3, FFTW includes distributed-memory parallel
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transforms using MPI.
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</li></ul>
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<p>We assume herein that you are familiar with the properties and uses of
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the DFT that are relevant to your application. Otherwise, see
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e.g. <cite>The Fast Fourier Transform and Its Applications</cite> by E. O. Brigham
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(Prentice-Hall, Englewood Cliffs, NJ, 1988).
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<a href="http://www.fftw.org">Our web page</a> also has links to FFT-related
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information online.
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<span id="index-FFTW"></span>
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</p>
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<p>In order to use FFTW effectively, you need to learn one basic concept
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of FFTW’s internal structure: FFTW does not use a fixed algorithm for
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computing the transform, but instead it adapts the DFT algorithm to
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details of the underlying hardware in order to maximize performance.
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Hence, the computation of the transform is split into two phases.
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First, FFTW’s <em>planner</em> “learns” the fastest way to compute the
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transform on your machine. The planner
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<span id="index-planner"></span>
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produces a data structure called a <em>plan</em> that contains this
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<span id="index-plan"></span>
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information. Subsequently, the plan is <em>executed</em>
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<span id="index-execute"></span>
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to transform the array of input data as dictated by the plan. The
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plan can be reused as many times as needed. In typical
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high-performance applications, many transforms of the same size are
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computed and, consequently, a relatively expensive initialization of
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this sort is acceptable. On the other hand, if you need a single
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transform of a given size, the one-time cost of the planner becomes
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significant. For this case, FFTW provides fast planners based on
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heuristics or on previously computed plans.
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</p>
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<p>FFTW supports transforms of data with arbitrary length, rank,
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multiplicity, and a general memory layout. In simple cases, however,
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this generality may be unnecessary and confusing. Consequently, we
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organized the interface to FFTW into three levels of increasing
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generality.
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</p><ul>
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<li> The <em>basic interface</em> computes a single
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transform of contiguous data.
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</li><li> The <em>advanced interface</em> computes transforms
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of multiple or strided arrays.
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</li><li> The <em>guru interface</em> supports the most general data
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layouts, multiplicities, and strides.
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</li></ul>
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<p>We expect that most users will be best served by the basic interface,
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whereas the guru interface requires careful attention to the
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documentation to avoid problems.
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<span id="index-basic-interface"></span>
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<span id="index-advanced-interface"></span>
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<span id="index-guru-interface"></span>
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</p>
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<p>Besides the automatic performance adaptation performed by the planner,
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it is also possible for advanced users to customize FFTW manually. For
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example, if code space is a concern, we provide a tool that links only
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the subset of FFTW needed by your application. Conversely, you may need
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to extend FFTW because the standard distribution is not sufficient for
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your needs. For example, the standard FFTW distribution works most
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efficiently for arrays whose size can be factored into small primes
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(<em>2</em>, <em>3</em>, <em>5</em>, and <em>7</em>), and otherwise it uses a
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slower general-purpose routine. If you need efficient transforms of
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other sizes, you can use FFTW’s code generator, which produces fast C
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programs (“codelets”) for any particular array size you may care
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about.
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<span id="index-code-generator"></span>
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<span id="index-codelet"></span>
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For example, if you need transforms of size
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513 = 19*3<sup>3</sup>,
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you can customize FFTW to support the factor <em>19</em> efficiently.
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</p>
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<p>For more information regarding FFTW, see the paper, “The Design and
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Implementation of FFTW3,” by M. Frigo and S. G. Johnson, which was an
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invited paper in <cite>Proc. IEEE</cite> <b>93</b> (2), p. 216 (2005). The
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code generator is described in the paper “A fast Fourier transform
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compiler”,
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<span id="index-compiler"></span>
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by M. Frigo, in the <cite>Proceedings of the 1999 ACM SIGPLAN Conference
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on Programming Language Design and Implementation (PLDI), Atlanta,
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Georgia, May 1999</cite>. These papers, along with the latest version of
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FFTW, the FAQ, benchmarks, and other links, are available at
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<a href="http://www.fftw.org">the FFTW home page</a>.
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</p>
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<p>The current version of FFTW incorporates many good ideas from the past
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thirty years of FFT literature. In one way or another, FFTW uses the
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Cooley-Tukey algorithm, the prime factor algorithm, Rader’s algorithm
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for prime sizes, and a split-radix algorithm (with a
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“conjugate-pair” variation pointed out to us by Dan Bernstein).
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FFTW’s code generator also produces new algorithms that we do not
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completely understand.
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<span id="index-algorithm"></span>
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The reader is referred to the cited papers for the appropriate
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references.
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</p>
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<p>The rest of this manual is organized as follows. We first discuss the
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sequential (single-processor) implementation. We start by describing
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the basic interface/features of FFTW in <a href="Tutorial.html">Tutorial</a>.
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Next, <a href="Other-Important-Topics.html">Other Important Topics</a> discusses data alignment
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(see <a href="SIMD-alignment-and-fftw_005fmalloc.html">SIMD alignment and fftw_malloc</a>),
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the storage scheme of multi-dimensional arrays
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(see <a href="Multi_002ddimensional-Array-Format.html">Multi-dimensional Array Format</a>), and FFTW’s mechanism for
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storing plans on disk (see <a href="Words-of-Wisdom_002dSaving-Plans.html">Words of Wisdom-Saving Plans</a>). Next,
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<a href="FFTW-Reference.html">FFTW Reference</a> provides comprehensive documentation of all
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FFTW’s features. Parallel transforms are discussed in their own
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chapters: <a href="Multi_002dthreaded-FFTW.html">Multi-threaded FFTW</a> and <a href="Distributed_002dmemory-FFTW-with-MPI.html">Distributed-memory FFTW with MPI</a>. Fortran programmers can also use FFTW, as described in
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<a href="Calling-FFTW-from-Legacy-Fortran.html">Calling FFTW from Legacy Fortran</a> and <a href="Calling-FFTW-from-Modern-Fortran.html">Calling FFTW from Modern Fortran</a>. <a href="Installation-and-Customization.html">Installation and Customization</a> explains how to
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install FFTW in your computer system and how to adapt FFTW to your
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needs. License and copyright information is given in <a href="License-and-Copyright.html">License and Copyright</a>. Finally, we thank all the people who helped us in
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<a href="Acknowledgments.html">Acknowledgments</a>.
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</p>
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<hr>
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<div class="header">
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<p>
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Next: <a href="Tutorial.html" accesskey="n" rel="next">Tutorial</a>, Previous: <a href="index.html" accesskey="p" rel="prev">Top</a>, Up: <a href="index.html" accesskey="u" rel="up">Top</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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</div>
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</body>
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</html>
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