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This manual documents version 3.3.3 of FFTW, the Chris@10: Fastest Fourier Transform in the West. FFTW is a comprehensive Chris@10: collection of fast C routines for computing the discrete Fourier Chris@10: transform (DFT) and various special cases thereof. Chris@10: Chris@10:
We assume herein that you are familiar with the properties and uses of Chris@10: the DFT that are relevant to your application. Otherwise, see Chris@10: e.g. The Fast Fourier Transform and Its Applications by E. O. Brigham Chris@10: (Prentice-Hall, Englewood Cliffs, NJ, 1988). Chris@10: Our web page also has links to FFT-related Chris@10: information online. Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10: Chris@10:
In order to use FFTW effectively, you need to learn one basic concept Chris@10: of FFTW's internal structure: FFTW does not use a fixed algorithm for Chris@10: computing the transform, but instead it adapts the DFT algorithm to Chris@10: details of the underlying hardware in order to maximize performance. Chris@10: Hence, the computation of the transform is split into two phases. Chris@10: First, FFTW's planner “learns” the fastest way to compute the Chris@10: transform on your machine. The planner Chris@10: produces a data structure called a plan that contains this Chris@10: information. Subsequently, the plan is executed Chris@10: to transform the array of input data as dictated by the plan. The Chris@10: plan can be reused as many times as needed. In typical Chris@10: high-performance applications, many transforms of the same size are Chris@10: computed and, consequently, a relatively expensive initialization of Chris@10: this sort is acceptable. On the other hand, if you need a single Chris@10: transform of a given size, the one-time cost of the planner becomes Chris@10: significant. For this case, FFTW provides fast planners based on Chris@10: heuristics or on previously computed plans. Chris@10: Chris@10:
FFTW supports transforms of data with arbitrary length, rank, Chris@10: multiplicity, and a general memory layout. In simple cases, however, Chris@10: this generality may be unnecessary and confusing. Consequently, we Chris@10: organized the interface to FFTW into three levels of increasing Chris@10: generality. Chris@10:
Besides the automatic performance adaptation performed by the planner, Chris@10: it is also possible for advanced users to customize FFTW manually. For Chris@10: example, if code space is a concern, we provide a tool that links only Chris@10: the subset of FFTW needed by your application. Conversely, you may need Chris@10: to extend FFTW because the standard distribution is not sufficient for Chris@10: your needs. For example, the standard FFTW distribution works most Chris@10: efficiently for arrays whose size can be factored into small primes Chris@10: (2, 3, 5, and 7), and otherwise it uses a Chris@10: slower general-purpose routine. If you need efficient transforms of Chris@10: other sizes, you can use FFTW's code generator, which produces fast C Chris@10: programs (“codelets”) for any particular array size you may care Chris@10: about. Chris@10: For example, if you need transforms of size Chris@10: 513 = 19*33,you can customize FFTW to support the factor 19 efficiently. Chris@10: Chris@10:
For more information regarding FFTW, see the paper, “The Design and Chris@10: Implementation of FFTW3,” by M. Frigo and S. G. Johnson, which was an Chris@10: invited paper in Proc. IEEE 93 (2), p. 216 (2005). The Chris@10: code generator is described in the paper “A fast Fourier transform Chris@10: compiler”, Chris@10: by M. Frigo, in the Proceedings of the 1999 ACM SIGPLAN Conference Chris@10: on Programming Language Design and Implementation (PLDI), Atlanta, Chris@10: Georgia, May 1999. These papers, along with the latest version of Chris@10: FFTW, the FAQ, benchmarks, and other links, are available at Chris@10: the FFTW home page. Chris@10: Chris@10:
The current version of FFTW incorporates many good ideas from the past Chris@10: thirty years of FFT literature. In one way or another, FFTW uses the Chris@10: Cooley-Tukey algorithm, the prime factor algorithm, Rader's algorithm Chris@10: for prime sizes, and a split-radix algorithm (with a Chris@10: “conjugate-pair” variation pointed out to us by Dan Bernstein). Chris@10: FFTW's code generator also produces new algorithms that we do not Chris@10: completely understand. Chris@10: The reader is referred to the cited papers for the appropriate Chris@10: references. Chris@10: Chris@10:
The rest of this manual is organized as follows. We first discuss the Chris@10: sequential (single-processor) implementation. We start by describing Chris@10: the basic interface/features of FFTW in Tutorial. Chris@10: Next, Other Important Topics discusses data alignment Chris@10: (see SIMD alignment and fftw_malloc), Chris@10: the storage scheme of multi-dimensional arrays Chris@10: (see Multi-dimensional Array Format), and FFTW's mechanism for Chris@10: storing plans on disk (see Words of Wisdom-Saving Plans). Next, Chris@10: FFTW Reference provides comprehensive documentation of all Chris@10: FFTW's features. Parallel transforms are discussed in their own Chris@10: chapters: Multi-threaded FFTW and Distributed-memory FFTW with MPI. Fortran programmers can also use FFTW, as described in Chris@10: Calling FFTW from Legacy Fortran and Calling FFTW from Modern Fortran. Installation and Customization explains how to Chris@10: install FFTW in your computer system and how to adapt FFTW to your Chris@10: needs. License and copyright information is given in License and Copyright. Finally, we thank all the people who helped us in Chris@10: Acknowledgments. Chris@10: Chris@10: Chris@10: