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## 1 Introduction

This manual documents version 3.3.8 of FFTW, the Fastest Fourier Transform in the West. FFTW is a comprehensive collection of fast C routines for computing the discrete Fourier transform (DFT) and various special cases thereof.

- FFTW computes the DFT of complex data, real data, even- or odd-symmetric real data (these symmetric transforms are usually known as the discrete cosine or sine transform, respectively), and the discrete Hartley transform (DHT) of real data.
- The input data can have arbitrary length. FFTW employs O(n log n) algorithms for all lengths, including prime numbers.
- FFTW supports arbitrary multi-dimensional data.
- FFTW supports the SSE, SSE2, AVX, AVX2, AVX512, KCVI, Altivec, VSX, and NEON vector instruction sets.
- FFTW includes parallel (multi-threaded) transforms for shared-memory systems.
- Starting with version 3.3, FFTW includes distributed-memory parallel transforms using MPI.

We assume herein that you are familiar with the properties and uses of the DFT that are relevant to your application. Otherwise, see e.g. The Fast Fourier Transform and Its Applications by E. O. Brigham (Prentice-Hall, Englewood Cliffs, NJ, 1988). Our web page also has links to FFT-related information online.

In order to use FFTW effectively, you need to learn one basic concept of FFTW’s internal structure: FFTW does not use a fixed algorithm for computing the transform, but instead it adapts the DFT algorithm to details of the underlying hardware in order to maximize performance. Hence, the computation of the transform is split into two phases. First, FFTW’s planner “learns” the fastest way to compute the transform on your machine. The planner produces a data structure called a plan that contains this information. Subsequently, the plan is executed to transform the array of input data as dictated by the plan. The plan can be reused as many times as needed. In typical high-performance applications, many transforms of the same size are computed and, consequently, a relatively expensive initialization of this sort is acceptable. On the other hand, if you need a single transform of a given size, the one-time cost of the planner becomes significant. For this case, FFTW provides fast planners based on heuristics or on previously computed plans.

FFTW supports transforms of data with arbitrary length, rank, multiplicity, and a general memory layout. In simple cases, however, this generality may be unnecessary and confusing. Consequently, we organized the interface to FFTW into three levels of increasing generality.

- The basic interface computes a single transform of contiguous data.
- The advanced interface computes transforms of multiple or strided arrays.
- The guru interface supports the most general data layouts, multiplicities, and strides.

We expect that most users will be best served by the basic interface, whereas the guru interface requires careful attention to the documentation to avoid problems.

Besides the automatic performance adaptation performed by the planner, it is also possible for advanced users to customize FFTW manually. For example, if code space is a concern, we provide a tool that links only the subset of FFTW needed by your application. Conversely, you may need to extend FFTW because the standard distribution is not sufficient for your needs. For example, the standard FFTW distribution works most efficiently for arrays whose size can be factored into small primes (2, 3, 5, and 7), and otherwise it uses a slower general-purpose routine. If you need efficient transforms of other sizes, you can use FFTW’s code generator, which produces fast C programs (“codelets”) for any particular array size you may care about. For example, if you need transforms of size 513 = 19*33, you can customize FFTW to support the factor 19 efficiently.

For more information regarding FFTW, see the paper, “The Design and Implementation of FFTW3,” by M. Frigo and S. G. Johnson, which was an invited paper in Proc. IEEE 93 (2), p. 216 (2005). The code generator is described in the paper “A fast Fourier transform compiler”, by M. Frigo, in the Proceedings of the 1999 ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI), Atlanta, Georgia, May 1999. These papers, along with the latest version of FFTW, the FAQ, benchmarks, and other links, are available at the FFTW home page.

The current version of FFTW incorporates many good ideas from the past thirty years of FFT literature. In one way or another, FFTW uses the Cooley-Tukey algorithm, the prime factor algorithm, Rader’s algorithm for prime sizes, and a split-radix algorithm (with a “conjugate-pair” variation pointed out to us by Dan Bernstein). FFTW’s code generator also produces new algorithms that we do not completely understand. The reader is referred to the cited papers for the appropriate references.

The rest of this manual is organized as follows. We first discuss the sequential (single-processor) implementation. We start by describing the basic interface/features of FFTW in Tutorial. Next, Other Important Topics discusses data alignment (see SIMD alignment and fftw_malloc), the storage scheme of multi-dimensional arrays (see Multi-dimensional Array Format), and FFTW’s mechanism for storing plans on disk (see Words of Wisdom-Saving Plans). Next, FFTW Reference provides comprehensive documentation of all FFTW’s features. Parallel transforms are discussed in their own chapters: Multi-threaded FFTW and Distributed-memory FFTW with MPI. Fortran programmers can also use FFTW, as described in Calling FFTW from Legacy Fortran and Calling FFTW from Modern Fortran. Installation and Customization explains how to install FFTW in your computer system and how to adapt FFTW to your needs. License and copyright information is given in License and Copyright. Finally, we thank all the people who helped us in Acknowledgments.

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