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# 1 Introduction

This manual documents version 3.3.1 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, Altivec, and MIPS PS instruction sets.
- FFTW 3.3.1 includes parallel (multi-threaded) transforms for shared-memory systems. FFTW 3.3.1 does not include distributed-memory parallel transforms, but we plan to implement an MPI version soon. (Meanwhile, you can use the MPI implementation from FFTW 2.1.3.)

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*3^{3},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 section SIMD alignment and fftw_malloc), the storage scheme of multi-dimensional arrays (see section Multi-dimensional Array Format), and FFTW’s mechanism for storing plans on disk (see section 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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