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<html>
<head>
<title>
FFTPACK5.1 - Fast Fourier Transforms
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FFTPACK5.1 <br> Fast Fourier Transform
</h1>
<hr>
<p>
<b>FFTPACK5.1</b>
is a FORTRAN90 library which
computes the Fast Fourier Transform,
by Paul Swarztrauber and Dick Valent;
</p>
<p>
Special features include:
<ul>
<li>
real or complex data can be handled;
</li>
<li>
separate routines for forward analysis (data => Fourier coefficients)
and backward analysis (Fourier coefficients => data);
</li>
<li>
sine and cosine transform routines;
</li>
<li>
quarter wave sine and cosine transform routines;
</li>
<li>
the amount of data is NOT required to be a power of 2.
</li>
</ul>
</p>
<p>
Routines in the library come in groups of three:
<ul>
<li>
an initialization routine;
</li>
<li>
the forward computational routine (data to FFT coefficients);
</li>
<li>
the backward computational routine (FFT coefficients to data).
</li>
</ul>
</p>
<p>
The following table shows how the names of the routines in each group:
<table border = 1>
<tr><th>Initialization</th><th>Forward</th><th>Backward</th><th>Purpose</th></tr>
<tr><td>CFFT1I</td><td>CFFT1F</td><td>CFFT1B</td><td>FFT for complex 1D data</td></tr>
<tr><td>CFFT2I</td><td>CFFT2F</td><td>CFFT2B</td><td>FFT for complex 2D data</td></tr>
<tr><td>CFFTMI</td><td>CFFTMF</td><td>CFFTMB</td><td>FFT for complex multi-D data</td></tr>
<tr><td>COSQ1I</td><td>COSQ1F</td><td>COSQ1B</td><td>Even function 1D quarter-cosine FFT</td></tr>
<tr><td>COSQMI</td><td>COSQMF</td><td>COSQMB</td><td>Even function 1D quarter-cosine FFT</td></tr>
<tr><td>COST1I</td><td>COST1F</td><td>COST1B</td><td>Even function 1D cosine FFT</td></tr>
<tr><td>COSTMI</td><td>COSTMF</td><td>COSTMB</td><td>Even function 1D cosine FFT</td></tr>
<tr><td>RFFT1I</td><td>RFFT1F</td><td>RFFT1B</td><td>FFT for real 1D data</td></tr>
<tr><td>RFFT2I</td><td>RFFT2F</td><td>RFFT2B</td><td>FFT for real 2D data</td></tr>
<tr><td>RFFTMI</td><td>RFFTMF</td><td>RFFTMB</td><td>FFT for real multi-D data</td></tr>
<tr><td>SINQ1I</td><td>SINQ1F</td><td>SINQ1B</td><td>Even function 1D quarter-sine FFT</td></tr>
<tr><td>SINQMI</td><td>SINQMF</td><td>SINQMB</td><td>Even function 1D quarter-sine FFT</td></tr>
<tr><td>SINT1I</td><td>SINT1F</td><td>SINT1B</td><td>Even function 1D sine FFT</td></tr>
<tr><td>SINTMI</td><td>SINTMF</td><td>SINTMB</td><td>Even function 1D sine FFT</td></tr>
</table>
</p>
<h3 align = "center">
Bugfixes from FFTPACK 5.0 to FFTPACK 5.1
</h3>
<p>
The functionality of FFTPACK 5.1 is identical with the previous version 5.0.
</p>
<p>
The following bugfixes have been applied:
<ol>
<li>
Corrected index error for high-level routines [CR]FFT[12][BFI] requiring
WSAVE array, and all dependent routines to which LWSAVE is passed.
Namely, in LWSAV = L + LOG(REAL(INT(L))) + 4 definitions, the summand term
LOG(REAL(INT(L))) has been corrected to LOG(REAL(INT(L)/LOG(2.)).
</li>
<li>
Corrected index error in routime C1FM1F at label 56, where array offset
was declared 1 that should have been 2. This resulted in C1FFTF transforms
of length N calculated incorrectly, where N is any prime .GT. 5, or N=7*M,
M .GE. 6.
</li>
<li>
Corrected RFFT2x routines by rewriting them. The backward transform
followed by forward now returns identity. Input argument LENSAV must be
be at least
L + 3*M + INT(LOG(REAL(L))/LOG(2.)) + 2*INT(LOG(REAL(M))/LOG(2.)) +12.
Previously, the required value was smaller.
</li>
</ol>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
Licensed under the GNU General Public License (GPL).<br>
Copyright (C) 1995-2004, Scientific Computing Division,<br>
University Corporation for Atmospheric Research.
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>FFTPACK5.1</b> is available in
<a href = "../../f77_src/fftpack5.1/fftpack5.1.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fftpack5.1/fftpack5.1.html">a FORTRAN90 version</a>.
</p>
<h3 align = "center">
Related Programs:
</h3>
<p>
<a href = "../../f_src/fftpack5/fftpack5.html">
FFTPACK5</a>,
a FORTRAN90 library which
implements the Fast Fourier Transform.
</p>
<p>
<a href = "../../f_src/fftpack5.1d/fftpack5.1d.html">
FFTPACK5.1D</a>,
a FORTRAN90 library which
implements the Fast Fourier Transform
using double precision arithmetic,
by Paul Swarztrauber and Dick Valent;
</p>
<p>
<a href = "../../f_src/fftw3/fftw3.html">
FFTW3</a>,
FORTRAN90 programs which
illustrate the use of the FFTW3 library for Fast Fourier Transforms,
by Matteo Frigo and Steven Johnson.
</p>
<p>
<a href = "../../cpp_src/gsl/gsl.html">
GSL</a>,
a C++ library which
embodies the "GNU Scientific Library", and includes FFT routines.
</p>
<p>
<a href = "../../f_src/sftpack/sftpack.html">
SFTPACK</a>,
a FORTRAN90 library which
implements the "slow" Fourier transform, intended as a teaching
tool and comparison with the fast Fourier transform.
</p>
<p>
<a href = "../../f_src/sine_transform/sine_transform.html">
SINE_TRANSFORM</a>,
a FORTRAN90 library which
demonstrates some simple properties of the discrete sine transform.
</p>
<p>
<a href = "../../f_src/walsh/walsh.html">
WALSH</a>,
a FORTRAN90 library which
implements versions of the Walsh and Haar transforms.
</p>
<p>
<a href = "../../f_src/xerror/xerror.html">
XERROR</a>,
a FORTRAN90 library which
is used for error reporting.
</p>
<h3 align = "center">
Author:
</h3>
<p>
The original Fortran77 version of this code was written by Dick Valent and
Paul Swarztrauber, of the National Center for Atmospheric Research (NCAR).
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
William Briggs, Van Emden Henson,<br>
The DFT: An Owner's Manual for the Discrete Fourier Transform,<br>
SIAM, 1995,<br>
ISBN13: 978-0-898713-42-8,<br>
LC: QA403.5.B75.
</li>
<li>
Elbert Brigham,<br>
The Fast Fourier Transform and Its Applications,<br>
Prentice-Hall, 1988,<br>
ISBN: 0133075052,<br>
LC: QA403.B74.
</li>
<li>
Bill Buzbee,<br>
The SLATEC Common Math Library,<br>
in Sources and Development of Mathematical Software,<br>
edited by Wayne Cowell,<br>
Prentice-Hall, 1984,<br>
ISBN: 0-13-823501-5,<br>
LC: QA76.95.S68.
</li>
<li>
Eleanor Chu, Alan George,<br>
Inside the FFT Black Box,<br>
CRC Press,<br>
ISBN: 0849302706,<br>
LC: QA403.5C5.
</li>
<li>
David Kahaner, Cleve Moler, Steven Nash,<br>
Numerical Methods and Software,<br>
Prentice Hall, 1989,<br>
ISBN: 0-13-627258-4,<br>
LC: TA345.K34.
</li>
<li>
Paul Swarztrauber,<br>
Symmetric FFT's,<br>
Mathematics of Computation,<br>
Volume 47, Number 175, July 1986, pages 323-346.
</li>
<li>
Paul Swarztrauber,<br>
Vectorizing the Fast Fourier Transforms,<br>
in Parallel Computations,<br>
edited by Garry Rodrigue,<br>
Academic Press, 1982,<br>
ISBN: 0125921012,<br>
LC: QA76.6.P348.
</li>
<li>
Paul Swarztrauber,<br>
Fast Fourier Transform Algorithms for Vector Computers,<br>
Parallel Computing,<br>
Volume 1, Number 1, August 1984, pages 45-63.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "fftpack5.1.f90">fftpack5.1.f90</a>, the source code.
</li>
<li>
<a href = "fftpack5.1.sh">fftpack5.1.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "fftpack5.1_prb.f90">fftpack5.1_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "fftpack5.1_prb.sh">fftpack5.1_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "fftpack5.1_prb_output.txt">fftpack5.1_prb_output.txt</a>,
the output from a run of the sample program.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CFFT1B:</b> complex backward fast Fourier transform, 1D.
</li>
<li>
<b>CFFT1F:</b> complex forward fast Fourier transform, 1D.
</li>
<li>
<b>CFFT1I:</b> initialization for CFFT1B and CFFT1F.
</li>
<li>
<b>CFFT2B:</b> complex backward fast Fourier transform, 2D.
</li>
<li>
<b>CFFT2F:</b> complex forward fast Fourier transform, 2D.
</li>
<li>
<b>CFFT2I:</b> initialization for CFFT2B and CFFT2F.
</li>
<li>
<b>CFFTMB:</b> complex backward fast Fourier transform, 1D, multiple vectors.
</li>
<li>
<b>CFFTMF:</b> complex forward fast Fourier transform, 1D, multiple vectors.
</li>
<li>
<b>CFFTMI:</b> initialization for CFFTMB and CFFTMF.
</li>
<li>
<b>COSQ1B</b>: complex backward cosine quarter wave transform, 1D.
</li>
<li>
<b>COSQ1F</b>: complex forward cosine quarter wave transform, 1D.
</li>
<li>
<b>COSQ1I</b>: initialization for COSQ1B and COSQ1F.
</li>
<li>
<b>COSQMB</b>: complex backward cosine quarter wave transform, multiple vectors.
</li>
<li>
<b>COSQMF</b>: complex forward cosine quarter wave transform, multiple vectors.
</li>
<li>
<b>COSQMI</b>: initialization for COSQMB and COSQMF.
</li>
<li>
<b>COST1B</b>: complex backward cosine transform, 1D.
</li>
<li>
<b>COST1F</b>: complex forward cosine transform, 1D.
</li>
<li>
<b>COST1I</b>: initialization for COST1B and COST1F.
</li>
<li>
<b>COSTMB</b>: complex backward cosine transform, multiple vectors.
</li>
<li>
<b>COSTMF</b>: complex forward cosine transform, multiple vectors.
</li>
<li>
<b>COSTMI</b>: initialization for COSTMB and COSTMF.
</li>
<li>
<b>FACTOR</b> determines the factors of an integer.
</li>
<li>
<b>RFFT1B:</b> real backward fast Fourier transform, 1D.
</li>
<li>
<b>RFFT1F:</b> real forward fast Fourier transform, 1D.
</li>
<li>
<b>RFFT1I:</b> initialization for RFFT1B and RFFT1F.
</li>
<li>
<b>RFFT2B:</b> real backward fast Fourier transform, 2D.
</li>
<li>
<b>RFFT2F:</b> real forward fast Fourier transform, 2D.
</li>
<li>
<b>RFFT2I:</b> initialization for RFFT2B and RFFT2F.
</li>
<li>
<b>RFFTMB</b>: real backward fast Fourier transform, 1D, multiple vectors.
</li>
<li>
<b>RFFTMF</b>: real forward fast Fourier transform, 1D, multiple vectors.
</li>
<li>
<b>RFFTMI</b>: initialization for RFFTMB and RFFTMF.
</li>
<li>
<b>SINQ1B</b>: complex backward sine quarter wave transform, 1D.
</li>
<li>
<b>SINQ1F</b>: complex forward sine quarter wave transform, 1D.
</li>
<li>
<b>SINQ1I</b>: initialization for SINQ1B and SINQ1F.
</li>
<li>
<b>SINQMB</b>: complex backward sine quarter wave transform, multiple vectors.
</li>
<li>
<b>SINQMF</b>: complex forward sine quarter wave transform, multiple vectors.
</li>
<li>
<b>SINQMI</b>: initialization for SINQMB and SINQMF.
</li>
<li>
<b>SINT1B</b>: complex backward sine transform, 1D.
</li>
<li>
<b>SINT1F</b>: complex forward sine transform, 1D.
</li>
<li>
<b>SINT1I</b>: initialization for SINT1B and SINT1F.
</li>
<li>
<b>SINTMB</b>: complex backward sine transform, multiple vectors.
</li>
<li>
<b>SINTMF</b>: complex forward sine transform, multiple vectors.
</li>
<li>
<b>SINTMI</b>: initialization for SINTMB and SINTMF.
</li>
<li>
<b>TABLES</b> computes trigonometric tables needed by the FFT routines.
</li>
<li>
<b>XERCON</b> checks INC, JUMP, N and LOT for consistency.
</li>
<li>
<b>XERFFT</b> is an error handler for the FFTPACK routines.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 31 July 2011.
</i>
<!-- John Burkardt -->
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