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<html>
<head>
<title>
HAAR - The Haar Transform
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
HAAR <br> The Haar Transform
</h1>
<hr>
<p>
<b>HAAR</b>
is a FORTRAN90 library which
computes the Haar transform of data.
</p>
<p>
In the simplest case, one is given a vector X whose length N is a power of 2.
We now consider consecutive pairs of entries of X, and for I from 0 to (N/2)-1
we define:
<pre>
S[I] = ( X[2*I] + X[2*I+1] ) / sqrt ( 2 )
D[I] = ( X[2*I] - X[2*I+1] ) / sqrt ( 2 )
</pre>
We now replace X by the vector S concatenated with D. Assuming that (N/2)
is greater than 1, we repeat the operation on the (N/2) entries of S, and
so on, until we have reached a stage where our resultant S and D each contain
one entry.
</p>
<p>
For data in the form of a 2D array, the transform is applied to the columns
and then the rows.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>HAAR</b> is available in
<a href = "../../c_src/haar/haar.html">a C version</a> and
<a href = "../../cpp_src/haar/haar.html">a C++ version</a> and
<a href = "../../f77_src/haar/haar.html">a FORTRAN77 version</a> and
<a href = "../../f_src/haar/haar.html">a FORTRAN90 version</a> and
<a href = "../../m_src/haar/haar.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/fftpack5/fftpack5.html">
FFTPACK5</a>,
a FORTRAN90 library which
implements the Fast Fourier Transform
by Paul Swarztrauber and Dick Valent;
</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/wavelet/wavelet.html">
WAVELET</a>,
a FORTRAN90 library which
does some simple wavelet calculations;
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Ken Beauchamp,<br>
Walsh functions and their applications,<br>
Academic Press, 1975,<br>
ISBN: 0-12-084050-2,<br>
LC: QA404.5.B33.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "haar.f90">haar.f90</a>, the source code.
</li>
<li>
<a href = "haar.sh">haar.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "haar_prb.f90">haar_prb.f90</a>
a sample calling program.
</li>
<li>
<a href = "haar_prb.sh">haar_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "haar_prb_output.txt">haar_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>HAAR_1D</b> computes the Haar transform of a vector.
</li>
<li>
<b>HAAR_1D_INVERSE</b> computes the inverse Haar transform of a vector.
</li>
<li>
<b>HAAR_2D</b> computes the Haar transform of an array.
</li>
<li>
<b>HAAR_2D_INVERSE</b> inverts the Haar transform of an array.
</li>
<li>
<b>R8MAT_PRINT</b> prints an R8MAT.
</li>
<li>
<b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
</li>
<li>
<b>R8MAT_UNIFORM_01</b> fills an R8MAT with unit pseudorandom numbers.
</li>
<li>
<b>R8VEC_LINSPACE</b> returns a vector of linearly spaced values.
</li>
<li>
<b>R8VEC_UNIFORM_01</b> returns a unit pseudorandom R8VEC.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</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 14 March 2011.
</i>
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</body>
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</html>