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<td align="left"><h1>SLRA: Software for weighted structured low-rank approximation</h1>
</td></tr></table>
<h2>Overview</h2>
<p>SLRA is a C<tt></tt> software with interface to MATLAB, Octave, and R for solving low-rank approximation problems</p>

<div class="eqwl"><img class="eqwl" src="eqs/1629822693081112490-150.png" alt=" text{minimize} quad text{over}  widehat p quad |p - widehat p|_w quad text{subject to} quad text{rank}({cal S}(widehat p)) leq r, " />
<br /></div><p>with the following features:</p>
<ul>
<li><p><a href="examples.html">mosaic Hankel structured</a> approximating matrix <img class="eq" src="eqs/9133601666111744340-150.png" alt="{cal S}(widehat p)" style="vertical-align: -6px" />,</p>
</li>
<li><p>weighted 2-norm approximation criterion </p>
</li>
</ul>

<div class="eqwl"><img class="eqwl" src="eqs/1062761975630790230-150.png" alt=" |p|_w = textstylesqrt{sum_{i=1}^{n_p} w_i p_i^2}, " />
<br /></div><ul>
<li><p>fixed elements in the approximating matrix,</p>
</li>
<li><p>missing elements in the data matrix,</p>
</li>
<li><p>linear constraints on an approximating matrix's left kernel basis.</p>
</li>
</ul>
<p>For an <img class="eq" src="eqs/3187889088634215596-150.png" alt="mtimes n" style="vertical-align: -1px" /> matrix <img class="eq" src="eqs/9133601666111744340-150.png" alt="{cal S}(widehat p)" style="vertical-align: -6px" />, with <img class="eq" src="eqs/310184670480442714-150.png" alt="m &lt; n" style="vertical-align: -1px" />, the computational complexity of the cost function and derivative evaluation is <img class="eq" src="eqs/5400137134684481330-150.png" alt="O(m^2n)" style="vertical-align: -6px" />, so that the package is suitable for applications with <img class="eq" src="eqs/3312475274443383293-150.png" alt="ngg m" style="vertical-align: -2px" />. Problems with <img class="eq" src="eqs/4307665416819844441-150.png" alt="n_p leq n(m - r)" style="vertical-align: -7px" /> can not be solved by the method, used in the software.</p>
<h2>Usage</h2>
<p>In MATLAB/Octave, once the structure and approximation criterion are specified in a variable <tt>s</tt> (see <a href="examples.html">examples</a> and the <a href="https://github.com/slra/slra/raw/master/slra.pdf">manual</a>), the problem is solved by a call of the <tt>slra</tt> function: </p>
<div class="codeblock">
<div class="blockcontent"><pre>
[ph, info] = slra(p, s, r, opt);
</pre></div></div>
<ul>
<li><p>The optional parameter <tt>opt</tt> is used to specify an initial approximation and options for the optimization method. </p>
</li>
<li><p>The output parameter <tt>info</tt> contains information about the convergence of the optimization method.</p>
</li>
</ul>
<h3>System identification </h3>
<p>A major application area of mosaic Hankel structured low-rank approximation methods is system identification. The <tt>slra</tt> package includes a wrapper function <tt>ident</tt> for identification of linear time-invariant systems. For details see the <a href="software-ident.html">ident package</a>. </p>
<h2>Download and installation</h2>
<h3>Latest version</h3>
<ul>
<li><p>Download <a href="https://github.com/slra/slra/zipball/master">ZIP File</a></p>
</li>
<li><p>Download <a href="https://github.com/slra/slra/tarball/master">TAR Ball</a></p>
</li>
<li><p>View on <a href="https://github.com/slra/slra">GitHub</a></p>
</li>
</ul>
<h3>Installation</h3>
<ul>
<li><p>For MATLAB/Octave users:  unpack the archive and use <tt>addpath</tt> in MATLAB/Octave</p>
</li>
<li><p>For R users (Linux):  unpack the archive and run <tt>make R</tt> in console
</p>
</li>
</ul>
<h2>For advanced users/developers</h2>
<ul>
<li><p>The instructions how to compile from source are contained <a href="https://github.com/slra/slra/">here</a>
</p>
</li>
</ul>
<h2>Citing</h2>
<div class="codeblock">
<div class="blockcontent"><pre>
@Article{slra-software,
  author = 	 {I. Markovsky and K. Usevich},
  title = 	 {Software for weighted structured low-rank approximation},
  journal =      {J. Comput. Appl. Math.},
  volume =       {256},
  pages =        {278--292},
  year = 	 {2014},
}
</pre></div></div>
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