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<div class="menu-item"><a href="index.html">Overview</a></div>
<div class="menu-item"><a href="software.html" class="current">Software</a></div>
<div class="menu-item"><a href="software-slra.html"> SLRA package</a></div>
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<div class="menu-item"><a href="lectures.html">Mini courses</a></div>
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<h1>Examples</h1>
<h2>Hankel matrix</h2>
<div class="eqwl"><img class="eqwl" src="eqs/4353269141702604213-150.png" alt=" {cal H}_{m,n}(p) := left[begin{array}{ccccc} p_1 & p_2 & p_3 & cdots & p_n p_2 & p_3 & iddots & & p_{n+1} p_3 & iddots & & & vdots vdots & & & & p_m & p_{m+1} & cdots & & p_{m+n-1} end{array}right] in mathbb{R}^{mtimes n} " />
<br /></div><p>is specified in MATLAB/Octave by a structure variable with fields <tt>m</tt> (number of rows) and, optionally, <tt>n</tt> (number of columns):</p>
<div class="codeblock">
<div class="blockcontent"><pre>
s.m = m; s.n = n;
</pre></div></div>
<p>If <tt>s.n</tt> is missing, the number of columns is determined from the length <img class="eq" src="eqs/5261102140379497868-150.png" alt="n_p" style="vertical-align: -7px" /> of <img class="eq" src="eqs/14336043121-150.png" alt="p" style="vertical-align: -5px" /> as <img class="eq" src="eqs/3577289840459820554-150.png" alt="n = n_p - m + 1" style="vertical-align: -7px" />.</p>
<h2>Mosaic Hankel matrix</h2>
<p>A mosaic Hankel matrix</p>
<div class="eqwl"><img class="eqwl" src="eqs/3310238954953334795-150.png" alt=" {cal H}_{mathbf{m},mathbf{n}} := left[begin{array}{ccc} {cal H}_{m_1,n_1}(p^{(11)}) & cdots & {cal H}_{m_1,n_N}(p^{(1N)}) vdots & & vdots {cal H}_{m_q,n_1}(p^{(q1)}) & cdots & {cal H}_{m_q,n_N}(p^{(qN)}) end{array}right] " />
<br /></div><p>is specified by the vectors </p>
<div class="eqwl"><img class="eqwl" src="eqs/384591629377686689-150.png" alt=" mathbf{m} = begin{bmatrix} m_1& cdots & m_q end{bmatrix}, qquad mathbf{n} = begin{bmatrix} n_1& cdots& n_N end{bmatrix}. " />
<br /></div><p>For example, with <img class="eq" src="eqs/7046377602793978756-150.png" alt="q = N = 2" style="vertical-align: -5px" />,</p>
<div class="codeblock">
<div class="blockcontent"><pre>
s.m = [m1; m2]; s.n = [n1; n2];
</pre></div></div>
<h2>Unstructured matrix</h2>
<p>An <img class="eq" src="eqs/3187889088634215596-150.png" alt="mtimes n" style="vertical-align: -1px" /> unstructured matrix is a special case of a mosaic Hankel matrix and is specified by</p>
<div class="codeblock">
<div class="blockcontent"><pre>
s.m = ones(m, 1); s.n = ones(n, 1);
</pre></div></div>
<h2>Weight vector</h2>
<p>The vector <img class="eq" src="eqs/15232045814-150.png" alt="w" style="vertical-align: -1px" /> of weights is considered part of the structure and is specified by a filed <tt>w</tt> of <tt>s</tt>:</p>
<div class="codeblock">
<div class="blockcontent"><pre>
s.w = w;
</pre></div></div>
<h2>Exact elements</h2>
<p>Exact elements of <img class="eq" src="eqs/14336043121-150.png" alt="p" style="vertical-align: -5px" />, i.e., constraints <img class="eq" src="eqs/4655768527924275747-150.png" alt="widehat p_i = p_i" style="vertical-align: -5px" />, are specifies by infinite weights:</p>
<div class="codeblock">
<div class="blockcontent"><pre>
s.w(I_exact) = inf;
</pre></div></div>
<p>where <tt>I_exct</tt> are the indeces of the exact elements of <img class="eq" src="eqs/14336043121-150.png" alt="p" style="vertical-align: -5px" />.</p>
<h2>Missing elements</h2>
<p>Missing elements of <img class="eq" src="eqs/14336043121-150.png" alt="p" style="vertical-align: -5px" /> are specified by setting them to <tt>NaN</tt>:</p>
<div class="codeblock">
<div class="blockcontent"><pre>
p(I_missing) = NaN;
</pre></div></div>
<p>where <tt>I_imssing</tt> are the indeces of the missing elements of <img class="eq" src="eqs/14336043121-150.png" alt="p" style="vertical-align: -5px" />.</p>
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