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<div id="projectname">ViennaCL - The Vienna Computing Library
 <span id="projectnumber">1.7.1</span>
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<div id="projectbrief">Free open-source GPU-accelerated linear algebra and solver library.</div>
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<div class="title">Structured Matrix Types </div> </div>
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<div class="textblock"><dl class="section warning"><dt>Warning</dt><dd>Structured matrix types are experimental in ViennaCL. Interface changes as well as considerable performance improvements may be included in future releases!</dd></dl>
<p>There are a number of structured dense matrices for which some algorithms such as matrix-vector products can be computed with much lower computational effort than for the general dense matrix case. In the following, four structured dense matrix types included in ViennaCL are discussed. Example code can be found in <code><a class="el" href="examples_2tutorial_2structured-matrices_8cpp.html">examples/tutorial/structured-matrices.cpp</a></code>.</p>
<h1><a class="anchor" id="manual-structured-matrix-circulant"></a>
Circulant Matrix</h1>
<p>A circulant matrix is a matrix of the form </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \left( \begin{array}{ccccc} c_0 & c_{n-1} & \ldots & c_2 & c_1 \\ c_1 & c_0 & c_{n-1} & & c_2 \\ \vdots & c_1 & c_0 & \ddots & \vdots \\ c_{n-2} & & \ddots & \ddots & c_{n-1} \\ c_{n-1} & c_{n-2} & \hdots & c_1 & c_0 \\ \end{array} \right) \]" src="form_97.png"/>
</p>
<p> and available in ViennaCL via </p>
<div class="fragment"><div class="line"><span class="preprocessor">#include "<a class="code" href="circulant__matrix_8hpp.html">viennacl/circulant_matrix.hpp</a>"</span></div>
<div class="line"></div>
<div class="line">std::size_t s = 42;</div>
<div class="line"><a class="code" href="classviennacl_1_1circulant__matrix.html">viennacl::circulant_matrix</a> circ_mat(s, s);</div>
</div><!-- fragment --><p> The <code>circulant_matrix</code> type can be manipulated in the same way as the dense matrix type <code>matrix</code>. Note that writing to a single element of the matrix is structure-preserving, e.g. changing <code>circ_mat(1,2)</code> will automatically update <code>circ_mat(0,1)</code>, <code>circ_mat(2,3)</code> and so on.</p>
<h1><a class="anchor" id="manual-structured-matrix-hankel"></a>
Hankel Matrix</h1>
<p>A Hankel matrix is a matrix of the form </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \left( \begin{array}{cccc} a & b & c & d \\ b & c & d & e \\ c & d & e & f \\ d & e & f & g \\ \end{array} \right) \]" src="form_98.png"/>
</p>
<p> and available in ViennaCL via </p>
<div class="fragment"><div class="line"><span class="preprocessor">#include "<a class="code" href="hankel__matrix_8hpp.html">viennacl/hankel_matrix.hpp</a>"</span></div>
<div class="line"></div>
<div class="line">std::size_t s = 42;</div>
<div class="line"><a class="code" href="classviennacl_1_1hankel__matrix.html">viennacl::hankel_matrix</a> hank_mat(s, s);</div>
</div><!-- fragment --><p> The <code>hankel_matrix</code> type can be manipulated in the same way as the dense matrix type <code>matrix</code>. Note that writing to a single element of the matrix is structure-preserving, e.g. changing <code>hank_mat(1,2)</code> in the example above will also update <code>hank_mat(0,3)</code>, <code>hank_mat(2,1)</code>, <code>hank_mat(3,0)</code>, etc.</p>
<h1><a class="anchor" id="manual-structured-matrix-toeplitz"></a>
Toeplitz Matrix</h1>
<p>A Toeplitz matrix is a matrix of the form </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \left( \begin{array}{cccc} a & b & c & d \\ e & a & b & c \\ f & e & a & b \\ g & f & e & a \\ \end{array} \right) \]" src="form_99.png"/>
</p>
<p> and available in ViennaCL via </p>
<div class="fragment"><div class="line"><span class="preprocessor">#include "<a class="code" href="toeplitz__matrix_8hpp.html">viennacl/toeplitz_matrix.hpp</a>"</span></div>
<div class="line"></div>
<div class="line">std::size_t s = 42;</div>
<div class="line"><a class="code" href="classviennacl_1_1toeplitz__matrix.html">viennacl::toeplitz_matrix</a> toep_mat(s, s);</div>
</div><!-- fragment --><p> The <code>toeplitz_matrix</code> type can be manipulated in the same way as the dense matrix type <code>matrix</code>. Note that writing to a single element of the matrix is structure-preserving, e.g. changing <code>toep_mat(1,2)</code> in the example above will also update <code>toep_mat(0,1)</code>, <code>toep_mat(2,3)</code>, etc.</p>
<h1><a class="anchor" id="manual-structured-matrix-vandermonde"></a>
Vandermonde Matrix</h1>
<p>A Vandermonde matrix is a matrix of the form </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \left( \begin{array}{ccccc} 1 & \alpha_1 & \alpha_1^2 & \ldots & \alpha_1^{n-1} \\ 1 & \alpha_2 & \alpha_2^2 & \ldots & \alpha_2^{n-1} \\ 1 & \vdots & \vdots & \vdots \\ 1 & \alpha_m & \alpha_m^2 & \ldots & \alpha_m^{n-1} \\ \end{array} \right) \]" src="form_100.png"/>
</p>
<p> and available in ViennaCL via </p>
<div class="fragment"><div class="line"><span class="preprocessor">#include "<a class="code" href="vandermonde__matrix_8hpp.html">viennacl/vandermonde_matrix.hpp</a>"</span></div>
<div class="line"></div>
<div class="line">std::size_t s = 42;</div>
<div class="line"><a class="code" href="classviennacl_1_1vandermonde__matrix.html">viennacl::vandermonde_matrix</a> vand_mat(s, s);</div>
</div><!-- fragment --><p> The <code>vandermonde_matrix</code> type can be manipulated in the same way as the dense matrix type <code>matrix</code>, but restrictions apply. For example, the addition or subtraction of two Vandermonde matrices does not yield another Vandermonde matrix. Note that writing to a single element of the matrix is structure-preserving, e.g. changing <code>vand_mat(1,2)</code> in the example above will automatically update <code>vand_mat(1,3)</code>, <code>vand_mat(1,4)</code>, etc. </p>
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