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d2/d3b/namespaceqr__algorithm.html
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<head>
<meta http-equiv="Content-Type" content="text/xhtml;charset=UTF-8"/>
<meta http-equiv="X-UA-Compatible" content="IE=11"/>
<meta name="generator" content="Doxygen 1.9.3"/>
<meta name="generator" content="Doxygen 1.9.4"/>
<meta name="viewport" content="width=device-width, initial-scale=1"/>
<title>Algorithms_in_C++: qr_algorithm Namespace Reference</title>
<link href="../../tabs.css" rel="stylesheet" type="text/css"/>
@@ -41,7 +41,7 @@ MathJax.Hub.Config({
</table>
</div>
<!-- end header part -->
<!-- Generated by Doxygen 1.9.3 -->
<!-- Generated by Doxygen 1.9.4 -->
<script type="text/javascript">
/* @license magnet:?xt=urn:btih:d3d9a9a6595521f9666a5e94cc830dab83b65699&amp;dn=expat.txt MIT */
var searchBox = new SearchBox("searchBox", "../../search",'Search','.html');
@@ -398,12 +398,31 @@ template&lt;typename T &gt; </div>
</div><div class="memdoc">
<p >Decompose matrix \(A\) using <a href="https://en.wikipedia.org/wiki/QR_decomposition" target="_blank">Gram-Schmidt process</a>.</p>
<p class="formulaDsp">
\begin{eqnarray*} \text{given that}\quad A &amp;=&amp; *\left[\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_{N-1},\right]\\ \text{where}\quad\mathbf{a}_i &amp;=&amp; \left[a_{0i},a_{1i},a_{2i},\ldots,a_{(M-1)i}\right]^T\quad\ldots\mbox{(column vectors)}\\ \text{then}\quad\mathbf{u}_i &amp;=&amp; \mathbf{a}_i *-\sum_{j=0}^{i-1}\text{proj}_{\mathbf{u}_j}\mathbf{a}_i\\ \mathbf{e}_i &amp;=&amp;\frac{\mathbf{u}_i}{\left|\mathbf{u}_i\right|}\\ Q &amp;=&amp; \begin{bmatrix}\mathbf{e}_0 &amp; \mathbf{e}_1 &amp; \mathbf{e}_2 &amp; \dots &amp; \mathbf{e}_{N-1}\end{bmatrix}\\ R &amp;=&amp; \begin{bmatrix}\langle\mathbf{e}_0\,,\mathbf{a}_0\rangle &amp; \langle\mathbf{e}_1\,,\mathbf{a}_1\rangle &amp; \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle &amp; \dots \\ 0 &amp; \langle\mathbf{e}_1\,,\mathbf{a}_1\rangle &amp; \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle &amp; \dots\\ 0 &amp; 0 &amp; \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle &amp; \dots\\ \vdots &amp; \vdots &amp; \vdots &amp; \ddots \end{bmatrix}\\ \end{eqnarray*}
\begin{eqnarray*}
\text{given that}\quad A &amp;=&amp;
*\left[\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_{N-1},\right]\\
\text{where}\quad\mathbf{a}_i &amp;=&amp;
\left[a_{0i},a_{1i},a_{2i},\ldots,a_{(M-1)i}\right]^T\quad\ldots\mbox{(column
vectors)}\\
\text{then}\quad\mathbf{u}_i &amp;=&amp; \mathbf{a}_i
*-\sum_{j=0}^{i-1}\text{proj}_{\mathbf{u}_j}\mathbf{a}_i\\
\mathbf{e}_i &amp;=&amp;\frac{\mathbf{u}_i}{\left|\mathbf{u}_i\right|}\\
Q &amp;=&amp; \begin{bmatrix}\mathbf{e}_0 &amp; \mathbf{e}_1 &amp; \mathbf{e}_2 &amp; \dots &amp;
\mathbf{e}_{N-1}\end{bmatrix}\\
R &amp;=&amp; \begin{bmatrix}\langle\mathbf{e}_0\,,\mathbf{a}_0\rangle &amp;
\langle\mathbf{e}_1\,,\mathbf{a}_1\rangle &amp;
\langle\mathbf{e}_2\,,\mathbf{a}_2\rangle &amp; \dots \\
0 &amp; \langle\mathbf{e}_1\,,\mathbf{a}_1\rangle &amp;
\langle\mathbf{e}_2\,,\mathbf{a}_2\rangle &amp; \dots\\
0 &amp; 0 &amp; \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle &amp;
\dots\\ \vdots &amp; \vdots &amp; \vdots &amp; \ddots
\end{bmatrix}\\
\end{eqnarray*}
</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">A</td><td>input matrix to decompose </td></tr>
<tr><td class="paramname">Q</td><td>output decomposed matrix </td></tr>
<tr><td class="paramname">A</td><td>input matrix to decompose</td></tr>
<tr><td class="paramname">Q</td><td>output decomposed matrix</td></tr>
<tr><td class="paramname">R</td><td>output decomposed matrix </td></tr>
</table>
</dd>
@@ -632,7 +651,7 @@ Here is the call graph for this function:</div>
<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
<ul>
<li class="navelem"><a class="el" href="../../d2/d3b/namespaceqr__algorithm.html">qr_algorithm</a></li>
<li class="footer">Generated by <a href="https://www.doxygen.org/index.html"><img class="footer" src="../../doxygen.svg" width="104" height="31" alt="doxygen"/></a> 1.9.3 </li>
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