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<p>Compute real eigen values and eigen vectors of a symmetric matrix using <a href="https://en.wikipedia.org/wiki/QR_decomposition" target="_blank">QR decomposition</a> method.
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;cassert&gt;</code><br />
<code>#include &lt;cmath&gt;</code><br />
<code>#include &lt;cstdlib&gt;</code><br />
<code>#include &lt;ctime&gt;</code><br />
<code>#include &lt;iostream&gt;</code><br />
<code>#include &quot;<a class="el" href="../../d4/d68/qr__decompose_8h_source.html">./qr_decompose.h</a>&quot;</code><br />
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Include dependency graph for qr_eigen_values.cpp:</div>
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Namespaces</h2></td></tr>
<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">namespace &#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d2/d3b/namespaceqr__algorithm.html">qr_algorithm</a></td></tr>
<tr class="memdesc:d2/d3b/namespaceqr__algorithm"><td class="mdescLeft">&#160;</td><td class="mdescRight">Functions to compute <a href="https://en.wikipedia.org/wiki/QR_decomposition" target="_blank">QR decomposition</a> of any rectangular matrix. <br /></td></tr>
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Functions</h2></td></tr>
<tr class="memitem:a9bbf469d5525a816b0d6ca812119093d" id="r_a9bbf469d5525a816b0d6ca812119093d"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a9bbf469d5525a816b0d6ca812119093d">create_matrix</a> (<a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; &gt; *A)</td></tr>
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<tr class="memitem:abb8bf4c55e10685a5eb2ad3797fde1ae" id="r_abb8bf4c55e10685a5eb2ad3797fde1ae"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#abb8bf4c55e10685a5eb2ad3797fde1ae">mat_mul</a> (const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; &gt; &amp;A, const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; &gt; &amp;B, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; &gt; *OUT)</td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>Compute real eigen values and eigen vectors of a symmetric matrix using <a href="https://en.wikipedia.org/wiki/QR_decomposition" target="_blank">QR decomposition</a> method. </p>
<dl class="section author"><dt>Author</dt><dd><a href="https://github.com/kvedala" target="_blank">Krishna Vedala</a> </dd></dl>
</div><h2 class="groupheader">Macro Definition Documentation</h2>
<a id="aee57a411f07599034f5ceb8cc7d65b40" name="aee57a411f07599034f5ceb8cc7d65b40"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aee57a411f07599034f5ceb8cc7d65b40">&#9670;&#160;</a></span>LIMS</h2>
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<td class="memname">#define LIMS&#160;&#160;&#160;9</td>
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<p>limit of range of matrix values </p>
</div>
</div>
<h2 class="groupheader">Function Documentation</h2>
<a id="a9bbf469d5525a816b0d6ca812119093d" name="a9bbf469d5525a816b0d6ca812119093d"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a9bbf469d5525a816b0d6ca812119093d">&#9670;&#160;</a></span>create_matrix()</h2>
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<td class="memname">void create_matrix </td>
<td>(</td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; &gt; *</td> <td class="paramname"><span class="paramname"><em>A</em></span></td><td>)</td>
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<p>create a symmetric square matrix of given size with random elements. A symmetric square matrix will <em>always</em> have real eigen values.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[out]</td><td class="paramname">A</td><td>matrix to create (must be pre-allocated in memory) </td></tr>
</table>
</dd>
</dl>
<div class="fragment"><div class="line"><span class="lineno"> 28</span> {</div>
<div class="line"><span class="lineno"> 29</span> <span class="keywordtype">int</span> i, j, tmp, lim2 = <a class="code hl_define" href="#aee57a411f07599034f5ceb8cc7d65b40">LIMS</a> &gt;&gt; 1;</div>
<div class="line"><span class="lineno"> 30</span> <span class="keywordtype">int</span> <a class="code hl_variable" href="../../d8/dab/sparse__table_8cpp.html#a10f3ffb3f6f7e1b83d556b9c8de89a5d">N</a> = A-&gt;size();</div>
<div class="line"><span class="lineno"> 31</span> </div>
<div class="line"><span class="lineno"> 32</span><span class="preprocessor">#ifdef _OPENMP</span></div>
<div class="line"><span class="lineno"> 33</span><span class="preprocessor">#pragma omp for</span></div>
<div class="line"><span class="lineno"> 34</span><span class="preprocessor">#endif</span></div>
<div class="line"><span class="lineno"> 35</span> <span class="keywordflow">for</span> (i = 0; i &lt; <a class="code hl_variable" href="../../d8/dab/sparse__table_8cpp.html#a10f3ffb3f6f7e1b83d556b9c8de89a5d">N</a>; i++) {</div>
<div class="line"><span class="lineno"> 36</span> A[0][i][i] = (<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/rand.html">std::rand</a>() % <a class="code hl_define" href="#aee57a411f07599034f5ceb8cc7d65b40">LIMS</a>) - lim2;</div>
<div class="line"><span class="lineno"> 37</span> <span class="keywordflow">for</span> (j = i + 1; j &lt; <a class="code hl_variable" href="../../d8/dab/sparse__table_8cpp.html#a10f3ffb3f6f7e1b83d556b9c8de89a5d">N</a>; j++) {</div>
<div class="line"><span class="lineno"> 38</span> tmp = (<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/rand.html">std::rand</a>() % <a class="code hl_define" href="#aee57a411f07599034f5ceb8cc7d65b40">LIMS</a>) - lim2;</div>
<div class="line"><span class="lineno"> 39</span> A[0][i][j] = tmp; <span class="comment">// summetrically distribute random values</span></div>
<div class="line"><span class="lineno"> 40</span> A[0][j][i] = tmp;</div>
<div class="line"><span class="lineno"> 41</span> }</div>
<div class="line"><span class="lineno"> 42</span> }</div>
<div class="line"><span class="lineno"> 43</span>}</div>
<div class="ttc" id="aqr__eigen__values_8cpp_html_aee57a411f07599034f5ceb8cc7d65b40"><div class="ttname"><a href="#aee57a411f07599034f5ceb8cc7d65b40">LIMS</a></div><div class="ttdeci">#define LIMS</div><div class="ttdef"><b>Definition</b> qr_eigen_values.cpp:20</div></div>
<div class="ttc" id="arand_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/random/rand.html">std::rand</a></div><div class="ttdeci">T rand(T... args)</div></div>
<div class="ttc" id="asparse__table_8cpp_html_a10f3ffb3f6f7e1b83d556b9c8de89a5d"><div class="ttname"><a href="../../d8/dab/sparse__table_8cpp.html#a10f3ffb3f6f7e1b83d556b9c8de89a5d">data_structures::sparse_table::N</a></div><div class="ttdeci">constexpr uint32_t N</div><div class="ttdoc">A struct to represent sparse table for min() as their invariant function, for the given array A....</div><div class="ttdef"><b>Definition</b> sparse_table.cpp:48</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a3c04138a5bfe5d72780bb7e82a18e627">&#9670;&#160;</a></span>main()</h2>
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<td class="memname">int main </td>
<td>(</td>
<td class="paramtype">int</td> <td class="paramname"><span class="paramname"><em>argc</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">char **</td> <td class="paramname"><span class="paramname"><em>argv</em></span>&#160;)</td>
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<p>main function </p>
<div class="fragment"><div class="line"><span class="lineno"> 243</span> {</div>
<div class="line"><span class="lineno"> 244</span> <span class="keywordtype">int</span> <a class="code hl_variable" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9977ad12548c4a49dee9dc3f0685aa54">mat_size</a> = 5;</div>
<div class="line"><span class="lineno"> 245</span> <span class="keywordflow">if</span> (argc == 2) {</div>
<div class="line"><span class="lineno"> 246</span> <a class="code hl_variable" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9977ad12548c4a49dee9dc3f0685aa54">mat_size</a> = <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/byte/atoi.html">atoi</a>(argv[1]);</div>
<div class="line"><span class="lineno"> 247</span> } <span class="keywordflow">else</span> { <span class="comment">// if invalid input argument is given run tests</span></div>
<div class="line"><span class="lineno"> 248</span> <a class="code hl_function" href="#a1440a7779ac56f47a3f355ce4a8c7da0">test1</a>();</div>
<div class="line"><span class="lineno"> 249</span> <a class="code hl_function" href="#a0283886819c7c140a023582b7269e2d0">test2</a>();</div>
<div class="line"><span class="lineno"> 250</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Usage: ./qr_eigen_values [mat_size]\n&quot;</span>;</div>
<div class="line"><span class="lineno"> 251</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><span class="lineno"> 252</span> }</div>
<div class="line"><span class="lineno"> 253</span> </div>
<div class="line"><span class="lineno"> 254</span> <span class="keywordflow">if</span> (<a class="code hl_variable" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9977ad12548c4a49dee9dc3f0685aa54">mat_size</a> &lt; 2) {</div>
<div class="line"><span class="lineno"> 255</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/c/fprintf.html">fprintf</a>(stderr, <span class="stringliteral">&quot;Matrix size should be &gt; 2\n&quot;</span>);</div>
<div class="line"><span class="lineno"> 256</span> <span class="keywordflow">return</span> -1;</div>
<div class="line"><span class="lineno"> 257</span> }</div>
<div class="line"><span class="lineno"> 258</span> </div>
<div class="line"><span class="lineno"> 259</span> <span class="comment">// initialize random number generator</span></div>
<div class="line"><span class="lineno"> 260</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/srand.html">std::srand</a>(<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/time.html">std::time</a>(<span class="keyword">nullptr</span>));</div>
<div class="line"><span class="lineno"> 261</span> </div>
<div class="line"><span class="lineno"> 262</span> <span class="keywordtype">int</span> i, rows = <a class="code hl_variable" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9977ad12548c4a49dee9dc3f0685aa54">mat_size</a>, columns = <a class="code hl_variable" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9977ad12548c4a49dee9dc3f0685aa54">mat_size</a>;</div>
<div class="line"><span class="lineno"> 263</span> </div>
<div class="line"><span class="lineno"> 264</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;std::valarray&lt;double&gt;</a>&gt; A(rows);</div>
<div class="line"><span class="lineno"> 265</span> </div>
<div class="line"><span class="lineno"> 266</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; rows; i++) {</div>
<div class="line"><span class="lineno"> 267</span> A[i] = <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a>(columns);</div>
<div class="line"><span class="lineno"> 268</span> }</div>
<div class="line"><span class="lineno"> 269</span> </div>
<div class="line"><span class="lineno"> 270</span> <span class="comment">/* create a random matrix */</span></div>
<div class="line"><span class="lineno"> 271</span> <a class="code hl_function" href="#a9bbf469d5525a816b0d6ca812119093d">create_matrix</a>(&amp;A);</div>
<div class="line"><span class="lineno"> 272</span> </div>
<div class="line"><span class="lineno"> 273</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; A &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><span class="lineno"> 274</span> </div>
<div class="line"><span class="lineno"> 275</span> clock_t t1 = <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">clock</a>();</div>
<div class="line"><span class="lineno"> 276</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a> eigen_vals = <a class="code hl_function" href="../../d2/d3b/namespaceqr__algorithm.html#a28e2fa3e803abaea6c568dc45d69d8cc">qr_algorithm::eigen_values</a>(&amp;A);</div>
<div class="line"><span class="lineno"> 277</span> <span class="keywordtype">double</span> dtime = <span class="keyword">static_cast&lt;</span><span class="keywordtype">double</span><span class="keyword">&gt;</span>(<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">clock</a>() - t1) / CLOCKS_PER_SEC;</div>
<div class="line"><span class="lineno"> 278</span> </div>
<div class="line"><span class="lineno"> 279</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Eigen vals: &quot;</span>;</div>
<div class="line"><span class="lineno"> 280</span> <span class="keywordflow">for</span> (i = 0; i &lt; <a class="code hl_variable" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9977ad12548c4a49dee9dc3f0685aa54">mat_size</a>; i++) <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; eigen_vals[i] &lt;&lt; <span class="stringliteral">&quot;\t&quot;</span>;</div>
<div class="line"><span class="lineno"> 281</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;\nTime taken to compute: &quot;</span> &lt;&lt; dtime &lt;&lt; <span class="stringliteral">&quot; sec\n&quot;</span>;</div>
<div class="line"><span class="lineno"> 282</span> </div>
<div class="line"><span class="lineno"> 283</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><span class="lineno"> 284</span>}</div>
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<div class="ttc" id="amatrix__exponentiation_8cpp_html_a9977ad12548c4a49dee9dc3f0685aa54"><div class="ttname"><a href="../../d7/d35/matrix__exponentiation_8cpp.html#a9977ad12548c4a49dee9dc3f0685aa54">mat_size</a></div><div class="ttdeci">ll mat_size</div><div class="ttdef"><b>Definition</b> matrix_exponentiation.cpp:45</div></div>
<div class="ttc" id="anamespaceqr__algorithm_html_a28e2fa3e803abaea6c568dc45d69d8cc"><div class="ttname"><a href="../../d2/d3b/namespaceqr__algorithm.html#a28e2fa3e803abaea6c568dc45d69d8cc">qr_algorithm::eigen_values</a></div><div class="ttdeci">std::valarray&lt; double &gt; eigen_values(std::valarray&lt; std::valarray&lt; double &gt; &gt; *A, bool print_intermediates=false)</div><div class="ttdef"><b>Definition</b> qr_eigen_values.cpp:98</div></div>
<div class="ttc" id="aqr__eigen__values_8cpp_html_a0283886819c7c140a023582b7269e2d0"><div class="ttname"><a href="#a0283886819c7c140a023582b7269e2d0">test2</a></div><div class="ttdeci">void test2()</div><div class="ttdef"><b>Definition</b> qr_eigen_values.cpp:210</div></div>
<div class="ttc" id="aqr__eigen__values_8cpp_html_a1440a7779ac56f47a3f355ce4a8c7da0"><div class="ttname"><a href="#a1440a7779ac56f47a3f355ce4a8c7da0">test1</a></div><div class="ttdeci">void test1()</div><div class="ttdef"><b>Definition</b> qr_eigen_values.cpp:177</div></div>
<div class="ttc" id="aqr__eigen__values_8cpp_html_a9bbf469d5525a816b0d6ca812119093d"><div class="ttname"><a href="#a9bbf469d5525a816b0d6ca812119093d">create_matrix</a></div><div class="ttdeci">void create_matrix(std::valarray&lt; std::valarray&lt; double &gt; &gt; *A)</div><div class="ttdef"><b>Definition</b> qr_eigen_values.cpp:28</div></div>
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<div class="ttc" id="avalarray_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a></div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#abb8bf4c55e10685a5eb2ad3797fde1ae">&#9670;&#160;</a></span>mat_mul()</h2>
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<td class="memname">void mat_mul </td>
<td>(</td>
<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; &gt; &amp;</td> <td class="paramname"><span class="paramname"><em>A</em></span>, </td>
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<td class="paramkey"></td>
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<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; &gt; &amp;</td> <td class="paramname"><span class="paramname"><em>B</em></span>, </td>
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<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; &gt; *</td> <td class="paramname"><span class="paramname"><em>OUT</em></span>&#160;)</td>
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<p>Perform multiplication of two matrices.</p><ul>
<li>R2 must be equal to C1</li>
<li>Resultant matrix size should be R1xC2 <dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">A</td><td>first matrix to multiply </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">B</td><td>second matrix to multiply </td></tr>
<tr><td class="paramdir">[out]</td><td class="paramname">OUT</td><td>output matrix (must be pre-allocated) </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>pointer to resultant matrix </dd></dl>
</li>
</ul>
<div class="fragment"><div class="line"><span class="lineno"> 56</span> {</div>
<div class="line"><span class="lineno"> 57</span> <span class="keywordtype">int</span> R1 = A.size();</div>
<div class="line"><span class="lineno"> 58</span> <span class="keywordtype">int</span> C1 = A[0].size();</div>
<div class="line"><span class="lineno"> 59</span> <span class="keywordtype">int</span> R2 = B.size();</div>
<div class="line"><span class="lineno"> 60</span> <span class="keywordtype">int</span> C2 = B[0].size();</div>
<div class="line"><span class="lineno"> 61</span> <span class="keywordflow">if</span> (C1 != R2) {</div>
<div class="line"><span class="lineno"> 62</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/c/perror.html">perror</a>(<span class="stringliteral">&quot;Matrix dimensions mismatch!&quot;</span>);</div>
<div class="line"><span class="lineno"> 63</span> <span class="keywordflow">return</span>;</div>
<div class="line"><span class="lineno"> 64</span> }</div>
<div class="line"><span class="lineno"> 65</span> </div>
<div class="line"><span class="lineno"> 66</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; R1; i++) {</div>
<div class="line"><span class="lineno"> 67</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = 0; j &lt; C2; j++) {</div>
<div class="line"><span class="lineno"> 68</span> OUT[0][i][j] = 0.f;</div>
<div class="line"><span class="lineno"> 69</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> k = 0; <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a1b74d828b33760094906797042b89442">k</a> &lt; C1; <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a1b74d828b33760094906797042b89442">k</a>++) {</div>
<div class="line"><span class="lineno"> 70</span> OUT[0][i][j] += A[i][<a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a1b74d828b33760094906797042b89442">k</a>] * B[<a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a1b74d828b33760094906797042b89442">k</a>][j];</div>
<div class="line"><span class="lineno"> 71</span> }</div>
<div class="line"><span class="lineno"> 72</span> }</div>
<div class="line"><span class="lineno"> 73</span> }</div>
<div class="line"><span class="lineno"> 74</span>}</div>
<div class="ttc" id="acomposite__simpson__rule_8cpp_html_a1b74d828b33760094906797042b89442"><div class="ttname"><a href="../../d4/d18/composite__simpson__rule_8cpp.html#a1b74d828b33760094906797042b89442">numerical_methods::simpson_method::k</a></div><div class="ttdeci">double k(double x)</div><div class="ttdoc">Another test function.</div><div class="ttdef"><b>Definition</b> composite_simpson_rule.cpp:117</div></div>
<div class="ttc" id="aperror_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/c/perror.html">std::perror</a></div><div class="ttdeci">T perror(T... args)</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a1440a7779ac56f47a3f355ce4a8c7da0">&#9670;&#160;</a></span>test1()</h2>
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<td class="memname">void test1 </td>
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<p>test function to compute eigen values of a 2x2 matrix </p><p class="formulaDsp">
\[\begin{bmatrix}
5 &amp; 7\\
7 &amp; 11
\end{bmatrix}\]
</p>
<p> which are approximately, {15.56158, 0.384227} </p>
<div class="fragment"><div class="line"><span class="lineno"> 177</span> {</div>
<div class="line"><span class="lineno"> 178</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;std::valarray&lt;double&gt;</a>&gt; X = {{5, 7}, {7, 11}};</div>
<div class="line"><span class="lineno"> 179</span> <span class="keywordtype">double</span> y[] = {15.56158, 0.384227}; <span class="comment">// corresponding y-values</span></div>
<div class="line"><span class="lineno"> 180</span> </div>
<div class="line"><span class="lineno"> 181</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;------- Test 1 -------&quot;</span> &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><span class="lineno"> 182</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a> eig_vals = <a class="code hl_function" href="../../d2/d3b/namespaceqr__algorithm.html#a28e2fa3e803abaea6c568dc45d69d8cc">qr_algorithm::eigen_values</a>(&amp;X);</div>
<div class="line"><span class="lineno"> 183</span> </div>
<div class="line"><span class="lineno"> 184</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; 2; i++) {</div>
<div class="line"><span class="lineno"> 185</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; i + 1 &lt;&lt; <span class="stringliteral">&quot;/2 Checking for &quot;</span> &lt;&lt; y[i] &lt;&lt; <span class="stringliteral">&quot; --&gt; &quot;</span>;</div>
<div class="line"><span class="lineno"> 186</span> <span class="keywordtype">bool</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a> = <span class="keyword">false</span>;</div>
<div class="line"><span class="lineno"> 187</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = 0; j &lt; 2 &amp;&amp; !<a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>; j++) {</div>
<div class="line"><span class="lineno"> 188</span> <span class="keywordflow">if</span> (std::abs(y[i] - eig_vals[j]) &lt; 0.1) {</div>
<div class="line"><span class="lineno"> 189</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a> = <span class="keyword">true</span>;</div>
<div class="line"><span class="lineno"> 190</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;(&quot;</span> &lt;&lt; eig_vals[j] &lt;&lt; <span class="stringliteral">&quot;) &quot;</span>;</div>
<div class="line"><span class="lineno"> 191</span> }</div>
<div class="line"><span class="lineno"> 192</span> }</div>
<div class="line"><span class="lineno"> 193</span> assert(result); <span class="comment">// ensure that i^th expected eigen value was computed</span></div>
<div class="line"><span class="lineno"> 194</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;found\n&quot;</span>;</div>
<div class="line"><span class="lineno"> 195</span> }</div>
<div class="line"><span class="lineno"> 196</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Test 1 Passed\n\n&quot;</span>;</div>
<div class="line"><span class="lineno"> 197</span>}</div>
<div class="ttc" id="aendl_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a></div><div class="ttdeci">T endl(T... args)</div></div>
<div class="ttc" id="afibonacci__sum_8cpp_html_aadb40ac4c74a7efc0680b83eeee138aa"><div class="ttname"><a href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">math::fibonacci_sum::result</a></div><div class="ttdeci">uint64_t result(uint64_t n)</div><div class="ttdef"><b>Definition</b> fibonacci_sum.cpp:77</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a0283886819c7c140a023582b7269e2d0">&#9670;&#160;</a></span>test2()</h2>
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<td class="memname">void test2 </td>
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<p>test function to compute eigen values of a 2x2 matrix </p><p class="formulaDsp">
\[\begin{bmatrix}
-4&amp; 4&amp; 2&amp; 0&amp; -3\\
4&amp; -4&amp; 4&amp; -3&amp; -1\\
2&amp; 4&amp; 4&amp; 3&amp; -3\\
0&amp; -3&amp; 3&amp; -1&amp;-1\\
-3&amp; -1&amp; -3&amp; -3&amp; 0
\end{bmatrix}\]
</p>
<p> which are approximately, {9.27648, -9.26948, 2.0181, -1.03516, -5.98994} </p>
<div class="fragment"><div class="line"><span class="lineno"> 210</span> {</div>
<div class="line"><span class="lineno"> 211</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;std::valarray&lt;double&gt;</a>&gt; X = {{-4, 4, 2, 0, -3},</div>
<div class="line"><span class="lineno"> 212</span> {4, -4, 4, -3, -1},</div>
<div class="line"><span class="lineno"> 213</span> {2, 4, 4, 3, -3},</div>
<div class="line"><span class="lineno"> 214</span> {0, -3, 3, -1, -3},</div>
<div class="line"><span class="lineno"> 215</span> {-3, -1, -3, -3, 0}};</div>
<div class="line"><span class="lineno"> 216</span> <span class="keywordtype">double</span> y[] = {9.27648, -9.26948, 2.0181, -1.03516,</div>
<div class="line"><span class="lineno"> 217</span> -5.98994}; <span class="comment">// corresponding y-values</span></div>
<div class="line"><span class="lineno"> 218</span> </div>
<div class="line"><span class="lineno"> 219</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;------- Test 2 -------&quot;</span> &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><span class="lineno"> 220</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a> eig_vals = <a class="code hl_function" href="../../d2/d3b/namespaceqr__algorithm.html#a28e2fa3e803abaea6c568dc45d69d8cc">qr_algorithm::eigen_values</a>(&amp;X);</div>
<div class="line"><span class="lineno"> 221</span> </div>
<div class="line"><span class="lineno"> 222</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; X &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span></div>
<div class="line"><span class="lineno"> 223</span> &lt;&lt; <span class="stringliteral">&quot;Eigen values: &quot;</span> &lt;&lt; eig_vals &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><span class="lineno"> 224</span> </div>
<div class="line"><span class="lineno"> 225</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; 5; i++) {</div>
<div class="line"><span class="lineno"> 226</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; i + 1 &lt;&lt; <span class="stringliteral">&quot;/5 Checking for &quot;</span> &lt;&lt; y[i] &lt;&lt; <span class="stringliteral">&quot; --&gt; &quot;</span>;</div>
<div class="line"><span class="lineno"> 227</span> <span class="keywordtype">bool</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a> = <span class="keyword">false</span>;</div>
<div class="line"><span class="lineno"> 228</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = 0; j &lt; 5 &amp;&amp; !<a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>; j++) {</div>
<div class="line"><span class="lineno"> 229</span> <span class="keywordflow">if</span> (std::abs(y[i] - eig_vals[j]) &lt; 0.1) {</div>
<div class="line"><span class="lineno"> 230</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a> = <span class="keyword">true</span>;</div>
<div class="line"><span class="lineno"> 231</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;(&quot;</span> &lt;&lt; eig_vals[j] &lt;&lt; <span class="stringliteral">&quot;) &quot;</span>;</div>
<div class="line"><span class="lineno"> 232</span> }</div>
<div class="line"><span class="lineno"> 233</span> }</div>
<div class="line"><span class="lineno"> 234</span> assert(result); <span class="comment">// ensure that i^th expected eigen value was computed</span></div>
<div class="line"><span class="lineno"> 235</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;found\n&quot;</span>;</div>
<div class="line"><span class="lineno"> 236</span> }</div>
<div class="line"><span class="lineno"> 237</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Test 2 Passed\n\n&quot;</span>;</div>
<div class="line"><span class="lineno"> 238</span>}</div>
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