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 <span id="projectnumber">1.0.0</span>
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<div class="title">qr_decompose.h</div> </div>
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<a href="../../d4/d68/qr__decompose_8h.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno"> 1</span> <span class="comment">/**</span></div>
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<div class="line"><a name="l00002"></a><span class="lineno"> 2</span> <span class="comment"> * @file</span></div>
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<div class="line"><a name="l00003"></a><span class="lineno"> 3</span> <span class="comment"> * \brief Library functions to compute [QR</span></div>
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<div class="line"><a name="l00004"></a><span class="lineno"> 4</span> <span class="comment"> * decomposition](https://en.wikipedia.org/wiki/QR_decomposition) of a given</span></div>
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<div class="line"><a name="l00005"></a><span class="lineno"> 5</span> <span class="comment"> * matrix.</span></div>
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<div class="line"><a name="l00006"></a><span class="lineno"> 6</span> <span class="comment"> * \author [Krishna Vedala](https://github.com/kvedala)</span></div>
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<div class="line"><a name="l00007"></a><span class="lineno"> 7</span> <span class="comment"> */</span></div>
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<div class="line"><a name="l00008"></a><span class="lineno"> 8</span>  </div>
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<div class="line"><a name="l00009"></a><span class="lineno"> 9</span> <span class="preprocessor">#ifndef NUMERICAL_METHODS_QR_DECOMPOSE_H_</span></div>
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<div class="line"><a name="l00010"></a><span class="lineno"> 10</span> <span class="preprocessor">#define NUMERICAL_METHODS_QR_DECOMPOSE_H_</span></div>
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<div class="line"><a name="l00011"></a><span class="lineno"> 11</span>  </div>
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<div class="line"><a name="l00012"></a><span class="lineno"> 12</span> <span class="preprocessor">#include <cmath></span></div>
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<div class="line"><a name="l00013"></a><span class="lineno"> 13</span> <span class="preprocessor">#include <cstdlib></span></div>
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<div class="line"><a name="l00014"></a><span class="lineno"> 14</span> <span class="preprocessor">#include <iomanip></span></div>
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<div class="line"><a name="l00015"></a><span class="lineno"> 15</span> <span class="preprocessor">#include <iostream></span></div>
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<div class="line"><a name="l00016"></a><span class="lineno"> 16</span> <span class="preprocessor">#include <limits></span></div>
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<div class="line"><a name="l00017"></a><span class="lineno"> 17</span> <span class="preprocessor">#include <numeric></span></div>
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<div class="line"><a name="l00018"></a><span class="lineno"> 18</span> <span class="preprocessor">#include <valarray></span></div>
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<div class="line"><a name="l00019"></a><span class="lineno"> 19</span> <span class="preprocessor">#ifdef _OPENMP</span></div>
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<div class="line"><a name="l00020"></a><span class="lineno"> 20</span> <span class="preprocessor">#include <omp.h></span></div>
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<div class="line"><a name="l00021"></a><span class="lineno"> 21</span> <span class="preprocessor">#endif</span></div>
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<div class="line"><a name="l00022"></a><span class="lineno"> 22</span> <span class="comment"></span> </div>
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<div class="line"><a name="l00023"></a><span class="lineno"><a class="line" href="../../d2/d3b/namespaceqr__algorithm.html"> 23</a></span> <span class="comment">/** \namespace qr_algorithm</span></div>
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<div class="line"><a name="l00024"></a><span class="lineno"> 24</span> <span class="comment"> * \brief Functions to compute [QR</span></div>
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<div class="line"><a name="l00025"></a><span class="lineno"> 25</span> <span class="comment"> * decomposition](https://en.wikipedia.org/wiki/QR_decomposition) of any</span></div>
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<div class="line"><a name="l00026"></a><span class="lineno"> 26</span> <span class="comment"> * rectangular matrix</span></div>
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<div class="line"><a name="l00027"></a><span class="lineno"> 27</span> <span class="comment"> */</span></div>
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<div class="line"><a name="l00028"></a><span class="lineno"> 28</span> <span class="keyword">namespace </span><a class="code" href="../../d2/d3b/namespaceqr__algorithm.html">qr_algorithm</a> {<span class="comment"></span></div>
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<div class="line"><a name="l00029"></a><span class="lineno"> 29</span> <span class="comment">/**</span></div>
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<div class="line"><a name="l00030"></a><span class="lineno"> 30</span> <span class="comment"> * operator to print a matrix</span></div>
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<div class="line"><a name="l00031"></a><span class="lineno"> 31</span> <span class="comment"> */</span></div>
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<div class="line"><a name="l00032"></a><span class="lineno"> 32</span> <span class="keyword">template</span> <<span class="keyword">typename</span> T></div>
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<div class="line"><a name="l00033"></a><span class="lineno"><a class="line" href="../../d2/d3b/namespaceqr__algorithm.html#a7c63c78d6839b8ad73b77c161210b514"> 33</a></span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> &<a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a7c63c78d6839b8ad73b77c161210b514">operator<<</a>(<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> &out,</div>
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<div class="line"><a name="l00034"></a><span class="lineno"> 34</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a><<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a>> <span class="keyword">const</span> &v) {</div>
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<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>  <span class="keyword">const</span> <span class="keywordtype">int</span> width = 12;</div>
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<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>  <span class="keyword">const</span> <span class="keywordtype">char</span> separator = <span class="charliteral">' '</span>;</div>
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<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>  </div>
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<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>  out.precision(4);</div>
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<div class="line"><a name="l00039"></a><span class="lineno"> 39</span>  <span class="keywordflow">for</span> (<span class="keywordtype">size_t</span> row = 0; row < v.size(); row++) {</div>
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<div class="line"><a name="l00040"></a><span class="lineno"> 40</span>  <span class="keywordflow">for</span> (<span class="keywordtype">size_t</span> col = 0; col < v[row].size(); col++)</div>
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<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>  out << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/left.html">std::right</a> << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setw.html">std::setw</a>(width) << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setfill.html">std::setfill</a>(separator)</div>
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<div class="line"><a name="l00042"></a><span class="lineno"> 42</span>  << v[row][col];</div>
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<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>  out << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
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<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>  }</div>
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<div class="line"><a name="l00045"></a><span class="lineno"> 45</span>  </div>
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<div class="line"><a name="l00046"></a><span class="lineno"> 46</span>  <span class="keywordflow">return</span> out;</div>
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<div class="line"><a name="l00047"></a><span class="lineno"> 47</span> }</div>
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<div class="line"><a name="l00048"></a><span class="lineno"> 48</span> <span class="comment"></span> </div>
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<div class="line"><a name="l00049"></a><span class="lineno"> 49</span> <span class="comment">/**</span></div>
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<div class="line"><a name="l00050"></a><span class="lineno"> 50</span> <span class="comment"> * operator to print a vector</span></div>
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<div class="line"><a name="l00051"></a><span class="lineno"> 51</span> <span class="comment"> */</span></div>
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<div class="line"><a name="l00052"></a><span class="lineno"> 52</span> <span class="keyword">template</span> <<span class="keyword">typename</span> T></div>
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<div class="line"><a name="l00053"></a><span class="lineno"><a class="line" href="../../d2/d3b/namespaceqr__algorithm.html#a240edf2b6b5942c0e5f2c9d3e326797f"> 53</a></span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> &<a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a7c63c78d6839b8ad73b77c161210b514">operator<<</a>(<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> &out, <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a> <span class="keyword">const</span> &v) {</div>
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<div class="line"><a name="l00054"></a><span class="lineno"> 54</span>  <span class="keyword">const</span> <span class="keywordtype">int</span> width = 10;</div>
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<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>  <span class="keyword">const</span> <span class="keywordtype">char</span> separator = <span class="charliteral">' '</span>;</div>
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<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>  </div>
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<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>  out.precision(4);</div>
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<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>  <span class="keywordflow">for</span> (<span class="keywordtype">size_t</span> row = 0; row < v.size(); row++) {</div>
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<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>  out << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/left.html">std::right</a> << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setw.html">std::setw</a>(width) << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setfill.html">std::setfill</a>(separator)</div>
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<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>  << v[row];</div>
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<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>  }</div>
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<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>  </div>
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<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>  <span class="keywordflow">return</span> out;</div>
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<div class="line"><a name="l00064"></a><span class="lineno"> 64</span> }</div>
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<div class="line"><a name="l00065"></a><span class="lineno"> 65</span> <span class="comment"></span> </div>
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<div class="line"><a name="l00066"></a><span class="lineno"> 66</span> <span class="comment">/**</span></div>
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<div class="line"><a name="l00067"></a><span class="lineno"> 67</span> <span class="comment"> * Compute dot product of two vectors of equal lengths</span></div>
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<div class="line"><a name="l00068"></a><span class="lineno"> 68</span> <span class="comment"> *</span></div>
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<div class="line"><a name="l00069"></a><span class="lineno"> 69</span> <span class="comment"> * If \f$\vec{a}=\left[a_0,a_1,a_2,...,a_L\right]\f$ and</span></div>
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<div class="line"><a name="l00070"></a><span class="lineno"> 70</span> <span class="comment"> * \f$\vec{b}=\left[b_0,b_1,b_1,...,b_L\right]\f$ then</span></div>
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<div class="line"><a name="l00071"></a><span class="lineno"> 71</span> <span class="comment"> * \f$\vec{a}\cdot\vec{b}=\displaystyle\sum_{i=0}^L a_i\times b_i\f$</span></div>
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<div class="line"><a name="l00072"></a><span class="lineno"> 72</span> <span class="comment"> *</span></div>
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<div class="line"><a name="l00073"></a><span class="lineno"> 73</span> <span class="comment"> * \returns \f$\vec{a}\cdot\vec{b}\f$</span></div>
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<div class="line"><a name="l00074"></a><span class="lineno"> 74</span> <span class="comment"> */</span></div>
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<div class="line"><a name="l00075"></a><span class="lineno"> 75</span> <span class="keyword">template</span> <<span class="keyword">typename</span> T></div>
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<div class="line"><a name="l00076"></a><span class="lineno"><a class="line" href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec"> 76</a></span> <span class="keyword">inline</span> <span class="keywordtype">double</span> <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(<span class="keyword">const</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a> &a, <span class="keyword">const</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a> &b) {</div>
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<div class="line"><a name="l00077"></a><span class="lineno"> 77</span>  <span class="keywordflow">return</span> (a * b).sum();</div>
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<div class="line"><a name="l00078"></a><span class="lineno"> 78</span>  <span class="comment">// could also use following</span></div>
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<div class="line"><a name="l00079"></a><span class="lineno"> 79</span>  <span class="comment">// return std::inner_product(std::begin(a), std::end(a), std::begin(b),</span></div>
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<div class="line"><a name="l00080"></a><span class="lineno"> 80</span>  <span class="comment">// 0.f);</span></div>
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<div class="line"><a name="l00081"></a><span class="lineno"> 81</span> }</div>
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<div class="line"><a name="l00082"></a><span class="lineno"> 82</span> <span class="comment"></span> </div>
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<div class="line"><a name="l00083"></a><span class="lineno"> 83</span> <span class="comment">/**</span></div>
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<div class="line"><a name="l00084"></a><span class="lineno"> 84</span> <span class="comment"> * Compute magnitude of vector.</span></div>
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<div class="line"><a name="l00085"></a><span class="lineno"> 85</span> <span class="comment"> *</span></div>
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<div class="line"><a name="l00086"></a><span class="lineno"> 86</span> <span class="comment"> * If \f$\vec{a}=\left[a_0,a_1,a_2,...,a_L\right]\f$ then</span></div>
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<div class="line"><a name="l00087"></a><span class="lineno"> 87</span> <span class="comment"> * \f$\left|\vec{a}\right|=\sqrt{\displaystyle\sum_{i=0}^L a_i^2}\f$</span></div>
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<div class="line"><a name="l00088"></a><span class="lineno"> 88</span> <span class="comment"> *</span></div>
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<div class="line"><a name="l00089"></a><span class="lineno"> 89</span> <span class="comment"> * \returns \f$\left|\vec{a}\right|\f$</span></div>
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<div class="line"><a name="l00090"></a><span class="lineno"> 90</span> <span class="comment"> */</span></div>
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<div class="line"><a name="l00091"></a><span class="lineno"> 91</span> <span class="keyword">template</span> <<span class="keyword">typename</span> T></div>
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<div class="line"><a name="l00092"></a><span class="lineno"><a class="line" href="../../d2/d3b/namespaceqr__algorithm.html#ad16da2183db22378435042f26af43d5f"> 92</a></span> <span class="keyword">inline</span> <span class="keywordtype">double</span> <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#ad16da2183db22378435042f26af43d5f">vector_mag</a>(<span class="keyword">const</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a> &a) {</div>
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<div class="line"><a name="l00093"></a><span class="lineno"> 93</span>  <span class="keywordtype">double</span> dot = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(a, a);</div>
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<div class="line"><a name="l00094"></a><span class="lineno"> 94</span>  <span class="keywordflow">return</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a>(dot);</div>
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<div class="line"><a name="l00095"></a><span class="lineno"> 95</span> }</div>
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<div class="line"><a name="l00096"></a><span class="lineno"> 96</span> <span class="comment"></span> </div>
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<div class="line"><a name="l00097"></a><span class="lineno"> 97</span> <span class="comment">/**</span></div>
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<div class="line"><a name="l00098"></a><span class="lineno"> 98</span> <span class="comment"> * Compute projection of vector \f$\vec{a}\f$ on \f$\vec{b}\f$ defined as</span></div>
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<div class="line"><a name="l00099"></a><span class="lineno"> 99</span> <span class="comment"> * \f[\text{proj}_\vec{b}\vec{a}=\frac{\vec{a}\cdot\vec{b}}{\left|\vec{b}\right|^2}\vec{b}\f]</span></div>
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<div class="line"><a name="l00100"></a><span class="lineno"> 100</span> <span class="comment"> *</span></div>
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<div class="line"><a name="l00101"></a><span class="lineno"> 101</span> <span class="comment"> * \returns NULL if error, otherwise pointer to output</span></div>
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<div class="line"><a name="l00102"></a><span class="lineno"> 102</span> <span class="comment"> */</span></div>
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<div class="line"><a name="l00103"></a><span class="lineno"> 103</span> <span class="keyword">template</span> <<span class="keyword">typename</span> T></div>
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<div class="line"><a name="l00104"></a><span class="lineno"><a class="line" href="../../d2/d3b/namespaceqr__algorithm.html#a5422c76bfa322620de611083bd518d71"> 104</a></span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a> <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a5422c76bfa322620de611083bd518d71">vector_proj</a>(<span class="keyword">const</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a> &a,</div>
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<div class="line"><a name="l00105"></a><span class="lineno"> 105</span>  <span class="keyword">const</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a> &b) {</div>
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<div class="line"><a name="l00106"></a><span class="lineno"> 106</span>  <span class="keywordtype">double</span> num = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(a, b);</div>
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<div class="line"><a name="l00107"></a><span class="lineno"> 107</span>  <span class="keywordtype">double</span> deno = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(b, b);</div>
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<div class="line"><a name="l00108"></a><span class="lineno"> 108</span> <span class="comment"></span> </div>
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<div class="line"><a name="l00109"></a><span class="lineno"> 109</span> <span class="comment"> /*! check for division by zero using machine epsilon */</span></div>
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<div class="line"><a name="l00110"></a><span class="lineno"> 110</span>  <span class="keywordflow">if</span> (deno <= <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/numeric_limits.html">std::numeric_limits<double>::epsilon</a>()) {</div>
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<div class="line"><a name="l00111"></a><span class="lineno"> 111</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cerr</a> << <span class="stringliteral">"["</span> << __func__ << <span class="stringliteral">"] Possible division by zero\n"</span>;</div>
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<div class="line"><a name="l00112"></a><span class="lineno"> 112</span>  <span class="keywordflow">return</span> a; <span class="comment">// return vector a back</span></div>
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<div class="line"><a name="l00113"></a><span class="lineno"> 113</span>  }</div>
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<div class="line"><a name="l00114"></a><span class="lineno"> 114</span>  </div>
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<div class="line"><a name="l00115"></a><span class="lineno"> 115</span>  <span class="keywordtype">double</span> scalar = num / deno;</div>
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<div class="line"><a name="l00116"></a><span class="lineno"> 116</span>  </div>
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<div class="line"><a name="l00117"></a><span class="lineno"> 117</span>  <span class="keywordflow">return</span> b * scalar;</div>
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<div class="line"><a name="l00118"></a><span class="lineno"> 118</span> }</div>
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<div class="line"><a name="l00119"></a><span class="lineno"> 119</span> <span class="comment"></span> </div>
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<div class="line"><a name="l00120"></a><span class="lineno"> 120</span> <span class="comment">/**</span></div>
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<div class="line"><a name="l00121"></a><span class="lineno"> 121</span> <span class="comment"> * Decompose matrix \f$A\f$ using [Gram-Schmidt</span></div>
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<div class="line"><a name="l00122"></a><span class="lineno"> 122</span> <span class="comment"> *process](https://en.wikipedia.org/wiki/QR_decomposition).</span></div>
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<div class="line"><a name="l00123"></a><span class="lineno"> 123</span> <span class="comment"> *</span></div>
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<div class="line"><a name="l00124"></a><span class="lineno"> 124</span> <span class="comment"> * \f{eqnarray*}{</span></div>
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<div class="line"><a name="l00125"></a><span class="lineno"> 125</span> <span class="comment"> * \text{given that}\quad A &=&</span></div>
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<div class="line"><a name="l00126"></a><span class="lineno"> 126</span> <span class="comment"> *\left[\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_{N-1},\right]\\</span></div>
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<div class="line"><a name="l00127"></a><span class="lineno"> 127</span> <span class="comment"> * \text{where}\quad\mathbf{a}_i &=&</span></div>
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<div class="line"><a name="l00128"></a><span class="lineno"> 128</span> <span class="comment"> * \left[a_{0i},a_{1i},a_{2i},\ldots,a_{(M-1)i}\right]^T\quad\ldots\mbox{(column</span></div>
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<div class="line"><a name="l00129"></a><span class="lineno"> 129</span> <span class="comment"> * vectors)}\\</span></div>
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<div class="line"><a name="l00130"></a><span class="lineno"> 130</span> <span class="comment"> * \text{then}\quad\mathbf{u}_i &=& \mathbf{a}_i</span></div>
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<div class="line"><a name="l00131"></a><span class="lineno"> 131</span> <span class="comment"> *-\sum_{j=0}^{i-1}\text{proj}_{\mathbf{u}_j}\mathbf{a}_i\\</span></div>
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<div class="line"><a name="l00132"></a><span class="lineno"> 132</span> <span class="comment"> * \mathbf{e}_i &=&\frac{\mathbf{u}_i}{\left|\mathbf{u}_i\right|}\\</span></div>
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<div class="line"><a name="l00133"></a><span class="lineno"> 133</span> <span class="comment"> * Q &=& \begin{bmatrix}\mathbf{e}_0 & \mathbf{e}_1 & \mathbf{e}_2 & \dots &</span></div>
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<div class="line"><a name="l00134"></a><span class="lineno"> 134</span> <span class="comment"> * \mathbf{e}_{N-1}\end{bmatrix}\\</span></div>
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<div class="line"><a name="l00135"></a><span class="lineno"> 135</span> <span class="comment"> * R &=& \begin{bmatrix}\langle\mathbf{e}_0\,,\mathbf{a}_0\rangle &</span></div>
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<div class="line"><a name="l00136"></a><span class="lineno"> 136</span> <span class="comment"> * \langle\mathbf{e}_1\,,\mathbf{a}_1\rangle &</span></div>
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<div class="line"><a name="l00137"></a><span class="lineno"> 137</span> <span class="comment"> * \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle & \dots \\</span></div>
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<div class="line"><a name="l00138"></a><span class="lineno"> 138</span> <span class="comment"> * 0 & \langle\mathbf{e}_1\,,\mathbf{a}_1\rangle &</span></div>
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<div class="line"><a name="l00139"></a><span class="lineno"> 139</span> <span class="comment"> * \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle & \dots\\</span></div>
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<div class="line"><a name="l00140"></a><span class="lineno"> 140</span> <span class="comment"> * 0 & 0 & \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle &</span></div>
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<div class="line"><a name="l00141"></a><span class="lineno"> 141</span> <span class="comment"> * \dots\\ \vdots & \vdots & \vdots & \ddots</span></div>
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<div class="line"><a name="l00142"></a><span class="lineno"> 142</span> <span class="comment"> * \end{bmatrix}\\</span></div>
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<div class="line"><a name="l00143"></a><span class="lineno"> 143</span> <span class="comment"> * \f}</span></div>
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<div class="line"><a name="l00144"></a><span class="lineno"> 144</span> <span class="comment"> */</span></div>
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<div class="line"><a name="l00145"></a><span class="lineno"> 145</span> <span class="keyword">template</span> <<span class="keyword">typename</span> T></div>
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<div class="line"><a name="l00146"></a><span class="lineno"><a class="line" href="../../d2/d3b/namespaceqr__algorithm.html#ab1ac74497ffb2101040cc1efe3546de8"> 146</a></span> <span class="keywordtype">void</span> <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#ab1ac74497ffb2101040cc1efe3546de8">qr_decompose</a>(</div>
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<div class="line"><a name="l00147"></a><span class="lineno"> 147</span>  <span class="keyword">const</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a><<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a>> &A, <span class="comment">/**< input matrix to decompose */</span></div>
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<div class="line"><a name="l00148"></a><span class="lineno"> 148</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a><<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a>> *Q, <span class="comment">/**< output decomposed matrix */</span></div>
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<div class="line"><a name="l00149"></a><span class="lineno"> 149</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a><<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a>> *R <span class="comment">/**< output decomposed matrix */</span></div>
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<div class="line"><a name="l00150"></a><span class="lineno"> 150</span> ) {</div>
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<div class="line"><a name="l00151"></a><span class="lineno"> 151</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/size_t.html">std::size_t</a> ROWS = A.size(); <span class="comment">// number of rows of A</span></div>
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<div class="line"><a name="l00152"></a><span class="lineno"> 152</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/size_t.html">std::size_t</a> COLUMNS = A[0].size(); <span class="comment">// number of columns of A</span></div>
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<div class="line"><a name="l00153"></a><span class="lineno"> 153</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a> col_vector(ROWS);</div>
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<div class="line"><a name="l00154"></a><span class="lineno"> 154</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a> col_vector2(ROWS);</div>
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<div class="line"><a name="l00155"></a><span class="lineno"> 155</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a> tmp_vector(ROWS);</div>
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<div class="line"><a name="l00156"></a><span class="lineno"> 156</span>  </div>
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<div class="line"><a name="l00157"></a><span class="lineno"> 157</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < COLUMNS; i++) {</div>
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<div class="line"><a name="l00158"></a><span class="lineno"> 158</span>  <span class="comment">/* for each column => R is a square matrix of NxN */</span></div>
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<div class="line"><a name="l00159"></a><span class="lineno"> 159</span>  <span class="keywordtype">int</span> j;</div>
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<div class="line"><a name="l00160"></a><span class="lineno"> 160</span>  R[0][i] = 0.; <span class="comment">/* make R upper triangular */</span></div>
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<div class="line"><a name="l00161"></a><span class="lineno"> 161</span>  </div>
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<div class="line"><a name="l00162"></a><span class="lineno"> 162</span>  <span class="comment">/* get corresponding Q vector */</span></div>
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<div class="line"><a name="l00163"></a><span class="lineno"> 163</span> <span class="preprocessor">#ifdef _OPENMP</span></div>
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<div class="line"><a name="l00164"></a><span class="lineno"> 164</span> <span class="comment">// parallelize on threads</span></div>
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<div class="line"><a name="l00165"></a><span class="lineno"> 165</span> <span class="preprocessor">#pragma omp for</span></div>
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<div class="line"><a name="l00166"></a><span class="lineno"> 166</span> <span class="preprocessor">#endif</span></div>
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<div class="line"><a name="l00167"></a><span class="lineno"> 167</span>  <span class="keywordflow">for</span> (j = 0; j < ROWS; j++) {</div>
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<div class="line"><a name="l00168"></a><span class="lineno"> 168</span>  tmp_vector[j] = A[j][i]; <span class="comment">/* accumulator for uk */</span></div>
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<div class="line"><a name="l00169"></a><span class="lineno"> 169</span>  col_vector[j] = A[j][i];</div>
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<div class="line"><a name="l00170"></a><span class="lineno"> 170</span>  }</div>
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<div class="line"><a name="l00171"></a><span class="lineno"> 171</span>  <span class="keywordflow">for</span> (j = 0; j < i; j++) {</div>
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<div class="line"><a name="l00172"></a><span class="lineno"> 172</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> k = 0; k < ROWS; k++) {</div>
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<div class="line"><a name="l00173"></a><span class="lineno"> 173</span>  col_vector2[k] = Q[0][k][j];</div>
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<div class="line"><a name="l00174"></a><span class="lineno"> 174</span>  }</div>
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<div class="line"><a name="l00175"></a><span class="lineno"> 175</span>  col_vector2 = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a5422c76bfa322620de611083bd518d71">vector_proj</a>(col_vector, col_vector2);</div>
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<div class="line"><a name="l00176"></a><span class="lineno"> 176</span>  tmp_vector -= col_vector2;</div>
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<div class="line"><a name="l00177"></a><span class="lineno"> 177</span>  }</div>
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<div class="line"><a name="l00178"></a><span class="lineno"> 178</span>  </div>
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<div class="line"><a name="l00179"></a><span class="lineno"> 179</span>  <span class="keywordtype">double</span> mag = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#ad16da2183db22378435042f26af43d5f">vector_mag</a>(tmp_vector);</div>
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<div class="line"><a name="l00180"></a><span class="lineno"> 180</span>  </div>
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<div class="line"><a name="l00181"></a><span class="lineno"> 181</span> <span class="preprocessor">#ifdef _OPENMP</span></div>
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<div class="line"><a name="l00182"></a><span class="lineno"> 182</span> <span class="comment">// parallelize on threads</span></div>
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<div class="line"><a name="l00183"></a><span class="lineno"> 183</span> <span class="preprocessor">#pragma omp for</span></div>
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<div class="line"><a name="l00184"></a><span class="lineno"> 184</span> <span class="preprocessor">#endif</span></div>
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<div class="line"><a name="l00185"></a><span class="lineno"> 185</span>  <span class="keywordflow">for</span> (j = 0; j < ROWS; j++) Q[0][j][i] = tmp_vector[j] / mag;</div>
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<div class="line"><a name="l00186"></a><span class="lineno"> 186</span>  </div>
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<div class="line"><a name="l00187"></a><span class="lineno"> 187</span>  <span class="comment">/* compute upper triangular values of R */</span></div>
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<div class="line"><a name="l00188"></a><span class="lineno"> 188</span> <span class="preprocessor">#ifdef _OPENMP</span></div>
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<div class="line"><a name="l00189"></a><span class="lineno"> 189</span> <span class="comment">// parallelize on threads</span></div>
|
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<div class="line"><a name="l00190"></a><span class="lineno"> 190</span> <span class="preprocessor">#pragma omp for</span></div>
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<div class="line"><a name="l00191"></a><span class="lineno"> 191</span> <span class="preprocessor">#endif</span></div>
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<div class="line"><a name="l00192"></a><span class="lineno"> 192</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> kk = 0; kk < ROWS; kk++) {</div>
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<div class="line"><a name="l00193"></a><span class="lineno"> 193</span>  col_vector[kk] = Q[0][kk][i];</div>
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<div class="line"><a name="l00194"></a><span class="lineno"> 194</span>  }</div>
|
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<div class="line"><a name="l00195"></a><span class="lineno"> 195</span>  </div>
|
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<div class="line"><a name="l00196"></a><span class="lineno"> 196</span> <span class="preprocessor">#ifdef _OPENMP</span></div>
|
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<div class="line"><a name="l00197"></a><span class="lineno"> 197</span> <span class="comment">// parallelize on threads</span></div>
|
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<div class="line"><a name="l00198"></a><span class="lineno"> 198</span> <span class="preprocessor">#pragma omp for</span></div>
|
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<div class="line"><a name="l00199"></a><span class="lineno"> 199</span> <span class="preprocessor">#endif</span></div>
|
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<div class="line"><a name="l00200"></a><span class="lineno"> 200</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> k = i; k < COLUMNS; k++) {</div>
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<div class="line"><a name="l00201"></a><span class="lineno"> 201</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> kk = 0; kk < ROWS; kk++) {</div>
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<div class="line"><a name="l00202"></a><span class="lineno"> 202</span>  col_vector2[kk] = A[kk][k];</div>
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<div class="line"><a name="l00203"></a><span class="lineno"> 203</span>  }</div>
|
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<div class="line"><a name="l00204"></a><span class="lineno"> 204</span>  R[0][i][k] = (col_vector * col_vector2).sum();</div>
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<div class="line"><a name="l00205"></a><span class="lineno"> 205</span>  }</div>
|
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<div class="line"><a name="l00206"></a><span class="lineno"> 206</span>  }</div>
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<div class="line"><a name="l00207"></a><span class="lineno"> 207</span> }</div>
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<div class="line"><a name="l00208"></a><span class="lineno"> 208</span> } <span class="comment">// namespace qr_algorithm</span></div>
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<div class="line"><a name="l00209"></a><span class="lineno"> 209</span>  </div>
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<div class="line"><a name="l00210"></a><span class="lineno"> 210</span> <span class="preprocessor">#endif // NUMERICAL_METHODS_QR_DECOMPOSE_H_</span></div>
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</div><!-- fragment --></div><!-- contents -->
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</div><!-- doc-content -->
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<div class="ttc" id="aode__midpoint__euler_8cpp_html_aa13517b8e5de1b75592052db7f7e237f"><div class="ttname"><a href="../../d6/dd3/ode__midpoint__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a></div><div class="ttdeci">void save_exact_solution(const double &X0, const double &X_MAX, const double &step_size, const std::valarray< double > &Y0)</div><div class="ttdef"><b>Definition:</b> ode_midpoint_euler.cpp:156</div></div>
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<div class="ttc" id="asrand_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/random/srand.html">std::srand</a></div><div class="ttdeci">T srand(T... args)</div></div>
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<div class="ttc" id="agroup__ode_html_ga3874d294ab00fe9ce8731c5b7991a089"><div class="ttname"><a href="../../d6/d60/group__ode.html#ga3874d294ab00fe9ce8731c5b7991a089">semi_implicit_euler</a></div><div class="ttdeci">double semi_implicit_euler(double dx, double x0, double x_max, std::valarray< double > *y, bool save_to_file=false)</div><div class="ttdoc">Compute approximation using the semi-implicit-Euler method in the given limits.</div><div class="ttdef"><b>Definition:</b> ode_semi_implicit_euler.cpp:103</div></div>
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<div class="ttc" id="agroup__ode_html_ga827bf009831ddc477c5fa8891d5cb35f"><div class="ttname"><a href="../../d6/d60/group__ode.html#ga827bf009831ddc477c5fa8891d5cb35f">semi_implicit_euler_step</a></div><div class="ttdeci">void semi_implicit_euler_step(const double dx, const double &x, std::valarray< double > *y, std::valarray< double > *dy)</div><div class="ttdoc">Compute next step approximation using the semi-implicit-Euler method.</div><div class="ttdef"><b>Definition:</b> ode_semi_implicit_euler.cpp:82</div></div>
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<div class="ttc" id="agroup__ode_html_ga195d23bbdfcb80e83c9cda45c9ad5723"><div class="ttname"><a href="../../d6/d60/group__ode.html#ga195d23bbdfcb80e83c9cda45c9ad5723">forward_euler_step</a></div><div class="ttdeci">void forward_euler_step(const double dx, const double x, std::valarray< double > *y, std::valarray< double > *dy)</div><div class="ttdoc">Compute next step approximation using the forward-Euler method.</div><div class="ttdef"><b>Definition:</b> ode_forward_euler.cpp:86</div></div>
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<div class="ttc" id="acos_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/cos.html">std::cos</a></div><div class="ttdeci">T cos(T... args)</div></div>
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<div class="ttc" id="anewton__raphson__method_8cpp_html_a3d3f7f41977394680af6ebbed96f3386"><div class="ttname"><a href="../../de/dd3/newton__raphson__method_8cpp.html#a3d3f7f41977394680af6ebbed96f3386">eq_der</a></div><div class="ttdeci">static double eq_der(double i)</div><div class="ttdef"><b>Definition:</b> newton_raphson_method.cpp:39</div></div>
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<div class="ttc" id="aode__forward__euler_8cpp_html_af3adf7b092a87868917ee5fb4255192b"><div class="ttname"><a href="../../db/dd3/ode__forward__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a></div><div class="ttdeci">void exact_solution(const double &x, std::valarray< double > *y)</div><div class="ttdoc">Exact solution of the problem. Used for solution comparison.</div><div class="ttdef"><b>Definition:</b> ode_forward_euler.cpp:67</div></div>
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<div class="ttc" id="aatof_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/string/byte/atof.html">std::atof</a></div><div class="ttdeci">T atof(T... args)</div></div>
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<div class="ttc" id="aode__forward__euler_8cpp_html_a0ddf1224851353fc92bfbff6f499fa97"><div class="ttname"><a href="../../db/dd3/ode__forward__euler_8cpp.html#a0ddf1224851353fc92bfbff6f499fa97">main</a></div><div class="ttdeci">int main(int argc, char *argv[])</div><div class="ttdef"><b>Definition:</b> ode_forward_euler.cpp:189</div></div>
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<div class="ttc" id="asetfill_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/manip/setfill.html">std::setfill</a></div><div class="ttdeci">T setfill(T... args)</div></div>
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<div class="ttc" id="agroup__ode_html_ga8c319db420c3d97a83e9dcca803b6812"><div class="ttname"><a href="../../d6/d60/group__ode.html#ga8c319db420c3d97a83e9dcca803b6812">midpoint_euler_step</a></div><div class="ttdeci">void midpoint_euler_step(const double dx, const double &x, std::valarray< double > *y, std::valarray< double > *dy)</div><div class="ttdoc">Compute next step approximation using the midpoint-Euler method.</div><div class="ttdef"><b>Definition:</b> ode_midpoint_euler.cpp:85</div></div>
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<div class="ttc" id="asqrt_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a></div><div class="ttdeci">T sqrt(T... args)</div></div>
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<div class="ttc" id="anamespaceqr__algorithm_html"><div class="ttname"><a href="../../d2/d3b/namespaceqr__algorithm.html">qr_algorithm</a></div><div class="ttdoc">Functions to compute QR decomposition of any rectangular matrix.</div></div>
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<div class="ttc" id="aclock_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">std::clock</a></div><div class="ttdeci">T clock(T... args)</div></div>
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<div class="ttc" id="aode__forward__euler_8cpp_html_aa13517b8e5de1b75592052db7f7e237f"><div class="ttname"><a href="../../db/dd3/ode__forward__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a></div><div class="ttdeci">void save_exact_solution(const double &X0, const double &X_MAX, const double &step_size, const std::valarray< double > &Y0)</div><div class="ttdef"><b>Definition:</b> ode_forward_euler.cpp:153</div></div>
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<div class="ttc" id="anewton__raphson__method_8cpp_html_a75f355966e0a97665eaf5a4c79baa7af"><div class="ttname"><a href="../../de/dd3/newton__raphson__method_8cpp.html#a75f355966e0a97665eaf5a4c79baa7af">EPSILON</a></div><div class="ttdeci">constexpr double EPSILON</div><div class="ttdoc">system accuracy limit</div><div class="ttdef"><b>Definition:</b> newton_raphson_method.cpp:20</div></div>
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<div class="ttc" id="anamespaceqr__algorithm_html_ad16da2183db22378435042f26af43d5f"><div class="ttname"><a href="../../d2/d3b/namespaceqr__algorithm.html#ad16da2183db22378435042f26af43d5f">qr_algorithm::vector_mag</a></div><div class="ttdeci">double vector_mag(const std::valarray< T > &a)</div><div class="ttdef"><b>Definition:</b> qr_decompose.h:92</div></div>
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<div class="ttc" id="anewton__raphson__method_8cpp_html_a5c5ff05b9f37ae59dad67e1d4f6cd51d"><div class="ttname"><a href="../../de/dd3/newton__raphson__method_8cpp.html#a5c5ff05b9f37ae59dad67e1d4f6cd51d">MAX_ITERATIONS</a></div><div class="ttdeci">constexpr int16_t MAX_ITERATIONS</div><div class="ttdoc">Maximum number of iterations.</div><div class="ttdef"><b>Definition:</b> newton_raphson_method.cpp:21</div></div>
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<div class="ttc" id="aclose_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_ofstream/close.html">std::ofstream::close</a></div><div class="ttdeci">T close(T... args)</div></div>
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<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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<div class="ttc" id="anamespaceqr__algorithm_html_a7c63c78d6839b8ad73b77c161210b514"><div class="ttname"><a href="../../d2/d3b/namespaceqr__algorithm.html#a7c63c78d6839b8ad73b77c161210b514">qr_algorithm::operator<<</a></div><div class="ttdeci">std::ostream & operator<<(std::ostream &out, std::valarray< std::valarray< T >> const &v)</div><div class="ttdef"><b>Definition:</b> qr_decompose.h:33</div></div>
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<div class="ttc" id="agroup__ode_html_gae0509f8843e2bc42de2abbd00a14b7b9"><div class="ttname"><a href="../../d6/d60/group__ode.html#gae0509f8843e2bc42de2abbd00a14b7b9">forward_euler</a></div><div class="ttdeci">double forward_euler(double dx, double x0, double x_max, std::valarray< double > *y, bool save_to_file=false)</div><div class="ttdoc">Compute approximation using the forward-Euler method in the given limits.</div><div class="ttdef"><b>Definition:</b> ode_forward_euler.cpp:102</div></div>
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<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>
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<div class="ttc" id="aode__semi__implicit__euler_8cpp_html_abaeae8f62a018d197f0187a1c80a90fe"><div class="ttname"><a href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a></div><div class="ttdeci">void problem(const double &x, std::valarray< double > *y, std::valarray< double > *dy)</div><div class="ttdoc">Problem statement for a system with first-order differential equations. Updates the system differenti...</div><div class="ttdef"><b>Definition:</b> ode_semi_implicit_euler.cpp:53</div></div>
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<div class="ttc" id="asin_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/sin.html">std::sin</a></div><div class="ttdeci">T sin(T... args)</div></div>
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<div class="ttc" id="aode__forward__euler_8cpp_html_abaeae8f62a018d197f0187a1c80a90fe"><div class="ttname"><a href="../../db/dd3/ode__forward__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a></div><div class="ttdeci">void problem(const double &x, std::valarray< double > *y, std::valarray< double > *dy)</div><div class="ttdoc">Problem statement for a system with first-order differential equations. Updates the system differenti...</div><div class="ttdef"><b>Definition:</b> ode_forward_euler.cpp:54</div></div>
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<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>
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<div class="ttc" id="anamespaceqr__algorithm_html_a8ea313a1a1b5f9d0e3e332c29c6446ec"><div class="ttname"><a href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec">qr_algorithm::vector_dot</a></div><div class="ttdeci">double vector_dot(const std::valarray< T > &a, const std::valarray< T > &b)</div><div class="ttdef"><b>Definition:</b> qr_decompose.h:76</div></div>
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<div class="ttc" id="aleft_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/manip/left.html">std::right</a></div><div class="ttdeci">T right(T... args)</div></div>
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<div class="ttc" id="aode__midpoint__euler_8cpp_html_abaeae8f62a018d197f0187a1c80a90fe"><div class="ttname"><a href="../../d6/dd3/ode__midpoint__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a></div><div class="ttdeci">void problem(const double &x, std::valarray< double > *y, std::valarray< double > *dy)</div><div class="ttdoc">Problem statement for a system with first-order differential equations. Updates the system differenti...</div><div class="ttdef"><b>Definition:</b> ode_midpoint_euler.cpp:53</div></div>
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<div class="ttc" id="anewton__raphson__method_8cpp_html_ae66f6b31b5ad750f1fe042a706a4e3d4"><div class="ttname"><a href="../../de/dd3/newton__raphson__method_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a></div><div class="ttdeci">int main()</div><div class="ttdef"><b>Definition:</b> newton_raphson_method.cpp:44</div></div>
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<div class="ttc" id="agroup__ode_html_ga36ff0710861ca39d957046c0b09b2985"><div class="ttname"><a href="../../d6/d60/group__ode.html#ga36ff0710861ca39d957046c0b09b2985">midpoint_euler</a></div><div class="ttdeci">double midpoint_euler(double dx, double x0, double x_max, std::valarray< double > *y, bool save_to_file=false)</div><div class="ttdoc">Compute approximation using the midpoint-Euler method in the given limits.</div><div class="ttdef"><b>Definition:</b> ode_midpoint_euler.cpp:107</div></div>
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<div class="ttc" id="anamespaceqr__algorithm_html_ab1ac74497ffb2101040cc1efe3546de8"><div class="ttname"><a href="../../d2/d3b/namespaceqr__algorithm.html#ab1ac74497ffb2101040cc1efe3546de8">qr_algorithm::qr_decompose</a></div><div class="ttdeci">void qr_decompose(const std::valarray< std::valarray< T >> &A, std::valarray< std::valarray< T >> *Q, std::valarray< std::valarray< T >> *R)</div><div class="ttdef"><b>Definition:</b> qr_decompose.h:146</div></div>
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<div class="ttc" id="aode__semi__implicit__euler_8cpp_html_a0ddf1224851353fc92bfbff6f499fa97"><div class="ttname"><a href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#a0ddf1224851353fc92bfbff6f499fa97">main</a></div><div class="ttdeci">int main(int argc, char *argv[])</div><div class="ttdef"><b>Definition:</b> ode_semi_implicit_euler.cpp:189</div></div>
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<div class="ttc" id="aode__semi__implicit__euler_8cpp_html_aa13517b8e5de1b75592052db7f7e237f"><div class="ttname"><a href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a></div><div class="ttdeci">void save_exact_solution(const double &X0, const double &X_MAX, const double &step_size, const std::valarray< double > &Y0)</div><div class="ttdef"><b>Definition:</b> ode_semi_implicit_euler.cpp:153</div></div>
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<div class="ttc" id="aode__semi__implicit__euler_8cpp_html_af3adf7b092a87868917ee5fb4255192b"><div class="ttname"><a href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a></div><div class="ttdeci">void exact_solution(const double &x, std::valarray< double > *y)</div><div class="ttdoc">Exact solution of the problem. Used for solution comparison.</div><div class="ttdef"><b>Definition:</b> ode_semi_implicit_euler.cpp:66</div></div>
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<div class="ttc" id="asetw_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/manip/setw.html">std::setw</a></div><div class="ttdeci">T setw(T... args)</div></div>
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<div class="ttc" id="aode__midpoint__euler_8cpp_html_af3adf7b092a87868917ee5fb4255192b"><div class="ttname"><a href="../../d6/dd3/ode__midpoint__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a></div><div class="ttdeci">void exact_solution(const double &x, std::valarray< double > *y)</div><div class="ttdoc">Exact solution of the problem. Used for solution comparison.</div><div class="ttdef"><b>Definition:</b> ode_midpoint_euler.cpp:66</div></div>
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<div class="ttc" id="aode__midpoint__euler_8cpp_html_a0ddf1224851353fc92bfbff6f499fa97"><div class="ttname"><a href="../../d6/dd3/ode__midpoint__euler_8cpp.html#a0ddf1224851353fc92bfbff6f499fa97">main</a></div><div class="ttdeci">int main(int argc, char *argv[])</div><div class="ttdef"><b>Definition:</b> ode_midpoint_euler.cpp:192</div></div>
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<div class="ttc" id="apow_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a></div><div class="ttdeci">T pow(T... args)</div></div>
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