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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 id="l00001" name="l00001"></a><span class="lineno"> 1</span><span class="comment">/**</span></div>
<div class="line"><a id="l00002" name="l00002"></a><span class="lineno"> 2</span><span class="comment"> * @file</span></div>
<div class="line"><a id="l00003" name="l00003"></a><span class="lineno"> 3</span><span class="comment"> * \brief Library functions to compute [QR</span></div>
<div class="line"><a id="l00004" name="l00004"></a><span class="lineno"> 4</span><span class="comment"> * decomposition](https://en.wikipedia.org/wiki/QR_decomposition) of a given</span></div>
<div class="line"><a id="l00005" name="l00005"></a><span class="lineno"> 5</span><span class="comment"> * matrix.</span></div>
<div class="line"><a id="l00006" name="l00006"></a><span class="lineno"> 6</span><span class="comment"> * \author [Krishna Vedala](https://github.com/kvedala)</span></div>
<div class="line"><a id="l00007" name="l00007"></a><span class="lineno"> 7</span><span class="comment"> */</span></div>
<div class="line"><a id="l00008" name="l00008"></a><span class="lineno"> 8</span> </div>
<div class="line"><a id="l00009" name="l00009"></a><span class="lineno"> 9</span><span class="preprocessor">#ifndef NUMERICAL_METHODS_QR_DECOMPOSE_H_</span></div>
<div class="line"><a id="l00010" name="l00010"></a><span class="lineno"> 10</span><span class="preprocessor">#define NUMERICAL_METHODS_QR_DECOMPOSE_H_</span></div>
<div class="line"><a id="l00011" name="l00011"></a><span class="lineno"> 11</span> </div>
<div class="line"><a id="l00012" name="l00012"></a><span class="lineno"> 12</span><span class="preprocessor">#include &lt;cmath&gt;</span></div>
<div class="line"><a id="l00013" name="l00013"></a><span class="lineno"> 13</span><span class="preprocessor">#include &lt;cstdlib&gt;</span></div>
<div class="line"><a id="l00014" name="l00014"></a><span class="lineno"> 14</span><span class="preprocessor">#include &lt;iomanip&gt;</span></div>
<div class="line"><a id="l00015" name="l00015"></a><span class="lineno"> 15</span><span class="preprocessor">#include &lt;iostream&gt;</span></div>
<div class="line"><a id="l00016" name="l00016"></a><span class="lineno"> 16</span><span class="preprocessor">#include &lt;limits&gt;</span></div>
<div class="line"><a id="l00017" name="l00017"></a><span class="lineno"> 17</span><span class="preprocessor">#include &lt;numeric&gt;</span></div>
<div class="line"><a id="l00018" name="l00018"></a><span class="lineno"> 18</span><span class="preprocessor">#include &lt;valarray&gt;</span></div>
<div class="line"><a id="l00019" name="l00019"></a><span class="lineno"> 19</span><span class="preprocessor">#ifdef _OPENMP</span></div>
<div class="line"><a id="l00020" name="l00020"></a><span class="lineno"> 20</span><span class="preprocessor">#include &lt;omp.h&gt;</span></div>
<div class="line"><a id="l00021" name="l00021"></a><span class="lineno"> 21</span><span class="preprocessor">#endif</span></div>
<div class="line"><a id="l00022" name="l00022"></a><span class="lineno"> 22</span><span class="comment"></span> </div>
<div class="line"><a id="l00023" 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>
<div class="line"><a id="l00024" name="l00024"></a><span class="lineno"> 24</span><span class="comment"> * \brief Functions to compute [QR</span></div>
<div class="line"><a id="l00025" name="l00025"></a><span class="lineno"> 25</span><span class="comment"> * decomposition](https://en.wikipedia.org/wiki/QR_decomposition) of any</span></div>
<div class="line"><a id="l00026" name="l00026"></a><span class="lineno"> 26</span><span class="comment"> * rectangular matrix</span></div>
<div class="line"><a id="l00027" name="l00027"></a><span class="lineno"> 27</span><span class="comment"> */</span></div>
<div class="line"><a id="l00028" name="l00028"></a><span class="lineno"> 28</span><span class="keyword">namespace </span><a class="code hl_namespace" href="../../d2/d3b/namespaceqr__algorithm.html">qr_algorithm</a> {<span class="comment"></span></div>
<div class="line"><a id="l00029" name="l00029"></a><span class="lineno"> 29</span><span class="comment">/**</span></div>
<div class="line"><a id="l00030" name="l00030"></a><span class="lineno"> 30</span><span class="comment"> * operator to print a matrix</span></div>
<div class="line"><a id="l00031" name="l00031"></a><span class="lineno"> 31</span><span class="comment"> */</span></div>
<div class="line"><a id="l00032" name="l00032"></a><span class="lineno"> 32</span><span class="keyword">template</span> &lt;<span class="keyword">typename</span> T&gt;</div>
<div class="foldopen" id="foldopen00033" data-start="{" data-end="}">
<div class="line"><a id="l00033" name="l00033"></a><span class="lineno"><a class="line" href="../../d2/d3b/namespaceqr__algorithm.html#adfbdf47277c8cfee229b05b72f1f7834"> 33</a></span><a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> &amp;<a class="code hl_function" href="../../d2/d3b/namespaceqr__algorithm.html#adfbdf47277c8cfee229b05b72f1f7834">operator&lt;&lt;</a>(<a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> &amp;out,</div>
<div class="line"><a id="l00034" name="l00034"></a><span class="lineno"> 34</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt;<a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;T&gt;</a>&gt; <span class="keyword">const</span> &amp;v) {</div>
<div class="line"><a id="l00035" name="l00035"></a><span class="lineno"> 35</span> <span class="keyword">const</span> <span class="keywordtype">int</span> width = 12;</div>
<div class="line"><a id="l00036" name="l00036"></a><span class="lineno"> 36</span> <span class="keyword">const</span> <span class="keywordtype">char</span> separator = <span class="charliteral">&#39; &#39;</span>;</div>
<div class="line"><a id="l00037" name="l00037"></a><span class="lineno"> 37</span> </div>
<div class="line"><a id="l00038" name="l00038"></a><span class="lineno"> 38</span> out.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/ios_base/precision.html">precision</a>(4);</div>
<div class="line"><a id="l00039" name="l00039"></a><span class="lineno"> 39</span> <span class="keywordflow">for</span> (<span class="keywordtype">size_t</span> row = 0; row &lt; v.size(); row++) {</div>
<div class="line"><a id="l00040" name="l00040"></a><span class="lineno"> 40</span> <span class="keywordflow">for</span> (<span class="keywordtype">size_t</span> col = 0; col &lt; v[row].size(); col++)</div>
<div class="line"><a id="l00041" name="l00041"></a><span class="lineno"> 41</span> out &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/left.html">std::right</a> &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setw.html">std::setw</a>(width) &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setfill.html">std::setfill</a>(separator)</div>
<div class="line"><a id="l00042" name="l00042"></a><span class="lineno"> 42</span> &lt;&lt; v[row][col];</div>
<div class="line"><a id="l00043" name="l00043"></a><span class="lineno"> 43</span> out &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"><a id="l00044" name="l00044"></a><span class="lineno"> 44</span> }</div>
<div class="line"><a id="l00045" name="l00045"></a><span class="lineno"> 45</span> </div>
<div class="line"><a id="l00046" name="l00046"></a><span class="lineno"> 46</span> <span class="keywordflow">return</span> out;</div>
<div class="line"><a id="l00047" name="l00047"></a><span class="lineno"> 47</span>}</div>
</div>
<div class="line"><a id="l00048" name="l00048"></a><span class="lineno"> 48</span><span class="comment"></span> </div>
<div class="line"><a id="l00049" name="l00049"></a><span class="lineno"> 49</span><span class="comment">/**</span></div>
<div class="line"><a id="l00050" name="l00050"></a><span class="lineno"> 50</span><span class="comment"> * operator to print a vector</span></div>
<div class="line"><a id="l00051" name="l00051"></a><span class="lineno"> 51</span><span class="comment"> */</span></div>
<div class="line"><a id="l00052" name="l00052"></a><span class="lineno"> 52</span><span class="keyword">template</span> &lt;<span class="keyword">typename</span> T&gt;</div>
<div class="foldopen" id="foldopen00053" data-start="{" data-end="}">
<div class="line"><a id="l00053" name="l00053"></a><span class="lineno"><a class="line" href="../../d2/d3b/namespaceqr__algorithm.html#a257425cb2365359da51c6fe6741834d8"> 53</a></span><a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> &amp;<a class="code hl_function" href="../../d2/d3b/namespaceqr__algorithm.html#adfbdf47277c8cfee229b05b72f1f7834">operator&lt;&lt;</a>(<a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> &amp;out, <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;T&gt;</a> <span class="keyword">const</span> &amp;v) {</div>
<div class="line"><a id="l00054" name="l00054"></a><span class="lineno"> 54</span> <span class="keyword">const</span> <span class="keywordtype">int</span> width = 10;</div>
<div class="line"><a id="l00055" name="l00055"></a><span class="lineno"> 55</span> <span class="keyword">const</span> <span class="keywordtype">char</span> separator = <span class="charliteral">&#39; &#39;</span>;</div>
<div class="line"><a id="l00056" name="l00056"></a><span class="lineno"> 56</span> </div>
<div class="line"><a id="l00057" name="l00057"></a><span class="lineno"> 57</span> out.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/ios_base/precision.html">precision</a>(4);</div>
<div class="line"><a id="l00058" name="l00058"></a><span class="lineno"> 58</span> <span class="keywordflow">for</span> (<span class="keywordtype">size_t</span> row = 0; row &lt; v.size(); row++) {</div>
<div class="line"><a id="l00059" name="l00059"></a><span class="lineno"> 59</span> out &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/left.html">std::right</a> &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setw.html">std::setw</a>(width) &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setfill.html">std::setfill</a>(separator)</div>
<div class="line"><a id="l00060" name="l00060"></a><span class="lineno"> 60</span> &lt;&lt; v[row];</div>
<div class="line"><a id="l00061" name="l00061"></a><span class="lineno"> 61</span> }</div>
<div class="line"><a id="l00062" name="l00062"></a><span class="lineno"> 62</span> </div>
<div class="line"><a id="l00063" name="l00063"></a><span class="lineno"> 63</span> <span class="keywordflow">return</span> out;</div>
<div class="line"><a id="l00064" name="l00064"></a><span class="lineno"> 64</span>}</div>
</div>
<div class="line"><a id="l00065" name="l00065"></a><span class="lineno"> 65</span><span class="comment"></span> </div>
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"> 66</span><span class="comment">/**</span></div>
<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"> 67</span><span class="comment"> * Compute dot product of two vectors of equal lengths</span></div>
<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span><span class="comment"> *</span></div>
<div class="line"><a id="l00069" 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>
<div class="line"><a id="l00070" 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>
<div class="line"><a id="l00071" 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>
<div class="line"><a id="l00072" name="l00072"></a><span class="lineno"> 72</span><span class="comment"> *</span></div>
<div class="line"><a id="l00073" name="l00073"></a><span class="lineno"> 73</span><span class="comment"> * \returns \f$\vec{a}\cdot\vec{b}\f$</span></div>
<div class="line"><a id="l00074" name="l00074"></a><span class="lineno"> 74</span><span class="comment"> */</span></div>
<div class="line"><a id="l00075" name="l00075"></a><span class="lineno"> 75</span><span class="keyword">template</span> &lt;<span class="keyword">typename</span> T&gt;</div>
<div class="foldopen" id="foldopen00076" data-start="{" data-end="}">
<div class="line"><a id="l00076" 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 hl_function" href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(<span class="keyword">const</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;T&gt;</a> &amp;a, <span class="keyword">const</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;T&gt;</a> &amp;b) {</div>
<div class="line"><a id="l00077" name="l00077"></a><span class="lineno"> 77</span> <span class="keywordflow">return</span> (a * b).sum();</div>
<div class="line"><a id="l00078" name="l00078"></a><span class="lineno"> 78</span> <span class="comment">// could also use following</span></div>
<div class="line"><a id="l00079" 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>
<div class="line"><a id="l00080" name="l00080"></a><span class="lineno"> 80</span> <span class="comment">// 0.f);</span></div>
<div class="line"><a id="l00081" name="l00081"></a><span class="lineno"> 81</span>}</div>
</div>
<div class="line"><a id="l00082" name="l00082"></a><span class="lineno"> 82</span><span class="comment"></span> </div>
<div class="line"><a id="l00083" name="l00083"></a><span class="lineno"> 83</span><span class="comment">/**</span></div>
<div class="line"><a id="l00084" name="l00084"></a><span class="lineno"> 84</span><span class="comment"> * Compute magnitude of vector.</span></div>
<div class="line"><a id="l00085" name="l00085"></a><span class="lineno"> 85</span><span class="comment"> *</span></div>
<div class="line"><a id="l00086" 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>
<div class="line"><a id="l00087" 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>
<div class="line"><a id="l00088" name="l00088"></a><span class="lineno"> 88</span><span class="comment"> *</span></div>
<div class="line"><a id="l00089" name="l00089"></a><span class="lineno"> 89</span><span class="comment"> * \returns \f$\left|\vec{a}\right|\f$</span></div>
<div class="line"><a id="l00090" name="l00090"></a><span class="lineno"> 90</span><span class="comment"> */</span></div>
<div class="line"><a id="l00091" name="l00091"></a><span class="lineno"> 91</span><span class="keyword">template</span> &lt;<span class="keyword">typename</span> T&gt;</div>
<div class="foldopen" id="foldopen00092" data-start="{" data-end="}">
<div class="line"><a id="l00092" 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 hl_function" href="../../d2/d3b/namespaceqr__algorithm.html#ad16da2183db22378435042f26af43d5f">vector_mag</a>(<span class="keyword">const</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;T&gt;</a> &amp;a) {</div>
<div class="line"><a id="l00093" name="l00093"></a><span class="lineno"> 93</span> <span class="keywordtype">double</span> dot = <a class="code hl_function" href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(a, a);</div>
<div class="line"><a id="l00094" name="l00094"></a><span class="lineno"> 94</span> <span class="keywordflow">return</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a>(dot);</div>
<div class="line"><a id="l00095" name="l00095"></a><span class="lineno"> 95</span>}</div>
</div>
<div class="line"><a id="l00096" name="l00096"></a><span class="lineno"> 96</span><span class="comment"></span> </div>
<div class="line"><a id="l00097" name="l00097"></a><span class="lineno"> 97</span><span class="comment">/**</span></div>
<div class="line"><a id="l00098" 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>
<div class="line"><a id="l00099" 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>
<div class="line"><a id="l00100" name="l00100"></a><span class="lineno"> 100</span><span class="comment"> *</span></div>
<div class="line"><a id="l00101" name="l00101"></a><span class="lineno"> 101</span><span class="comment"> * \returns NULL if error, otherwise pointer to output</span></div>
<div class="line"><a id="l00102" name="l00102"></a><span class="lineno"> 102</span><span class="comment"> */</span></div>
<div class="line"><a id="l00103" name="l00103"></a><span class="lineno"> 103</span><span class="keyword">template</span> &lt;<span class="keyword">typename</span> T&gt;</div>
<div class="foldopen" id="foldopen00104" data-start="{" data-end="}">
<div class="line"><a id="l00104" name="l00104"></a><span class="lineno"><a class="line" href="../../d2/d3b/namespaceqr__algorithm.html#a6d3c7dce1f142141f509d09f6c0e25dc"> 104</a></span><a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;T&gt;</a> <a class="code hl_function" href="../../d2/d3b/namespaceqr__algorithm.html#a6d3c7dce1f142141f509d09f6c0e25dc">vector_proj</a>(<span class="keyword">const</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;T&gt;</a> &amp;a,</div>
<div class="line"><a id="l00105" name="l00105"></a><span class="lineno"> 105</span> <span class="keyword">const</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;T&gt;</a> &amp;b) {</div>
<div class="line"><a id="l00106" name="l00106"></a><span class="lineno"> 106</span> <span class="keywordtype">double</span> num = <a class="code hl_function" href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(a, b);</div>
<div class="line"><a id="l00107" name="l00107"></a><span class="lineno"> 107</span> <span class="keywordtype">double</span> deno = <a class="code hl_function" href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(b, b);</div>
<div class="line"><a id="l00108" name="l00108"></a><span class="lineno"> 108</span><span class="comment"></span> </div>
<div class="line"><a id="l00109" name="l00109"></a><span class="lineno"> 109</span><span class="comment"> /*! check for division by zero using machine epsilon */</span></div>
<div class="line"><a id="l00110" name="l00110"></a><span class="lineno"> 110</span> <span class="keywordflow">if</span> (deno &lt;= <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/numeric_limits.html">std::numeric_limits&lt;double&gt;::epsilon</a>()) {</div>
<div class="line"><a id="l00111" name="l00111"></a><span class="lineno"> 111</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cerr</a> &lt;&lt; <span class="stringliteral">&quot;[&quot;</span> &lt;&lt; __func__ &lt;&lt; <span class="stringliteral">&quot;] Possible division by zero\n&quot;</span>;</div>
<div class="line"><a id="l00112" name="l00112"></a><span class="lineno"> 112</span> <span class="keywordflow">return</span> a; <span class="comment">// return vector a back</span></div>
<div class="line"><a id="l00113" name="l00113"></a><span class="lineno"> 113</span> }</div>
<div class="line"><a id="l00114" name="l00114"></a><span class="lineno"> 114</span> </div>
<div class="line"><a id="l00115" name="l00115"></a><span class="lineno"> 115</span> <span class="keywordtype">double</span> scalar = num / deno;</div>
<div class="line"><a id="l00116" name="l00116"></a><span class="lineno"> 116</span> </div>
<div class="line"><a id="l00117" name="l00117"></a><span class="lineno"> 117</span> <span class="keywordflow">return</span> b * scalar;</div>
<div class="line"><a id="l00118" name="l00118"></a><span class="lineno"> 118</span>}</div>
</div>
<div class="line"><a id="l00119" name="l00119"></a><span class="lineno"> 119</span><span class="comment"></span> </div>
<div class="line"><a id="l00120" name="l00120"></a><span class="lineno"> 120</span><span class="comment">/**</span></div>
<div class="line"><a id="l00121" name="l00121"></a><span class="lineno"> 121</span><span class="comment"> * Decompose matrix \f$A\f$ using [Gram-Schmidt</span></div>
<div class="line"><a id="l00122" name="l00122"></a><span class="lineno"> 122</span><span class="comment"> *process](https://en.wikipedia.org/wiki/QR_decomposition).</span></div>
<div class="line"><a id="l00123" name="l00123"></a><span class="lineno"> 123</span><span class="comment"> *</span></div>
<div class="line"><a id="l00124" name="l00124"></a><span class="lineno"> 124</span><span class="comment"> * \f{eqnarray*}{</span></div>
<div class="line"><a id="l00125" name="l00125"></a><span class="lineno"> 125</span><span class="comment"> * \text{given that}\quad A &amp;=&amp;</span></div>
<div class="line"><a id="l00126" 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>
<div class="line"><a id="l00127" name="l00127"></a><span class="lineno"> 127</span><span class="comment"> * \text{where}\quad\mathbf{a}_i &amp;=&amp;</span></div>
<div class="line"><a id="l00128" 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>
<div class="line"><a id="l00129" name="l00129"></a><span class="lineno"> 129</span><span class="comment"> * vectors)}\\</span></div>
<div class="line"><a id="l00130" name="l00130"></a><span class="lineno"> 130</span><span class="comment"> * \text{then}\quad\mathbf{u}_i &amp;=&amp; \mathbf{a}_i</span></div>
<div class="line"><a id="l00131" 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>
<div class="line"><a id="l00132" name="l00132"></a><span class="lineno"> 132</span><span class="comment"> * \mathbf{e}_i &amp;=&amp;\frac{\mathbf{u}_i}{\left|\mathbf{u}_i\right|}\\</span></div>
<div class="line"><a id="l00133" name="l00133"></a><span class="lineno"> 133</span><span class="comment"> * Q &amp;=&amp; \begin{bmatrix}\mathbf{e}_0 &amp; \mathbf{e}_1 &amp; \mathbf{e}_2 &amp; \dots &amp;</span></div>
<div class="line"><a id="l00134" name="l00134"></a><span class="lineno"> 134</span><span class="comment"> * \mathbf{e}_{N-1}\end{bmatrix}\\</span></div>
<div class="line"><a id="l00135" name="l00135"></a><span class="lineno"> 135</span><span class="comment"> * R &amp;=&amp; \begin{bmatrix}\langle\mathbf{e}_0\,,\mathbf{a}_0\rangle &amp;</span></div>
<div class="line"><a id="l00136" name="l00136"></a><span class="lineno"> 136</span><span class="comment"> * \langle\mathbf{e}_1\,,\mathbf{a}_1\rangle &amp;</span></div>
<div class="line"><a id="l00137" name="l00137"></a><span class="lineno"> 137</span><span class="comment"> * \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle &amp; \dots \\</span></div>
<div class="line"><a id="l00138" name="l00138"></a><span class="lineno"> 138</span><span class="comment"> * 0 &amp; \langle\mathbf{e}_1\,,\mathbf{a}_1\rangle &amp;</span></div>
<div class="line"><a id="l00139" name="l00139"></a><span class="lineno"> 139</span><span class="comment"> * \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle &amp; \dots\\</span></div>
<div class="line"><a id="l00140" name="l00140"></a><span class="lineno"> 140</span><span class="comment"> * 0 &amp; 0 &amp; \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle &amp;</span></div>
<div class="line"><a id="l00141" name="l00141"></a><span class="lineno"> 141</span><span class="comment"> * \dots\\ \vdots &amp; \vdots &amp; \vdots &amp; \ddots</span></div>
<div class="line"><a id="l00142" name="l00142"></a><span class="lineno"> 142</span><span class="comment"> * \end{bmatrix}\\</span></div>
<div class="line"><a id="l00143" name="l00143"></a><span class="lineno"> 143</span><span class="comment"> * \f}</span></div>
<div class="line"><a id="l00144" name="l00144"></a><span class="lineno"> 144</span><span class="comment"> */</span></div>
<div class="line"><a id="l00145" name="l00145"></a><span class="lineno"> 145</span><span class="keyword">template</span> &lt;<span class="keyword">typename</span> T&gt;</div>
<div class="foldopen" id="foldopen00146" data-start="{" data-end="}">
<div class="line"><a id="l00146" name="l00146"></a><span class="lineno"><a class="line" href="../../d2/d3b/namespaceqr__algorithm.html#a73ce637634fc49e1d10d190eb388ebf1"> 146</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../d2/d3b/namespaceqr__algorithm.html#a73ce637634fc49e1d10d190eb388ebf1">qr_decompose</a>(</div>
<div class="line"><a id="l00147" name="l00147"></a><span class="lineno"> 147</span> <span class="keyword">const</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt;<a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;T&gt;</a>&gt; &amp;A, <span class="comment">/**&lt; input matrix to decompose */</span></div>
<div class="line"><a id="l00148" name="l00148"></a><span class="lineno"> 148</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt;<a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;T&gt;</a>&gt; *Q, <span class="comment">/**&lt; output decomposed matrix */</span></div>
<div class="line"><a id="l00149" name="l00149"></a><span class="lineno"> 149</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt;<a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;T&gt;</a>&gt; *R <span class="comment">/**&lt; output decomposed matrix */</span></div>
<div class="line"><a id="l00150" name="l00150"></a><span class="lineno"> 150</span>) {</div>
<div class="line"><a id="l00151" name="l00151"></a><span class="lineno"> 151</span> <a class="code hl_classRef" 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>
<div class="line"><a id="l00152" name="l00152"></a><span class="lineno"> 152</span> <a class="code hl_classRef" 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>
<div class="line"><a id="l00153" name="l00153"></a><span class="lineno"> 153</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;T&gt;</a> col_vector(ROWS);</div>
<div class="line"><a id="l00154" name="l00154"></a><span class="lineno"> 154</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;T&gt;</a> col_vector2(ROWS);</div>
<div class="line"><a id="l00155" name="l00155"></a><span class="lineno"> 155</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;T&gt;</a> tmp_vector(ROWS);</div>
<div class="line"><a id="l00156" name="l00156"></a><span class="lineno"> 156</span> </div>
<div class="line"><a id="l00157" name="l00157"></a><span class="lineno"> 157</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; COLUMNS; i++) {</div>
<div class="line"><a id="l00158" name="l00158"></a><span class="lineno"> 158</span> <span class="comment">/* for each column =&gt; R is a square matrix of NxN */</span></div>
<div class="line"><a id="l00159" name="l00159"></a><span class="lineno"> 159</span> <span class="keywordtype">int</span> j;</div>
<div class="line"><a id="l00160" name="l00160"></a><span class="lineno"> 160</span> R[0][i] = 0.; <span class="comment">/* make R upper triangular */</span></div>
<div class="line"><a id="l00161" name="l00161"></a><span class="lineno"> 161</span> </div>
<div class="line"><a id="l00162" name="l00162"></a><span class="lineno"> 162</span> <span class="comment">/* get corresponding Q vector */</span></div>
<div class="line"><a id="l00163" name="l00163"></a><span class="lineno"> 163</span><span class="preprocessor">#ifdef _OPENMP</span></div>
<div class="line"><a id="l00164" name="l00164"></a><span class="lineno"> 164</span><span class="comment">// parallelize on threads</span></div>
<div class="line"><a id="l00165" name="l00165"></a><span class="lineno"> 165</span><span class="preprocessor">#pragma omp for</span></div>
<div class="line"><a id="l00166" name="l00166"></a><span class="lineno"> 166</span><span class="preprocessor">#endif</span></div>
<div class="line"><a id="l00167" name="l00167"></a><span class="lineno"> 167</span> <span class="keywordflow">for</span> (j = 0; j &lt; ROWS; j++) {</div>
<div class="line"><a id="l00168" name="l00168"></a><span class="lineno"> 168</span> tmp_vector[j] = A[j][i]; <span class="comment">/* accumulator for uk */</span></div>
<div class="line"><a id="l00169" name="l00169"></a><span class="lineno"> 169</span> col_vector[j] = A[j][i];</div>
<div class="line"><a id="l00170" name="l00170"></a><span class="lineno"> 170</span> }</div>
<div class="line"><a id="l00171" name="l00171"></a><span class="lineno"> 171</span> <span class="keywordflow">for</span> (j = 0; j &lt; i; j++) {</div>
<div class="line"><a id="l00172" name="l00172"></a><span class="lineno"> 172</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> k = 0; k &lt; ROWS; k++) {</div>
<div class="line"><a id="l00173" name="l00173"></a><span class="lineno"> 173</span> col_vector2[k] = Q[0][k][j];</div>
<div class="line"><a id="l00174" name="l00174"></a><span class="lineno"> 174</span> }</div>
<div class="line"><a id="l00175" name="l00175"></a><span class="lineno"> 175</span> col_vector2 = <a class="code hl_function" href="../../d2/d3b/namespaceqr__algorithm.html#a6d3c7dce1f142141f509d09f6c0e25dc">vector_proj</a>(col_vector, col_vector2);</div>
<div class="line"><a id="l00176" name="l00176"></a><span class="lineno"> 176</span> tmp_vector -= col_vector2;</div>
<div class="line"><a id="l00177" name="l00177"></a><span class="lineno"> 177</span> }</div>
<div class="line"><a id="l00178" name="l00178"></a><span class="lineno"> 178</span> </div>
<div class="line"><a id="l00179" name="l00179"></a><span class="lineno"> 179</span> <span class="keywordtype">double</span> mag = <a class="code hl_function" href="../../d2/d3b/namespaceqr__algorithm.html#ad16da2183db22378435042f26af43d5f">vector_mag</a>(tmp_vector);</div>
<div class="line"><a id="l00180" name="l00180"></a><span class="lineno"> 180</span> </div>
<div class="line"><a id="l00181" name="l00181"></a><span class="lineno"> 181</span><span class="preprocessor">#ifdef _OPENMP</span></div>
<div class="line"><a id="l00182" name="l00182"></a><span class="lineno"> 182</span><span class="comment">// parallelize on threads</span></div>
<div class="line"><a id="l00183" name="l00183"></a><span class="lineno"> 183</span><span class="preprocessor">#pragma omp for</span></div>
<div class="line"><a id="l00184" name="l00184"></a><span class="lineno"> 184</span><span class="preprocessor">#endif</span></div>
<div class="line"><a id="l00185" name="l00185"></a><span class="lineno"> 185</span> <span class="keywordflow">for</span> (j = 0; j &lt; ROWS; j++) Q[0][j][i] = tmp_vector[j] / mag;</div>
<div class="line"><a id="l00186" name="l00186"></a><span class="lineno"> 186</span> </div>
<div class="line"><a id="l00187" name="l00187"></a><span class="lineno"> 187</span> <span class="comment">/* compute upper triangular values of R */</span></div>
<div class="line"><a id="l00188" name="l00188"></a><span class="lineno"> 188</span><span class="preprocessor">#ifdef _OPENMP</span></div>
<div class="line"><a id="l00189" name="l00189"></a><span class="lineno"> 189</span><span class="comment">// parallelize on threads</span></div>
<div class="line"><a id="l00190" name="l00190"></a><span class="lineno"> 190</span><span class="preprocessor">#pragma omp for</span></div>
<div class="line"><a id="l00191" name="l00191"></a><span class="lineno"> 191</span><span class="preprocessor">#endif</span></div>
<div class="line"><a id="l00192" name="l00192"></a><span class="lineno"> 192</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> kk = 0; kk &lt; ROWS; kk++) {</div>
<div class="line"><a id="l00193" name="l00193"></a><span class="lineno"> 193</span> col_vector[kk] = Q[0][kk][i];</div>
<div class="line"><a id="l00194" name="l00194"></a><span class="lineno"> 194</span> }</div>
<div class="line"><a id="l00195" name="l00195"></a><span class="lineno"> 195</span> </div>
<div class="line"><a id="l00196" name="l00196"></a><span class="lineno"> 196</span><span class="preprocessor">#ifdef _OPENMP</span></div>
<div class="line"><a id="l00197" name="l00197"></a><span class="lineno"> 197</span><span class="comment">// parallelize on threads</span></div>
<div class="line"><a id="l00198" name="l00198"></a><span class="lineno"> 198</span><span class="preprocessor">#pragma omp for</span></div>
<div class="line"><a id="l00199" name="l00199"></a><span class="lineno"> 199</span><span class="preprocessor">#endif</span></div>
<div class="line"><a id="l00200" name="l00200"></a><span class="lineno"> 200</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> k = i; k &lt; COLUMNS; k++) {</div>
<div class="line"><a id="l00201" name="l00201"></a><span class="lineno"> 201</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> kk = 0; kk &lt; ROWS; kk++) {</div>
<div class="line"><a id="l00202" name="l00202"></a><span class="lineno"> 202</span> col_vector2[kk] = A[kk][k];</div>
<div class="line"><a id="l00203" name="l00203"></a><span class="lineno"> 203</span> }</div>
<div class="line"><a id="l00204" name="l00204"></a><span class="lineno"> 204</span> R[0][i][k] = (col_vector * col_vector2).sum();</div>
<div class="line"><a id="l00205" name="l00205"></a><span class="lineno"> 205</span> }</div>
<div class="line"><a id="l00206" name="l00206"></a><span class="lineno"> 206</span> }</div>
<div class="line"><a id="l00207" name="l00207"></a><span class="lineno"> 207</span>}</div>
</div>
<div class="line"><a id="l00208" name="l00208"></a><span class="lineno"> 208</span>} <span class="comment">// namespace qr_algorithm</span></div>
<div class="line"><a id="l00209" name="l00209"></a><span class="lineno"> 209</span> </div>
<div class="line"><a id="l00210" name="l00210"></a><span class="lineno"> 210</span><span class="preprocessor">#endif </span><span class="comment">// NUMERICAL_METHODS_QR_DECOMPOSE_H_</span></div>
<div class="ttc" id="abasic_ostream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a></div></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="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>
<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>
<div class="ttc" id="anamespaceqr__algorithm_html_a6d3c7dce1f142141f509d09f6c0e25dc"><div class="ttname"><a href="../../d2/d3b/namespaceqr__algorithm.html#a6d3c7dce1f142141f509d09f6c0e25dc">qr_algorithm::vector_proj</a></div><div class="ttdeci">std::valarray&lt; T &gt; vector_proj(const std::valarray&lt; T &gt; &amp;a, const std::valarray&lt; T &gt; &amp;b)</div><div class="ttdef"><b>Definition</b> qr_decompose.h:104</div></div>
<div class="ttc" id="anamespaceqr__algorithm_html_a73ce637634fc49e1d10d190eb388ebf1"><div class="ttname"><a href="../../d2/d3b/namespaceqr__algorithm.html#a73ce637634fc49e1d10d190eb388ebf1">qr_algorithm::qr_decompose</a></div><div class="ttdeci">void qr_decompose(const std::valarray&lt; std::valarray&lt; T &gt; &gt; &amp;A, std::valarray&lt; std::valarray&lt; T &gt; &gt; *Q, std::valarray&lt; std::valarray&lt; T &gt; &gt; *R)</div><div class="ttdef"><b>Definition</b> qr_decompose.h:146</div></div>
<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&lt; T &gt; &amp;a, const std::valarray&lt; T &gt; &amp;b)</div><div class="ttdef"><b>Definition</b> qr_decompose.h:76</div></div>
<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&lt; T &gt; &amp;a)</div><div class="ttdef"><b>Definition</b> qr_decompose.h:92</div></div>
<div class="ttc" id="anamespaceqr__algorithm_html_adfbdf47277c8cfee229b05b72f1f7834"><div class="ttname"><a href="../../d2/d3b/namespaceqr__algorithm.html#adfbdf47277c8cfee229b05b72f1f7834">qr_algorithm::operator&lt;&lt;</a></div><div class="ttdeci">std::ostream &amp; operator&lt;&lt;(std::ostream &amp;out, std::valarray&lt; std::valarray&lt; T &gt; &gt; const &amp;v)</div><div class="ttdef"><b>Definition</b> qr_decompose.h:33</div></div>
<div class="ttc" id="anumeric_limits_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/types/numeric_limits.html">std::numeric_limits</a></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>
<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="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>
<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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