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<div class="contents">
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<p>Functions to compute <a href="https://en.wikipedia.org/wiki/QR_decomposition" target="_blank">QR decomposition</a> of any rectangular matrix.
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<a href="#details">More...</a></p>
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<table class="memberdecls">
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<tr class="heading"><td colspan="2"><h2 id="header-func-members" class="groupheader"><a id="func-members" name="func-members"></a>
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Functions</h2></td></tr>
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<tr class="memitem:adfbdf47277c8cfee229b05b72f1f7834" id="r_adfbdf47277c8cfee229b05b72f1f7834"><td class="memTemplParams" colspan="2">template<typename T> </td></tr>
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<tr class="memitem:adfbdf47277c8cfee229b05b72f1f7834 template"><td class="memItemLeft" align="right" valign="top">std::ostream & </td><td class="memItemRight" valign="bottom"><a class="el" href="#adfbdf47277c8cfee229b05b72f1f7834">operator<<</a> (std::ostream &out, std::valarray< std::valarray< T > > const &v)</td></tr>
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<tr class="memitem:a257425cb2365359da51c6fe6741834d8" id="r_a257425cb2365359da51c6fe6741834d8"><td class="memTemplParams" colspan="2">template<typename T> </td></tr>
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<tr class="memitem:a257425cb2365359da51c6fe6741834d8 template"><td class="memItemLeft" align="right" valign="top">std::ostream & </td><td class="memItemRight" valign="bottom"><a class="el" href="#a257425cb2365359da51c6fe6741834d8">operator<<</a> (std::ostream &out, std::valarray< T > const &v)</td></tr>
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<tr class="memitem:a8ea313a1a1b5f9d0e3e332c29c6446ec" id="r_a8ea313a1a1b5f9d0e3e332c29c6446ec"><td class="memTemplParams" colspan="2">template<typename T> </td></tr>
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<tr class="memitem:a8ea313a1a1b5f9d0e3e332c29c6446ec template"><td class="memItemLeft" align="right" valign="top">double </td><td class="memItemRight" valign="bottom"><a class="el" href="#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a> (const std::valarray< T > &a, const std::valarray< T > &b)</td></tr>
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<tr class="memitem:ad16da2183db22378435042f26af43d5f" id="r_ad16da2183db22378435042f26af43d5f"><td class="memTemplParams" colspan="2">template<typename T> </td></tr>
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<tr class="memitem:ad16da2183db22378435042f26af43d5f template"><td class="memItemLeft" align="right" valign="top">double </td><td class="memItemRight" valign="bottom"><a class="el" href="#ad16da2183db22378435042f26af43d5f">vector_mag</a> (const std::valarray< T > &a)</td></tr>
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<tr class="memitem:a6d3c7dce1f142141f509d09f6c0e25dc" id="r_a6d3c7dce1f142141f509d09f6c0e25dc"><td class="memTemplParams" colspan="2">template<typename T> </td></tr>
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<tr class="memitem:a6d3c7dce1f142141f509d09f6c0e25dc template"><td class="memItemLeft" align="right" valign="top">std::valarray< T > </td><td class="memItemRight" valign="bottom"><a class="el" href="#a6d3c7dce1f142141f509d09f6c0e25dc">vector_proj</a> (const std::valarray< T > &a, const std::valarray< T > &b)</td></tr>
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<tr class="memitem:a73ce637634fc49e1d10d190eb388ebf1" id="r_a73ce637634fc49e1d10d190eb388ebf1"><td class="memTemplParams" colspan="2">template<typename T> </td></tr>
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<tr class="memitem:a73ce637634fc49e1d10d190eb388ebf1 template"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="#a73ce637634fc49e1d10d190eb388ebf1">qr_decompose</a> (const std::valarray< std::valarray< T > > &A, std::valarray< std::valarray< T > > *Q, std::valarray< std::valarray< T > > *R)</td></tr>
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<tr class="memitem:a28e2fa3e803abaea6c568dc45d69d8cc" id="r_a28e2fa3e803abaea6c568dc45d69d8cc"><td class="memItemLeft" align="right" valign="top">std::valarray< double > </td><td class="memItemRight" valign="bottom"><a class="el" href="#a28e2fa3e803abaea6c568dc45d69d8cc">eigen_values</a> (std::valarray< std::valarray< double > > *A, bool print_intermediates=false)</td></tr>
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</table>
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<a name="details" id="details"></a><h2 id="header-details" class="groupheader">Detailed Description</h2>
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<div class="textblock"><p>Functions to compute <a href="https://en.wikipedia.org/wiki/QR_decomposition" target="_blank">QR decomposition</a> of any rectangular matrix. </p>
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</div><a name="doc-func-members" id="doc-func-members"></a><h2 id="header-doc-func-members" class="groupheader">Function Documentation</h2>
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<a id="a28e2fa3e803abaea6c568dc45d69d8cc" name="a28e2fa3e803abaea6c568dc45d69d8cc"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a28e2fa3e803abaea6c568dc45d69d8cc">◆ </a></span>eigen_values()</h2>
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<div class="memitem">
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<div class="memproto">
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<table class="memname">
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<tr>
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<td class="memname">std::valarray< double > qr_algorithm::eigen_values </td>
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<td>(</td>
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<td class="paramtype">std::valarray< std::valarray< double > > *</td> <td class="paramname"><span class="paramname"><em>A</em></span>, </td>
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</tr>
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<tr>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">bool</td> <td class="paramname"><span class="paramname"><em>print_intermediates</em></span><span class="paramdefsep"> = </span><span class="paramdefval">false</span> )</td>
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</tr>
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</table>
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</div><div class="memdoc">
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<p>Compute eigen values using iterative shifted QR decomposition algorithm as follows:</p><ol type="1">
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<li>Use last diagonal element of A as eigen value approximation \(c\)</li>
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<li>Shift diagonals of matrix \(A' = A - cI\)</li>
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<li>Decompose matrix \(A'=QR\)</li>
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<li>Compute next approximation \(A'_1 = RQ \)</li>
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<li>Shift diagonals back \(A_1 = A'_1 + cI\)</li>
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<li>Termination condition check: last element below diagonal is almost 0<ol type="a">
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<li>If not 0, go back to step 1 with the new approximation \(A_1\)</li>
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<li>If 0, continue to step 7</li>
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</ol>
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</li>
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<li>Save last known \(c\) as the eigen value.</li>
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<li>Are all eigen values found?<ol type="a">
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<li>If not, remove last row and column of \(A_1\) and go back to step 1.</li>
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<li>If yes, stop.</li>
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</ol>
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</li>
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</ol>
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<dl class="section note"><dt>Note</dt><dd>The matrix \(A\) gets modified</dd></dl>
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<dl class="params"><dt>Parameters</dt><dd>
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<table class="params">
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<tr><td class="paramdir">[in,out]</td><td class="paramname">A</td><td>matrix to compute eigen values for </td></tr>
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<tr><td class="paramdir">[in]</td><td class="paramname">print_intermediates</td><td>(optional) whether to print intermediate A, Q and R matrices (default = <span class="tt">false</span>) </td></tr>
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</table>
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</dd>
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</dl>
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<p class="definition">Definition at line <a class="el" href="../../de/d75/qr__eigen__values_8cpp_source.html#l00098">98</a> of file <a class="el" href="../../de/d75/qr__eigen__values_8cpp_source.html">qr_eigen_values.cpp</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 99</span> {</div>
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<div class="line"><span class="lineno"> 100</span> <span class="keywordtype">int</span> rows = A->size();</div>
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<div class="line"><span class="lineno"> 101</span> <span class="keywordtype">int</span> columns = rows;</div>
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<div class="line"><span class="lineno"> 102</span> <span class="keywordtype">int</span> counter = 0, num_eigs = rows - 1;</div>
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<div class="line"><span class="lineno"> 103</span> <span class="keywordtype">double</span> last_eig = 0;</div>
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<div class="line"><span class="lineno"> 104</span> </div>
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<div class="line"><span class="lineno"> 105</span> std::valarray<std::valarray<double>> Q(rows);</div>
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<div class="line"><span class="lineno"> 106</span> std::valarray<std::valarray<double>> R(columns);</div>
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<div class="line"><span class="lineno"> 107</span> </div>
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<div class="line"><span class="lineno"> 108</span> <span class="comment">/* number of eigen values = matrix size */</span></div>
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<div class="line"><span class="lineno"> 109</span> std::valarray<double> eigen_vals(rows);</div>
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<div class="line"><span class="lineno"> 110</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < rows; i++) {</div>
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<div class="line"><span class="lineno"> 111</span> Q[i] = std::valarray<double>(columns);</div>
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<div class="line"><span class="lineno"> 112</span> R[i] = std::valarray<double>(columns);</div>
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<div class="line"><span class="lineno"> 113</span> }</div>
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<div class="line"><span class="lineno"> 114</span> </div>
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<div class="line"><span class="lineno"> 115</span> <span class="comment">/* continue till all eigen values are found */</span></div>
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<div class="line"><span class="lineno"> 116</span> <span class="keywordflow">while</span> (num_eigs > 0) {</div>
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<div class="line"><span class="lineno"> 117</span> <span class="comment">/* iterate with QR decomposition */</span></div>
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<div class="line"><span class="lineno"> 118</span> <span class="keywordflow">while</span> (std::abs(A[0][num_eigs][num_eigs - 1]) ></div>
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<div class="line"><span class="lineno"> 119</span> std::numeric_limits<double>::epsilon()) {</div>
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<div class="line"><span class="lineno"> 120</span> <span class="comment">// initial approximation = last diagonal element</span></div>
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<div class="line"><span class="lineno"> 121</span> last_eig = A[0][num_eigs][num_eigs];</div>
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<div class="line"><span class="lineno"> 122</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < rows; i++) {</div>
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<div class="line"><span class="lineno"> 123</span> A[0][i][i] -= last_eig; <span class="comment">/* A - cI */</span></div>
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<div class="line"><span class="lineno"> 124</span> }</div>
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<div class="line"><span class="lineno"> 125</span> </div>
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<div class="line"><span class="lineno"> 126</span> <a class="code hl_function" href="#a73ce637634fc49e1d10d190eb388ebf1">qr_decompose</a>(*A, &Q, &R);</div>
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<div class="line"><span class="lineno"> 127</span> </div>
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<div class="line"><span class="lineno"> 128</span> <span class="keywordflow">if</span> (print_intermediates) {</div>
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<div class="line"><span class="lineno"> 129</span> std::cout << *A << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><span class="lineno"> 130</span> std::cout << Q << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><span class="lineno"> 131</span> std::cout << R << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><span class="lineno"> 132</span> printf(<span class="stringliteral">"-------------------- %d ---------------------\n"</span>,</div>
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<div class="line"><span class="lineno"> 133</span> ++counter);</div>
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<div class="line"><span class="lineno"> 134</span> }</div>
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<div class="line"><span class="lineno"> 135</span> </div>
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<div class="line"><span class="lineno"> 136</span> <span class="comment">// new approximation A' = R * Q</span></div>
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<div class="line"><span class="lineno"> 137</span> <a class="code hl_function" href="../../de/d75/qr__eigen__values_8cpp.html#abb8bf4c55e10685a5eb2ad3797fde1ae">mat_mul</a>(R, Q, A);</div>
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<div class="line"><span class="lineno"> 138</span> </div>
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<div class="line"><span class="lineno"> 139</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < rows; i++) {</div>
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<div class="line"><span class="lineno"> 140</span> A[0][i][i] += last_eig; <span class="comment">/* A + cI */</span></div>
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<div class="line"><span class="lineno"> 141</span> }</div>
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<div class="line"><span class="lineno"> 142</span> }</div>
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<div class="line"><span class="lineno"> 143</span> </div>
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<div class="line"><span class="lineno"> 144</span> <span class="comment">/* store the converged eigen value */</span></div>
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<div class="line"><span class="lineno"> 145</span> eigen_vals[num_eigs] = last_eig;</div>
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<div class="line"><span class="lineno"> 146</span> <span class="comment">// A[0][num_eigs][num_eigs];</span></div>
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<div class="line"><span class="lineno"> 147</span> <span class="keywordflow">if</span> (print_intermediates) {</div>
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<div class="line"><span class="lineno"> 148</span> std::cout << <span class="stringliteral">"========================\n"</span>;</div>
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<div class="line"><span class="lineno"> 149</span> std::cout << <span class="stringliteral">"Eigen value: "</span> << last_eig << <span class="stringliteral">",\n"</span>;</div>
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<div class="line"><span class="lineno"> 150</span> std::cout << <span class="stringliteral">"========================\n"</span>;</div>
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<div class="line"><span class="lineno"> 151</span> }</div>
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<div class="line"><span class="lineno"> 152</span> </div>
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<div class="line"><span class="lineno"> 153</span> num_eigs--;</div>
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<div class="line"><span class="lineno"> 154</span> rows--;</div>
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<div class="line"><span class="lineno"> 155</span> columns--;</div>
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<div class="line"><span class="lineno"> 156</span> }</div>
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<div class="line"><span class="lineno"> 157</span> eigen_vals[0] = A[0][0][0];</div>
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<div class="line"><span class="lineno"> 158</span> </div>
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<div class="line"><span class="lineno"> 159</span> <span class="keywordflow">if</span> (print_intermediates) {</div>
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<div class="line"><span class="lineno"> 160</span> std::cout << Q << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><span class="lineno"> 161</span> std::cout << R << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><span class="lineno"> 162</span> }</div>
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<div class="line"><span class="lineno"> 163</span> </div>
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<div class="line"><span class="lineno"> 164</span> <span class="keywordflow">return</span> eigen_vals;</div>
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<div class="line"><span class="lineno"> 165</span>}</div>
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<div class="ttc" id="anamespaceqr__algorithm_html_a73ce637634fc49e1d10d190eb388ebf1"><div class="ttname"><a href="#a73ce637634fc49e1d10d190eb388ebf1">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> <a href="../../d4/d68/qr__decompose_8h_source.html#l00146">qr_decompose.h:146</a></div></div>
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<div class="ttc" id="aqr__eigen__values_8cpp_html_abb8bf4c55e10685a5eb2ad3797fde1ae"><div class="ttname"><a href="../../de/d75/qr__eigen__values_8cpp.html#abb8bf4c55e10685a5eb2ad3797fde1ae">mat_mul</a></div><div class="ttdeci">void mat_mul(const std::valarray< std::valarray< double > > &A, const std::valarray< std::valarray< double > > &B, std::valarray< std::valarray< double > > *OUT)</div><div class="ttdef"><b>Definition</b> <a href="../../de/d75/qr__eigen__values_8cpp_source.html#l00054">qr_eigen_values.cpp:54</a></div></div>
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<a id="adfbdf47277c8cfee229b05b72f1f7834" name="adfbdf47277c8cfee229b05b72f1f7834"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#adfbdf47277c8cfee229b05b72f1f7834">◆ </a></span>operator<<() <span class="overload">[1/2]</span></h2>
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<td class="paramtype">std::valarray< std::valarray< T > > const &</td> <td class="paramname"><span class="paramname"><em>v</em></span> )</td>
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<p>operator to print a matrix </p>
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<p class="definition">Definition at line <a class="el" href="../../d4/d68/qr__decompose_8h_source.html#l00033">33</a> of file <a class="el" href="../../d4/d68/qr__decompose_8h_source.html">qr_decompose.h</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 34</span> {</div>
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<div class="line"><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"><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"><span class="lineno"> 37</span> </div>
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<div class="line"><span class="lineno"> 38</span> out.precision(4);</div>
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<div class="line"><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"><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"><span class="lineno"> 41</span> out << std::right << std::setw(width) << std::setfill(separator)</div>
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<div class="line"><span class="lineno"> 42</span> << v[row][col];</div>
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<div class="line"><span class="lineno"> 43</span> out << std::endl;</div>
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<div class="line"><span class="lineno"> 44</span> }</div>
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<div class="line"><span class="lineno"> 45</span> </div>
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<div class="line"><span class="lineno"> 46</span> <span class="keywordflow">return</span> out;</div>
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<div class="line"><span class="lineno"> 47</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a257425cb2365359da51c6fe6741834d8">◆ </a></span>operator<<() <span class="overload">[2/2]</span></h2>
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<p>operator to print a vector </p>
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<p class="definition">Definition at line <a class="el" href="../../d4/d68/qr__decompose_8h_source.html#l00053">53</a> of file <a class="el" href="../../d4/d68/qr__decompose_8h_source.html">qr_decompose.h</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 53</span> {</div>
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<div class="line"><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"><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"><span class="lineno"> 56</span> </div>
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<div class="line"><span class="lineno"> 57</span> out.precision(4);</div>
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<div class="line"><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"><span class="lineno"> 59</span> out << std::right << std::setw(width) << std::setfill(separator)</div>
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<div class="line"><span class="lineno"> 60</span> << v[row];</div>
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<div class="line"><span class="lineno"> 61</span> }</div>
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<div class="line"><span class="lineno"> 62</span> </div>
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<div class="line"><span class="lineno"> 63</span> <span class="keywordflow">return</span> out;</div>
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<div class="line"><span class="lineno"> 64</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a73ce637634fc49e1d10d190eb388ebf1">◆ </a></span>qr_decompose()</h2>
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template<typename T> </div>
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<td class="memname">void qr_algorithm::qr_decompose </td>
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<td>(</td>
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<td class="paramtype">const std::valarray< std::valarray< T > > &</td> <td class="paramname"><span class="paramname"><em>A</em></span>, </td>
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<td class="paramtype">std::valarray< std::valarray< T > > *</td> <td class="paramname"><span class="paramname"><em>Q</em></span>, </td>
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<td class="paramtype">std::valarray< std::valarray< T > > *</td> <td class="paramname"><span class="paramname"><em>R</em></span> )</td>
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<p>Decompose matrix \(A\) using <a href="https://en.wikipedia.org/wiki/QR_decomposition" target="_blank">Gram-Schmidt process</a>.</p>
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<p class="formulaDsp">
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\begin{eqnarray*} \text{given that}\quad A &=&
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*\left[\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_{N-1},\right]\\
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\text{where}\quad\mathbf{a}_i &=&
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\left[a_{0i},a_{1i},a_{2i},\ldots,a_{(M-1)i}\right]^T\quad\ldots\mbox{(column
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vectors)}\\
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\text{then}\quad\mathbf{u}_i &=& \mathbf{a}_i
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*-\sum_{j=0}^{i-1}\text{proj}_{\mathbf{u}_j}\mathbf{a}_i\\
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\mathbf{e}_i &=&\frac{\mathbf{u}_i}{\left|\mathbf{u}_i\right|}\\
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Q &=& \begin{bmatrix}\mathbf{e}_0 & \mathbf{e}_1 & \mathbf{e}_2 & \dots &
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\mathbf{e}_{N-1}\end{bmatrix}\\
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R &=& \begin{bmatrix}\langle\mathbf{e}_0\,,\mathbf{a}_0\rangle &
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\langle\mathbf{e}_1\,,\mathbf{a}_1\rangle &
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\langle\mathbf{e}_2\,,\mathbf{a}_2\rangle & \dots \\
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0 & \langle\mathbf{e}_1\,,\mathbf{a}_1\rangle &
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\langle\mathbf{e}_2\,,\mathbf{a}_2\rangle & \dots\\
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0 & 0 & \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle &
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\dots\\ \vdots & \vdots & \vdots & \ddots
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\end{bmatrix}\\
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\end{eqnarray*}
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</p>
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<dl class="params"><dt>Parameters</dt><dd>
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<table class="params">
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<tr><td class="paramname">A</td><td>input matrix to decompose </td></tr>
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<tr><td class="paramname">Q</td><td>output decomposed matrix </td></tr>
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<tr><td class="paramname">R</td><td>output decomposed matrix </td></tr>
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<p class="definition">Definition at line <a class="el" href="../../d4/d68/qr__decompose_8h_source.html#l00146">146</a> of file <a class="el" href="../../d4/d68/qr__decompose_8h_source.html">qr_decompose.h</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 150</span> {</div>
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<div class="line"><span class="lineno"> 151</span> std::size_t ROWS = A.size(); <span class="comment">// number of rows of A</span></div>
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<div class="line"><span class="lineno"> 152</span> std::size_t COLUMNS = A[0].size(); <span class="comment">// number of columns of A</span></div>
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<div class="line"><span class="lineno"> 153</span> std::valarray<T> col_vector(ROWS);</div>
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<div class="line"><span class="lineno"> 154</span> std::valarray<T> col_vector2(ROWS);</div>
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<div class="line"><span class="lineno"> 155</span> std::valarray<T> tmp_vector(ROWS);</div>
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<div class="line"><span class="lineno"> 156</span> </div>
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<div class="line"><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"><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"><span class="lineno"> 159</span> <span class="keywordtype">int</span> j;</div>
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<div class="line"><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"><span class="lineno"> 161</span> </div>
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<div class="line"><span class="lineno"> 162</span> <span class="comment">/* get corresponding Q vector */</span></div>
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<div class="line"><span class="lineno"> 163</span><span class="preprocessor">#ifdef _OPENMP</span></div>
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<div class="line"><span class="lineno"> 164</span><span class="comment">// parallelize on threads</span></div>
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<div class="line"><span class="lineno"> 165</span><span class="preprocessor">#pragma omp for</span></div>
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<div class="line"><span class="lineno"> 166</span><span class="preprocessor">#endif</span></div>
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<div class="line"><span class="lineno"> 167</span> <span class="keywordflow">for</span> (j = 0; j < ROWS; j++) {</div>
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<div class="line"><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"><span class="lineno"> 169</span> col_vector[j] = A[j][i];</div>
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<div class="line"><span class="lineno"> 170</span> }</div>
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<div class="line"><span class="lineno"> 171</span> <span class="keywordflow">for</span> (j = 0; j < i; j++) {</div>
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<div class="line"><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"><span class="lineno"> 173</span> col_vector2[k] = Q[0][k][j];</div>
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<div class="line"><span class="lineno"> 174</span> }</div>
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<div class="line"><span class="lineno"> 175</span> col_vector2 = <a class="code hl_function" href="#a6d3c7dce1f142141f509d09f6c0e25dc">vector_proj</a>(col_vector, col_vector2);</div>
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<div class="line"><span class="lineno"> 176</span> tmp_vector -= col_vector2;</div>
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<div class="line"><span class="lineno"> 177</span> }</div>
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<div class="line"><span class="lineno"> 178</span> </div>
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<div class="line"><span class="lineno"> 179</span> <span class="keywordtype">double</span> <a class="code hl_function" href="../../df/d66/vector__cross__product_8cpp.html#a4b2a9757a87c18e1642d72410ecfaba8">mag</a> = <a class="code hl_function" href="#ad16da2183db22378435042f26af43d5f">vector_mag</a>(tmp_vector);</div>
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<div class="line"><span class="lineno"> 180</span> </div>
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<div class="line"><span class="lineno"> 181</span><span class="preprocessor">#ifdef _OPENMP</span></div>
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<div class="line"><span class="lineno"> 182</span><span class="comment">// parallelize on threads</span></div>
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<div class="line"><span class="lineno"> 183</span><span class="preprocessor">#pragma omp for</span></div>
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<div class="line"><span class="lineno"> 184</span><span class="preprocessor">#endif</span></div>
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<div class="line"><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"><span class="lineno"> 186</span> </div>
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<div class="line"><span class="lineno"> 187</span> <span class="comment">/* compute upper triangular values of R */</span></div>
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<div class="line"><span class="lineno"> 188</span><span class="preprocessor">#ifdef _OPENMP</span></div>
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<div class="line"><span class="lineno"> 189</span><span class="comment">// parallelize on threads</span></div>
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<div class="line"><span class="lineno"> 190</span><span class="preprocessor">#pragma omp for</span></div>
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<div class="line"><span class="lineno"> 191</span><span class="preprocessor">#endif</span></div>
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<div class="line"><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"><span class="lineno"> 193</span> col_vector[kk] = Q[0][kk][i];</div>
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<div class="line"><span class="lineno"> 194</span> }</div>
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<div class="line"><span class="lineno"> 195</span> </div>
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<div class="line"><span class="lineno"> 196</span><span class="preprocessor">#ifdef _OPENMP</span></div>
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<div class="line"><span class="lineno"> 197</span><span class="comment">// parallelize on threads</span></div>
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<div class="line"><span class="lineno"> 198</span><span class="preprocessor">#pragma omp for</span></div>
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<div class="line"><span class="lineno"> 199</span><span class="preprocessor">#endif</span></div>
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<div class="line"><span class="lineno"> 200</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> k = i; <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a1b74d828b33760094906797042b89442">k</a> < COLUMNS; <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a1b74d828b33760094906797042b89442">k</a>++) {</div>
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<div class="line"><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"><span class="lineno"> 202</span> col_vector2[kk] = A[kk][<a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a1b74d828b33760094906797042b89442">k</a>];</div>
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<div class="line"><span class="lineno"> 203</span> }</div>
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<div class="line"><span class="lineno"> 204</span> R[0][i][<a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a1b74d828b33760094906797042b89442">k</a>] = (col_vector * col_vector2).<a class="code hl_function" href="../../d8/d77/namespacemachine__learning.html#a6f1c98c016ad34ff3d9f39372161bd35">sum</a>();</div>
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<div class="line"><span class="lineno"> 205</span> }</div>
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<div class="line"><span class="lineno"> 206</span> }</div>
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<div class="line"><span class="lineno"> 207</span>}</div>
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<div class="ttc" id="acomposite__simpson__rule_8cpp_html_a1b74d828b33760094906797042b89442"><div class="ttname"><a href="../../d4/d18/composite__simpson__rule_8cpp.html#a1b74d828b33760094906797042b89442">numerical_methods::simpson_method::k</a></div><div class="ttdeci">double k(double x)</div><div class="ttdoc">Another test function.</div><div class="ttdef"><b>Definition</b> <a href="../../d4/d18/composite__simpson__rule_8cpp_source.html#l00117">composite_simpson_rule.cpp:117</a></div></div>
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<div class="ttc" id="anamespacemachine__learning_html_a6f1c98c016ad34ff3d9f39372161bd35"><div class="ttname"><a href="../../d8/d77/namespacemachine__learning.html#a6f1c98c016ad34ff3d9f39372161bd35">machine_learning::sum</a></div><div class="ttdeci">T sum(const std::vector< std::valarray< T > > &A)</div><div class="ttdef"><b>Definition</b> <a href="../../d8/d95/vector__ops_8hpp_source.html#l00232">vector_ops.hpp:232</a></div></div>
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<div class="ttc" id="anamespaceqr__algorithm_html_a6d3c7dce1f142141f509d09f6c0e25dc"><div class="ttname"><a href="#a6d3c7dce1f142141f509d09f6c0e25dc">qr_algorithm::vector_proj</a></div><div class="ttdeci">std::valarray< T > vector_proj(const std::valarray< T > &a, const std::valarray< T > &b)</div><div class="ttdef"><b>Definition</b> <a href="../../d4/d68/qr__decompose_8h_source.html#l00104">qr_decompose.h:104</a></div></div>
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<div class="ttc" id="anamespaceqr__algorithm_html_ad16da2183db22378435042f26af43d5f"><div class="ttname"><a href="#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> <a href="../../d4/d68/qr__decompose_8h_source.html#l00092">qr_decompose.h:92</a></div></div>
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<div class="ttc" id="avector__cross__product_8cpp_html_a4b2a9757a87c18e1642d72410ecfaba8"><div class="ttname"><a href="../../df/d66/vector__cross__product_8cpp.html#a4b2a9757a87c18e1642d72410ecfaba8">math::vector_cross::mag</a></div><div class="ttdeci">double mag(const std::array< double, 3 > &vec)</div><div class="ttdoc">Calculates the magnitude of the mathematical vector from it's direction ratios.</div><div class="ttdef"><b>Definition</b> <a href="../../df/d66/vector__cross__product_8cpp_source.html#l00083">vector_cross_product.cpp:83</a></div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a8ea313a1a1b5f9d0e3e332c29c6446ec">◆ </a></span>vector_dot()</h2>
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<td class="memname">double qr_algorithm::vector_dot </td>
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<p>Compute dot product of two vectors of equal lengths</p>
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<p>If \(\vec{a}=\left[a_0,a_1,a_2,...,a_L\right]\) and \(\vec{b}=\left[b_0,b_1,b_1,...,b_L\right]\) then \(\vec{a}\cdot\vec{b}=\displaystyle\sum_{i=0}^L a_i\times b_i\)</p>
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<dl class="section return"><dt>Returns</dt><dd>\(\vec{a}\cdot\vec{b}\) </dd></dl>
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<p class="definition">Definition at line <a class="el" href="../../d4/d68/qr__decompose_8h_source.html#l00076">76</a> of file <a class="el" href="../../d4/d68/qr__decompose_8h_source.html">qr_decompose.h</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 76</span> {</div>
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<div class="line"><span class="lineno"> 77</span> <span class="keywordflow">return</span> (a * b).sum();</div>
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<div class="line"><span class="lineno"> 78</span> <span class="comment">// could also use following</span></div>
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<div class="line"><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"><span class="lineno"> 80</span> <span class="comment">// 0.f);</span></div>
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<div class="line"><span class="lineno"> 81</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ad16da2183db22378435042f26af43d5f">◆ </a></span>vector_mag()</h2>
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<p>Compute magnitude of vector.</p>
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<p>If \(\vec{a}=\left[a_0,a_1,a_2,...,a_L\right]\) then \(\left|\vec{a}\right|=\sqrt{\displaystyle\sum_{i=0}^L a_i^2}\)</p>
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<dl class="section return"><dt>Returns</dt><dd>\(\left|\vec{a}\right|\) </dd></dl>
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<p class="definition">Definition at line <a class="el" href="../../d4/d68/qr__decompose_8h_source.html#l00092">92</a> of file <a class="el" href="../../d4/d68/qr__decompose_8h_source.html">qr_decompose.h</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 92</span> {</div>
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<div class="line"><span class="lineno"> 93</span> <span class="keywordtype">double</span> dot = <a class="code hl_function" href="#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(a, a);</div>
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<div class="line"><span class="lineno"> 94</span> <span class="keywordflow">return</span> std::sqrt(dot);</div>
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<div class="line"><span class="lineno"> 95</span>}</div>
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<div class="ttc" id="anamespaceqr__algorithm_html_a8ea313a1a1b5f9d0e3e332c29c6446ec"><div class="ttname"><a href="#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> <a href="../../d4/d68/qr__decompose_8h_source.html#l00076">qr_decompose.h:76</a></div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a6d3c7dce1f142141f509d09f6c0e25dc">◆ </a></span>vector_proj()</h2>
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<p>Compute projection of vector \(\vec{a}\) on \(\vec{b}\) defined as </p><p class="formulaDsp">
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\[\text{proj}_\vec{b}\vec{a}=\frac{\vec{a}\cdot\vec{b}}{\left|\vec{b}\right|^2}\vec{b}\]
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<dl class="section return"><dt>Returns</dt><dd>NULL if error, otherwise pointer to output </dd></dl>
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<p>check for division by zero using machine epsilon</p>
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<p class="definition">Definition at line <a class="el" href="../../d4/d68/qr__decompose_8h_source.html#l00104">104</a> of file <a class="el" href="../../d4/d68/qr__decompose_8h_source.html">qr_decompose.h</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 105</span> {</div>
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<div class="line"><span class="lineno"> 106</span> <span class="keywordtype">double</span> num = <a class="code hl_function" href="#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(a, b);</div>
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<div class="line"><span class="lineno"> 107</span> <span class="keywordtype">double</span> deno = <a class="code hl_function" href="#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(b, b);</div>
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<div class="line"><span class="lineno"> 108</span></div>
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<div class="line"><span class="lineno"> 110</span> <span class="keywordflow">if</span> (deno <= std::numeric_limits<double>::epsilon()) {</div>
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<div class="line"><span class="lineno"> 111</span> std::cerr << <span class="stringliteral">"["</span> << __func__ << <span class="stringliteral">"] Possible division by zero\n"</span>;</div>
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<div class="line"><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"><span class="lineno"> 113</span> }</div>
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<div class="line"><span class="lineno"> 114</span> </div>
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<div class="line"><span class="lineno"> 115</span> <span class="keywordtype">double</span> scalar = num / deno;</div>
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<div class="line"><span class="lineno"> 116</span> </div>
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<div class="line"><span class="lineno"> 117</span> <span class="keywordflow">return</span> b * scalar;</div>
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<div class="line"><span class="lineno"> 118</span>}</div>
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