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<p>Functions to compute <a href="https://en.wikipedia.org/wiki/QR_decomposition">QR decomposition</a> of any rectangular matrix.
<p>Functions to compute <a href="https://en.wikipedia.org/wiki/QR_decomposition" target="_blank">QR decomposition</a> of any rectangular matrix.
<a href="../../d2/d3b/namespaceqr__algorithm.html#details">More...</a></p>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>Functions to compute <a href="https://en.wikipedia.org/wiki/QR_decomposition">QR decomposition</a> of any rectangular matrix. </p>
<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>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="a763896dbb4a7e95c4a1e614ac0819d66"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a763896dbb4a7e95c4a1e614ac0819d66">&#9670;&nbsp;</a></span>eigen_values()</h2>
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<h2 class="memtitle"><span class="permalink"><a href="#a28e2fa3e803abaea6c568dc45d69d8cc">&#9670;&nbsp;</a></span>eigen_values()</h2>
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<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt;double&gt; qr_algorithm::eigen_values </td>
<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; qr_algorithm::eigen_values </td>
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<p>Compute eigen values using iterative shifted QR decomposition algorithm as follows:</p><ol type="1">
<p >Compute eigen values using iterative shifted QR decomposition algorithm as follows:</p><ol type="1">
<li>Use last diagonal element of A as eigen value approximation \(c\)</li>
<li>Shift diagonals of matrix \(A&#39; = A - cI\)</li>
<li>Decompose matrix \(A&#39;=QR\)</li>
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<div class="fragment"><div class="line"><a name="l00099"></a><span class="lineno"> 99</span>&#160; {</div>
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<div class="line"><a name="l00104"></a><span class="lineno"> 104</span>&#160; </div>
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<div class="line"><a name="l00106"></a><span class="lineno"> 106</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;std::valarray&lt;double&gt;</a>&gt; R(columns);</div>
<div class="line"><a name="l00107"></a><span class="lineno"> 107</span>&#160; </div>
<div class="line"><a name="l00108"></a><span class="lineno"> 108</span>&#160; <span class="comment">/* number of eigen values = matrix size */</span></div>
<div class="line"><a name="l00109"></a><span class="lineno"> 109</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a> eigen_vals(rows);</div>
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<div class="line"><a name="l00112"></a><span class="lineno"> 112</span>&#160; R[i] = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a>(columns);</div>
<div class="line"><a name="l00113"></a><span class="lineno"> 113</span>&#160; }</div>
<div class="line"><a name="l00114"></a><span class="lineno"> 114</span>&#160; </div>
<div class="line"><a name="l00115"></a><span class="lineno"> 115</span>&#160; <span class="comment">/* continue till all eigen values are found */</span></div>
<div class="line"><a name="l00116"></a><span class="lineno"> 116</span>&#160; <span class="keywordflow">while</span> (num_eigs &gt; 0) {</div>
<div class="line"><a name="l00117"></a><span class="lineno"> 117</span>&#160; <span class="comment">/* iterate with QR decomposition */</span></div>
<div class="line"><a name="l00118"></a><span class="lineno"> 118</span>&#160; <span class="keywordflow">while</span> (std::abs(A[0][num_eigs][num_eigs - 1]) &gt;</div>
<div class="line"><a name="l00119"></a><span class="lineno"> 119</span>&#160; <a class="codeRef" 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 name="l00120"></a><span class="lineno"> 120</span>&#160; <span class="comment">// initial approximation = last diagonal element</span></div>
<div class="line"><a name="l00121"></a><span class="lineno"> 121</span>&#160; last_eig = A[0][num_eigs][num_eigs];</div>
<div class="line"><a name="l00122"></a><span class="lineno"> 122</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; rows; i++) {</div>
<div class="line"><a name="l00123"></a><span class="lineno"> 123</span>&#160; A[0][i][i] -= last_eig; <span class="comment">/* A - cI */</span></div>
<div class="line"><a name="l00124"></a><span class="lineno"> 124</span>&#160; }</div>
<div class="line"><a name="l00125"></a><span class="lineno"> 125</span>&#160; </div>
<div class="line"><a name="l00126"></a><span class="lineno"> 126</span>&#160; <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#ab1ac74497ffb2101040cc1efe3546de8">qr_decompose</a>(*A, &amp;Q, &amp;R);</div>
<div class="line"><a name="l00127"></a><span class="lineno"> 127</span>&#160; </div>
<div class="line"><a name="l00128"></a><span class="lineno"> 128</span>&#160; <span class="keywordflow">if</span> (print_intermediates) {</div>
<div class="line"><a name="l00129"></a><span class="lineno"> 129</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; *A &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a name="l00130"></a><span class="lineno"> 130</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; Q &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a name="l00131"></a><span class="lineno"> 131</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; R &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a name="l00132"></a><span class="lineno"> 132</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/c/fprintf.html">printf</a>(<span class="stringliteral">&quot;-------------------- %d ---------------------\n&quot;</span>,</div>
<div class="line"><a name="l00133"></a><span class="lineno"> 133</span>&#160; ++counter);</div>
<div class="line"><a name="l00134"></a><span class="lineno"> 134</span>&#160; }</div>
<div class="line"><a name="l00135"></a><span class="lineno"> 135</span>&#160; </div>
<div class="line"><a name="l00136"></a><span class="lineno"> 136</span>&#160; <span class="comment">// new approximation A&#39; = R * Q</span></div>
<div class="line"><a name="l00137"></a><span class="lineno"> 137</span>&#160; <a class="code" href="../../de/d75/qr__eigen__values_8cpp.html#a7558fd7779bf207157ced66f275951a0">mat_mul</a>(R, Q, A);</div>
<div class="line"><a name="l00138"></a><span class="lineno"> 138</span>&#160; </div>
<div class="line"><a name="l00139"></a><span class="lineno"> 139</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; rows; i++) {</div>
<div class="line"><a name="l00140"></a><span class="lineno"> 140</span>&#160; A[0][i][i] += last_eig; <span class="comment">/* A + cI */</span></div>
<div class="line"><a name="l00141"></a><span class="lineno"> 141</span>&#160; }</div>
<div class="line"><a name="l00142"></a><span class="lineno"> 142</span>&#160; }</div>
<div class="line"><a name="l00143"></a><span class="lineno"> 143</span>&#160; </div>
<div class="line"><a name="l00144"></a><span class="lineno"> 144</span>&#160; <span class="comment">/* store the converged eigen value */</span></div>
<div class="line"><a name="l00145"></a><span class="lineno"> 145</span>&#160; eigen_vals[num_eigs] = last_eig;</div>
<div class="line"><a name="l00146"></a><span class="lineno"> 146</span>&#160; <span class="comment">// A[0][num_eigs][num_eigs];</span></div>
<div class="line"><a name="l00147"></a><span class="lineno"> 147</span>&#160; <span class="keywordflow">if</span> (print_intermediates) {</div>
<div class="line"><a name="l00148"></a><span class="lineno"> 148</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;========================\n&quot;</span>;</div>
<div class="line"><a name="l00149"></a><span class="lineno"> 149</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Eigen value: &quot;</span> &lt;&lt; last_eig &lt;&lt; <span class="stringliteral">&quot;,\n&quot;</span>;</div>
<div class="line"><a name="l00150"></a><span class="lineno"> 150</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;========================\n&quot;</span>;</div>
<div class="line"><a name="l00151"></a><span class="lineno"> 151</span>&#160; }</div>
<div class="line"><a name="l00152"></a><span class="lineno"> 152</span>&#160; </div>
<div class="line"><a name="l00153"></a><span class="lineno"> 153</span>&#160; num_eigs--;</div>
<div class="line"><a name="l00154"></a><span class="lineno"> 154</span>&#160; rows--;</div>
<div class="line"><a name="l00155"></a><span class="lineno"> 155</span>&#160; columns--;</div>
<div class="line"><a name="l00156"></a><span class="lineno"> 156</span>&#160; }</div>
<div class="line"><a name="l00157"></a><span class="lineno"> 157</span>&#160; eigen_vals[0] = A[0][0][0];</div>
<div class="line"><a name="l00158"></a><span class="lineno"> 158</span>&#160; </div>
<div class="line"><a name="l00159"></a><span class="lineno"> 159</span>&#160; <span class="keywordflow">if</span> (print_intermediates) {</div>
<div class="line"><a name="l00160"></a><span class="lineno"> 160</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; Q &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a name="l00161"></a><span class="lineno"> 161</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; R &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a name="l00162"></a><span class="lineno"> 162</span>&#160; }</div>
<div class="line"><a name="l00163"></a><span class="lineno"> 163</span>&#160; </div>
<div class="line"><a name="l00164"></a><span class="lineno"> 164</span>&#160; <span class="keywordflow">return</span> eigen_vals;</div>
<div class="line"><a name="l00165"></a><span class="lineno"> 165</span>&#160;}</div>
<div class="fragment"><div class="line"><a id="l00099" name="l00099"></a><span class="lineno"> 99</span> {</div>
<div class="line"><a id="l00100" name="l00100"></a><span class="lineno"> 100</span> <span class="keywordtype">int</span> rows = A-&gt;size();</div>
<div class="line"><a id="l00101" name="l00101"></a><span class="lineno"> 101</span> <span class="keywordtype">int</span> columns = rows;</div>
<div class="line"><a id="l00102" name="l00102"></a><span class="lineno"> 102</span> <span class="keywordtype">int</span> counter = 0, num_eigs = rows - 1;</div>
<div class="line"><a id="l00103" name="l00103"></a><span class="lineno"> 103</span> <span class="keywordtype">double</span> last_eig = 0;</div>
<div class="line"><a id="l00104" name="l00104"></a><span class="lineno"> 104</span> </div>
<div class="line"><a id="l00105" name="l00105"></a><span class="lineno"> 105</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;std::valarray&lt;double&gt;</a>&gt; Q(rows);</div>
<div class="line"><a id="l00106" name="l00106"></a><span class="lineno"> 106</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;std::valarray&lt;double&gt;</a>&gt; R(columns);</div>
<div class="line"><a id="l00107" name="l00107"></a><span class="lineno"> 107</span> </div>
<div class="line"><a id="l00108" name="l00108"></a><span class="lineno"> 108</span> <span class="comment">/* number of eigen values = matrix size */</span></div>
<div class="line"><a id="l00109" name="l00109"></a><span class="lineno"> 109</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a> eigen_vals(rows);</div>
<div class="line"><a id="l00110" name="l00110"></a><span class="lineno"> 110</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; rows; i++) {</div>
<div class="line"><a id="l00111" name="l00111"></a><span class="lineno"> 111</span> Q[i] = <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a>(columns);</div>
<div class="line"><a id="l00112" name="l00112"></a><span class="lineno"> 112</span> R[i] = <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a>(columns);</div>
<div class="line"><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="comment">/* continue till all eigen values are found */</span></div>
<div class="line"><a id="l00116" name="l00116"></a><span class="lineno"> 116</span> <span class="keywordflow">while</span> (num_eigs &gt; 0) {</div>
<div class="line"><a id="l00117" name="l00117"></a><span class="lineno"> 117</span> <span class="comment">/* iterate with QR decomposition */</span></div>
<div class="line"><a id="l00118" name="l00118"></a><span class="lineno"> 118</span> <span class="keywordflow">while</span> (std::abs(A[0][num_eigs][num_eigs - 1]) &gt;</div>
<div class="line"><a id="l00119" name="l00119"></a><span class="lineno"> 119</span> <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="l00120" name="l00120"></a><span class="lineno"> 120</span> <span class="comment">// initial approximation = last diagonal element</span></div>
<div class="line"><a id="l00121" name="l00121"></a><span class="lineno"> 121</span> last_eig = A[0][num_eigs][num_eigs];</div>
<div class="line"><a id="l00122" name="l00122"></a><span class="lineno"> 122</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; rows; i++) {</div>
<div class="line"><a id="l00123" name="l00123"></a><span class="lineno"> 123</span> A[0][i][i] -= last_eig; <span class="comment">/* A - cI */</span></div>
<div class="line"><a id="l00124" name="l00124"></a><span class="lineno"> 124</span> }</div>
<div class="line"><a id="l00125" name="l00125"></a><span class="lineno"> 125</span> </div>
<div class="line"><a id="l00126" name="l00126"></a><span class="lineno"> 126</span> <a class="code hl_function" href="../../d2/d3b/namespaceqr__algorithm.html#a73ce637634fc49e1d10d190eb388ebf1">qr_decompose</a>(*A, &amp;Q, &amp;R);</div>
<div class="line"><a id="l00127" name="l00127"></a><span class="lineno"> 127</span> </div>
<div class="line"><a id="l00128" name="l00128"></a><span class="lineno"> 128</span> <span class="keywordflow">if</span> (print_intermediates) {</div>
<div class="line"><a id="l00129" name="l00129"></a><span class="lineno"> 129</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; *A &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a id="l00130" name="l00130"></a><span class="lineno"> 130</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; Q &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a id="l00131" name="l00131"></a><span class="lineno"> 131</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; R &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a id="l00132" name="l00132"></a><span class="lineno"> 132</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/c/fprintf.html">printf</a>(<span class="stringliteral">&quot;-------------------- %d ---------------------\n&quot;</span>,</div>
<div class="line"><a id="l00133" name="l00133"></a><span class="lineno"> 133</span> ++counter);</div>
<div class="line"><a id="l00134" name="l00134"></a><span class="lineno"> 134</span> }</div>
<div class="line"><a id="l00135" name="l00135"></a><span class="lineno"> 135</span> </div>
<div class="line"><a id="l00136" name="l00136"></a><span class="lineno"> 136</span> <span class="comment">// new approximation A&#39; = R * Q</span></div>
<div class="line"><a id="l00137" name="l00137"></a><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>
<div class="line"><a id="l00138" name="l00138"></a><span class="lineno"> 138</span> </div>
<div class="line"><a id="l00139" name="l00139"></a><span class="lineno"> 139</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; rows; i++) {</div>
<div class="line"><a id="l00140" name="l00140"></a><span class="lineno"> 140</span> A[0][i][i] += last_eig; <span class="comment">/* A + cI */</span></div>
<div class="line"><a id="l00141" name="l00141"></a><span class="lineno"> 141</span> }</div>
<div class="line"><a id="l00142" name="l00142"></a><span class="lineno"> 142</span> }</div>
<div class="line"><a id="l00143" name="l00143"></a><span class="lineno"> 143</span> </div>
<div class="line"><a id="l00144" name="l00144"></a><span class="lineno"> 144</span> <span class="comment">/* store the converged eigen value */</span></div>
<div class="line"><a id="l00145" name="l00145"></a><span class="lineno"> 145</span> eigen_vals[num_eigs] = last_eig;</div>
<div class="line"><a id="l00146" name="l00146"></a><span class="lineno"> 146</span> <span class="comment">// A[0][num_eigs][num_eigs];</span></div>
<div class="line"><a id="l00147" name="l00147"></a><span class="lineno"> 147</span> <span class="keywordflow">if</span> (print_intermediates) {</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/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;========================\n&quot;</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/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Eigen value: &quot;</span> &lt;&lt; last_eig &lt;&lt; <span class="stringliteral">&quot;,\n&quot;</span>;</div>
<div class="line"><a id="l00150" name="l00150"></a><span class="lineno"> 150</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;========================\n&quot;</span>;</div>
<div class="line"><a id="l00151" name="l00151"></a><span class="lineno"> 151</span> }</div>
<div class="line"><a id="l00152" name="l00152"></a><span class="lineno"> 152</span> </div>
<div class="line"><a id="l00153" name="l00153"></a><span class="lineno"> 153</span> num_eigs--;</div>
<div class="line"><a id="l00154" name="l00154"></a><span class="lineno"> 154</span> rows--;</div>
<div class="line"><a id="l00155" name="l00155"></a><span class="lineno"> 155</span> columns--;</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> eigen_vals[0] = A[0][0][0];</div>
<div class="line"><a id="l00158" name="l00158"></a><span class="lineno"> 158</span> </div>
<div class="line"><a id="l00159" name="l00159"></a><span class="lineno"> 159</span> <span class="keywordflow">if</span> (print_intermediates) {</div>
<div class="line"><a id="l00160" name="l00160"></a><span class="lineno"> 160</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; Q &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a id="l00161" name="l00161"></a><span class="lineno"> 161</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; R &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a id="l00162" name="l00162"></a><span class="lineno"> 162</span> }</div>
<div class="line"><a id="l00163" name="l00163"></a><span class="lineno"> 163</span> </div>
<div class="line"><a id="l00164" name="l00164"></a><span class="lineno"> 164</span> <span class="keywordflow">return</span> eigen_vals;</div>
<div class="line"><a id="l00165" name="l00165"></a><span class="lineno"> 165</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::cout</a></div></div>
<div class="ttc" id="afprintf_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/c/fprintf.html">std::printf</a></div><div class="ttdeci">T printf(T... args)</div></div>
<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&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_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="anumeric_limits_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/types/numeric_limits.html">std::numeric_limits</a></div></div>
<div class="ttc" id="aqr__eigen__values_8cpp_html_a7558fd7779bf207157ced66f275951a0"><div class="ttname"><a href="../../de/d75/qr__eigen__values_8cpp.html#a7558fd7779bf207157ced66f275951a0">mat_mul</a></div><div class="ttdeci">void mat_mul(const std::valarray&lt; std::valarray&lt; double &gt;&gt; &amp;A, const std::valarray&lt; std::valarray&lt; double &gt;&gt; &amp;B, std::valarray&lt; std::valarray&lt; double &gt;&gt; *OUT)</div><div class="ttdef"><b>Definition:</b> qr_eigen_values.cpp:54</div></div>
<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&lt; std::valarray&lt; double &gt; &gt; &amp;A, const std::valarray&lt; std::valarray&lt; double &gt; &gt; &amp;B, std::valarray&lt; std::valarray&lt; double &gt; &gt; *OUT)</div><div class="ttdef"><b>Definition:</b> qr_eigen_values.cpp:54</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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<h2 class="memtitle"><span class="permalink"><a href="#a7c63c78d6839b8ad73b77c161210b514">&#9670;&nbsp;</a></span>operator&lt;&lt;() <span class="overload">[1/2]</span></h2>
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<h2 class="memtitle"><span class="permalink"><a href="#adfbdf47277c8cfee229b05b72f1f7834">&#9670;&nbsp;</a></span>operator&lt;&lt;() <span class="overload">[1/2]</span></h2>
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template&lt;typename T &gt; </div>
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<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a>&amp; qr_algorithm::operator&lt;&lt; </td>
<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> &amp; qr_algorithm::operator&lt;&lt; </td>
<td>(</td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> &amp;&#160;</td>
<td class="paramname"><em>out</em>, </td>
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<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; T &gt;&gt; const &amp;&#160;</td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; T &gt; &gt; const &amp;&#160;</td>
<td class="paramname"><em>v</em>&#160;</td>
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<p>operator to print a matrix </p>
<div class="fragment"><div class="line"><a name="l00034"></a><span class="lineno"> 34</span>&#160; {</div>
<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>&#160; <span class="keyword">const</span> <span class="keywordtype">int</span> width = 12;</div>
<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>&#160; <span class="keyword">const</span> <span class="keywordtype">char</span> separator = <span class="charliteral">&#39; &#39;</span>;</div>
<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>&#160; </div>
<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>&#160; out.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/ios_base/precision.html">precision</a>(4);</div>
<div class="line"><a name="l00039"></a><span class="lineno"> 39</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">size_t</span> row = 0; row &lt; v.size(); row++) {</div>
<div class="line"><a name="l00040"></a><span class="lineno"> 40</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">size_t</span> col = 0; col &lt; v[row].size(); col++)</div>
<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>&#160; out &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/left.html">std::right</a> &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setw.html">std::setw</a>(width) &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setfill.html">std::setfill</a>(separator)</div>
<div class="line"><a name="l00042"></a><span class="lineno"> 42</span>&#160; &lt;&lt; v[row][col];</div>
<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>&#160; out &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>&#160; }</div>
<div class="line"><a name="l00045"></a><span class="lineno"> 45</span>&#160; </div>
<div class="line"><a name="l00046"></a><span class="lineno"> 46</span>&#160; <span class="keywordflow">return</span> out;</div>
<div class="line"><a name="l00047"></a><span class="lineno"> 47</span>&#160;}</div>
<p >operator to print a matrix </p>
<div class="fragment"><div class="line"><a id="l00034" name="l00034"></a><span class="lineno"> 34</span> {</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 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>
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<h2 class="memtitle"><span class="permalink"><a href="#a240edf2b6b5942c0e5f2c9d3e326797f">&#9670;&nbsp;</a></span>operator&lt;&lt;() <span class="overload">[2/2]</span></h2>
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<h2 class="memtitle"><span class="permalink"><a href="#a257425cb2365359da51c6fe6741834d8">&#9670;&nbsp;</a></span>operator&lt;&lt;() <span class="overload">[2/2]</span></h2>
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template&lt;typename T &gt; </div>
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<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a>&amp; qr_algorithm::operator&lt;&lt; </td>
<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> &amp; qr_algorithm::operator&lt;&lt; </td>
<td>(</td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> &amp;&#160;</td>
<td class="paramname"><em>out</em>, </td>
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<p>operator to print a vector </p>
<div class="fragment"><div class="line"><a name="l00053"></a><span class="lineno"> 53</span>&#160; {</div>
<div class="line"><a name="l00054"></a><span class="lineno"> 54</span>&#160; <span class="keyword">const</span> <span class="keywordtype">int</span> width = 10;</div>
<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>&#160; <span class="keyword">const</span> <span class="keywordtype">char</span> separator = <span class="charliteral">&#39; &#39;</span>;</div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>&#160; </div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>&#160; out.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/ios_base/precision.html">precision</a>(4);</div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">size_t</span> row = 0; row &lt; v.size(); row++) {</div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>&#160; out &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/left.html">std::right</a> &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setw.html">std::setw</a>(width) &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setfill.html">std::setfill</a>(separator)</div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>&#160; &lt;&lt; v[row];</div>
<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>&#160; }</div>
<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>&#160; </div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>&#160; <span class="keywordflow">return</span> out;</div>
<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>&#160;}</div>
<p >operator to print a vector </p>
<div class="fragment"><div class="line"><a id="l00053" name="l00053"></a><span class="lineno"> 53</span> {</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>
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<h2 class="memtitle"><span class="permalink"><a href="#ab1ac74497ffb2101040cc1efe3546de8">&#9670;&nbsp;</a></span>qr_decompose()</h2>
<a id="a73ce637634fc49e1d10d190eb388ebf1" name="a73ce637634fc49e1d10d190eb388ebf1"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a73ce637634fc49e1d10d190eb388ebf1">&#9670;&nbsp;</a></span>qr_decompose()</h2>
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@@ -375,19 +374,19 @@ template&lt;typename T &gt; </div>
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<td class="memname">void qr_algorithm::qr_decompose </td>
<td>(</td>
<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; T &gt;&gt; &amp;&#160;</td>
<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; T &gt; &gt; &amp;&#160;</td>
<td class="paramname"><em>A</em>, </td>
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<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; T &gt;&gt; *&#160;</td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; T &gt; &gt; *&#160;</td>
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<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; T &gt;&gt; *&#160;</td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; T &gt; &gt; *&#160;</td>
<td class="paramname"><em>R</em>&#160;</td>
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@@ -397,7 +396,7 @@ template&lt;typename T &gt; </div>
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<p>Decompose matrix \(A\) using <a href="https://en.wikipedia.org/wiki/QR_decomposition">Gram-Schmidt process</a>.</p>
<p >Decompose matrix \(A\) using <a href="https://en.wikipedia.org/wiki/QR_decomposition" target="_blank">Gram-Schmidt process</a>.</p>
<p class="formulaDsp">
\begin{eqnarray*} \text{given that}\quad A &amp;=&amp; *\left[\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_{N-1},\right]\\ \text{where}\quad\mathbf{a}_i &amp;=&amp; \left[a_{0i},a_{1i},a_{2i},\ldots,a_{(M-1)i}\right]^T\quad\ldots\mbox{(column vectors)}\\ \text{then}\quad\mathbf{u}_i &amp;=&amp; \mathbf{a}_i *-\sum_{j=0}^{i-1}\text{proj}_{\mathbf{u}_j}\mathbf{a}_i\\ \mathbf{e}_i &amp;=&amp;\frac{\mathbf{u}_i}{\left|\mathbf{u}_i\right|}\\ Q &amp;=&amp; \begin{bmatrix}\mathbf{e}_0 &amp; \mathbf{e}_1 &amp; \mathbf{e}_2 &amp; \dots &amp; \mathbf{e}_{N-1}\end{bmatrix}\\ R &amp;=&amp; \begin{bmatrix}\langle\mathbf{e}_0\,,\mathbf{a}_0\rangle &amp; \langle\mathbf{e}_1\,,\mathbf{a}_1\rangle &amp; \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle &amp; \dots \\ 0 &amp; \langle\mathbf{e}_1\,,\mathbf{a}_1\rangle &amp; \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle &amp; \dots\\ 0 &amp; 0 &amp; \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle &amp; \dots\\ \vdots &amp; \vdots &amp; \vdots &amp; \ddots \end{bmatrix}\\ \end{eqnarray*}
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@@ -409,79 +408,79 @@ template&lt;typename T &gt; </div>
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<div class="fragment"><div class="line"><a name="l00150"></a><span class="lineno"> 150</span>&#160; {</div>
<div class="line"><a name="l00151"></a><span class="lineno"> 151</span>&#160; <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>
<div class="line"><a name="l00152"></a><span class="lineno"> 152</span>&#160; <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>
<div class="line"><a name="l00153"></a><span class="lineno"> 153</span>&#160; <a class="codeRef" 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 name="l00154"></a><span class="lineno"> 154</span>&#160; <a class="codeRef" 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 name="l00155"></a><span class="lineno"> 155</span>&#160; <a class="codeRef" 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 name="l00156"></a><span class="lineno"> 156</span>&#160; </div>
<div class="line"><a name="l00157"></a><span class="lineno"> 157</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; COLUMNS; i++) {</div>
<div class="line"><a name="l00158"></a><span class="lineno"> 158</span>&#160; <span class="comment">/* for each column =&gt; R is a square matrix of NxN */</span></div>
<div class="line"><a name="l00159"></a><span class="lineno"> 159</span>&#160; <span class="keywordtype">int</span> j;</div>
<div class="line"><a name="l00160"></a><span class="lineno"> 160</span>&#160; R[0][i] = 0.; <span class="comment">/* make R upper triangular */</span></div>
<div class="line"><a name="l00161"></a><span class="lineno"> 161</span>&#160; </div>
<div class="line"><a name="l00162"></a><span class="lineno"> 162</span>&#160; <span class="comment">/* get corresponding Q vector */</span></div>
<div class="line"><a name="l00163"></a><span class="lineno"> 163</span>&#160;<span class="preprocessor">#ifdef _OPENMP</span></div>
<div class="line"><a name="l00164"></a><span class="lineno"> 164</span>&#160;<span class="comment">// parallelize on threads</span></div>
<div class="line"><a name="l00165"></a><span class="lineno"> 165</span>&#160;<span class="preprocessor">#pragma omp for</span></div>
<div class="line"><a name="l00166"></a><span class="lineno"> 166</span>&#160;<span class="preprocessor">#endif</span></div>
<div class="line"><a name="l00167"></a><span class="lineno"> 167</span>&#160; <span class="keywordflow">for</span> (j = 0; j &lt; ROWS; j++) {</div>
<div class="line"><a name="l00168"></a><span class="lineno"> 168</span>&#160; tmp_vector[j] = A[j][i]; <span class="comment">/* accumulator for uk */</span></div>
<div class="line"><a name="l00169"></a><span class="lineno"> 169</span>&#160; col_vector[j] = A[j][i];</div>
<div class="line"><a name="l00170"></a><span class="lineno"> 170</span>&#160; }</div>
<div class="line"><a name="l00171"></a><span class="lineno"> 171</span>&#160; <span class="keywordflow">for</span> (j = 0; j &lt; i; j++) {</div>
<div class="line"><a name="l00172"></a><span class="lineno"> 172</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> k = 0; k &lt; ROWS; k++) {</div>
<div class="line"><a name="l00173"></a><span class="lineno"> 173</span>&#160; col_vector2[k] = Q[0][k][j];</div>
<div class="line"><a name="l00174"></a><span class="lineno"> 174</span>&#160; }</div>
<div class="line"><a name="l00175"></a><span class="lineno"> 175</span>&#160; col_vector2 = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a5422c76bfa322620de611083bd518d71">vector_proj</a>(col_vector, col_vector2);</div>
<div class="line"><a name="l00176"></a><span class="lineno"> 176</span>&#160; tmp_vector -= col_vector2;</div>
<div class="line"><a name="l00177"></a><span class="lineno"> 177</span>&#160; }</div>
<div class="line"><a name="l00178"></a><span class="lineno"> 178</span>&#160; </div>
<div class="line"><a name="l00179"></a><span class="lineno"> 179</span>&#160; <span class="keywordtype">double</span> <a class="code" href="../../df/d66/vector__cross__product_8cpp.html#a4b2a9757a87c18e1642d72410ecfaba8">mag</a> = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#ad16da2183db22378435042f26af43d5f">vector_mag</a>(tmp_vector);</div>
<div class="line"><a name="l00180"></a><span class="lineno"> 180</span>&#160; </div>
<div class="line"><a name="l00181"></a><span class="lineno"> 181</span>&#160;<span class="preprocessor">#ifdef _OPENMP</span></div>
<div class="line"><a name="l00182"></a><span class="lineno"> 182</span>&#160;<span class="comment">// parallelize on threads</span></div>
<div class="line"><a name="l00183"></a><span class="lineno"> 183</span>&#160;<span class="preprocessor">#pragma omp for</span></div>
<div class="line"><a name="l00184"></a><span class="lineno"> 184</span>&#160;<span class="preprocessor">#endif</span></div>
<div class="line"><a name="l00185"></a><span class="lineno"> 185</span>&#160; <span class="keywordflow">for</span> (j = 0; j &lt; ROWS; j++) Q[0][j][i] = tmp_vector[j] / mag;</div>
<div class="line"><a name="l00186"></a><span class="lineno"> 186</span>&#160; </div>
<div class="line"><a name="l00187"></a><span class="lineno"> 187</span>&#160; <span class="comment">/* compute upper triangular values of R */</span></div>
<div class="line"><a name="l00188"></a><span class="lineno"> 188</span>&#160;<span class="preprocessor">#ifdef _OPENMP</span></div>
<div class="line"><a name="l00189"></a><span class="lineno"> 189</span>&#160;<span class="comment">// parallelize on threads</span></div>
<div class="line"><a name="l00190"></a><span class="lineno"> 190</span>&#160;<span class="preprocessor">#pragma omp for</span></div>
<div class="line"><a name="l00191"></a><span class="lineno"> 191</span>&#160;<span class="preprocessor">#endif</span></div>
<div class="line"><a name="l00192"></a><span class="lineno"> 192</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> kk = 0; kk &lt; ROWS; kk++) {</div>
<div class="line"><a name="l00193"></a><span class="lineno"> 193</span>&#160; col_vector[kk] = Q[0][kk][i];</div>
<div class="line"><a name="l00194"></a><span class="lineno"> 194</span>&#160; }</div>
<div class="line"><a name="l00195"></a><span class="lineno"> 195</span>&#160; </div>
<div class="line"><a name="l00196"></a><span class="lineno"> 196</span>&#160;<span class="preprocessor">#ifdef _OPENMP</span></div>
<div class="line"><a name="l00197"></a><span class="lineno"> 197</span>&#160;<span class="comment">// parallelize on threads</span></div>
<div class="line"><a name="l00198"></a><span class="lineno"> 198</span>&#160;<span class="preprocessor">#pragma omp for</span></div>
<div class="line"><a name="l00199"></a><span class="lineno"> 199</span>&#160;<span class="preprocessor">#endif</span></div>
<div class="line"><a name="l00200"></a><span class="lineno"> 200</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> k = i; k &lt; COLUMNS; k++) {</div>
<div class="line"><a name="l00201"></a><span class="lineno"> 201</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> kk = 0; kk &lt; ROWS; kk++) {</div>
<div class="line"><a name="l00202"></a><span class="lineno"> 202</span>&#160; col_vector2[kk] = A[kk][k];</div>
<div class="line"><a name="l00203"></a><span class="lineno"> 203</span>&#160; }</div>
<div class="line"><a name="l00204"></a><span class="lineno"> 204</span>&#160; R[0][i][k] = (col_vector * col_vector2).<a class="code" href="../../d8/d77/namespacemachine__learning.html#a5305cc18d11943d3dc0d66e70d7e9c44">sum</a>();</div>
<div class="line"><a name="l00205"></a><span class="lineno"> 205</span>&#160; }</div>
<div class="line"><a name="l00206"></a><span class="lineno"> 206</span>&#160; }</div>
<div class="line"><a name="l00207"></a><span class="lineno"> 207</span>&#160;}</div>
<div class="ttc" id="anamespacemachine__learning_html_a5305cc18d11943d3dc0d66e70d7e9c44"><div class="ttname"><a href="../../d8/d77/namespacemachine__learning.html#a5305cc18d11943d3dc0d66e70d7e9c44">machine_learning::sum</a></div><div class="ttdeci">T sum(const std::vector&lt; std::valarray&lt; T &gt;&gt; &amp;A)</div><div class="ttdef"><b>Definition:</b> vector_ops.hpp:232</div></div>
<div class="ttc" id="anamespaceqr__algorithm_html_a5422c76bfa322620de611083bd518d71"><div class="ttname"><a href="../../d2/d3b/namespaceqr__algorithm.html#a5422c76bfa322620de611083bd518d71">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="fragment"><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> <a class="code hl_function" href="../../df/d66/vector__cross__product_8cpp.html#a4b2a9757a87c18e1642d72410ecfaba8">mag</a> = <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).<a class="code hl_function" href="../../d8/d77/namespacemachine__learning.html#a6f1c98c016ad34ff3d9f39372161bd35">sum</a>();</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 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&lt; std::valarray&lt; T &gt; &gt; &amp;A)</div><div class="ttdef"><b>Definition:</b> vector_ops.hpp:232</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_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="asize_t_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/types/size_t.html">std::size_t</a></div></div>
<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&lt; double, 3 &gt; &amp;vec)</div><div class="ttdoc">Calculates the magnitude of the mathematical vector from it's direction ratios.</div><div class="ttdef"><b>Definition:</b> vector_cross_product.cpp:83</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a8ea313a1a1b5f9d0e3e332c29c6446ec">&#9670;&nbsp;</a></span>vector_dot()</h2>
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<p>Compute dot product of two vectors of equal lengths</p>
<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>
<p >Compute dot product of two vectors of equal lengths</p>
<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>
<dl class="section return"><dt>Returns</dt><dd>\(\vec{a}\cdot\vec{b}\) </dd></dl>
<div class="fragment"><div class="line"><a name="l00076"></a><span class="lineno"> 76</span>&#160; {</div>
<div class="line"><a name="l00077"></a><span class="lineno"> 77</span>&#160; <span class="keywordflow">return</span> (a * b).sum();</div>
<div class="line"><a name="l00078"></a><span class="lineno"> 78</span>&#160; <span class="comment">// could also use following</span></div>
<div class="line"><a name="l00079"></a><span class="lineno"> 79</span>&#160; <span class="comment">// return std::inner_product(std::begin(a), std::end(a), std::begin(b),</span></div>
<div class="line"><a name="l00080"></a><span class="lineno"> 80</span>&#160; <span class="comment">// 0.f);</span></div>
<div class="line"><a name="l00081"></a><span class="lineno"> 81</span>&#160;}</div>
<div class="fragment"><div class="line"><a id="l00076" name="l00076"></a><span class="lineno"> 76</span> {</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>
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<h2 class="memtitle"><span class="permalink"><a href="#ad16da2183db22378435042f26af43d5f">&#9670;&nbsp;</a></span>vector_mag()</h2>
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<p>Compute magnitude of vector.</p>
<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>
<p >Compute magnitude of vector.</p>
<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>
<dl class="section return"><dt>Returns</dt><dd>\(\left|\vec{a}\right|\) </dd></dl>
<div class="fragment"><div class="line"><a name="l00092"></a><span class="lineno"> 92</span>&#160; {</div>
<div class="line"><a name="l00093"></a><span class="lineno"> 93</span>&#160; <span class="keywordtype">double</span> dot = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(a, a);</div>
<div class="line"><a name="l00094"></a><span class="lineno"> 94</span>&#160; <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>
<div class="line"><a name="l00095"></a><span class="lineno"> 95</span>&#160;}</div>
<div class="fragment"><div class="line"><a id="l00092" name="l00092"></a><span class="lineno"> 92</span> {</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 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="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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template&lt;typename T &gt; </div>
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<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt;T&gt; qr_algorithm::vector_proj </td>
<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; T &gt; qr_algorithm::vector_proj </td>
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<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; T &gt; &amp;&#160;</td>
<td class="paramname"><em>a</em>, </td>
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<p>Compute projection of vector \(\vec{a}\) on \(\vec{b}\) defined as </p><p class="formulaDsp">
<p >Compute projection of vector \(\vec{a}\) on \(\vec{b}\) defined as </p><p class="formulaDsp">
\[\text{proj}_\vec{b}\vec{a}=\frac{\vec{a}\cdot\vec{b}}{\left|\vec{b}\right|^2}\vec{b}\]
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<dl class="section return"><dt>Returns</dt><dd>NULL if error, otherwise pointer to output </dd></dl>
<p>check for division by zero using machine epsilon</p>
<div class="fragment"><div class="line"><a name="l00105"></a><span class="lineno"> 105</span>&#160; {</div>
<div class="line"><a name="l00106"></a><span class="lineno"> 106</span>&#160; <span class="keywordtype">double</span> num = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(a, b);</div>
<div class="line"><a name="l00107"></a><span class="lineno"> 107</span>&#160; <span class="keywordtype">double</span> deno = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(b, b);</div>
<div class="line"><a name="l00108"></a><span class="lineno"> 108</span>&#160;<span class="comment"></span> </div>
<div class="line"><a name="l00109"></a><span class="lineno"> 109</span>&#160;<span class="comment"> /*! check for division by zero using machine epsilon */</span></div>
<div class="line"><a name="l00110"></a><span class="lineno"> 110</span>&#160; <span class="keywordflow">if</span> (deno &lt;= <a class="codeRef" 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 name="l00111"></a><span class="lineno"> 111</span>&#160; <a class="codeRef" 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 name="l00112"></a><span class="lineno"> 112</span>&#160; <span class="keywordflow">return</span> a; <span class="comment">// return vector a back</span></div>
<div class="line"><a name="l00113"></a><span class="lineno"> 113</span>&#160; }</div>
<div class="line"><a name="l00114"></a><span class="lineno"> 114</span>&#160; </div>
<div class="line"><a name="l00115"></a><span class="lineno"> 115</span>&#160; <span class="keywordtype">double</span> scalar = num / deno;</div>
<div class="line"><a name="l00116"></a><span class="lineno"> 116</span>&#160; </div>
<div class="line"><a name="l00117"></a><span class="lineno"> 117</span>&#160; <span class="keywordflow">return</span> b * scalar;</div>
<div class="line"><a name="l00118"></a><span class="lineno"> 118</span>&#160;}</div>
<p >check for division by zero using machine epsilon</p>
<div class="fragment"><div class="line"><a id="l00105" name="l00105"></a><span class="lineno"> 105</span> {</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>
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