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<p>Functions to compute <a href="https://en.wikipedia.org/wiki/QR_decomposition">QR decomposition</a> of any rectangular matrix.
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<a href="../../d2/d3b/namespaceqr__algorithm.html#details">More...</a></p>
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<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="func-members"></a>
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Functions</h2></td></tr>
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<tr class="memitem:a7c63c78d6839b8ad73b77c161210b514"><td class="memTemplParams" colspan="2">template<typename T > </td></tr>
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<tr class="memitem:a7c63c78d6839b8ad73b77c161210b514"><td class="memTemplItemLeft" align="right" valign="top"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> & </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="../../d2/d3b/namespaceqr__algorithm.html#a7c63c78d6839b8ad73b77c161210b514">operator<<</a> (<a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> &out, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< T >> const &v)</td></tr>
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<tr class="separator:a7c63c78d6839b8ad73b77c161210b514"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a240edf2b6b5942c0e5f2c9d3e326797f"><td class="memTemplParams" colspan="2">template<typename T > </td></tr>
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<tr class="memitem:a240edf2b6b5942c0e5f2c9d3e326797f"><td class="memTemplItemLeft" align="right" valign="top"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> & </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="../../d2/d3b/namespaceqr__algorithm.html#a240edf2b6b5942c0e5f2c9d3e326797f">operator<<</a> (<a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> &out, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< T > const &v)</td></tr>
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<tr class="separator:a240edf2b6b5942c0e5f2c9d3e326797f"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a8ea313a1a1b5f9d0e3e332c29c6446ec"><td class="memTemplParams" colspan="2">template<typename T > </td></tr>
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<tr class="memitem:a8ea313a1a1b5f9d0e3e332c29c6446ec"><td class="memTemplItemLeft" align="right" valign="top">double </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a> (const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< T > &<a class="el" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< T > &b)</td></tr>
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<tr class="separator:a8ea313a1a1b5f9d0e3e332c29c6446ec"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:ad16da2183db22378435042f26af43d5f"><td class="memTemplParams" colspan="2">template<typename T > </td></tr>
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<tr class="memitem:ad16da2183db22378435042f26af43d5f"><td class="memTemplItemLeft" align="right" valign="top">double </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="../../d2/d3b/namespaceqr__algorithm.html#ad16da2183db22378435042f26af43d5f">vector_mag</a> (const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< T > &<a class="el" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>)</td></tr>
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<tr class="memitem:a5422c76bfa322620de611083bd518d71"><td class="memTemplParams" colspan="2">template<typename T > </td></tr>
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<tr class="memitem:a5422c76bfa322620de611083bd518d71"><td class="memTemplItemLeft" align="right" valign="top"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< T > </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="../../d2/d3b/namespaceqr__algorithm.html#a5422c76bfa322620de611083bd518d71">vector_proj</a> (const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< T > &<a class="el" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< T > &b)</td></tr>
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<tr class="memitem:ab1ac74497ffb2101040cc1efe3546de8"><td class="memTemplParams" colspan="2">template<typename T > </td></tr>
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<tr class="memitem:ab1ac74497ffb2101040cc1efe3546de8"><td class="memTemplItemLeft" align="right" valign="top">void </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="../../d2/d3b/namespaceqr__algorithm.html#ab1ac74497ffb2101040cc1efe3546de8">qr_decompose</a> (const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< T >> &A, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< T >> *Q, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< T >> *R)</td></tr>
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<tr class="separator:ab1ac74497ffb2101040cc1efe3546de8"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a763896dbb4a7e95c4a1e614ac0819d66"><td class="memItemLeft" align="right" valign="top"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d2/d3b/namespaceqr__algorithm.html#a763896dbb4a7e95c4a1e614ac0819d66">eigen_values</a> (<a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double >> *A, bool print_intermediates=false)</td></tr>
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</table>
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<a name="details" id="details"></a><h2 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">QR decomposition</a> of any rectangular matrix. </p>
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</div><h2 class="groupheader">Function Documentation</h2>
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<a id="a763896dbb4a7e95c4a1e614ac0819d66"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a763896dbb4a7e95c4a1e614ac0819d66">◆ </a></span>eigen_values()</h2>
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<div class="memitem">
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<div class="memproto">
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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><double> qr_algorithm::eigen_values </td>
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<td>(</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>< <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double >> * </td>
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<td class="paramname"><em>A</em>, </td>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">bool </td>
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<td class="paramname"><em>print_intermediates</em> = <code>false</code> </td>
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<tr>
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<td></td>
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<td>)</td>
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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 = <code>false</code>) </td></tr>
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</dd>
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</dl>
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<div class="fragment"><div class="line"><a name="l00099"></a><span class="lineno"> 99</span>  {</div>
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<div class="line"><a name="l00100"></a><span class="lineno"> 100</span>  <span class="keywordtype">int</span> rows = A->size();</div>
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<div class="line"><a name="l00101"></a><span class="lineno"> 101</span>  <span class="keywordtype">int</span> columns = rows;</div>
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<div class="line"><a name="l00102"></a><span class="lineno"> 102</span>  <span class="keywordtype">int</span> counter = 0, num_eigs = rows - 1;</div>
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<div class="line"><a name="l00103"></a><span class="lineno"> 103</span>  <span class="keywordtype">double</span> last_eig = 0;</div>
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<div class="line"><a name="l00104"></a><span class="lineno"> 104</span>  </div>
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<div class="line"><a name="l00105"></a><span class="lineno"> 105</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<std::valarray<double></a>> Q(rows);</div>
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<div class="line"><a name="l00106"></a><span class="lineno"> 106</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<std::valarray<double></a>> R(columns);</div>
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<div class="line"><a name="l00107"></a><span class="lineno"> 107</span>  </div>
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<div class="line"><a name="l00108"></a><span class="lineno"> 108</span>  <span class="comment">/* number of eigen values = matrix size */</span></div>
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<div class="line"><a name="l00109"></a><span class="lineno"> 109</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<double></a> eigen_vals(rows);</div>
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<div class="line"><a name="l00110"></a><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"><a name="l00111"></a><span class="lineno"> 111</span>  Q[i] = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<double></a>(columns);</div>
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<div class="line"><a name="l00112"></a><span class="lineno"> 112</span>  R[i] = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<double></a>(columns);</div>
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<div class="line"><a name="l00113"></a><span class="lineno"> 113</span>  }</div>
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<div class="line"><a name="l00114"></a><span class="lineno"> 114</span>  </div>
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<div class="line"><a name="l00115"></a><span class="lineno"> 115</span>  <span class="comment">/* continue till all eigen values are found */</span></div>
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<div class="line"><a name="l00116"></a><span class="lineno"> 116</span>  <span class="keywordflow">while</span> (num_eigs > 0) {</div>
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<div class="line"><a name="l00117"></a><span class="lineno"> 117</span>  <span class="comment">/* iterate with QR decomposition */</span></div>
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<div class="line"><a name="l00118"></a><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"><a name="l00119"></a><span class="lineno"> 119</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/numeric_limits.html">std::numeric_limits<double>::epsilon</a>()) {</div>
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<div class="line"><a name="l00120"></a><span class="lineno"> 120</span>  <span class="comment">// initial approximation = last diagonal element</span></div>
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<div class="line"><a name="l00121"></a><span class="lineno"> 121</span>  last_eig = A[0][num_eigs][num_eigs];</div>
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<div class="line"><a name="l00122"></a><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"><a name="l00123"></a><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"><a name="l00124"></a><span class="lineno"> 124</span>  }</div>
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<div class="line"><a name="l00125"></a><span class="lineno"> 125</span>  </div>
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<div class="line"><a name="l00126"></a><span class="lineno"> 126</span>  <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#ab1ac74497ffb2101040cc1efe3546de8">qr_decompose</a>(*A, &Q, &R);</div>
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<div class="line"><a name="l00127"></a><span class="lineno"> 127</span>  </div>
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<div class="line"><a name="l00128"></a><span class="lineno"> 128</span>  <span class="keywordflow">if</span> (print_intermediates) {</div>
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<div class="line"><a name="l00129"></a><span class="lineno"> 129</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << *A << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><a name="l00130"></a><span class="lineno"> 130</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << Q << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><a name="l00131"></a><span class="lineno"> 131</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << R << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><a name="l00132"></a><span class="lineno"> 132</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/c/fprintf.html">printf</a>(<span class="stringliteral">"-------------------- %d ---------------------\n"</span>,</div>
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<div class="line"><a name="l00133"></a><span class="lineno"> 133</span>  ++counter);</div>
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<div class="line"><a name="l00134"></a><span class="lineno"> 134</span>  }</div>
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<div class="line"><a name="l00135"></a><span class="lineno"> 135</span>  </div>
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<div class="line"><a name="l00136"></a><span class="lineno"> 136</span>  <span class="comment">// new approximation A' = R * Q</span></div>
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<div class="line"><a name="l00137"></a><span class="lineno"> 137</span>  <a class="code" href="../../de/d75/qr__eigen__values_8cpp.html#a7558fd7779bf207157ced66f275951a0">mat_mul</a>(R, Q, A);</div>
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<div class="line"><a name="l00138"></a><span class="lineno"> 138</span>  </div>
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<div class="line"><a name="l00139"></a><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"><a name="l00140"></a><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"><a name="l00141"></a><span class="lineno"> 141</span>  }</div>
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<div class="line"><a name="l00142"></a><span class="lineno"> 142</span>  }</div>
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<div class="line"><a name="l00143"></a><span class="lineno"> 143</span>  </div>
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<div class="line"><a name="l00144"></a><span class="lineno"> 144</span>  <span class="comment">/* store the converged eigen value */</span></div>
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<div class="line"><a name="l00145"></a><span class="lineno"> 145</span>  eigen_vals[num_eigs] = last_eig;</div>
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<div class="line"><a name="l00146"></a><span class="lineno"> 146</span>  <span class="comment">// A[0][num_eigs][num_eigs];</span></div>
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<div class="line"><a name="l00147"></a><span class="lineno"> 147</span>  <span class="keywordflow">if</span> (print_intermediates) {</div>
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<div class="line"><a name="l00148"></a><span class="lineno"> 148</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"========================\n"</span>;</div>
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<div class="line"><a name="l00149"></a><span class="lineno"> 149</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"Eigen value: "</span> << last_eig << <span class="stringliteral">",\n"</span>;</div>
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<div class="line"><a name="l00150"></a><span class="lineno"> 150</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"========================\n"</span>;</div>
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<div class="line"><a name="l00151"></a><span class="lineno"> 151</span>  }</div>
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<div class="line"><a name="l00152"></a><span class="lineno"> 152</span>  </div>
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<div class="line"><a name="l00153"></a><span class="lineno"> 153</span>  num_eigs--;</div>
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<div class="line"><a name="l00154"></a><span class="lineno"> 154</span>  rows--;</div>
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<div class="line"><a name="l00155"></a><span class="lineno"> 155</span>  columns--;</div>
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<div class="line"><a name="l00156"></a><span class="lineno"> 156</span>  }</div>
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<div class="line"><a name="l00157"></a><span class="lineno"> 157</span>  eigen_vals[0] = A[0][0][0];</div>
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<div class="line"><a name="l00158"></a><span class="lineno"> 158</span>  </div>
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<div class="line"><a name="l00159"></a><span class="lineno"> 159</span>  <span class="keywordflow">if</span> (print_intermediates) {</div>
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<div class="line"><a name="l00160"></a><span class="lineno"> 160</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << Q << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><a name="l00161"></a><span class="lineno"> 161</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << R << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><a name="l00162"></a><span class="lineno"> 162</span>  }</div>
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<div class="line"><a name="l00163"></a><span class="lineno"> 163</span>  </div>
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<div class="line"><a name="l00164"></a><span class="lineno"> 164</span>  <span class="keywordflow">return</span> eigen_vals;</div>
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<div class="line"><a name="l00165"></a><span class="lineno"> 165</span> }</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a7c63c78d6839b8ad73b77c161210b514">◆ </a></span>operator<<() <span class="overload">[1/2]</span></h2>
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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>& qr_algorithm::operator<< </td>
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<td>(</td>
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<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> & </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>< <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< T >> const & </td>
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<td class="paramname"><em>v</em> </td>
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<p>operator to print a matrix </p>
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<div class="fragment"><div class="line"><a name="l00034"></a><span class="lineno"> 34</span>  {</div>
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<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>  <span class="keyword">const</span> <span class="keywordtype">int</span> width = 12;</div>
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<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>  <span class="keyword">const</span> <span class="keywordtype">char</span> separator = <span class="charliteral">' '</span>;</div>
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<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>  </div>
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<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>  out.precision(4);</div>
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<div class="line"><a name="l00039"></a><span class="lineno"> 39</span>  <span class="keywordflow">for</span> (<span class="keywordtype">size_t</span> row = 0; row < v.size(); row++) {</div>
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<div class="line"><a name="l00040"></a><span class="lineno"> 40</span>  <span class="keywordflow">for</span> (<span class="keywordtype">size_t</span> col = 0; col < v[row].size(); col++)</div>
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<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>  out << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/left.html">std::right</a> << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setw.html">std::setw</a>(width) << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setfill.html">std::setfill</a>(separator)</div>
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<div class="line"><a name="l00042"></a><span class="lineno"> 42</span>  << v[row][col];</div>
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<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>  out << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
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<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>  }</div>
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<div class="line"><a name="l00045"></a><span class="lineno"> 45</span>  </div>
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<div class="line"><a name="l00046"></a><span class="lineno"> 46</span>  <span class="keywordflow">return</span> out;</div>
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<div class="line"><a name="l00047"></a><span class="lineno"> 47</span> }</div>
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<a id="a240edf2b6b5942c0e5f2c9d3e326797f"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a240edf2b6b5942c0e5f2c9d3e326797f">◆ </a></span>operator<<() <span class="overload">[2/2]</span></h2>
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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>& qr_algorithm::operator<< </td>
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<td>(</td>
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<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::ostream</a> & </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>< T > const & </td>
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<td class="paramname"><em>v</em> </td>
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<p>operator to print a vector </p>
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<div class="fragment"><div class="line"><a name="l00053"></a><span class="lineno"> 53</span>  {</div>
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<div class="line"><a name="l00054"></a><span class="lineno"> 54</span>  <span class="keyword">const</span> <span class="keywordtype">int</span> width = 10;</div>
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<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>  <span class="keyword">const</span> <span class="keywordtype">char</span> separator = <span class="charliteral">' '</span>;</div>
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<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>  </div>
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<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>  out.precision(4);</div>
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<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>  <span class="keywordflow">for</span> (<span class="keywordtype">size_t</span> row = 0; row < v.size(); row++) {</div>
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<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>  out << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/left.html">std::right</a> << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setw.html">std::setw</a>(width) << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/setfill.html">std::setfill</a>(separator)</div>
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<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>  << v[row];</div>
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<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>  }</div>
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<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>  </div>
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<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>  <span class="keywordflow">return</span> out;</div>
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<div class="line"><a name="l00064"></a><span class="lineno"> 64</span> }</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ab1ac74497ffb2101040cc1efe3546de8">◆ </a></span>qr_decompose()</h2>
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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 <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< T >> & </td>
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<p>Decompose matrix \(A\) using <a href="https://en.wikipedia.org/wiki/QR_decomposition">Gram-Schmidt process</a>.</p>
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<p class="formulaDsp">
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\begin{eqnarray*} \text{given that}\quad A &=& *\left[\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_{N-1},\right]\\ \text{where}\quad\mathbf{a}_i &=& \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 &=& \mathbf{a}_i *-\sum_{j=0}^{i-1}\text{proj}_{\mathbf{u}_j}\mathbf{a}_i\\ \mathbf{e}_i &=&\frac{\mathbf{u}_i}{\left|\mathbf{u}_i\right|}\\ Q &=& \begin{bmatrix}\mathbf{e}_0 & \mathbf{e}_1 & \mathbf{e}_2 & \dots & \mathbf{e}_{N-1}\end{bmatrix}\\ R &=& \begin{bmatrix}\langle\mathbf{e}_0\,,\mathbf{a}_0\rangle & \langle\mathbf{e}_1\,,\mathbf{a}_1\rangle & \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle & \dots \\ 0 & \langle\mathbf{e}_1\,,\mathbf{a}_1\rangle & \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle & \dots\\ 0 & 0 & \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle & \dots\\ \vdots & \vdots & \vdots & \ddots \end{bmatrix}\\ \end{eqnarray*}
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</p>
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<dl class="params"><dt>Parameters</dt><dd>
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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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<div class="fragment"><div class="line"><a name="l00150"></a><span class="lineno"> 150</span>  {</div>
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<div class="line"><a name="l00151"></a><span class="lineno"> 151</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/size_t.html">std::size_t</a> ROWS = A.size(); <span class="comment">// number of rows of A</span></div>
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<div class="line"><a name="l00152"></a><span class="lineno"> 152</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/size_t.html">std::size_t</a> COLUMNS = A[0].size(); <span class="comment">// number of columns of A</span></div>
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<div class="line"><a name="l00153"></a><span class="lineno"> 153</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a> col_vector(ROWS);</div>
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<div class="line"><a name="l00154"></a><span class="lineno"> 154</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a> col_vector2(ROWS);</div>
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<div class="line"><a name="l00155"></a><span class="lineno"> 155</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<T></a> tmp_vector(ROWS);</div>
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<div class="line"><a name="l00156"></a><span class="lineno"> 156</span>  </div>
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<div class="line"><a name="l00157"></a><span class="lineno"> 157</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < COLUMNS; i++) {</div>
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<div class="line"><a name="l00158"></a><span class="lineno"> 158</span>  <span class="comment">/* for each column => R is a square matrix of NxN */</span></div>
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<div class="line"><a name="l00159"></a><span class="lineno"> 159</span>  <span class="keywordtype">int</span> j;</div>
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<div class="line"><a name="l00160"></a><span class="lineno"> 160</span>  R[0][i] = 0.; <span class="comment">/* make R upper triangular */</span></div>
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<div class="line"><a name="l00161"></a><span class="lineno"> 161</span>  </div>
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<div class="line"><a name="l00162"></a><span class="lineno"> 162</span>  <span class="comment">/* get corresponding Q vector */</span></div>
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<div class="line"><a name="l00163"></a><span class="lineno"> 163</span> <span class="preprocessor">#ifdef _OPENMP</span></div>
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<div class="line"><a name="l00164"></a><span class="lineno"> 164</span> <span class="comment">// parallelize on threads</span></div>
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<div class="line"><a name="l00165"></a><span class="lineno"> 165</span> <span class="preprocessor">#pragma omp for</span></div>
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<div class="line"><a name="l00166"></a><span class="lineno"> 166</span> <span class="preprocessor">#endif</span></div>
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<div class="line"><a name="l00167"></a><span class="lineno"> 167</span>  <span class="keywordflow">for</span> (j = 0; j < ROWS; j++) {</div>
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<div class="line"><a name="l00168"></a><span class="lineno"> 168</span>  tmp_vector[j] = A[j][i]; <span class="comment">/* accumulator for uk */</span></div>
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<div class="line"><a name="l00169"></a><span class="lineno"> 169</span>  col_vector[j] = A[j][i];</div>
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<div class="line"><a name="l00170"></a><span class="lineno"> 170</span>  }</div>
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<div class="line"><a name="l00171"></a><span class="lineno"> 171</span>  <span class="keywordflow">for</span> (j = 0; j < i; j++) {</div>
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<div class="line"><a name="l00172"></a><span class="lineno"> 172</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#afcf0d19607cd0edd8ae333d2b9c220c0">k</a> = 0; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#afcf0d19607cd0edd8ae333d2b9c220c0">k</a> < ROWS; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#afcf0d19607cd0edd8ae333d2b9c220c0">k</a>++) {</div>
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<div class="line"><a name="l00173"></a><span class="lineno"> 173</span>  col_vector2[<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#afcf0d19607cd0edd8ae333d2b9c220c0">k</a>] = Q[0][<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#afcf0d19607cd0edd8ae333d2b9c220c0">k</a>][j];</div>
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<div class="line"><a name="l00174"></a><span class="lineno"> 174</span>  }</div>
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<div class="line"><a name="l00175"></a><span class="lineno"> 175</span>  col_vector2 = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a5422c76bfa322620de611083bd518d71">vector_proj</a>(col_vector, col_vector2);</div>
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<div class="line"><a name="l00176"></a><span class="lineno"> 176</span>  tmp_vector -= col_vector2;</div>
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<div class="line"><a name="l00177"></a><span class="lineno"> 177</span>  }</div>
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<div class="line"><a name="l00178"></a><span class="lineno"> 178</span>  </div>
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<div class="line"><a name="l00179"></a><span class="lineno"> 179</span>  <span class="keywordtype">double</span> mag = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#ad16da2183db22378435042f26af43d5f">vector_mag</a>(tmp_vector);</div>
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<div class="line"><a name="l00180"></a><span class="lineno"> 180</span>  </div>
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<div class="line"><a name="l00181"></a><span class="lineno"> 181</span> <span class="preprocessor">#ifdef _OPENMP</span></div>
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<div class="line"><a name="l00182"></a><span class="lineno"> 182</span> <span class="comment">// parallelize on threads</span></div>
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<div class="line"><a name="l00183"></a><span class="lineno"> 183</span> <span class="preprocessor">#pragma omp for</span></div>
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<div class="line"><a name="l00184"></a><span class="lineno"> 184</span> <span class="preprocessor">#endif</span></div>
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<div class="line"><a name="l00185"></a><span class="lineno"> 185</span>  <span class="keywordflow">for</span> (j = 0; j < ROWS; j++) Q[0][j][i] = tmp_vector[j] / mag;</div>
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<div class="line"><a name="l00186"></a><span class="lineno"> 186</span>  </div>
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<div class="line"><a name="l00187"></a><span class="lineno"> 187</span>  <span class="comment">/* compute upper triangular values of R */</span></div>
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<div class="line"><a name="l00188"></a><span class="lineno"> 188</span> <span class="preprocessor">#ifdef _OPENMP</span></div>
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<div class="line"><a name="l00189"></a><span class="lineno"> 189</span> <span class="comment">// parallelize on threads</span></div>
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<div class="line"><a name="l00190"></a><span class="lineno"> 190</span> <span class="preprocessor">#pragma omp for</span></div>
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<div class="line"><a name="l00191"></a><span class="lineno"> 191</span> <span class="preprocessor">#endif</span></div>
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<div class="line"><a name="l00192"></a><span class="lineno"> 192</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> kk = 0; kk < ROWS; kk++) {</div>
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<div class="line"><a name="l00193"></a><span class="lineno"> 193</span>  col_vector[kk] = Q[0][kk][i];</div>
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<div class="line"><a name="l00194"></a><span class="lineno"> 194</span>  }</div>
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<div class="line"><a name="l00195"></a><span class="lineno"> 195</span>  </div>
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<div class="line"><a name="l00196"></a><span class="lineno"> 196</span> <span class="preprocessor">#ifdef _OPENMP</span></div>
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<div class="line"><a name="l00197"></a><span class="lineno"> 197</span> <span class="comment">// parallelize on threads</span></div>
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<div class="line"><a name="l00198"></a><span class="lineno"> 198</span> <span class="preprocessor">#pragma omp for</span></div>
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<div class="line"><a name="l00199"></a><span class="lineno"> 199</span> <span class="preprocessor">#endif</span></div>
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<div class="line"><a name="l00200"></a><span class="lineno"> 200</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#afcf0d19607cd0edd8ae333d2b9c220c0">k</a> = i; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#afcf0d19607cd0edd8ae333d2b9c220c0">k</a> < COLUMNS; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#afcf0d19607cd0edd8ae333d2b9c220c0">k</a>++) {</div>
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<div class="line"><a name="l00201"></a><span class="lineno"> 201</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> kk = 0; kk < ROWS; kk++) {</div>
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<div class="line"><a name="l00202"></a><span class="lineno"> 202</span>  col_vector2[kk] = A[kk][<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#afcf0d19607cd0edd8ae333d2b9c220c0">k</a>];</div>
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<div class="line"><a name="l00203"></a><span class="lineno"> 203</span>  }</div>
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<div class="line"><a name="l00204"></a><span class="lineno"> 204</span>  R[0][i][<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#afcf0d19607cd0edd8ae333d2b9c220c0">k</a>] = (col_vector * col_vector2).sum();</div>
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<div class="line"><a name="l00205"></a><span class="lineno"> 205</span>  }</div>
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<div class="line"><a name="l00206"></a><span class="lineno"> 206</span>  }</div>
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<div class="line"><a name="l00207"></a><span class="lineno"> 207</span> }</div>
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<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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<td>(</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>< T > & </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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<div class="fragment"><div class="line"><a name="l00076"></a><span class="lineno"> 76</span>  {</div>
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<div class="line"><a name="l00077"></a><span class="lineno"> 77</span>  <span class="keywordflow">return</span> (<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> * b).sum();</div>
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<div class="line"><a name="l00078"></a><span class="lineno"> 78</span>  <span class="comment">// could also use following</span></div>
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<div class="line"><a name="l00079"></a><span class="lineno"> 79</span>  <span class="comment">// return std::inner_product(std::begin(a), std::end(a), std::begin(b),</span></div>
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<div class="line"><a name="l00080"></a><span class="lineno"> 80</span>  <span class="comment">// 0.f);</span></div>
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<div class="line"><a name="l00081"></a><span class="lineno"> 81</span> }</div>
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<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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<div class="fragment"><div class="line"><a name="l00092"></a><span class="lineno"> 92</span>  {</div>
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<div class="line"><a name="l00093"></a><span class="lineno"> 93</span>  <span class="keywordtype">double</span> dot = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>);</div>
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<div class="line"><a name="l00094"></a><span class="lineno"> 94</span>  <span class="keywordflow">return</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a>(dot);</div>
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<div class="line"><a name="l00095"></a><span class="lineno"> 95</span> }</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a5422c76bfa322620de611083bd518d71">◆ </a></span>vector_proj()</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><T> qr_algorithm::vector_proj </td>
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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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<div class="fragment"><div class="line"><a name="l00105"></a><span class="lineno"> 105</span>  {</div>
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<div class="line"><a name="l00106"></a><span class="lineno"> 106</span>  <span class="keywordtype">double</span> num = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, b);</div>
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<div class="line"><a name="l00107"></a><span class="lineno"> 107</span>  <span class="keywordtype">double</span> deno = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a8ea313a1a1b5f9d0e3e332c29c6446ec">vector_dot</a>(b, b);</div>
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<div class="line"><a name="l00108"></a><span class="lineno"> 108</span> <span class="comment"></span> </div>
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<div class="line"><a name="l00109"></a><span class="lineno"> 109</span> <span class="comment"> /*! check for division by zero using machine epsilon */</span></div>
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<div class="line"><a name="l00110"></a><span class="lineno"> 110</span>  <span class="keywordflow">if</span> (deno <= <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/numeric_limits.html">std::numeric_limits<double>::epsilon</a>()) {</div>
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<div class="line"><a name="l00111"></a><span class="lineno"> 111</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cerr</a> << <span class="stringliteral">"["</span> << __func__ << <span class="stringliteral">"] Possible division by zero\n"</span>;</div>
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<div class="line"><a name="l00112"></a><span class="lineno"> 112</span>  <span class="keywordflow">return</span> <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>; <span class="comment">// return vector a back</span></div>
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<div class="line"><a name="l00113"></a><span class="lineno"> 113</span>  }</div>
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<div class="line"><a name="l00114"></a><span class="lineno"> 114</span>  </div>
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<div class="line"><a name="l00115"></a><span class="lineno"> 115</span>  <span class="keywordtype">double</span> scalar = num / deno;</div>
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<div class="line"><a name="l00116"></a><span class="lineno"> 116</span>  </div>
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<div class="line"><a name="l00117"></a><span class="lineno"> 117</span>  <span class="keywordflow">return</span> b * scalar;</div>
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<div class="line"><a name="l00118"></a><span class="lineno"> 118</span> }</div>
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<div class="ttc" id="asetfill_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/manip/setfill.html">std::setfill</a></div><div class="ttdeci">T setfill(T... args)</div></div>
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<div class="ttc" id="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< std::valarray< double >> &A, const std::valarray< std::valarray< double >> &B, std::valarray< std::valarray< double >> *OUT)</div><div class="ttdef"><b>Definition:</b> qr_eigen_values.cpp:54</div></div>
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<div class="ttc" id="asqrt_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a></div><div class="ttdeci">T sqrt(T... args)</div></div>
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<div class="ttc" id="amatrix__exponentiation_8cpp_html_afcf0d19607cd0edd8ae333d2b9c220c0"><div class="ttname"><a href="../../d7/d35/matrix__exponentiation_8cpp.html#afcf0d19607cd0edd8ae333d2b9c220c0">k</a></div><div class="ttdeci">ll k</div><div class="ttdef"><b>Definition:</b> matrix_exponentiation.cpp:48</div></div>
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<div class="ttc" id="avalarray_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a></div><div class="ttdoc">STL class.</div></div>
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