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<div class="title">RealSchur.h</div> </div>
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<div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno"> 1</span> <span class="comment">// This file is part of Eigen, a lightweight C++ template library</span></div>
<div class="line"><a name="l00002"></a><span class="lineno"> 2</span> <span class="comment">// for linear algebra.</span></div>
<div class="line"><a name="l00003"></a><span class="lineno"> 3</span> <span class="comment">//</span></div>
<div class="line"><a name="l00004"></a><span class="lineno"> 4</span> <span class="comment">// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr></span></div>
<div class="line"><a name="l00005"></a><span class="lineno"> 5</span> <span class="comment">// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk></span></div>
<div class="line"><a name="l00006"></a><span class="lineno"> 6</span> <span class="comment">//</span></div>
<div class="line"><a name="l00007"></a><span class="lineno"> 7</span> <span class="comment">// This Source Code Form is subject to the terms of the Mozilla</span></div>
<div class="line"><a name="l00008"></a><span class="lineno"> 8</span> <span class="comment">// Public License v. 2.0. If a copy of the MPL was not distributed</span></div>
<div class="line"><a name="l00009"></a><span class="lineno"> 9</span> <span class="comment">// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.</span></div>
<div class="line"><a name="l00010"></a><span class="lineno"> 10</span> </div>
<div class="line"><a name="l00011"></a><span class="lineno"> 11</span> <span class="preprocessor">#ifndef EIGEN_REAL_SCHUR_H</span></div>
<div class="line"><a name="l00012"></a><span class="lineno"> 12</span> <span class="preprocessor"></span><span class="preprocessor">#define EIGEN_REAL_SCHUR_H</span></div>
<div class="line"><a name="l00013"></a><span class="lineno"> 13</span> <span class="preprocessor"></span></div>
<div class="line"><a name="l00014"></a><span class="lineno"> 14</span> <span class="preprocessor">#include "./HessenbergDecomposition.h"</span></div>
<div class="line"><a name="l00015"></a><span class="lineno"> 15</span> </div>
<div class="line"><a name="l00016"></a><span class="lineno"> 16</span> <span class="keyword">namespace </span>Eigen { </div>
<div class="line"><a name="l00017"></a><span class="lineno"> 17</span> </div>
<div class="line"><a name="l00054"></a><span class="lineno"><a class="line" href="classEigen_1_1RealSchur.html"> 54</a></span> <span class="keyword">template</span><<span class="keyword">typename</span> _MatrixType> <span class="keyword">class </span><a class="code" href="classEigen_1_1RealSchur.html">RealSchur</a></div>
<div class="line"><a name="l00055"></a><span class="lineno"> 55</span> {</div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>  <span class="keyword">public</span>:</div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>  <span class="keyword">typedef</span> _MatrixType MatrixType;</div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>  <span class="keyword">enum</span> {</div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>  RowsAtCompileTime = MatrixType::RowsAtCompileTime,</div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>  ColsAtCompileTime = MatrixType::ColsAtCompileTime,</div>
<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>  Options = MatrixType::Options,</div>
<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>  MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,</div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>  MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime</div>
<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>  };</div>
<div class="line"><a name="l00065"></a><span class="lineno"> 65</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::Scalar Scalar;</div>
<div class="line"><a name="l00066"></a><span class="lineno"> 66</span>  <span class="keyword">typedef</span> std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;</div>
<div class="line"><a name="l00067"></a><span class="lineno"> 67</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::Index Index;</div>
<div class="line"><a name="l00068"></a><span class="lineno"> 68</span> </div>
<div class="line"><a name="l00069"></a><span class="lineno"> 69</span>  <span class="keyword">typedef</span> <a class="code" href="classEigen_1_1Matrix.html">Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1></a> <a class="code" href="classEigen_1_1Matrix.html">EigenvalueType</a>;</div>
<div class="line"><a name="l00070"></a><span class="lineno"> 70</span>  <span class="keyword">typedef</span> <a class="code" href="classEigen_1_1Matrix.html">Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1></a> <a class="code" href="classEigen_1_1Matrix.html">ColumnVectorType</a>;</div>
<div class="line"><a name="l00071"></a><span class="lineno"> 71</span> </div>
<div class="line"><a name="l00083"></a><span class="lineno"><a class="line" href="classEigen_1_1RealSchur.html#a782ab2c509de1deb484bbd12d6e863a0"> 83</a></span>  <a class="code" href="classEigen_1_1RealSchur.html#a782ab2c509de1deb484bbd12d6e863a0">RealSchur</a>(Index size = RowsAtCompileTime==<a class="code" href="namespaceEigen.html#adc9da5be31bdce40c25a92c27999c0e3">Dynamic</a> ? 1 : RowsAtCompileTime)</div>
<div class="line"><a name="l00084"></a><span class="lineno"> 84</span>  : m_matT(size, size),</div>
<div class="line"><a name="l00085"></a><span class="lineno"> 85</span>  m_matU(size, size),</div>
<div class="line"><a name="l00086"></a><span class="lineno"> 86</span>  m_workspaceVector(size),</div>
<div class="line"><a name="l00087"></a><span class="lineno"> 87</span>  m_hess(size),</div>
<div class="line"><a name="l00088"></a><span class="lineno"> 88</span>  m_isInitialized(false),</div>
<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>  m_matUisUptodate(false),</div>
<div class="line"><a name="l00090"></a><span class="lineno"> 90</span>  m_maxIters(-1)</div>
<div class="line"><a name="l00091"></a><span class="lineno"> 91</span>  { }</div>
<div class="line"><a name="l00092"></a><span class="lineno"> 92</span> </div>
<div class="line"><a name="l00103"></a><span class="lineno"><a class="line" href="classEigen_1_1RealSchur.html#a8d73d4e86d87bd2babf172909fc54198"> 103</a></span>  <a class="code" href="classEigen_1_1RealSchur.html#a8d73d4e86d87bd2babf172909fc54198">RealSchur</a>(<span class="keyword">const</span> MatrixType& matrix, <span class="keywordtype">bool</span> computeU = <span class="keyword">true</span>)</div>
<div class="line"><a name="l00104"></a><span class="lineno"> 104</span>  : m_matT(matrix.rows(),matrix.cols()),</div>
<div class="line"><a name="l00105"></a><span class="lineno"> 105</span>  m_matU(matrix.rows(),matrix.cols()),</div>
<div class="line"><a name="l00106"></a><span class="lineno"> 106</span>  m_workspaceVector(matrix.rows()),</div>
<div class="line"><a name="l00107"></a><span class="lineno"> 107</span>  m_hess(matrix.rows()),</div>
<div class="line"><a name="l00108"></a><span class="lineno"> 108</span>  m_isInitialized(false),</div>
<div class="line"><a name="l00109"></a><span class="lineno"> 109</span>  m_matUisUptodate(false),</div>
<div class="line"><a name="l00110"></a><span class="lineno"> 110</span>  m_maxIters(-1)</div>
<div class="line"><a name="l00111"></a><span class="lineno"> 111</span>  {</div>
<div class="line"><a name="l00112"></a><span class="lineno"> 112</span>  <a class="code" href="classEigen_1_1RealSchur.html#ace387c8cea391973ca2a99edc720671a">compute</a>(matrix, computeU);</div>
<div class="line"><a name="l00113"></a><span class="lineno"> 113</span>  }</div>
<div class="line"><a name="l00114"></a><span class="lineno"> 114</span> </div>
<div class="line"><a name="l00126"></a><span class="lineno"><a class="line" href="classEigen_1_1RealSchur.html#a7663c715ad9aaf8b57825646f5317166"> 126</a></span>  <span class="keyword">const</span> MatrixType& <a class="code" href="classEigen_1_1RealSchur.html#a7663c715ad9aaf8b57825646f5317166">matrixU</a>()<span class="keyword"> const</span></div>
<div class="line"><a name="l00127"></a><span class="lineno"> 127</span> <span class="keyword"> </span>{</div>
<div class="line"><a name="l00128"></a><span class="lineno"> 128</span>  eigen_assert(m_isInitialized && <span class="stringliteral">"RealSchur is not initialized."</span>);</div>
<div class="line"><a name="l00129"></a><span class="lineno"> 129</span>  eigen_assert(m_matUisUptodate && <span class="stringliteral">"The matrix U has not been computed during the RealSchur decomposition."</span>);</div>
<div class="line"><a name="l00130"></a><span class="lineno"> 130</span>  <span class="keywordflow">return</span> m_matU;</div>
<div class="line"><a name="l00131"></a><span class="lineno"> 131</span>  }</div>
<div class="line"><a name="l00132"></a><span class="lineno"> 132</span> </div>
<div class="line"><a name="l00143"></a><span class="lineno"><a class="line" href="classEigen_1_1RealSchur.html#a0d31900234ef9fea5751ce8ea693d71f"> 143</a></span>  <span class="keyword">const</span> MatrixType& <a class="code" href="classEigen_1_1RealSchur.html#a0d31900234ef9fea5751ce8ea693d71f">matrixT</a>()<span class="keyword"> const</span></div>
<div class="line"><a name="l00144"></a><span class="lineno"> 144</span> <span class="keyword"> </span>{</div>
<div class="line"><a name="l00145"></a><span class="lineno"> 145</span>  eigen_assert(m_isInitialized && <span class="stringliteral">"RealSchur is not initialized."</span>);</div>
<div class="line"><a name="l00146"></a><span class="lineno"> 146</span>  <span class="keywordflow">return</span> m_matT;</div>
<div class="line"><a name="l00147"></a><span class="lineno"> 147</span>  }</div>
<div class="line"><a name="l00148"></a><span class="lineno"> 148</span>  </div>
<div class="line"><a name="l00168"></a><span class="lineno"> 168</span>  <a class="code" href="classEigen_1_1RealSchur.html">RealSchur</a>& <a class="code" href="classEigen_1_1RealSchur.html#ace387c8cea391973ca2a99edc720671a">compute</a>(<span class="keyword">const</span> MatrixType& matrix, <span class="keywordtype">bool</span> computeU = <span class="keyword">true</span>);</div>
<div class="line"><a name="l00169"></a><span class="lineno"> 169</span> </div>
<div class="line"><a name="l00187"></a><span class="lineno"> 187</span>  <span class="keyword">template</span><<span class="keyword">typename</span> HessMatrixType, <span class="keyword">typename</span> OrthMatrixType></div>
<div class="line"><a name="l00188"></a><span class="lineno"> 188</span>  <a class="code" href="classEigen_1_1RealSchur.html">RealSchur</a>& <a class="code" href="classEigen_1_1RealSchur.html#a5b461a34397b36bb284ccfb0f3a4c498">computeFromHessenberg</a>(<span class="keyword">const</span> HessMatrixType& matrixH, <span class="keyword">const</span> OrthMatrixType& matrixQ, <span class="keywordtype">bool</span> computeU);</div>
<div class="line"><a name="l00193"></a><span class="lineno"><a class="line" href="classEigen_1_1RealSchur.html#a0c06d5c2034ebb329c54235369643ad2"> 193</a></span>  <a class="code" href="group__enums.html#ga51bc1ac16f26ebe51eae1abb77bd037b">ComputationInfo</a> <a class="code" href="classEigen_1_1RealSchur.html#a0c06d5c2034ebb329c54235369643ad2">info</a>()<span class="keyword"> const</span></div>
<div class="line"><a name="l00194"></a><span class="lineno"> 194</span> <span class="keyword"> </span>{</div>
<div class="line"><a name="l00195"></a><span class="lineno"> 195</span>  eigen_assert(m_isInitialized && <span class="stringliteral">"RealSchur is not initialized."</span>);</div>
<div class="line"><a name="l00196"></a><span class="lineno"> 196</span>  <span class="keywordflow">return</span> m_info;</div>
<div class="line"><a name="l00197"></a><span class="lineno"> 197</span>  }</div>
<div class="line"><a name="l00198"></a><span class="lineno"> 198</span> </div>
<div class="line"><a name="l00204"></a><span class="lineno"><a class="line" href="classEigen_1_1RealSchur.html#adcb20d95f17c74395f9a906e5a72ab6b"> 204</a></span>  <a class="code" href="classEigen_1_1RealSchur.html">RealSchur</a>& <a class="code" href="classEigen_1_1RealSchur.html#adcb20d95f17c74395f9a906e5a72ab6b">setMaxIterations</a>(Index maxIters)</div>
<div class="line"><a name="l00205"></a><span class="lineno"> 205</span>  {</div>
<div class="line"><a name="l00206"></a><span class="lineno"> 206</span>  m_maxIters = maxIters;</div>
<div class="line"><a name="l00207"></a><span class="lineno"> 207</span>  <span class="keywordflow">return</span> *<span class="keyword">this</span>;</div>
<div class="line"><a name="l00208"></a><span class="lineno"> 208</span>  }</div>
<div class="line"><a name="l00209"></a><span class="lineno"> 209</span> </div>
<div class="line"><a name="l00211"></a><span class="lineno"><a class="line" href="classEigen_1_1RealSchur.html#ab6f0a63ea1d26cef5e748207043eb43e"> 211</a></span>  Index <a class="code" href="classEigen_1_1RealSchur.html#ab6f0a63ea1d26cef5e748207043eb43e">getMaxIterations</a>()</div>
<div class="line"><a name="l00212"></a><span class="lineno"> 212</span>  {</div>
<div class="line"><a name="l00213"></a><span class="lineno"> 213</span>  <span class="keywordflow">return</span> m_maxIters;</div>
<div class="line"><a name="l00214"></a><span class="lineno"> 214</span>  }</div>
<div class="line"><a name="l00215"></a><span class="lineno"> 215</span> </div>
<div class="line"><a name="l00221"></a><span class="lineno"><a class="line" href="classEigen_1_1RealSchur.html#afdafb24d67af7529bb903a4c9bff3ea4"> 221</a></span>  <span class="keyword">static</span> <span class="keyword">const</span> <span class="keywordtype">int</span> <a class="code" href="classEigen_1_1RealSchur.html#afdafb24d67af7529bb903a4c9bff3ea4">m_maxIterationsPerRow</a> = 40;</div>
<div class="line"><a name="l00222"></a><span class="lineno"> 222</span> </div>
<div class="line"><a name="l00223"></a><span class="lineno"> 223</span>  <span class="keyword">private</span>:</div>
<div class="line"><a name="l00224"></a><span class="lineno"> 224</span>  </div>
<div class="line"><a name="l00225"></a><span class="lineno"> 225</span>  MatrixType m_matT;</div>
<div class="line"><a name="l00226"></a><span class="lineno"> 226</span>  MatrixType m_matU;</div>
<div class="line"><a name="l00227"></a><span class="lineno"> 227</span>  <a class="code" href="classEigen_1_1Matrix.html">ColumnVectorType</a> m_workspaceVector;</div>
<div class="line"><a name="l00228"></a><span class="lineno"> 228</span>  <a class="code" href="classEigen_1_1HessenbergDecomposition.html">HessenbergDecomposition<MatrixType></a> m_hess;</div>
<div class="line"><a name="l00229"></a><span class="lineno"> 229</span>  <a class="code" href="group__enums.html#ga51bc1ac16f26ebe51eae1abb77bd037b">ComputationInfo</a> m_info;</div>
<div class="line"><a name="l00230"></a><span class="lineno"> 230</span>  <span class="keywordtype">bool</span> m_isInitialized;</div>
<div class="line"><a name="l00231"></a><span class="lineno"> 231</span>  <span class="keywordtype">bool</span> m_matUisUptodate;</div>
<div class="line"><a name="l00232"></a><span class="lineno"> 232</span>  Index m_maxIters;</div>
<div class="line"><a name="l00233"></a><span class="lineno"> 233</span> </div>
<div class="line"><a name="l00234"></a><span class="lineno"> 234</span>  <span class="keyword">typedef</span> <a class="code" href="classEigen_1_1Matrix.html">Matrix<Scalar,3,1></a> <a class="code" href="classEigen_1_1Matrix.html">Vector3s</a>;</div>
<div class="line"><a name="l00235"></a><span class="lineno"> 235</span> </div>
<div class="line"><a name="l00236"></a><span class="lineno"> 236</span>  Scalar computeNormOfT();</div>
<div class="line"><a name="l00237"></a><span class="lineno"> 237</span>  Index findSmallSubdiagEntry(Index iu, <span class="keyword">const</span> Scalar& norm);</div>
<div class="line"><a name="l00238"></a><span class="lineno"> 238</span>  <span class="keywordtype">void</span> splitOffTwoRows(Index iu, <span class="keywordtype">bool</span> computeU, <span class="keyword">const</span> Scalar& exshift);</div>
<div class="line"><a name="l00239"></a><span class="lineno"> 239</span>  <span class="keywordtype">void</span> computeShift(Index iu, Index iter, Scalar& exshift, <a class="code" href="classEigen_1_1Matrix.html">Vector3s</a>& shiftInfo);</div>
<div class="line"><a name="l00240"></a><span class="lineno"> 240</span>  <span class="keywordtype">void</span> initFrancisQRStep(Index il, Index iu, <span class="keyword">const</span> <a class="code" href="classEigen_1_1Matrix.html">Vector3s</a>& shiftInfo, Index& im, <a class="code" href="classEigen_1_1Matrix.html">Vector3s</a>& firstHouseholderVector);</div>
<div class="line"><a name="l00241"></a><span class="lineno"> 241</span>  <span class="keywordtype">void</span> performFrancisQRStep(Index il, Index im, Index iu, <span class="keywordtype">bool</span> computeU, <span class="keyword">const</span> <a class="code" href="classEigen_1_1Matrix.html">Vector3s</a>& firstHouseholderVector, Scalar* workspace);</div>
<div class="line"><a name="l00242"></a><span class="lineno"> 242</span> };</div>
<div class="line"><a name="l00243"></a><span class="lineno"> 243</span> </div>
<div class="line"><a name="l00244"></a><span class="lineno"> 244</span> </div>
<div class="line"><a name="l00245"></a><span class="lineno"> 245</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00246"></a><span class="lineno"><a class="line" href="classEigen_1_1RealSchur.html#ace387c8cea391973ca2a99edc720671a"> 246</a></span> <a class="code" href="classEigen_1_1RealSchur.html">RealSchur<MatrixType></a>& <a class="code" href="classEigen_1_1RealSchur.html#ace387c8cea391973ca2a99edc720671a">RealSchur<MatrixType>::compute</a>(<span class="keyword">const</span> MatrixType& matrix, <span class="keywordtype">bool</span> computeU)</div>
<div class="line"><a name="l00247"></a><span class="lineno"> 247</span> {</div>
<div class="line"><a name="l00248"></a><span class="lineno"> 248</span>  eigen_assert(matrix.cols() == matrix.rows());</div>
<div class="line"><a name="l00249"></a><span class="lineno"> 249</span>  Index maxIters = m_maxIters;</div>
<div class="line"><a name="l00250"></a><span class="lineno"> 250</span>  <span class="keywordflow">if</span> (maxIters == -1)</div>
<div class="line"><a name="l00251"></a><span class="lineno"> 251</span>  maxIters = m_maxIterationsPerRow * matrix.rows();</div>
<div class="line"><a name="l00252"></a><span class="lineno"> 252</span> </div>
<div class="line"><a name="l00253"></a><span class="lineno"> 253</span>  <span class="comment">// Step 1. Reduce to Hessenberg form</span></div>
<div class="line"><a name="l00254"></a><span class="lineno"> 254</span>  m_hess.compute(matrix);</div>
<div class="line"><a name="l00255"></a><span class="lineno"> 255</span> </div>
<div class="line"><a name="l00256"></a><span class="lineno"> 256</span>  <span class="comment">// Step 2. Reduce to real Schur form </span></div>
<div class="line"><a name="l00257"></a><span class="lineno"> 257</span>  computeFromHessenberg(m_hess.matrixH(), m_hess.matrixQ(), computeU);</div>
<div class="line"><a name="l00258"></a><span class="lineno"> 258</span>  </div>
<div class="line"><a name="l00259"></a><span class="lineno"> 259</span>  <span class="keywordflow">return</span> *<span class="keyword">this</span>;</div>
<div class="line"><a name="l00260"></a><span class="lineno"> 260</span> }</div>
<div class="line"><a name="l00261"></a><span class="lineno"> 261</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00262"></a><span class="lineno"> 262</span> <span class="keyword">template</span><<span class="keyword">typename</span> HessMatrixType, <span class="keyword">typename</span> OrthMatrixType></div>
<div class="line"><a name="l00263"></a><span class="lineno"> 263</span> <a class="code" href="classEigen_1_1RealSchur.html">RealSchur<MatrixType></a>& <a class="code" href="classEigen_1_1RealSchur.html">RealSchur<MatrixType>::computeFromHessenberg</a>(<span class="keyword">const</span> HessMatrixType& matrixH, <span class="keyword">const</span> OrthMatrixType& matrixQ, <span class="keywordtype">bool</span> computeU)</div>
<div class="line"><a name="l00264"></a><span class="lineno"> 264</span> { </div>
<div class="line"><a name="l00265"></a><span class="lineno"> 265</span>  m_matT = matrixH; </div>
<div class="line"><a name="l00266"></a><span class="lineno"> 266</span>  <span class="keywordflow">if</span>(computeU)</div>
<div class="line"><a name="l00267"></a><span class="lineno"> 267</span>  m_matU = matrixQ;</div>
<div class="line"><a name="l00268"></a><span class="lineno"> 268</span>  </div>
<div class="line"><a name="l00269"></a><span class="lineno"> 269</span>  Index maxIters = m_maxIters;</div>
<div class="line"><a name="l00270"></a><span class="lineno"> 270</span>  <span class="keywordflow">if</span> (maxIters == -1)</div>
<div class="line"><a name="l00271"></a><span class="lineno"> 271</span>  maxIters = m_maxIterationsPerRow * matrixH.rows();</div>
<div class="line"><a name="l00272"></a><span class="lineno"> 272</span>  m_workspaceVector.resize(m_matT.cols());</div>
<div class="line"><a name="l00273"></a><span class="lineno"> 273</span>  Scalar* workspace = &m_workspaceVector.coeffRef(0);</div>
<div class="line"><a name="l00274"></a><span class="lineno"> 274</span> </div>
<div class="line"><a name="l00275"></a><span class="lineno"> 275</span>  <span class="comment">// The matrix m_matT is divided in three parts. </span></div>
<div class="line"><a name="l00276"></a><span class="lineno"> 276</span>  <span class="comment">// Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero. </span></div>
<div class="line"><a name="l00277"></a><span class="lineno"> 277</span>  <span class="comment">// Rows il,...,iu is the part we are working on (the active window).</span></div>
<div class="line"><a name="l00278"></a><span class="lineno"> 278</span>  <span class="comment">// Rows iu+1,...,end are already brought in triangular form.</span></div>
<div class="line"><a name="l00279"></a><span class="lineno"> 279</span>  Index iu = m_matT.cols() - 1;</div>
<div class="line"><a name="l00280"></a><span class="lineno"> 280</span>  Index iter = 0; <span class="comment">// iteration count for current eigenvalue</span></div>
<div class="line"><a name="l00281"></a><span class="lineno"> 281</span>  Index totalIter = 0; <span class="comment">// iteration count for whole matrix</span></div>
<div class="line"><a name="l00282"></a><span class="lineno"> 282</span>  Scalar exshift(0); <span class="comment">// sum of exceptional shifts</span></div>
<div class="line"><a name="l00283"></a><span class="lineno"> 283</span>  Scalar norm = computeNormOfT();</div>
<div class="line"><a name="l00284"></a><span class="lineno"> 284</span> </div>
<div class="line"><a name="l00285"></a><span class="lineno"> 285</span>  <span class="keywordflow">if</span>(norm!=0)</div>
<div class="line"><a name="l00286"></a><span class="lineno"> 286</span>  {</div>
<div class="line"><a name="l00287"></a><span class="lineno"> 287</span>  <span class="keywordflow">while</span> (iu >= 0)</div>
<div class="line"><a name="l00288"></a><span class="lineno"> 288</span>  {</div>
<div class="line"><a name="l00289"></a><span class="lineno"> 289</span>  Index il = findSmallSubdiagEntry(iu, norm);</div>
<div class="line"><a name="l00290"></a><span class="lineno"> 290</span> </div>
<div class="line"><a name="l00291"></a><span class="lineno"> 291</span>  <span class="comment">// Check for convergence</span></div>
<div class="line"><a name="l00292"></a><span class="lineno"> 292</span>  <span class="keywordflow">if</span> (il == iu) <span class="comment">// One root found</span></div>
<div class="line"><a name="l00293"></a><span class="lineno"> 293</span>  {</div>
<div class="line"><a name="l00294"></a><span class="lineno"> 294</span>  m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift;</div>
<div class="line"><a name="l00295"></a><span class="lineno"> 295</span>  <span class="keywordflow">if</span> (iu > 0)</div>
<div class="line"><a name="l00296"></a><span class="lineno"> 296</span>  m_matT.coeffRef(iu, iu-1) = Scalar(0);</div>
<div class="line"><a name="l00297"></a><span class="lineno"> 297</span>  iu--;</div>
<div class="line"><a name="l00298"></a><span class="lineno"> 298</span>  iter = 0;</div>
<div class="line"><a name="l00299"></a><span class="lineno"> 299</span>  }</div>
<div class="line"><a name="l00300"></a><span class="lineno"> 300</span>  <span class="keywordflow">else</span> <span class="keywordflow">if</span> (il == iu-1) <span class="comment">// Two roots found</span></div>
<div class="line"><a name="l00301"></a><span class="lineno"> 301</span>  {</div>
<div class="line"><a name="l00302"></a><span class="lineno"> 302</span>  splitOffTwoRows(iu, computeU, exshift);</div>
<div class="line"><a name="l00303"></a><span class="lineno"> 303</span>  iu -= 2;</div>
<div class="line"><a name="l00304"></a><span class="lineno"> 304</span>  iter = 0;</div>
<div class="line"><a name="l00305"></a><span class="lineno"> 305</span>  }</div>
<div class="line"><a name="l00306"></a><span class="lineno"> 306</span>  <span class="keywordflow">else</span> <span class="comment">// No convergence yet</span></div>
<div class="line"><a name="l00307"></a><span class="lineno"> 307</span>  {</div>
<div class="line"><a name="l00308"></a><span class="lineno"> 308</span>  <span class="comment">// The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC warning (-O1 -Wall -DNDEBUG )</span></div>
<div class="line"><a name="l00309"></a><span class="lineno"> 309</span>  Vector3s firstHouseholderVector(0,0,0), shiftInfo;</div>
<div class="line"><a name="l00310"></a><span class="lineno"> 310</span>  computeShift(iu, iter, exshift, shiftInfo);</div>
<div class="line"><a name="l00311"></a><span class="lineno"> 311</span>  iter = iter + 1;</div>
<div class="line"><a name="l00312"></a><span class="lineno"> 312</span>  totalIter = totalIter + 1;</div>
<div class="line"><a name="l00313"></a><span class="lineno"> 313</span>  <span class="keywordflow">if</span> (totalIter > maxIters) <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00314"></a><span class="lineno"> 314</span>  Index im;</div>
<div class="line"><a name="l00315"></a><span class="lineno"> 315</span>  initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);</div>
<div class="line"><a name="l00316"></a><span class="lineno"> 316</span>  performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);</div>
<div class="line"><a name="l00317"></a><span class="lineno"> 317</span>  }</div>
<div class="line"><a name="l00318"></a><span class="lineno"> 318</span>  }</div>
<div class="line"><a name="l00319"></a><span class="lineno"> 319</span>  }</div>
<div class="line"><a name="l00320"></a><span class="lineno"> 320</span>  <span class="keywordflow">if</span>(totalIter <= maxIters)</div>
<div class="line"><a name="l00321"></a><span class="lineno"> 321</span>  m_info = <a class="code" href="group__enums.html#gga51bc1ac16f26ebe51eae1abb77bd037bafdfbdf3247bd36a1f17270d5cec74c9c">Success</a>;</div>
<div class="line"><a name="l00322"></a><span class="lineno"> 322</span>  <span class="keywordflow">else</span></div>
<div class="line"><a name="l00323"></a><span class="lineno"> 323</span>  m_info = <a class="code" href="group__enums.html#gga51bc1ac16f26ebe51eae1abb77bd037ba4ff235bd185f3c5fceeec8d6540eb847">NoConvergence</a>;</div>
<div class="line"><a name="l00324"></a><span class="lineno"> 324</span> </div>
<div class="line"><a name="l00325"></a><span class="lineno"> 325</span>  m_isInitialized = <span class="keyword">true</span>;</div>
<div class="line"><a name="l00326"></a><span class="lineno"> 326</span>  m_matUisUptodate = computeU;</div>
<div class="line"><a name="l00327"></a><span class="lineno"> 327</span>  <span class="keywordflow">return</span> *<span class="keyword">this</span>;</div>
<div class="line"><a name="l00328"></a><span class="lineno"> 328</span> }</div>
<div class="line"><a name="l00329"></a><span class="lineno"> 329</span> </div>
<div class="line"><a name="l00331"></a><span class="lineno"> 331</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00332"></a><span class="lineno"> 332</span> <span class="keyword">inline</span> <span class="keyword">typename</span> MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT()</div>
<div class="line"><a name="l00333"></a><span class="lineno"> 333</span> {</div>
<div class="line"><a name="l00334"></a><span class="lineno"> 334</span>  <span class="keyword">const</span> Index size = m_matT.cols();</div>
<div class="line"><a name="l00335"></a><span class="lineno"> 335</span>  <span class="comment">// FIXME to be efficient the following would requires a triangular reduxion code</span></div>
<div class="line"><a name="l00336"></a><span class="lineno"> 336</span>  <span class="comment">// Scalar norm = m_matT.upper().cwiseAbs().sum() </span></div>
<div class="line"><a name="l00337"></a><span class="lineno"> 337</span>  <span class="comment">// + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum();</span></div>
<div class="line"><a name="l00338"></a><span class="lineno"> 338</span>  Scalar norm(0);</div>
<div class="line"><a name="l00339"></a><span class="lineno"> 339</span>  <span class="keywordflow">for</span> (Index j = 0; j < size; ++j)</div>
<div class="line"><a name="l00340"></a><span class="lineno"> 340</span>  norm += m_matT.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum();</div>
<div class="line"><a name="l00341"></a><span class="lineno"> 341</span>  <span class="keywordflow">return</span> norm;</div>
<div class="line"><a name="l00342"></a><span class="lineno"> 342</span> }</div>
<div class="line"><a name="l00343"></a><span class="lineno"> 343</span> </div>
<div class="line"><a name="l00345"></a><span class="lineno"> 345</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00346"></a><span class="lineno"> 346</span> <span class="keyword">inline</span> <span class="keyword">typename</span> MatrixType::Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu, <span class="keyword">const</span> Scalar& norm)</div>
<div class="line"><a name="l00347"></a><span class="lineno"> 347</span> {</div>
<div class="line"><a name="l00348"></a><span class="lineno"> 348</span>  <span class="keyword">using</span> std::abs;</div>
<div class="line"><a name="l00349"></a><span class="lineno"> 349</span>  Index res = iu;</div>
<div class="line"><a name="l00350"></a><span class="lineno"> 350</span>  <span class="keywordflow">while</span> (res > 0)</div>
<div class="line"><a name="l00351"></a><span class="lineno"> 351</span>  {</div>
<div class="line"><a name="l00352"></a><span class="lineno"> 352</span>  Scalar s = abs(m_matT.coeff(res-1,res-1)) + abs(m_matT.coeff(res,res));</div>
<div class="line"><a name="l00353"></a><span class="lineno"> 353</span>  <span class="keywordflow">if</span> (s == 0.0)</div>
<div class="line"><a name="l00354"></a><span class="lineno"> 354</span>  s = norm;</div>
<div class="line"><a name="l00355"></a><span class="lineno"> 355</span>  <span class="keywordflow">if</span> (abs(m_matT.coeff(res,res-1)) < NumTraits<Scalar>::epsilon() * s)</div>
<div class="line"><a name="l00356"></a><span class="lineno"> 356</span>  <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00357"></a><span class="lineno"> 357</span>  res--;</div>
<div class="line"><a name="l00358"></a><span class="lineno"> 358</span>  }</div>
<div class="line"><a name="l00359"></a><span class="lineno"> 359</span>  <span class="keywordflow">return</span> res;</div>
<div class="line"><a name="l00360"></a><span class="lineno"> 360</span> }</div>
<div class="line"><a name="l00361"></a><span class="lineno"> 361</span> </div>
<div class="line"><a name="l00363"></a><span class="lineno"> 363</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00364"></a><span class="lineno"> 364</span> <span class="keyword">inline</span> <span class="keywordtype">void</span> RealSchur<MatrixType>::splitOffTwoRows(Index iu, <span class="keywordtype">bool</span> computeU, <span class="keyword">const</span> Scalar& exshift)</div>
<div class="line"><a name="l00365"></a><span class="lineno"> 365</span> {</div>
<div class="line"><a name="l00366"></a><span class="lineno"> 366</span>  <span class="keyword">using</span> std::sqrt;</div>
<div class="line"><a name="l00367"></a><span class="lineno"> 367</span>  <span class="keyword">using</span> std::abs;</div>
<div class="line"><a name="l00368"></a><span class="lineno"> 368</span>  <span class="keyword">const</span> Index size = m_matT.cols();</div>
<div class="line"><a name="l00369"></a><span class="lineno"> 369</span> </div>
<div class="line"><a name="l00370"></a><span class="lineno"> 370</span>  <span class="comment">// The eigenvalues of the 2x2 matrix [a b; c d] are </span></div>
<div class="line"><a name="l00371"></a><span class="lineno"> 371</span>  <span class="comment">// trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc</span></div>
<div class="line"><a name="l00372"></a><span class="lineno"> 372</span>  Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu));</div>
<div class="line"><a name="l00373"></a><span class="lineno"> 373</span>  Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu); <span class="comment">// q = tr^2 / 4 - det = discr/4</span></div>
<div class="line"><a name="l00374"></a><span class="lineno"> 374</span>  m_matT.coeffRef(iu,iu) += exshift;</div>
<div class="line"><a name="l00375"></a><span class="lineno"> 375</span>  m_matT.coeffRef(iu-1,iu-1) += exshift;</div>
<div class="line"><a name="l00376"></a><span class="lineno"> 376</span> </div>
<div class="line"><a name="l00377"></a><span class="lineno"> 377</span>  <span class="keywordflow">if</span> (q >= Scalar(0)) <span class="comment">// Two real eigenvalues</span></div>
<div class="line"><a name="l00378"></a><span class="lineno"> 378</span>  {</div>
<div class="line"><a name="l00379"></a><span class="lineno"> 379</span>  Scalar z = sqrt(abs(q));</div>
<div class="line"><a name="l00380"></a><span class="lineno"> 380</span>  JacobiRotation<Scalar> rot;</div>
<div class="line"><a name="l00381"></a><span class="lineno"> 381</span>  <span class="keywordflow">if</span> (p >= Scalar(0))</div>
<div class="line"><a name="l00382"></a><span class="lineno"> 382</span>  rot.makeGivens(p + z, m_matT.coeff(iu, iu-1));</div>
<div class="line"><a name="l00383"></a><span class="lineno"> 383</span>  <span class="keywordflow">else</span></div>
<div class="line"><a name="l00384"></a><span class="lineno"> 384</span>  rot.makeGivens(p - z, m_matT.coeff(iu, iu-1));</div>
<div class="line"><a name="l00385"></a><span class="lineno"> 385</span> </div>
<div class="line"><a name="l00386"></a><span class="lineno"> 386</span>  m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint());</div>
<div class="line"><a name="l00387"></a><span class="lineno"> 387</span>  m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot);</div>
<div class="line"><a name="l00388"></a><span class="lineno"> 388</span>  m_matT.coeffRef(iu, iu-1) = Scalar(0); </div>
<div class="line"><a name="l00389"></a><span class="lineno"> 389</span>  <span class="keywordflow">if</span> (computeU)</div>
<div class="line"><a name="l00390"></a><span class="lineno"> 390</span>  m_matU.applyOnTheRight(iu-1, iu, rot);</div>
<div class="line"><a name="l00391"></a><span class="lineno"> 391</span>  }</div>
<div class="line"><a name="l00392"></a><span class="lineno"> 392</span> </div>
<div class="line"><a name="l00393"></a><span class="lineno"> 393</span>  <span class="keywordflow">if</span> (iu > 1) </div>
<div class="line"><a name="l00394"></a><span class="lineno"> 394</span>  m_matT.coeffRef(iu-1, iu-2) = Scalar(0);</div>
<div class="line"><a name="l00395"></a><span class="lineno"> 395</span> }</div>
<div class="line"><a name="l00396"></a><span class="lineno"> 396</span> </div>
<div class="line"><a name="l00398"></a><span class="lineno"> 398</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00399"></a><span class="lineno"> 399</span> <span class="keyword">inline</span> <span class="keywordtype">void</span> RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo)</div>
<div class="line"><a name="l00400"></a><span class="lineno"> 400</span> {</div>
<div class="line"><a name="l00401"></a><span class="lineno"> 401</span>  <span class="keyword">using</span> std::sqrt;</div>
<div class="line"><a name="l00402"></a><span class="lineno"> 402</span>  <span class="keyword">using</span> std::abs;</div>
<div class="line"><a name="l00403"></a><span class="lineno"> 403</span>  shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu);</div>
<div class="line"><a name="l00404"></a><span class="lineno"> 404</span>  shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1);</div>
<div class="line"><a name="l00405"></a><span class="lineno"> 405</span>  shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);</div>
<div class="line"><a name="l00406"></a><span class="lineno"> 406</span> </div>
<div class="line"><a name="l00407"></a><span class="lineno"> 407</span>  <span class="comment">// Wilkinson's original ad hoc shift</span></div>
<div class="line"><a name="l00408"></a><span class="lineno"> 408</span>  <span class="keywordflow">if</span> (iter == 10)</div>
<div class="line"><a name="l00409"></a><span class="lineno"> 409</span>  {</div>
<div class="line"><a name="l00410"></a><span class="lineno"> 410</span>  exshift += shiftInfo.coeff(0);</div>
<div class="line"><a name="l00411"></a><span class="lineno"> 411</span>  <span class="keywordflow">for</span> (Index i = 0; i <= iu; ++i)</div>
<div class="line"><a name="l00412"></a><span class="lineno"> 412</span>  m_matT.coeffRef(i,i) -= shiftInfo.coeff(0);</div>
<div class="line"><a name="l00413"></a><span class="lineno"> 413</span>  Scalar s = abs(m_matT.coeff(iu,iu-1)) + abs(m_matT.coeff(iu-1,iu-2));</div>
<div class="line"><a name="l00414"></a><span class="lineno"> 414</span>  shiftInfo.coeffRef(0) = Scalar(0.75) * s;</div>
<div class="line"><a name="l00415"></a><span class="lineno"> 415</span>  shiftInfo.coeffRef(1) = Scalar(0.75) * s;</div>
<div class="line"><a name="l00416"></a><span class="lineno"> 416</span>  shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;</div>
<div class="line"><a name="l00417"></a><span class="lineno"> 417</span>  }</div>
<div class="line"><a name="l00418"></a><span class="lineno"> 418</span> </div>
<div class="line"><a name="l00419"></a><span class="lineno"> 419</span>  <span class="comment">// MATLAB's new ad hoc shift</span></div>
<div class="line"><a name="l00420"></a><span class="lineno"> 420</span>  <span class="keywordflow">if</span> (iter == 30)</div>
<div class="line"><a name="l00421"></a><span class="lineno"> 421</span>  {</div>
<div class="line"><a name="l00422"></a><span class="lineno"> 422</span>  Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);</div>
<div class="line"><a name="l00423"></a><span class="lineno"> 423</span>  s = s * s + shiftInfo.coeff(2);</div>
<div class="line"><a name="l00424"></a><span class="lineno"> 424</span>  <span class="keywordflow">if</span> (s > Scalar(0))</div>
<div class="line"><a name="l00425"></a><span class="lineno"> 425</span>  {</div>
<div class="line"><a name="l00426"></a><span class="lineno"> 426</span>  s = sqrt(s);</div>
<div class="line"><a name="l00427"></a><span class="lineno"> 427</span>  <span class="keywordflow">if</span> (shiftInfo.coeff(1) < shiftInfo.coeff(0))</div>
<div class="line"><a name="l00428"></a><span class="lineno"> 428</span>  s = -s;</div>
<div class="line"><a name="l00429"></a><span class="lineno"> 429</span>  s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);</div>
<div class="line"><a name="l00430"></a><span class="lineno"> 430</span>  s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;</div>
<div class="line"><a name="l00431"></a><span class="lineno"> 431</span>  exshift += s;</div>
<div class="line"><a name="l00432"></a><span class="lineno"> 432</span>  <span class="keywordflow">for</span> (Index i = 0; i <= iu; ++i)</div>
<div class="line"><a name="l00433"></a><span class="lineno"> 433</span>  m_matT.coeffRef(i,i) -= s;</div>
<div class="line"><a name="l00434"></a><span class="lineno"> 434</span>  shiftInfo.setConstant(Scalar(0.964));</div>
<div class="line"><a name="l00435"></a><span class="lineno"> 435</span>  }</div>
<div class="line"><a name="l00436"></a><span class="lineno"> 436</span>  }</div>
<div class="line"><a name="l00437"></a><span class="lineno"> 437</span> }</div>
<div class="line"><a name="l00438"></a><span class="lineno"> 438</span> </div>
<div class="line"><a name="l00440"></a><span class="lineno"> 440</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00441"></a><span class="lineno"> 441</span> <span class="keyword">inline</span> <span class="keywordtype">void</span> RealSchur<MatrixType>::initFrancisQRStep(Index il, Index iu, <span class="keyword">const</span> Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector)</div>
<div class="line"><a name="l00442"></a><span class="lineno"> 442</span> {</div>
<div class="line"><a name="l00443"></a><span class="lineno"> 443</span>  <span class="keyword">using</span> std::abs;</div>
<div class="line"><a name="l00444"></a><span class="lineno"> 444</span>  Vector3s& v = firstHouseholderVector; <span class="comment">// alias to save typing</span></div>
<div class="line"><a name="l00445"></a><span class="lineno"> 445</span> </div>
<div class="line"><a name="l00446"></a><span class="lineno"> 446</span>  <span class="keywordflow">for</span> (im = iu-2; im >= il; --im)</div>
<div class="line"><a name="l00447"></a><span class="lineno"> 447</span>  {</div>
<div class="line"><a name="l00448"></a><span class="lineno"> 448</span>  <span class="keyword">const</span> Scalar Tmm = m_matT.coeff(im,im);</div>
<div class="line"><a name="l00449"></a><span class="lineno"> 449</span>  <span class="keyword">const</span> Scalar r = shiftInfo.coeff(0) - Tmm;</div>
<div class="line"><a name="l00450"></a><span class="lineno"> 450</span>  <span class="keyword">const</span> Scalar s = shiftInfo.coeff(1) - Tmm;</div>
<div class="line"><a name="l00451"></a><span class="lineno"> 451</span>  v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1);</div>
<div class="line"><a name="l00452"></a><span class="lineno"> 452</span>  v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s;</div>
<div class="line"><a name="l00453"></a><span class="lineno"> 453</span>  v.coeffRef(2) = m_matT.coeff(im+2,im+1);</div>
<div class="line"><a name="l00454"></a><span class="lineno"> 454</span>  <span class="keywordflow">if</span> (im == il) {</div>
<div class="line"><a name="l00455"></a><span class="lineno"> 455</span>  <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00456"></a><span class="lineno"> 456</span>  }</div>
<div class="line"><a name="l00457"></a><span class="lineno"> 457</span>  <span class="keyword">const</span> Scalar lhs = m_matT.coeff(im,im-1) * (abs(v.coeff(1)) + abs(v.coeff(2)));</div>
<div class="line"><a name="l00458"></a><span class="lineno"> 458</span>  <span class="keyword">const</span> Scalar rhs = v.coeff(0) * (abs(m_matT.coeff(im-1,im-1)) + abs(Tmm) + abs(m_matT.coeff(im+1,im+1)));</div>
<div class="line"><a name="l00459"></a><span class="lineno"> 459</span>  <span class="keywordflow">if</span> (abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)</div>
<div class="line"><a name="l00460"></a><span class="lineno"> 460</span>  {</div>
<div class="line"><a name="l00461"></a><span class="lineno"> 461</span>  <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00462"></a><span class="lineno"> 462</span>  }</div>
<div class="line"><a name="l00463"></a><span class="lineno"> 463</span>  }</div>
<div class="line"><a name="l00464"></a><span class="lineno"> 464</span> }</div>
<div class="line"><a name="l00465"></a><span class="lineno"> 465</span> </div>
<div class="line"><a name="l00467"></a><span class="lineno"> 467</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00468"></a><span class="lineno"> 468</span> <span class="keyword">inline</span> <span class="keywordtype">void</span> RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu, <span class="keywordtype">bool</span> computeU, <span class="keyword">const</span> Vector3s& firstHouseholderVector, Scalar* workspace)</div>
<div class="line"><a name="l00469"></a><span class="lineno"> 469</span> {</div>
<div class="line"><a name="l00470"></a><span class="lineno"> 470</span>  eigen_assert(im >= il);</div>
<div class="line"><a name="l00471"></a><span class="lineno"> 471</span>  eigen_assert(im <= iu-2);</div>
<div class="line"><a name="l00472"></a><span class="lineno"> 472</span> </div>
<div class="line"><a name="l00473"></a><span class="lineno"> 473</span>  <span class="keyword">const</span> Index size = m_matT.cols();</div>
<div class="line"><a name="l00474"></a><span class="lineno"> 474</span> </div>
<div class="line"><a name="l00475"></a><span class="lineno"> 475</span>  <span class="keywordflow">for</span> (Index k = im; k <= iu-2; ++k)</div>
<div class="line"><a name="l00476"></a><span class="lineno"> 476</span>  {</div>
<div class="line"><a name="l00477"></a><span class="lineno"> 477</span>  <span class="keywordtype">bool</span> firstIteration = (k == im);</div>
<div class="line"><a name="l00478"></a><span class="lineno"> 478</span> </div>
<div class="line"><a name="l00479"></a><span class="lineno"> 479</span>  Vector3s v;</div>
<div class="line"><a name="l00480"></a><span class="lineno"> 480</span>  <span class="keywordflow">if</span> (firstIteration)</div>
<div class="line"><a name="l00481"></a><span class="lineno"> 481</span>  v = firstHouseholderVector;</div>
<div class="line"><a name="l00482"></a><span class="lineno"> 482</span>  <span class="keywordflow">else</span></div>
<div class="line"><a name="l00483"></a><span class="lineno"> 483</span>  v = m_matT.template block<3,1>(k,k-1);</div>
<div class="line"><a name="l00484"></a><span class="lineno"> 484</span> </div>
<div class="line"><a name="l00485"></a><span class="lineno"> 485</span>  Scalar tau, beta;</div>
<div class="line"><a name="l00486"></a><span class="lineno"> 486</span>  Matrix<Scalar, 2, 1> ess;</div>
<div class="line"><a name="l00487"></a><span class="lineno"> 487</span>  v.makeHouseholder(ess, tau, beta);</div>
<div class="line"><a name="l00488"></a><span class="lineno"> 488</span>  </div>
<div class="line"><a name="l00489"></a><span class="lineno"> 489</span>  <span class="keywordflow">if</span> (beta != Scalar(0)) <span class="comment">// if v is not zero</span></div>
<div class="line"><a name="l00490"></a><span class="lineno"> 490</span>  {</div>
<div class="line"><a name="l00491"></a><span class="lineno"> 491</span>  <span class="keywordflow">if</span> (firstIteration && k > il)</div>
<div class="line"><a name="l00492"></a><span class="lineno"> 492</span>  m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1);</div>
<div class="line"><a name="l00493"></a><span class="lineno"> 493</span>  <span class="keywordflow">else</span> <span class="keywordflow">if</span> (!firstIteration)</div>
<div class="line"><a name="l00494"></a><span class="lineno"> 494</span>  m_matT.coeffRef(k,k-1) = beta;</div>
<div class="line"><a name="l00495"></a><span class="lineno"> 495</span> </div>
<div class="line"><a name="l00496"></a><span class="lineno"> 496</span>  <span class="comment">// These Householder transformations form the O(n^3) part of the algorithm</span></div>
<div class="line"><a name="l00497"></a><span class="lineno"> 497</span>  m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace);</div>
<div class="line"><a name="l00498"></a><span class="lineno"> 498</span>  m_matT.block(0, k, (std::min)(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace);</div>
<div class="line"><a name="l00499"></a><span class="lineno"> 499</span>  <span class="keywordflow">if</span> (computeU)</div>
<div class="line"><a name="l00500"></a><span class="lineno"> 500</span>  m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace);</div>
<div class="line"><a name="l00501"></a><span class="lineno"> 501</span>  }</div>
<div class="line"><a name="l00502"></a><span class="lineno"> 502</span>  }</div>
<div class="line"><a name="l00503"></a><span class="lineno"> 503</span> </div>
<div class="line"><a name="l00504"></a><span class="lineno"> 504</span>  Matrix<Scalar, 2, 1> v = m_matT.template block<2,1>(iu-1, iu-2);</div>
<div class="line"><a name="l00505"></a><span class="lineno"> 505</span>  Scalar tau, beta;</div>
<div class="line"><a name="l00506"></a><span class="lineno"> 506</span>  Matrix<Scalar, 1, 1> ess;</div>
<div class="line"><a name="l00507"></a><span class="lineno"> 507</span>  v.makeHouseholder(ess, tau, beta);</div>
<div class="line"><a name="l00508"></a><span class="lineno"> 508</span> </div>
<div class="line"><a name="l00509"></a><span class="lineno"> 509</span>  <span class="keywordflow">if</span> (beta != Scalar(0)) <span class="comment">// if v is not zero</span></div>
<div class="line"><a name="l00510"></a><span class="lineno"> 510</span>  {</div>
<div class="line"><a name="l00511"></a><span class="lineno"> 511</span>  m_matT.coeffRef(iu-1, iu-2) = beta;</div>
<div class="line"><a name="l00512"></a><span class="lineno"> 512</span>  m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace);</div>
<div class="line"><a name="l00513"></a><span class="lineno"> 513</span>  m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace);</div>
<div class="line"><a name="l00514"></a><span class="lineno"> 514</span>  <span class="keywordflow">if</span> (computeU)</div>
<div class="line"><a name="l00515"></a><span class="lineno"> 515</span>  m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace);</div>
<div class="line"><a name="l00516"></a><span class="lineno"> 516</span>  }</div>
<div class="line"><a name="l00517"></a><span class="lineno"> 517</span> </div>
<div class="line"><a name="l00518"></a><span class="lineno"> 518</span>  <span class="comment">// clean up pollution due to round-off errors</span></div>
<div class="line"><a name="l00519"></a><span class="lineno"> 519</span>  <span class="keywordflow">for</span> (Index i = im+2; i <= iu; ++i)</div>
<div class="line"><a name="l00520"></a><span class="lineno"> 520</span>  {</div>
<div class="line"><a name="l00521"></a><span class="lineno"> 521</span>  m_matT.coeffRef(i,i-2) = Scalar(0);</div>
<div class="line"><a name="l00522"></a><span class="lineno"> 522</span>  <span class="keywordflow">if</span> (i > im+2)</div>
<div class="line"><a name="l00523"></a><span class="lineno"> 523</span>  m_matT.coeffRef(i,i-3) = Scalar(0);</div>
<div class="line"><a name="l00524"></a><span class="lineno"> 524</span>  }</div>
<div class="line"><a name="l00525"></a><span class="lineno"> 525</span> }</div>
<div class="line"><a name="l00526"></a><span class="lineno"> 526</span> </div>
<div class="line"><a name="l00527"></a><span class="lineno"> 527</span> } <span class="comment">// end namespace Eigen</span></div>
<div class="line"><a name="l00528"></a><span class="lineno"> 528</span> </div>
<div class="line"><a name="l00529"></a><span class="lineno"> 529</span> <span class="preprocessor">#endif // EIGEN_REAL_SCHUR_H</span></div>
<div class="ttc" id="classEigen_1_1RealSchur_html_ace387c8cea391973ca2a99edc720671a"><div class="ttname"><a href="classEigen_1_1RealSchur.html#ace387c8cea391973ca2a99edc720671a">Eigen::RealSchur::compute</a></div><div class="ttdeci">RealSchur & compute(const MatrixType &matrix, bool computeU=true)</div><div class="ttdoc">Computes Schur decomposition of given matrix. </div><div class="ttdef"><b>Definition:</b> RealSchur.h:246</div></div>
<div class="ttc" id="classEigen_1_1RealSchur_html_a0c06d5c2034ebb329c54235369643ad2"><div class="ttname"><a href="classEigen_1_1RealSchur.html#a0c06d5c2034ebb329c54235369643ad2">Eigen::RealSchur::info</a></div><div class="ttdeci">ComputationInfo info() const </div><div class="ttdoc">Reports whether previous computation was successful. </div><div class="ttdef"><b>Definition:</b> RealSchur.h:193</div></div>
<div class="ttc" id="classEigen_1_1RealSchur_html"><div class="ttname"><a href="classEigen_1_1RealSchur.html">Eigen::RealSchur</a></div><div class="ttdoc">Performs a real Schur decomposition of a square matrix. </div><div class="ttdef"><b>Definition:</b> RealSchur.h:54</div></div>
<div class="ttc" id="classEigen_1_1RealSchur_html_a8d73d4e86d87bd2babf172909fc54198"><div class="ttname"><a href="classEigen_1_1RealSchur.html#a8d73d4e86d87bd2babf172909fc54198">Eigen::RealSchur::RealSchur</a></div><div class="ttdeci">RealSchur(const MatrixType &matrix, bool computeU=true)</div><div class="ttdoc">Constructor; computes real Schur decomposition of given matrix. </div><div class="ttdef"><b>Definition:</b> RealSchur.h:103</div></div>
<div class="ttc" id="classEigen_1_1RealSchur_html_afdafb24d67af7529bb903a4c9bff3ea4"><div class="ttname"><a href="classEigen_1_1RealSchur.html#afdafb24d67af7529bb903a4c9bff3ea4">Eigen::RealSchur::m_maxIterationsPerRow</a></div><div class="ttdeci">static const int m_maxIterationsPerRow</div><div class="ttdoc">Maximum number of iterations per row. </div><div class="ttdef"><b>Definition:</b> RealSchur.h:221</div></div>
<div class="ttc" id="namespaceEigen_html_adc9da5be31bdce40c25a92c27999c0e3"><div class="ttname"><a href="namespaceEigen.html#adc9da5be31bdce40c25a92c27999c0e3">Eigen::Dynamic</a></div><div class="ttdeci">const int Dynamic</div><div class="ttdef"><b>Definition:</b> Constants.h:21</div></div>
<div class="ttc" id="classEigen_1_1RealSchur_html_ab6f0a63ea1d26cef5e748207043eb43e"><div class="ttname"><a href="classEigen_1_1RealSchur.html#ab6f0a63ea1d26cef5e748207043eb43e">Eigen::RealSchur::getMaxIterations</a></div><div class="ttdeci">Index getMaxIterations()</div><div class="ttdoc">Returns the maximum number of iterations. </div><div class="ttdef"><b>Definition:</b> RealSchur.h:211</div></div>
<div class="ttc" id="classEigen_1_1RealSchur_html_adcb20d95f17c74395f9a906e5a72ab6b"><div class="ttname"><a href="classEigen_1_1RealSchur.html#adcb20d95f17c74395f9a906e5a72ab6b">Eigen::RealSchur::setMaxIterations</a></div><div class="ttdeci">RealSchur & setMaxIterations(Index maxIters)</div><div class="ttdoc">Sets the maximum number of iterations allowed. </div><div class="ttdef"><b>Definition:</b> RealSchur.h:204</div></div>
<div class="ttc" id="classEigen_1_1HessenbergDecomposition_html"><div class="ttname"><a href="classEigen_1_1HessenbergDecomposition.html">Eigen::HessenbergDecomposition< MatrixType ></a></div></div>
<div class="ttc" id="group__enums_html_gga51bc1ac16f26ebe51eae1abb77bd037ba4ff235bd185f3c5fceeec8d6540eb847"><div class="ttname"><a href="group__enums.html#gga51bc1ac16f26ebe51eae1abb77bd037ba4ff235bd185f3c5fceeec8d6540eb847">Eigen::NoConvergence</a></div><div class="ttdef"><b>Definition:</b> Constants.h:380</div></div>
<div class="ttc" id="classEigen_1_1RealSchur_html_a5b461a34397b36bb284ccfb0f3a4c498"><div class="ttname"><a href="classEigen_1_1RealSchur.html#a5b461a34397b36bb284ccfb0f3a4c498">Eigen::RealSchur::computeFromHessenberg</a></div><div class="ttdeci">RealSchur & computeFromHessenberg(const HessMatrixType &matrixH, const OrthMatrixType &matrixQ, bool computeU)</div><div class="ttdoc">Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T. </div></div>
<div class="ttc" id="group__enums_html_gga51bc1ac16f26ebe51eae1abb77bd037bafdfbdf3247bd36a1f17270d5cec74c9c"><div class="ttname"><a href="group__enums.html#gga51bc1ac16f26ebe51eae1abb77bd037bafdfbdf3247bd36a1f17270d5cec74c9c">Eigen::Success</a></div><div class="ttdef"><b>Definition:</b> Constants.h:376</div></div>
<div class="ttc" id="classEigen_1_1RealSchur_html_a782ab2c509de1deb484bbd12d6e863a0"><div class="ttname"><a href="classEigen_1_1RealSchur.html#a782ab2c509de1deb484bbd12d6e863a0">Eigen::RealSchur::RealSchur</a></div><div class="ttdeci">RealSchur(Index size=RowsAtCompileTime==Dynamic?1:RowsAtCompileTime)</div><div class="ttdoc">Default constructor. </div><div class="ttdef"><b>Definition:</b> RealSchur.h:83</div></div>
<div class="ttc" id="classEigen_1_1RealSchur_html_a0d31900234ef9fea5751ce8ea693d71f"><div class="ttname"><a href="classEigen_1_1RealSchur.html#a0d31900234ef9fea5751ce8ea693d71f">Eigen::RealSchur::matrixT</a></div><div class="ttdeci">const MatrixType & matrixT() const </div><div class="ttdoc">Returns the quasi-triangular matrix in the Schur decomposition. </div><div class="ttdef"><b>Definition:</b> RealSchur.h:143</div></div>
<div class="ttc" id="classEigen_1_1Matrix_html"><div class="ttname"><a href="classEigen_1_1Matrix.html">Eigen::Matrix< ComplexScalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 ></a></div></div>
<div class="ttc" id="group__enums_html_ga51bc1ac16f26ebe51eae1abb77bd037b"><div class="ttname"><a href="group__enums.html#ga51bc1ac16f26ebe51eae1abb77bd037b">Eigen::ComputationInfo</a></div><div class="ttdeci">ComputationInfo</div><div class="ttdef"><b>Definition:</b> Constants.h:374</div></div>
<div class="ttc" id="classEigen_1_1RealSchur_html_a7663c715ad9aaf8b57825646f5317166"><div class="ttname"><a href="classEigen_1_1RealSchur.html#a7663c715ad9aaf8b57825646f5317166">Eigen::RealSchur::matrixU</a></div><div class="ttdeci">const MatrixType & matrixU() const </div><div class="ttdoc">Returns the orthogonal matrix in the Schur decomposition. </div><div class="ttdef"><b>Definition:</b> RealSchur.h:126</div></div>
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