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<div class="title">RealQZ.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) 2012 Alexey Korepanov <kaikaikai@yandex.ru></span></div>
<div class="line"><a name="l00005"></a><span class="lineno"> 5</span> <span class="comment">//</span></div>
<div class="line"><a name="l00006"></a><span class="lineno"> 6</span> <span class="comment">// This Source Code Form is subject to the terms of the Mozilla</span></div>
<div class="line"><a name="l00007"></a><span class="lineno"> 7</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="l00008"></a><span class="lineno"> 8</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="l00009"></a><span class="lineno"> 9</span> </div>
<div class="line"><a name="l00010"></a><span class="lineno"> 10</span> <span class="preprocessor">#ifndef EIGEN_REAL_QZ_H</span></div>
<div class="line"><a name="l00011"></a><span class="lineno"> 11</span> <span class="preprocessor"></span><span class="preprocessor">#define EIGEN_REAL_QZ_H</span></div>
<div class="line"><a name="l00012"></a><span class="lineno"> 12</span> <span class="preprocessor"></span></div>
<div class="line"><a name="l00013"></a><span class="lineno"> 13</span> <span class="keyword">namespace </span>Eigen {</div>
<div class="line"><a name="l00014"></a><span class="lineno"> 14</span> </div>
<div class="line"><a name="l00057"></a><span class="lineno"><a class="line" href="classEigen_1_1RealQZ.html"> 57</a></span>  <span class="keyword">template</span><<span class="keyword">typename</span> _MatrixType> <span class="keyword">class </span><a class="code" href="classEigen_1_1RealQZ.html">RealQZ</a></div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>  {</div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>  <span class="keyword">public</span>:</div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>  <span class="keyword">typedef</span> _MatrixType MatrixType;</div>
<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>  <span class="keyword">enum</span> {</div>
<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>  RowsAtCompileTime = MatrixType::RowsAtCompileTime,</div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>  ColsAtCompileTime = MatrixType::ColsAtCompileTime,</div>
<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>  Options = MatrixType::Options,</div>
<div class="line"><a name="l00065"></a><span class="lineno"> 65</span>  MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,</div>
<div class="line"><a name="l00066"></a><span class="lineno"> 66</span>  MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime</div>
<div class="line"><a name="l00067"></a><span class="lineno"> 67</span>  };</div>
<div class="line"><a name="l00068"></a><span class="lineno"> 68</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::Scalar Scalar;</div>
<div class="line"><a name="l00069"></a><span class="lineno"> 69</span>  <span class="keyword">typedef</span> std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;</div>
<div class="line"><a name="l00070"></a><span class="lineno"> 70</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::Index Index;</div>
<div class="line"><a name="l00071"></a><span class="lineno"> 71</span> </div>
<div class="line"><a name="l00072"></a><span class="lineno"> 72</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="l00073"></a><span class="lineno"> 73</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="l00074"></a><span class="lineno"> 74</span> </div>
<div class="line"><a name="l00086"></a><span class="lineno"><a class="line" href="classEigen_1_1RealQZ.html#a4b2119ce39103693d003d8e434f00a3a"> 86</a></span>  <a class="code" href="classEigen_1_1RealQZ.html#a4b2119ce39103693d003d8e434f00a3a">RealQZ</a>(Index size = RowsAtCompileTime==<a class="code" href="namespaceEigen.html#adc9da5be31bdce40c25a92c27999c0e3">Dynamic</a> ? 1 : RowsAtCompileTime) : </div>
<div class="line"><a name="l00087"></a><span class="lineno"> 87</span>  m_S(size, size),</div>
<div class="line"><a name="l00088"></a><span class="lineno"> 88</span>  m_T(size, size),</div>
<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>  m_Q(size, size),</div>
<div class="line"><a name="l00090"></a><span class="lineno"> 90</span>  m_Z(size, size),</div>
<div class="line"><a name="l00091"></a><span class="lineno"> 91</span>  m_workspace(size*2),</div>
<div class="line"><a name="l00092"></a><span class="lineno"> 92</span>  m_maxIters(400),</div>
<div class="line"><a name="l00093"></a><span class="lineno"> 93</span>  m_isInitialized(false)</div>
<div class="line"><a name="l00094"></a><span class="lineno"> 94</span>  { }</div>
<div class="line"><a name="l00095"></a><span class="lineno"> 95</span> </div>
<div class="line"><a name="l00104"></a><span class="lineno"><a class="line" href="classEigen_1_1RealQZ.html#ab81ec305afdcbf94ed288d382710e8d7"> 104</a></span>  <a class="code" href="classEigen_1_1RealQZ.html#ab81ec305afdcbf94ed288d382710e8d7">RealQZ</a>(<span class="keyword">const</span> MatrixType& A, <span class="keyword">const</span> MatrixType& B, <span class="keywordtype">bool</span> computeQZ = <span class="keyword">true</span>) :</div>
<div class="line"><a name="l00105"></a><span class="lineno"> 105</span>  m_S(A.rows(),A.cols()),</div>
<div class="line"><a name="l00106"></a><span class="lineno"> 106</span>  m_T(A.rows(),A.cols()),</div>
<div class="line"><a name="l00107"></a><span class="lineno"> 107</span>  m_Q(A.rows(),A.cols()),</div>
<div class="line"><a name="l00108"></a><span class="lineno"> 108</span>  m_Z(A.rows(),A.cols()),</div>
<div class="line"><a name="l00109"></a><span class="lineno"> 109</span>  m_workspace(A.rows()*2),</div>
<div class="line"><a name="l00110"></a><span class="lineno"> 110</span>  m_maxIters(400),</div>
<div class="line"><a name="l00111"></a><span class="lineno"> 111</span>  m_isInitialized(false) {</div>
<div class="line"><a name="l00112"></a><span class="lineno"> 112</span>  <a class="code" href="classEigen_1_1RealQZ.html#a75cc509be2fa23cde7371b79ebd9f770">compute</a>(A, B, computeQZ);</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="l00119"></a><span class="lineno"><a class="line" href="classEigen_1_1RealQZ.html#a498541357c143f345f0af5d6a6b9b3c3"> 119</a></span>  <span class="keyword">const</span> MatrixType& <a class="code" href="classEigen_1_1RealQZ.html#a498541357c143f345f0af5d6a6b9b3c3">matrixQ</a>()<span class="keyword"> const </span>{</div>
<div class="line"><a name="l00120"></a><span class="lineno"> 120</span>  eigen_assert(m_isInitialized && <span class="stringliteral">"RealQZ is not initialized."</span>);</div>
<div class="line"><a name="l00121"></a><span class="lineno"> 121</span>  eigen_assert(m_computeQZ && <span class="stringliteral">"The matrices Q and Z have not been computed during the QZ decomposition."</span>);</div>
<div class="line"><a name="l00122"></a><span class="lineno"> 122</span>  <span class="keywordflow">return</span> m_Q;</div>
<div class="line"><a name="l00123"></a><span class="lineno"> 123</span>  }</div>
<div class="line"><a name="l00124"></a><span class="lineno"> 124</span> </div>
<div class="line"><a name="l00129"></a><span class="lineno"><a class="line" href="classEigen_1_1RealQZ.html#a0f5765f1177b790b663281635e15e73f"> 129</a></span>  <span class="keyword">const</span> MatrixType& <a class="code" href="classEigen_1_1RealQZ.html#a0f5765f1177b790b663281635e15e73f">matrixZ</a>()<span class="keyword"> const </span>{</div>
<div class="line"><a name="l00130"></a><span class="lineno"> 130</span>  eigen_assert(m_isInitialized && <span class="stringliteral">"RealQZ is not initialized."</span>);</div>
<div class="line"><a name="l00131"></a><span class="lineno"> 131</span>  eigen_assert(m_computeQZ && <span class="stringliteral">"The matrices Q and Z have not been computed during the QZ decomposition."</span>);</div>
<div class="line"><a name="l00132"></a><span class="lineno"> 132</span>  <span class="keywordflow">return</span> m_Z;</div>
<div class="line"><a name="l00133"></a><span class="lineno"> 133</span>  }</div>
<div class="line"><a name="l00134"></a><span class="lineno"> 134</span> </div>
<div class="line"><a name="l00139"></a><span class="lineno"><a class="line" href="classEigen_1_1RealQZ.html#af0356bea58e012ff54177cea8a340f64"> 139</a></span>  <span class="keyword">const</span> MatrixType& <a class="code" href="classEigen_1_1RealQZ.html#af0356bea58e012ff54177cea8a340f64">matrixS</a>()<span class="keyword"> const </span>{</div>
<div class="line"><a name="l00140"></a><span class="lineno"> 140</span>  eigen_assert(m_isInitialized && <span class="stringliteral">"RealQZ is not initialized."</span>);</div>
<div class="line"><a name="l00141"></a><span class="lineno"> 141</span>  <span class="keywordflow">return</span> m_S;</div>
<div class="line"><a name="l00142"></a><span class="lineno"> 142</span>  }</div>
<div class="line"><a name="l00143"></a><span class="lineno"> 143</span> </div>
<div class="line"><a name="l00148"></a><span class="lineno"><a class="line" href="classEigen_1_1RealQZ.html#a0d31900234ef9fea5751ce8ea693d71f"> 148</a></span>  <span class="keyword">const</span> MatrixType& <a class="code" href="classEigen_1_1RealQZ.html#a0d31900234ef9fea5751ce8ea693d71f">matrixT</a>()<span class="keyword"> const </span>{</div>
<div class="line"><a name="l00149"></a><span class="lineno"> 149</span>  eigen_assert(m_isInitialized && <span class="stringliteral">"RealQZ is not initialized."</span>);</div>
<div class="line"><a name="l00150"></a><span class="lineno"> 150</span>  <span class="keywordflow">return</span> m_T;</div>
<div class="line"><a name="l00151"></a><span class="lineno"> 151</span>  }</div>
<div class="line"><a name="l00152"></a><span class="lineno"> 152</span> </div>
<div class="line"><a name="l00160"></a><span class="lineno"> 160</span>  <a class="code" href="classEigen_1_1RealQZ.html">RealQZ</a>& <a class="code" href="classEigen_1_1RealQZ.html#a75cc509be2fa23cde7371b79ebd9f770">compute</a>(<span class="keyword">const</span> MatrixType& A, <span class="keyword">const</span> MatrixType& B, <span class="keywordtype">bool</span> computeQZ = <span class="keyword">true</span>);</div>
<div class="line"><a name="l00161"></a><span class="lineno"> 161</span> </div>
<div class="line"><a name="l00166"></a><span class="lineno"><a class="line" href="classEigen_1_1RealQZ.html#a0c06d5c2034ebb329c54235369643ad2"> 166</a></span>  <a class="code" href="group__enums.html#ga51bc1ac16f26ebe51eae1abb77bd037b">ComputationInfo</a> <a class="code" href="classEigen_1_1RealQZ.html#a0c06d5c2034ebb329c54235369643ad2">info</a>()<span class="keyword"> const</span></div>
<div class="line"><a name="l00167"></a><span class="lineno"> 167</span> <span class="keyword"> </span>{</div>
<div class="line"><a name="l00168"></a><span class="lineno"> 168</span>  eigen_assert(m_isInitialized && <span class="stringliteral">"RealQZ is not initialized."</span>);</div>
<div class="line"><a name="l00169"></a><span class="lineno"> 169</span>  <span class="keywordflow">return</span> m_info;</div>
<div class="line"><a name="l00170"></a><span class="lineno"> 170</span>  }</div>
<div class="line"><a name="l00171"></a><span class="lineno"> 171</span> </div>
<div class="line"><a name="l00174"></a><span class="lineno"><a class="line" href="classEigen_1_1RealQZ.html#ae231768dac8df88971dc1871f6493571"> 174</a></span>  Index <a class="code" href="classEigen_1_1RealQZ.html#ae231768dac8df88971dc1871f6493571">iterations</a>()<span class="keyword"> const</span></div>
<div class="line"><a name="l00175"></a><span class="lineno"> 175</span> <span class="keyword"> </span>{</div>
<div class="line"><a name="l00176"></a><span class="lineno"> 176</span>  eigen_assert(m_isInitialized && <span class="stringliteral">"RealQZ is not initialized."</span>);</div>
<div class="line"><a name="l00177"></a><span class="lineno"> 177</span>  <span class="keywordflow">return</span> m_global_iter;</div>
<div class="line"><a name="l00178"></a><span class="lineno"> 178</span>  }</div>
<div class="line"><a name="l00179"></a><span class="lineno"> 179</span> </div>
<div class="line"><a name="l00183"></a><span class="lineno"><a class="line" href="classEigen_1_1RealQZ.html#a1b369841b0e39a1ac80a6c32b721d242"> 183</a></span>  <a class="code" href="classEigen_1_1RealQZ.html">RealQZ</a>& <a class="code" href="classEigen_1_1RealQZ.html#a1b369841b0e39a1ac80a6c32b721d242">setMaxIterations</a>(Index maxIters)</div>
<div class="line"><a name="l00184"></a><span class="lineno"> 184</span>  {</div>
<div class="line"><a name="l00185"></a><span class="lineno"> 185</span>  m_maxIters = maxIters;</div>
<div class="line"><a name="l00186"></a><span class="lineno"> 186</span>  <span class="keywordflow">return</span> *<span class="keyword">this</span>;</div>
<div class="line"><a name="l00187"></a><span class="lineno"> 187</span>  }</div>
<div class="line"><a name="l00188"></a><span class="lineno"> 188</span> </div>
<div class="line"><a name="l00189"></a><span class="lineno"> 189</span>  <span class="keyword">private</span>:</div>
<div class="line"><a name="l00190"></a><span class="lineno"> 190</span> </div>
<div class="line"><a name="l00191"></a><span class="lineno"> 191</span>  MatrixType m_S, m_T, m_Q, m_Z;</div>
<div class="line"><a name="l00192"></a><span class="lineno"> 192</span>  <a class="code" href="classEigen_1_1Matrix.html">Matrix<Scalar,Dynamic,1></a> m_workspace;</div>
<div class="line"><a name="l00193"></a><span class="lineno"> 193</span>  <a class="code" href="group__enums.html#ga51bc1ac16f26ebe51eae1abb77bd037b">ComputationInfo</a> m_info;</div>
<div class="line"><a name="l00194"></a><span class="lineno"> 194</span>  Index m_maxIters;</div>
<div class="line"><a name="l00195"></a><span class="lineno"> 195</span>  <span class="keywordtype">bool</span> m_isInitialized;</div>
<div class="line"><a name="l00196"></a><span class="lineno"> 196</span>  <span class="keywordtype">bool</span> m_computeQZ;</div>
<div class="line"><a name="l00197"></a><span class="lineno"> 197</span>  Scalar m_normOfT, m_normOfS;</div>
<div class="line"><a name="l00198"></a><span class="lineno"> 198</span>  Index m_global_iter;</div>
<div class="line"><a name="l00199"></a><span class="lineno"> 199</span> </div>
<div class="line"><a name="l00200"></a><span class="lineno"> 200</span>  <span class="keyword">typedef</span> <a class="code" href="classEigen_1_1Matrix.html">Matrix<Scalar,3,1></a> Vector3s;</div>
<div class="line"><a name="l00201"></a><span class="lineno"> 201</span>  <span class="keyword">typedef</span> <a class="code" href="classEigen_1_1Matrix.html">Matrix<Scalar,2,1></a> Vector2s;</div>
<div class="line"><a name="l00202"></a><span class="lineno"> 202</span>  <span class="keyword">typedef</span> <a class="code" href="classEigen_1_1Matrix.html">Matrix<Scalar,2,2></a> Matrix2s;</div>
<div class="line"><a name="l00203"></a><span class="lineno"> 203</span>  <span class="keyword">typedef</span> <a class="code" href="classEigen_1_1JacobiRotation.html">JacobiRotation<Scalar></a> JRs;</div>
<div class="line"><a name="l00204"></a><span class="lineno"> 204</span> </div>
<div class="line"><a name="l00205"></a><span class="lineno"> 205</span>  <span class="keywordtype">void</span> hessenbergTriangular();</div>
<div class="line"><a name="l00206"></a><span class="lineno"> 206</span>  <span class="keywordtype">void</span> computeNorms();</div>
<div class="line"><a name="l00207"></a><span class="lineno"> 207</span>  Index findSmallSubdiagEntry(Index iu);</div>
<div class="line"><a name="l00208"></a><span class="lineno"> 208</span>  Index findSmallDiagEntry(Index f, Index l);</div>
<div class="line"><a name="l00209"></a><span class="lineno"> 209</span>  <span class="keywordtype">void</span> splitOffTwoRows(Index i);</div>
<div class="line"><a name="l00210"></a><span class="lineno"> 210</span>  <span class="keywordtype">void</span> pushDownZero(Index z, Index f, Index l);</div>
<div class="line"><a name="l00211"></a><span class="lineno"> 211</span>  <span class="keywordtype">void</span> step(Index f, Index l, Index iter);</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="comment">// RealQZ</span></div>
<div class="line"><a name="l00214"></a><span class="lineno"> 214</span> </div>
<div class="line"><a name="l00216"></a><span class="lineno"> 216</span>  <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00217"></a><span class="lineno"> 217</span>  <span class="keywordtype">void</span> RealQZ<MatrixType>::hessenbergTriangular()</div>
<div class="line"><a name="l00218"></a><span class="lineno"> 218</span>  {</div>
<div class="line"><a name="l00219"></a><span class="lineno"> 219</span> </div>
<div class="line"><a name="l00220"></a><span class="lineno"> 220</span>  <span class="keyword">const</span> Index dim = m_S.cols();</div>
<div class="line"><a name="l00221"></a><span class="lineno"> 221</span> </div>
<div class="line"><a name="l00222"></a><span class="lineno"> 222</span>  <span class="comment">// perform QR decomposition of T, overwrite T with R, save Q</span></div>
<div class="line"><a name="l00223"></a><span class="lineno"> 223</span>  HouseholderQR<MatrixType> qrT(m_T);</div>
<div class="line"><a name="l00224"></a><span class="lineno"> 224</span>  m_T = qrT.matrixQR();</div>
<div class="line"><a name="l00225"></a><span class="lineno"> 225</span>  m_T.template triangularView<StrictlyLower>().setZero();</div>
<div class="line"><a name="l00226"></a><span class="lineno"> 226</span>  m_Q = qrT.householderQ();</div>
<div class="line"><a name="l00227"></a><span class="lineno"> 227</span>  <span class="comment">// overwrite S with Q* S</span></div>
<div class="line"><a name="l00228"></a><span class="lineno"> 228</span>  m_S.applyOnTheLeft(m_Q.adjoint());</div>
<div class="line"><a name="l00229"></a><span class="lineno"> 229</span>  <span class="comment">// init Z as Identity</span></div>
<div class="line"><a name="l00230"></a><span class="lineno"> 230</span>  <span class="keywordflow">if</span> (m_computeQZ)</div>
<div class="line"><a name="l00231"></a><span class="lineno"> 231</span>  m_Z = MatrixType::Identity(dim,dim);</div>
<div class="line"><a name="l00232"></a><span class="lineno"> 232</span>  <span class="comment">// reduce S to upper Hessenberg with Givens rotations</span></div>
<div class="line"><a name="l00233"></a><span class="lineno"> 233</span>  <span class="keywordflow">for</span> (Index j=0; j<=dim-3; j++) {</div>
<div class="line"><a name="l00234"></a><span class="lineno"> 234</span>  <span class="keywordflow">for</span> (Index i=dim-1; i>=j+2; i--) {</div>
<div class="line"><a name="l00235"></a><span class="lineno"> 235</span>  JRs G;</div>
<div class="line"><a name="l00236"></a><span class="lineno"> 236</span>  <span class="comment">// kill S(i,j)</span></div>
<div class="line"><a name="l00237"></a><span class="lineno"> 237</span>  <span class="keywordflow">if</span>(m_S.coeff(i,j) != 0)</div>
<div class="line"><a name="l00238"></a><span class="lineno"> 238</span>  {</div>
<div class="line"><a name="l00239"></a><span class="lineno"> 239</span>  G.makeGivens(m_S.coeff(i-1,j), m_S.coeff(i,j), &m_S.coeffRef(i-1, j));</div>
<div class="line"><a name="l00240"></a><span class="lineno"> 240</span>  m_S.coeffRef(i,j) = Scalar(0.0);</div>
<div class="line"><a name="l00241"></a><span class="lineno"> 241</span>  m_S.rightCols(dim-j-1).applyOnTheLeft(i-1,i,G.adjoint());</div>
<div class="line"><a name="l00242"></a><span class="lineno"> 242</span>  m_T.rightCols(dim-i+1).applyOnTheLeft(i-1,i,G.adjoint());</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>  <span class="comment">// update Q</span></div>
<div class="line"><a name="l00245"></a><span class="lineno"> 245</span>  <span class="keywordflow">if</span> (m_computeQZ)</div>
<div class="line"><a name="l00246"></a><span class="lineno"> 246</span>  m_Q.applyOnTheRight(i-1,i,G);</div>
<div class="line"><a name="l00247"></a><span class="lineno"> 247</span>  <span class="comment">// kill T(i,i-1)</span></div>
<div class="line"><a name="l00248"></a><span class="lineno"> 248</span>  <span class="keywordflow">if</span>(m_T.coeff(i,i-1)!=Scalar(0))</div>
<div class="line"><a name="l00249"></a><span class="lineno"> 249</span>  {</div>
<div class="line"><a name="l00250"></a><span class="lineno"> 250</span>  G.makeGivens(m_T.coeff(i,i), m_T.coeff(i,i-1), &m_T.coeffRef(i,i));</div>
<div class="line"><a name="l00251"></a><span class="lineno"> 251</span>  m_T.coeffRef(i,i-1) = Scalar(0.0);</div>
<div class="line"><a name="l00252"></a><span class="lineno"> 252</span>  m_S.applyOnTheRight(i,i-1,G);</div>
<div class="line"><a name="l00253"></a><span class="lineno"> 253</span>  m_T.topRows(i).applyOnTheRight(i,i-1,G);</div>
<div class="line"><a name="l00254"></a><span class="lineno"> 254</span>  }</div>
<div class="line"><a name="l00255"></a><span class="lineno"> 255</span>  <span class="comment">// update Z</span></div>
<div class="line"><a name="l00256"></a><span class="lineno"> 256</span>  <span class="keywordflow">if</span> (m_computeQZ)</div>
<div class="line"><a name="l00257"></a><span class="lineno"> 257</span>  m_Z.applyOnTheLeft(i,i-1,G.adjoint());</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>  }</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> </div>
<div class="line"><a name="l00263"></a><span class="lineno"> 263</span>  <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00264"></a><span class="lineno"> 264</span>  <span class="keyword">inline</span> <span class="keywordtype">void</span> RealQZ<MatrixType>::computeNorms()</div>
<div class="line"><a name="l00265"></a><span class="lineno"> 265</span>  {</div>
<div class="line"><a name="l00266"></a><span class="lineno"> 266</span>  <span class="keyword">const</span> Index size = m_S.cols();</div>
<div class="line"><a name="l00267"></a><span class="lineno"> 267</span>  m_normOfS = Scalar(0.0);</div>
<div class="line"><a name="l00268"></a><span class="lineno"> 268</span>  m_normOfT = Scalar(0.0);</div>
<div class="line"><a name="l00269"></a><span class="lineno"> 269</span>  <span class="keywordflow">for</span> (Index j = 0; j < size; ++j)</div>
<div class="line"><a name="l00270"></a><span class="lineno"> 270</span>  {</div>
<div class="line"><a name="l00271"></a><span class="lineno"> 271</span>  m_normOfS += m_S.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum();</div>
<div class="line"><a name="l00272"></a><span class="lineno"> 272</span>  m_normOfT += m_T.row(j).segment(j, size - j).cwiseAbs().sum();</div>
<div class="line"><a name="l00273"></a><span class="lineno"> 273</span>  }</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> </div>
<div class="line"><a name="l00276"></a><span class="lineno"> 276</span> </div>
<div class="line"><a name="l00278"></a><span class="lineno"> 278</span>  <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00279"></a><span class="lineno"> 279</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> MatrixType::Index RealQZ<MatrixType>::findSmallSubdiagEntry(Index iu)</div>
<div class="line"><a name="l00280"></a><span class="lineno"> 280</span>  {</div>
<div class="line"><a name="l00281"></a><span class="lineno"> 281</span>  <span class="keyword">using</span> std::abs;</div>
<div class="line"><a name="l00282"></a><span class="lineno"> 282</span>  Index res = iu;</div>
<div class="line"><a name="l00283"></a><span class="lineno"> 283</span>  <span class="keywordflow">while</span> (res > 0)</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>  Scalar s = abs(m_S.coeff(res-1,res-1)) + abs(m_S.coeff(res,res));</div>
<div class="line"><a name="l00286"></a><span class="lineno"> 286</span>  <span class="keywordflow">if</span> (s == Scalar(0.0))</div>
<div class="line"><a name="l00287"></a><span class="lineno"> 287</span>  s = m_normOfS;</div>
<div class="line"><a name="l00288"></a><span class="lineno"> 288</span>  <span class="keywordflow">if</span> (abs(m_S.coeff(res,res-1)) < NumTraits<Scalar>::epsilon() * s)</div>
<div class="line"><a name="l00289"></a><span class="lineno"> 289</span>  <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00290"></a><span class="lineno"> 290</span>  res--;</div>
<div class="line"><a name="l00291"></a><span class="lineno"> 291</span>  }</div>
<div class="line"><a name="l00292"></a><span class="lineno"> 292</span>  <span class="keywordflow">return</span> res;</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> </div>
<div class="line"><a name="l00296"></a><span class="lineno"> 296</span>  <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00297"></a><span class="lineno"> 297</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> MatrixType::Index RealQZ<MatrixType>::findSmallDiagEntry(Index f, Index l)</div>
<div class="line"><a name="l00298"></a><span class="lineno"> 298</span>  {</div>
<div class="line"><a name="l00299"></a><span class="lineno"> 299</span>  <span class="keyword">using</span> std::abs;</div>
<div class="line"><a name="l00300"></a><span class="lineno"> 300</span>  Index res = l;</div>
<div class="line"><a name="l00301"></a><span class="lineno"> 301</span>  <span class="keywordflow">while</span> (res >= f) {</div>
<div class="line"><a name="l00302"></a><span class="lineno"> 302</span>  <span class="keywordflow">if</span> (abs(m_T.coeff(res,res)) <= NumTraits<Scalar>::epsilon() * m_normOfT)</div>
<div class="line"><a name="l00303"></a><span class="lineno"> 303</span>  <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00304"></a><span class="lineno"> 304</span>  res--;</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">return</span> res;</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> </div>
<div class="line"><a name="l00310"></a><span class="lineno"> 310</span>  <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00311"></a><span class="lineno"> 311</span>  <span class="keyword">inline</span> <span class="keywordtype">void</span> RealQZ<MatrixType>::splitOffTwoRows(Index i)</div>
<div class="line"><a name="l00312"></a><span class="lineno"> 312</span>  {</div>
<div class="line"><a name="l00313"></a><span class="lineno"> 313</span>  <span class="keyword">using</span> std::abs;</div>
<div class="line"><a name="l00314"></a><span class="lineno"> 314</span>  <span class="keyword">using</span> std::sqrt;</div>
<div class="line"><a name="l00315"></a><span class="lineno"> 315</span>  <span class="keyword">const</span> Index dim=m_S.cols();</div>
<div class="line"><a name="l00316"></a><span class="lineno"> 316</span>  <span class="keywordflow">if</span> (abs(m_S.coeff(i+1,i)==Scalar(0)))</div>
<div class="line"><a name="l00317"></a><span class="lineno"> 317</span>  <span class="keywordflow">return</span>;</div>
<div class="line"><a name="l00318"></a><span class="lineno"> 318</span>  Index z = findSmallDiagEntry(i,i+1);</div>
<div class="line"><a name="l00319"></a><span class="lineno"> 319</span>  <span class="keywordflow">if</span> (z==i-1)</div>
<div class="line"><a name="l00320"></a><span class="lineno"> 320</span>  {</div>
<div class="line"><a name="l00321"></a><span class="lineno"> 321</span>  <span class="comment">// block of (S T^{-1})</span></div>
<div class="line"><a name="l00322"></a><span class="lineno"> 322</span>  Matrix2s STi = m_T.template block<2,2>(i,i).<span class="keyword">template</span> triangularView<Upper>().</div>
<div class="line"><a name="l00323"></a><span class="lineno"> 323</span>  <span class="keyword">template</span> solve<OnTheRight>(m_S.template block<2,2>(i,i));</div>
<div class="line"><a name="l00324"></a><span class="lineno"> 324</span>  Scalar p = Scalar(0.5)*(STi(0,0)-STi(1,1));</div>
<div class="line"><a name="l00325"></a><span class="lineno"> 325</span>  Scalar q = p*p + STi(1,0)*STi(0,1);</div>
<div class="line"><a name="l00326"></a><span class="lineno"> 326</span>  <span class="keywordflow">if</span> (q>=0) {</div>
<div class="line"><a name="l00327"></a><span class="lineno"> 327</span>  Scalar z = sqrt(q);</div>
<div class="line"><a name="l00328"></a><span class="lineno"> 328</span>  <span class="comment">// one QR-like iteration for ABi - lambda I</span></div>
<div class="line"><a name="l00329"></a><span class="lineno"> 329</span>  <span class="comment">// is enough - when we know exact eigenvalue in advance,</span></div>
<div class="line"><a name="l00330"></a><span class="lineno"> 330</span>  <span class="comment">// convergence is immediate</span></div>
<div class="line"><a name="l00331"></a><span class="lineno"> 331</span>  JRs G;</div>
<div class="line"><a name="l00332"></a><span class="lineno"> 332</span>  <span class="keywordflow">if</span> (p>=0)</div>
<div class="line"><a name="l00333"></a><span class="lineno"> 333</span>  G.makeGivens(p + z, STi(1,0));</div>
<div class="line"><a name="l00334"></a><span class="lineno"> 334</span>  <span class="keywordflow">else</span></div>
<div class="line"><a name="l00335"></a><span class="lineno"> 335</span>  G.makeGivens(p - z, STi(1,0));</div>
<div class="line"><a name="l00336"></a><span class="lineno"> 336</span>  m_S.rightCols(dim-i).applyOnTheLeft(i,i+1,G.adjoint());</div>
<div class="line"><a name="l00337"></a><span class="lineno"> 337</span>  m_T.rightCols(dim-i).applyOnTheLeft(i,i+1,G.adjoint());</div>
<div class="line"><a name="l00338"></a><span class="lineno"> 338</span>  <span class="comment">// update Q</span></div>
<div class="line"><a name="l00339"></a><span class="lineno"> 339</span>  <span class="keywordflow">if</span> (m_computeQZ)</div>
<div class="line"><a name="l00340"></a><span class="lineno"> 340</span>  m_Q.applyOnTheRight(i,i+1,G);</div>
<div class="line"><a name="l00341"></a><span class="lineno"> 341</span> </div>
<div class="line"><a name="l00342"></a><span class="lineno"> 342</span>  G.makeGivens(m_T.coeff(i+1,i+1), m_T.coeff(i+1,i));</div>
<div class="line"><a name="l00343"></a><span class="lineno"> 343</span>  m_S.topRows(i+2).applyOnTheRight(i+1,i,G);</div>
<div class="line"><a name="l00344"></a><span class="lineno"> 344</span>  m_T.topRows(i+2).applyOnTheRight(i+1,i,G);</div>
<div class="line"><a name="l00345"></a><span class="lineno"> 345</span>  <span class="comment">// update Z</span></div>
<div class="line"><a name="l00346"></a><span class="lineno"> 346</span>  <span class="keywordflow">if</span> (m_computeQZ)</div>
<div class="line"><a name="l00347"></a><span class="lineno"> 347</span>  m_Z.applyOnTheLeft(i+1,i,G.adjoint());</div>
<div class="line"><a name="l00348"></a><span class="lineno"> 348</span> </div>
<div class="line"><a name="l00349"></a><span class="lineno"> 349</span>  m_S.coeffRef(i+1,i) = Scalar(0.0);</div>
<div class="line"><a name="l00350"></a><span class="lineno"> 350</span>  m_T.coeffRef(i+1,i) = Scalar(0.0);</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>  }</div>
<div class="line"><a name="l00353"></a><span class="lineno"> 353</span>  <span class="keywordflow">else</span></div>
<div class="line"><a name="l00354"></a><span class="lineno"> 354</span>  {</div>
<div class="line"><a name="l00355"></a><span class="lineno"> 355</span>  pushDownZero(z,i,i+1);</div>
<div class="line"><a name="l00356"></a><span class="lineno"> 356</span>  }</div>
<div class="line"><a name="l00357"></a><span class="lineno"> 357</span>  }</div>
<div class="line"><a name="l00358"></a><span class="lineno"> 358</span> </div>
<div class="line"><a name="l00360"></a><span class="lineno"> 360</span>  <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00361"></a><span class="lineno"> 361</span>  <span class="keyword">inline</span> <span class="keywordtype">void</span> RealQZ<MatrixType>::pushDownZero(Index z, Index f, Index l)</div>
<div class="line"><a name="l00362"></a><span class="lineno"> 362</span>  {</div>
<div class="line"><a name="l00363"></a><span class="lineno"> 363</span>  JRs G;</div>
<div class="line"><a name="l00364"></a><span class="lineno"> 364</span>  <span class="keyword">const</span> Index dim = m_S.cols();</div>
<div class="line"><a name="l00365"></a><span class="lineno"> 365</span>  <span class="keywordflow">for</span> (Index zz=z; zz<l; zz++)</div>
<div class="line"><a name="l00366"></a><span class="lineno"> 366</span>  {</div>
<div class="line"><a name="l00367"></a><span class="lineno"> 367</span>  <span class="comment">// push 0 down</span></div>
<div class="line"><a name="l00368"></a><span class="lineno"> 368</span>  Index firstColS = zz>f ? (zz-1) : zz;</div>
<div class="line"><a name="l00369"></a><span class="lineno"> 369</span>  G.makeGivens(m_T.coeff(zz, zz+1), m_T.coeff(zz+1, zz+1));</div>
<div class="line"><a name="l00370"></a><span class="lineno"> 370</span>  m_S.rightCols(dim-firstColS).applyOnTheLeft(zz,zz+1,G.adjoint());</div>
<div class="line"><a name="l00371"></a><span class="lineno"> 371</span>  m_T.rightCols(dim-zz).applyOnTheLeft(zz,zz+1,G.adjoint());</div>
<div class="line"><a name="l00372"></a><span class="lineno"> 372</span>  m_T.coeffRef(zz+1,zz+1) = Scalar(0.0);</div>
<div class="line"><a name="l00373"></a><span class="lineno"> 373</span>  <span class="comment">// update Q</span></div>
<div class="line"><a name="l00374"></a><span class="lineno"> 374</span>  <span class="keywordflow">if</span> (m_computeQZ)</div>
<div class="line"><a name="l00375"></a><span class="lineno"> 375</span>  m_Q.applyOnTheRight(zz,zz+1,G);</div>
<div class="line"><a name="l00376"></a><span class="lineno"> 376</span>  <span class="comment">// kill S(zz+1, zz-1)</span></div>
<div class="line"><a name="l00377"></a><span class="lineno"> 377</span>  <span class="keywordflow">if</span> (zz>f)</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>  G.makeGivens(m_S.coeff(zz+1, zz), m_S.coeff(zz+1,zz-1));</div>
<div class="line"><a name="l00380"></a><span class="lineno"> 380</span>  m_S.topRows(zz+2).applyOnTheRight(zz, zz-1,G);</div>
<div class="line"><a name="l00381"></a><span class="lineno"> 381</span>  m_T.topRows(zz+1).applyOnTheRight(zz, zz-1,G);</div>
<div class="line"><a name="l00382"></a><span class="lineno"> 382</span>  m_S.coeffRef(zz+1,zz-1) = Scalar(0.0);</div>
<div class="line"><a name="l00383"></a><span class="lineno"> 383</span>  <span class="comment">// update Z</span></div>
<div class="line"><a name="l00384"></a><span class="lineno"> 384</span>  <span class="keywordflow">if</span> (m_computeQZ)</div>
<div class="line"><a name="l00385"></a><span class="lineno"> 385</span>  m_Z.applyOnTheLeft(zz,zz-1,G.adjoint());</div>
<div class="line"><a name="l00386"></a><span class="lineno"> 386</span>  }</div>
<div class="line"><a name="l00387"></a><span class="lineno"> 387</span>  }</div>
<div class="line"><a name="l00388"></a><span class="lineno"> 388</span>  <span class="comment">// finally kill S(l,l-1)</span></div>
<div class="line"><a name="l00389"></a><span class="lineno"> 389</span>  G.makeGivens(m_S.coeff(l,l), m_S.coeff(l,l-1));</div>
<div class="line"><a name="l00390"></a><span class="lineno"> 390</span>  m_S.applyOnTheRight(l,l-1,G);</div>
<div class="line"><a name="l00391"></a><span class="lineno"> 391</span>  m_T.applyOnTheRight(l,l-1,G);</div>
<div class="line"><a name="l00392"></a><span class="lineno"> 392</span>  m_S.coeffRef(l,l-1)=Scalar(0.0);</div>
<div class="line"><a name="l00393"></a><span class="lineno"> 393</span>  <span class="comment">// update Z</span></div>
<div class="line"><a name="l00394"></a><span class="lineno"> 394</span>  <span class="keywordflow">if</span> (m_computeQZ)</div>
<div class="line"><a name="l00395"></a><span class="lineno"> 395</span>  m_Z.applyOnTheLeft(l,l-1,G.adjoint());</div>
<div class="line"><a name="l00396"></a><span class="lineno"> 396</span>  }</div>
<div class="line"><a name="l00397"></a><span class="lineno"> 397</span> </div>
<div class="line"><a name="l00399"></a><span class="lineno"> 399</span>  <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00400"></a><span class="lineno"> 400</span>  <span class="keyword">inline</span> <span class="keywordtype">void</span> RealQZ<MatrixType>::step(Index f, Index l, Index iter)</div>
<div class="line"><a name="l00401"></a><span class="lineno"> 401</span>  {</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>  <span class="keyword">const</span> Index dim = m_S.cols();</div>
<div class="line"><a name="l00404"></a><span class="lineno"> 404</span> </div>
<div class="line"><a name="l00405"></a><span class="lineno"> 405</span>  <span class="comment">// x, y, z</span></div>
<div class="line"><a name="l00406"></a><span class="lineno"> 406</span>  Scalar x, y, z;</div>
<div class="line"><a name="l00407"></a><span class="lineno"> 407</span>  <span class="keywordflow">if</span> (iter==10)</div>
<div class="line"><a name="l00408"></a><span class="lineno"> 408</span>  {</div>
<div class="line"><a name="l00409"></a><span class="lineno"> 409</span>  <span class="comment">// Wilkinson ad hoc shift</span></div>
<div class="line"><a name="l00410"></a><span class="lineno"> 410</span>  <span class="keyword">const</span> Scalar</div>
<div class="line"><a name="l00411"></a><span class="lineno"> 411</span>  a11=m_S.coeff(f+0,f+0), a12=m_S.coeff(f+0,f+1),</div>
<div class="line"><a name="l00412"></a><span class="lineno"> 412</span>  a21=m_S.coeff(f+1,f+0), a22=m_S.coeff(f+1,f+1), a32=m_S.coeff(f+2,f+1),</div>
<div class="line"><a name="l00413"></a><span class="lineno"> 413</span>  b12=m_T.coeff(f+0,f+1),</div>
<div class="line"><a name="l00414"></a><span class="lineno"> 414</span>  b11i=Scalar(1.0)/m_T.coeff(f+0,f+0),</div>
<div class="line"><a name="l00415"></a><span class="lineno"> 415</span>  b22i=Scalar(1.0)/m_T.coeff(f+1,f+1),</div>
<div class="line"><a name="l00416"></a><span class="lineno"> 416</span>  a87=m_S.coeff(l-1,l-2),</div>
<div class="line"><a name="l00417"></a><span class="lineno"> 417</span>  a98=m_S.coeff(l-0,l-1),</div>
<div class="line"><a name="l00418"></a><span class="lineno"> 418</span>  b77i=Scalar(1.0)/m_T.coeff(l-2,l-2),</div>
<div class="line"><a name="l00419"></a><span class="lineno"> 419</span>  b88i=Scalar(1.0)/m_T.coeff(l-1,l-1);</div>
<div class="line"><a name="l00420"></a><span class="lineno"> 420</span>  Scalar ss = abs(a87*b77i) + abs(a98*b88i),</div>
<div class="line"><a name="l00421"></a><span class="lineno"> 421</span>  lpl = Scalar(1.5)*ss,</div>
<div class="line"><a name="l00422"></a><span class="lineno"> 422</span>  ll = ss*ss;</div>
<div class="line"><a name="l00423"></a><span class="lineno"> 423</span>  x = ll + a11*a11*b11i*b11i - lpl*a11*b11i + a12*a21*b11i*b22i</div>
<div class="line"><a name="l00424"></a><span class="lineno"> 424</span>  - a11*a21*b12*b11i*b11i*b22i;</div>
<div class="line"><a name="l00425"></a><span class="lineno"> 425</span>  y = a11*a21*b11i*b11i - lpl*a21*b11i + a21*a22*b11i*b22i </div>
<div class="line"><a name="l00426"></a><span class="lineno"> 426</span>  - a21*a21*b12*b11i*b11i*b22i;</div>
<div class="line"><a name="l00427"></a><span class="lineno"> 427</span>  z = a21*a32*b11i*b22i;</div>
<div class="line"><a name="l00428"></a><span class="lineno"> 428</span>  }</div>
<div class="line"><a name="l00429"></a><span class="lineno"> 429</span>  <span class="keywordflow">else</span> <span class="keywordflow">if</span> (iter==16)</div>
<div class="line"><a name="l00430"></a><span class="lineno"> 430</span>  {</div>
<div class="line"><a name="l00431"></a><span class="lineno"> 431</span>  <span class="comment">// another exceptional shift</span></div>
<div class="line"><a name="l00432"></a><span class="lineno"> 432</span>  x = m_S.coeff(f,f)/m_T.coeff(f,f)-m_S.coeff(l,l)/m_T.coeff(l,l) + m_S.coeff(l,l-1)*m_T.coeff(l-1,l) /</div>
<div class="line"><a name="l00433"></a><span class="lineno"> 433</span>  (m_T.coeff(l-1,l-1)*m_T.coeff(l,l));</div>
<div class="line"><a name="l00434"></a><span class="lineno"> 434</span>  y = m_S.coeff(f+1,f)/m_T.coeff(f,f);</div>
<div class="line"><a name="l00435"></a><span class="lineno"> 435</span>  z = 0;</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>  <span class="keywordflow">else</span> <span class="keywordflow">if</span> (iter>23 && !(iter%8))</div>
<div class="line"><a name="l00438"></a><span class="lineno"> 438</span>  {</div>
<div class="line"><a name="l00439"></a><span class="lineno"> 439</span>  <span class="comment">// extremely exceptional shift</span></div>
<div class="line"><a name="l00440"></a><span class="lineno"> 440</span>  x = internal::random<Scalar>(-1.0,1.0);</div>
<div class="line"><a name="l00441"></a><span class="lineno"> 441</span>  y = internal::random<Scalar>(-1.0,1.0);</div>
<div class="line"><a name="l00442"></a><span class="lineno"> 442</span>  z = internal::random<Scalar>(-1.0,1.0);</div>
<div class="line"><a name="l00443"></a><span class="lineno"> 443</span>  }</div>
<div class="line"><a name="l00444"></a><span class="lineno"> 444</span>  <span class="keywordflow">else</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="comment">// Compute the shifts: (x,y,z,0...) = (AB^-1 - l1 I) (AB^-1 - l2 I) e1</span></div>
<div class="line"><a name="l00447"></a><span class="lineno"> 447</span>  <span class="comment">// where l1 and l2 are the eigenvalues of the 2x2 matrix C = U V^-1 where</span></div>
<div class="line"><a name="l00448"></a><span class="lineno"> 448</span>  <span class="comment">// U and V are 2x2 bottom right sub matrices of A and B. Thus:</span></div>
<div class="line"><a name="l00449"></a><span class="lineno"> 449</span>  <span class="comment">// = AB^-1AB^-1 + l1 l2 I - (l1+l2)(AB^-1)</span></div>
<div class="line"><a name="l00450"></a><span class="lineno"> 450</span>  <span class="comment">// = AB^-1AB^-1 + det(M) - tr(M)(AB^-1)</span></div>
<div class="line"><a name="l00451"></a><span class="lineno"> 451</span>  <span class="comment">// Since we are only interested in having x, y, z with a correct ratio, we have:</span></div>
<div class="line"><a name="l00452"></a><span class="lineno"> 452</span>  <span class="keyword">const</span> Scalar</div>
<div class="line"><a name="l00453"></a><span class="lineno"> 453</span>  a11 = m_S.coeff(f,f), a12 = m_S.coeff(f,f+1),</div>
<div class="line"><a name="l00454"></a><span class="lineno"> 454</span>  a21 = m_S.coeff(f+1,f), a22 = m_S.coeff(f+1,f+1),</div>
<div class="line"><a name="l00455"></a><span class="lineno"> 455</span>  a32 = m_S.coeff(f+2,f+1),</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>  a88 = m_S.coeff(l-1,l-1), a89 = m_S.coeff(l-1,l),</div>
<div class="line"><a name="l00458"></a><span class="lineno"> 458</span>  a98 = m_S.coeff(l,l-1), a99 = m_S.coeff(l,l),</div>
<div class="line"><a name="l00459"></a><span class="lineno"> 459</span> </div>
<div class="line"><a name="l00460"></a><span class="lineno"> 460</span>  b11 = m_T.coeff(f,f), b12 = m_T.coeff(f,f+1),</div>
<div class="line"><a name="l00461"></a><span class="lineno"> 461</span>  b22 = m_T.coeff(f+1,f+1),</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>  b88 = m_T.coeff(l-1,l-1), b89 = m_T.coeff(l-1,l),</div>
<div class="line"><a name="l00464"></a><span class="lineno"> 464</span>  b99 = m_T.coeff(l,l);</div>
<div class="line"><a name="l00465"></a><span class="lineno"> 465</span> </div>
<div class="line"><a name="l00466"></a><span class="lineno"> 466</span>  x = ( (a88/b88 - a11/b11)*(a99/b99 - a11/b11) - (a89/b99)*(a98/b88) + (a98/b88)*(b89/b99)*(a11/b11) ) * (b11/a21)</div>
<div class="line"><a name="l00467"></a><span class="lineno"> 467</span>  + a12/b22 - (a11/b11)*(b12/b22);</div>
<div class="line"><a name="l00468"></a><span class="lineno"> 468</span>  y = (a22/b22-a11/b11) - (a21/b11)*(b12/b22) - (a88/b88-a11/b11) - (a99/b99-a11/b11) + (a98/b88)*(b89/b99);</div>
<div class="line"><a name="l00469"></a><span class="lineno"> 469</span>  z = a32/b22;</div>
<div class="line"><a name="l00470"></a><span class="lineno"> 470</span>  }</div>
<div class="line"><a name="l00471"></a><span class="lineno"> 471</span> </div>
<div class="line"><a name="l00472"></a><span class="lineno"> 472</span>  JRs G;</div>
<div class="line"><a name="l00473"></a><span class="lineno"> 473</span> </div>
<div class="line"><a name="l00474"></a><span class="lineno"> 474</span>  <span class="keywordflow">for</span> (Index k=f; k<=l-2; k++)</div>
<div class="line"><a name="l00475"></a><span class="lineno"> 475</span>  {</div>
<div class="line"><a name="l00476"></a><span class="lineno"> 476</span>  <span class="comment">// variables for Householder reflections</span></div>
<div class="line"><a name="l00477"></a><span class="lineno"> 477</span>  Vector2s essential2;</div>
<div class="line"><a name="l00478"></a><span class="lineno"> 478</span>  Scalar tau, beta;</div>
<div class="line"><a name="l00479"></a><span class="lineno"> 479</span> </div>
<div class="line"><a name="l00480"></a><span class="lineno"> 480</span>  Vector3s hr(x,y,z);</div>
<div class="line"><a name="l00481"></a><span class="lineno"> 481</span> </div>
<div class="line"><a name="l00482"></a><span class="lineno"> 482</span>  <span class="comment">// Q_k to annihilate S(k+1,k-1) and S(k+2,k-1)</span></div>
<div class="line"><a name="l00483"></a><span class="lineno"> 483</span>  hr.makeHouseholderInPlace(tau, beta);</div>
<div class="line"><a name="l00484"></a><span class="lineno"> 484</span>  essential2 = hr.template bottomRows<2>();</div>
<div class="line"><a name="l00485"></a><span class="lineno"> 485</span>  Index fc=(std::max)(k-1,Index(0)); <span class="comment">// first col to update</span></div>
<div class="line"><a name="l00486"></a><span class="lineno"> 486</span>  m_S.template middleRows<3>(k).rightCols(dim-fc).applyHouseholderOnTheLeft(essential2, tau, m_workspace.data());</div>
<div class="line"><a name="l00487"></a><span class="lineno"> 487</span>  m_T.template middleRows<3>(k).rightCols(dim-fc).applyHouseholderOnTheLeft(essential2, tau, m_workspace.data());</div>
<div class="line"><a name="l00488"></a><span class="lineno"> 488</span>  <span class="keywordflow">if</span> (m_computeQZ)</div>
<div class="line"><a name="l00489"></a><span class="lineno"> 489</span>  m_Q.template middleCols<3>(k).applyHouseholderOnTheRight(essential2, tau, m_workspace.data());</div>
<div class="line"><a name="l00490"></a><span class="lineno"> 490</span>  <span class="keywordflow">if</span> (k>f)</div>
<div class="line"><a name="l00491"></a><span class="lineno"> 491</span>  m_S.coeffRef(k+2,k-1) = m_S.coeffRef(k+1,k-1) = Scalar(0.0);</div>
<div class="line"><a name="l00492"></a><span class="lineno"> 492</span> </div>
<div class="line"><a name="l00493"></a><span class="lineno"> 493</span>  <span class="comment">// Z_{k1} to annihilate T(k+2,k+1) and T(k+2,k)</span></div>
<div class="line"><a name="l00494"></a><span class="lineno"> 494</span>  hr << m_T.coeff(k+2,k+2),m_T.coeff(k+2,k),m_T.coeff(k+2,k+1);</div>
<div class="line"><a name="l00495"></a><span class="lineno"> 495</span>  hr.makeHouseholderInPlace(tau, beta);</div>
<div class="line"><a name="l00496"></a><span class="lineno"> 496</span>  essential2 = hr.template bottomRows<2>();</div>
<div class="line"><a name="l00497"></a><span class="lineno"> 497</span>  {</div>
<div class="line"><a name="l00498"></a><span class="lineno"> 498</span>  Index lr = (std::min)(k+4,dim); <span class="comment">// last row to update</span></div>
<div class="line"><a name="l00499"></a><span class="lineno"> 499</span>  Map<Matrix<Scalar,Dynamic,1> > tmp(m_workspace.data(),lr);</div>
<div class="line"><a name="l00500"></a><span class="lineno"> 500</span>  <span class="comment">// S</span></div>
<div class="line"><a name="l00501"></a><span class="lineno"> 501</span>  tmp = m_S.template middleCols<2>(k).topRows(lr) * essential2;</div>
<div class="line"><a name="l00502"></a><span class="lineno"> 502</span>  tmp += m_S.col(k+2).head(lr);</div>
<div class="line"><a name="l00503"></a><span class="lineno"> 503</span>  m_S.col(k+2).head(lr) -= tau*tmp;</div>
<div class="line"><a name="l00504"></a><span class="lineno"> 504</span>  m_S.template middleCols<2>(k).topRows(lr) -= (tau*tmp) * essential2.adjoint();</div>
<div class="line"><a name="l00505"></a><span class="lineno"> 505</span>  <span class="comment">// T</span></div>
<div class="line"><a name="l00506"></a><span class="lineno"> 506</span>  tmp = m_T.template middleCols<2>(k).topRows(lr) * essential2;</div>
<div class="line"><a name="l00507"></a><span class="lineno"> 507</span>  tmp += m_T.col(k+2).head(lr);</div>
<div class="line"><a name="l00508"></a><span class="lineno"> 508</span>  m_T.col(k+2).head(lr) -= tau*tmp;</div>
<div class="line"><a name="l00509"></a><span class="lineno"> 509</span>  m_T.template middleCols<2>(k).topRows(lr) -= (tau*tmp) * essential2.adjoint();</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>  <span class="keywordflow">if</span> (m_computeQZ)</div>
<div class="line"><a name="l00512"></a><span class="lineno"> 512</span>  {</div>
<div class="line"><a name="l00513"></a><span class="lineno"> 513</span>  <span class="comment">// Z</span></div>
<div class="line"><a name="l00514"></a><span class="lineno"> 514</span>  Map<Matrix<Scalar,1,Dynamic> > tmp(m_workspace.data(),dim);</div>
<div class="line"><a name="l00515"></a><span class="lineno"> 515</span>  tmp = essential2.adjoint()*(m_Z.template middleRows<2>(k));</div>
<div class="line"><a name="l00516"></a><span class="lineno"> 516</span>  tmp += m_Z.row(k+2);</div>
<div class="line"><a name="l00517"></a><span class="lineno"> 517</span>  m_Z.row(k+2) -= tau*tmp;</div>
<div class="line"><a name="l00518"></a><span class="lineno"> 518</span>  m_Z.template middleRows<2>(k) -= essential2 * (tau*tmp);</div>
<div class="line"><a name="l00519"></a><span class="lineno"> 519</span>  }</div>
<div class="line"><a name="l00520"></a><span class="lineno"> 520</span>  m_T.coeffRef(k+2,k) = m_T.coeffRef(k+2,k+1) = Scalar(0.0);</div>
<div class="line"><a name="l00521"></a><span class="lineno"> 521</span> </div>
<div class="line"><a name="l00522"></a><span class="lineno"> 522</span>  <span class="comment">// Z_{k2} to annihilate T(k+1,k)</span></div>
<div class="line"><a name="l00523"></a><span class="lineno"> 523</span>  G.makeGivens(m_T.coeff(k+1,k+1), m_T.coeff(k+1,k));</div>
<div class="line"><a name="l00524"></a><span class="lineno"> 524</span>  m_S.applyOnTheRight(k+1,k,G);</div>
<div class="line"><a name="l00525"></a><span class="lineno"> 525</span>  m_T.applyOnTheRight(k+1,k,G);</div>
<div class="line"><a name="l00526"></a><span class="lineno"> 526</span>  <span class="comment">// update Z</span></div>
<div class="line"><a name="l00527"></a><span class="lineno"> 527</span>  <span class="keywordflow">if</span> (m_computeQZ)</div>
<div class="line"><a name="l00528"></a><span class="lineno"> 528</span>  m_Z.applyOnTheLeft(k+1,k,G.adjoint());</div>
<div class="line"><a name="l00529"></a><span class="lineno"> 529</span>  m_T.coeffRef(k+1,k) = Scalar(0.0);</div>
<div class="line"><a name="l00530"></a><span class="lineno"> 530</span> </div>
<div class="line"><a name="l00531"></a><span class="lineno"> 531</span>  <span class="comment">// update x,y,z</span></div>
<div class="line"><a name="l00532"></a><span class="lineno"> 532</span>  x = m_S.coeff(k+1,k);</div>
<div class="line"><a name="l00533"></a><span class="lineno"> 533</span>  y = m_S.coeff(k+2,k);</div>
<div class="line"><a name="l00534"></a><span class="lineno"> 534</span>  <span class="keywordflow">if</span> (k < l-2)</div>
<div class="line"><a name="l00535"></a><span class="lineno"> 535</span>  z = m_S.coeff(k+3,k);</div>
<div class="line"><a name="l00536"></a><span class="lineno"> 536</span>  } <span class="comment">// loop over k</span></div>
<div class="line"><a name="l00537"></a><span class="lineno"> 537</span> </div>
<div class="line"><a name="l00538"></a><span class="lineno"> 538</span>  <span class="comment">// Q_{n-1} to annihilate y = S(l,l-2)</span></div>
<div class="line"><a name="l00539"></a><span class="lineno"> 539</span>  G.makeGivens(x,y);</div>
<div class="line"><a name="l00540"></a><span class="lineno"> 540</span>  m_S.applyOnTheLeft(l-1,l,G.adjoint());</div>
<div class="line"><a name="l00541"></a><span class="lineno"> 541</span>  m_T.applyOnTheLeft(l-1,l,G.adjoint());</div>
<div class="line"><a name="l00542"></a><span class="lineno"> 542</span>  <span class="keywordflow">if</span> (m_computeQZ)</div>
<div class="line"><a name="l00543"></a><span class="lineno"> 543</span>  m_Q.applyOnTheRight(l-1,l,G);</div>
<div class="line"><a name="l00544"></a><span class="lineno"> 544</span>  m_S.coeffRef(l,l-2) = Scalar(0.0);</div>
<div class="line"><a name="l00545"></a><span class="lineno"> 545</span> </div>
<div class="line"><a name="l00546"></a><span class="lineno"> 546</span>  <span class="comment">// Z_{n-1} to annihilate T(l,l-1)</span></div>
<div class="line"><a name="l00547"></a><span class="lineno"> 547</span>  G.makeGivens(m_T.coeff(l,l),m_T.coeff(l,l-1));</div>
<div class="line"><a name="l00548"></a><span class="lineno"> 548</span>  m_S.applyOnTheRight(l,l-1,G);</div>
<div class="line"><a name="l00549"></a><span class="lineno"> 549</span>  m_T.applyOnTheRight(l,l-1,G);</div>
<div class="line"><a name="l00550"></a><span class="lineno"> 550</span>  <span class="keywordflow">if</span> (m_computeQZ)</div>
<div class="line"><a name="l00551"></a><span class="lineno"> 551</span>  m_Z.applyOnTheLeft(l,l-1,G.adjoint());</div>
<div class="line"><a name="l00552"></a><span class="lineno"> 552</span>  m_T.coeffRef(l,l-1) = Scalar(0.0);</div>
<div class="line"><a name="l00553"></a><span class="lineno"> 553</span>  }</div>
<div class="line"><a name="l00554"></a><span class="lineno"> 554</span> </div>
<div class="line"><a name="l00555"></a><span class="lineno"> 555</span> </div>
<div class="line"><a name="l00556"></a><span class="lineno"> 556</span>  <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00557"></a><span class="lineno"><a class="line" href="classEigen_1_1RealQZ.html#a75cc509be2fa23cde7371b79ebd9f770"> 557</a></span>  <a class="code" href="classEigen_1_1RealQZ.html">RealQZ<MatrixType></a>& <a class="code" href="classEigen_1_1RealQZ.html">RealQZ<MatrixType>::compute</a>(<span class="keyword">const</span> MatrixType& A_in, <span class="keyword">const</span> MatrixType& B_in, <span class="keywordtype">bool</span> computeQZ)</div>
<div class="line"><a name="l00558"></a><span class="lineno"> 558</span>  {</div>
<div class="line"><a name="l00559"></a><span class="lineno"> 559</span> </div>
<div class="line"><a name="l00560"></a><span class="lineno"> 560</span>  <span class="keyword">const</span> Index dim = A_in.cols();</div>
<div class="line"><a name="l00561"></a><span class="lineno"> 561</span> </div>
<div class="line"><a name="l00562"></a><span class="lineno"> 562</span>  eigen_assert (A_in.rows()==dim && A_in.cols()==dim </div>
<div class="line"><a name="l00563"></a><span class="lineno"> 563</span>  && B_in.rows()==dim && B_in.cols()==dim </div>
<div class="line"><a name="l00564"></a><span class="lineno"> 564</span>  && <span class="stringliteral">"Need square matrices of the same dimension"</span>);</div>
<div class="line"><a name="l00565"></a><span class="lineno"> 565</span> </div>
<div class="line"><a name="l00566"></a><span class="lineno"> 566</span>  m_isInitialized = <span class="keyword">true</span>;</div>
<div class="line"><a name="l00567"></a><span class="lineno"> 567</span>  m_computeQZ = computeQZ;</div>
<div class="line"><a name="l00568"></a><span class="lineno"> 568</span>  m_S = A_in; m_T = B_in;</div>
<div class="line"><a name="l00569"></a><span class="lineno"> 569</span>  m_workspace.resize(dim*2);</div>
<div class="line"><a name="l00570"></a><span class="lineno"> 570</span>  m_global_iter = 0;</div>
<div class="line"><a name="l00571"></a><span class="lineno"> 571</span> </div>
<div class="line"><a name="l00572"></a><span class="lineno"> 572</span>  <span class="comment">// entrance point: hessenberg triangular decomposition</span></div>
<div class="line"><a name="l00573"></a><span class="lineno"> 573</span>  hessenbergTriangular();</div>
<div class="line"><a name="l00574"></a><span class="lineno"> 574</span>  <span class="comment">// compute L1 vector norms of T, S into m_normOfS, m_normOfT</span></div>
<div class="line"><a name="l00575"></a><span class="lineno"> 575</span>  computeNorms();</div>
<div class="line"><a name="l00576"></a><span class="lineno"> 576</span> </div>
<div class="line"><a name="l00577"></a><span class="lineno"> 577</span>  Index l = dim-1, </div>
<div class="line"><a name="l00578"></a><span class="lineno"> 578</span>  f, </div>
<div class="line"><a name="l00579"></a><span class="lineno"> 579</span>  local_iter = 0;</div>
<div class="line"><a name="l00580"></a><span class="lineno"> 580</span> </div>
<div class="line"><a name="l00581"></a><span class="lineno"> 581</span>  <span class="keywordflow">while</span> (l>0 && local_iter<m_maxIters)</div>
<div class="line"><a name="l00582"></a><span class="lineno"> 582</span>  {</div>
<div class="line"><a name="l00583"></a><span class="lineno"> 583</span>  f = findSmallSubdiagEntry(l);</div>
<div class="line"><a name="l00584"></a><span class="lineno"> 584</span>  <span class="comment">// now rows and columns f..l (including) decouple from the rest of the problem</span></div>
<div class="line"><a name="l00585"></a><span class="lineno"> 585</span>  <span class="keywordflow">if</span> (f>0) m_S.coeffRef(f,f-1) = Scalar(0.0);</div>
<div class="line"><a name="l00586"></a><span class="lineno"> 586</span>  <span class="keywordflow">if</span> (f == l) <span class="comment">// One root found</span></div>
<div class="line"><a name="l00587"></a><span class="lineno"> 587</span>  {</div>
<div class="line"><a name="l00588"></a><span class="lineno"> 588</span>  l--;</div>
<div class="line"><a name="l00589"></a><span class="lineno"> 589</span>  local_iter = 0;</div>
<div class="line"><a name="l00590"></a><span class="lineno"> 590</span>  }</div>
<div class="line"><a name="l00591"></a><span class="lineno"> 591</span>  <span class="keywordflow">else</span> <span class="keywordflow">if</span> (f == l-1) <span class="comment">// Two roots found</span></div>
<div class="line"><a name="l00592"></a><span class="lineno"> 592</span>  {</div>
<div class="line"><a name="l00593"></a><span class="lineno"> 593</span>  splitOffTwoRows(f);</div>
<div class="line"><a name="l00594"></a><span class="lineno"> 594</span>  l -= 2;</div>
<div class="line"><a name="l00595"></a><span class="lineno"> 595</span>  local_iter = 0;</div>
<div class="line"><a name="l00596"></a><span class="lineno"> 596</span>  }</div>
<div class="line"><a name="l00597"></a><span class="lineno"> 597</span>  <span class="keywordflow">else</span> <span class="comment">// No convergence yet</span></div>
<div class="line"><a name="l00598"></a><span class="lineno"> 598</span>  {</div>
<div class="line"><a name="l00599"></a><span class="lineno"> 599</span>  <span class="comment">// if there's zero on diagonal of T, we can isolate an eigenvalue with Givens rotations</span></div>
<div class="line"><a name="l00600"></a><span class="lineno"> 600</span>  Index z = findSmallDiagEntry(f,l);</div>
<div class="line"><a name="l00601"></a><span class="lineno"> 601</span>  <span class="keywordflow">if</span> (z>=f)</div>
<div class="line"><a name="l00602"></a><span class="lineno"> 602</span>  {</div>
<div class="line"><a name="l00603"></a><span class="lineno"> 603</span>  <span class="comment">// zero found</span></div>
<div class="line"><a name="l00604"></a><span class="lineno"> 604</span>  pushDownZero(z,f,l);</div>
<div class="line"><a name="l00605"></a><span class="lineno"> 605</span>  }</div>
<div class="line"><a name="l00606"></a><span class="lineno"> 606</span>  <span class="keywordflow">else</span></div>
<div class="line"><a name="l00607"></a><span class="lineno"> 607</span>  {</div>
<div class="line"><a name="l00608"></a><span class="lineno"> 608</span>  <span class="comment">// We are sure now that S.block(f,f, l-f+1,l-f+1) is underuced upper-Hessenberg </span></div>
<div class="line"><a name="l00609"></a><span class="lineno"> 609</span>  <span class="comment">// and T.block(f,f, l-f+1,l-f+1) is invertible uper-triangular, which allows to</span></div>
<div class="line"><a name="l00610"></a><span class="lineno"> 610</span>  <span class="comment">// apply a QR-like iteration to rows and columns f..l.</span></div>
<div class="line"><a name="l00611"></a><span class="lineno"> 611</span>  step(f,l, local_iter);</div>
<div class="line"><a name="l00612"></a><span class="lineno"> 612</span>  local_iter++;</div>
<div class="line"><a name="l00613"></a><span class="lineno"> 613</span>  m_global_iter++;</div>
<div class="line"><a name="l00614"></a><span class="lineno"> 614</span>  }</div>
<div class="line"><a name="l00615"></a><span class="lineno"> 615</span>  }</div>
<div class="line"><a name="l00616"></a><span class="lineno"> 616</span>  }</div>
<div class="line"><a name="l00617"></a><span class="lineno"> 617</span>  <span class="comment">// check if we converged before reaching iterations limit</span></div>
<div class="line"><a name="l00618"></a><span class="lineno"> 618</span>  m_info = (local_iter<m_maxIters) ? <a class="code" href="group__enums.html#gga51bc1ac16f26ebe51eae1abb77bd037bafdfbdf3247bd36a1f17270d5cec74c9c">Success</a> : <a class="code" href="group__enums.html#gga51bc1ac16f26ebe51eae1abb77bd037ba4ff235bd185f3c5fceeec8d6540eb847">NoConvergence</a>;</div>
<div class="line"><a name="l00619"></a><span class="lineno"> 619</span>  <span class="keywordflow">return</span> *<span class="keyword">this</span>;</div>
<div class="line"><a name="l00620"></a><span class="lineno"> 620</span>  } <span class="comment">// end compute</span></div>
<div class="line"><a name="l00621"></a><span class="lineno"> 621</span> </div>
<div class="line"><a name="l00622"></a><span class="lineno"> 622</span> } <span class="comment">// end namespace Eigen</span></div>
<div class="line"><a name="l00623"></a><span class="lineno"> 623</span> </div>
<div class="line"><a name="l00624"></a><span class="lineno"> 624</span> <span class="preprocessor">#endif //EIGEN_REAL_QZ</span></div>
<div class="ttc" id="classEigen_1_1RealQZ_html_a0f5765f1177b790b663281635e15e73f"><div class="ttname"><a href="classEigen_1_1RealQZ.html#a0f5765f1177b790b663281635e15e73f">Eigen::RealQZ::matrixZ</a></div><div class="ttdeci">const MatrixType & matrixZ() const </div><div class="ttdoc">Returns matrix Z in the QZ decomposition. </div><div class="ttdef"><b>Definition:</b> RealQZ.h:129</div></div>
<div class="ttc" id="classEigen_1_1JacobiRotation_html"><div class="ttname"><a href="classEigen_1_1JacobiRotation.html">Eigen::JacobiRotation</a></div><div class="ttdoc">Rotation given by a cosine-sine pair. </div><div class="ttdef"><b>Definition:</b> ForwardDeclarations.h:228</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_1RealQZ_html_a75cc509be2fa23cde7371b79ebd9f770"><div class="ttname"><a href="classEigen_1_1RealQZ.html#a75cc509be2fa23cde7371b79ebd9f770">Eigen::RealQZ::compute</a></div><div class="ttdeci">RealQZ & compute(const MatrixType &A, const MatrixType &B, bool computeQZ=true)</div><div class="ttdoc">Computes QZ decomposition of given matrix. </div><div class="ttdef"><b>Definition:</b> RealQZ.h:557</div></div>
<div class="ttc" id="classEigen_1_1RealQZ_html_af0356bea58e012ff54177cea8a340f64"><div class="ttname"><a href="classEigen_1_1RealQZ.html#af0356bea58e012ff54177cea8a340f64">Eigen::RealQZ::matrixS</a></div><div class="ttdeci">const MatrixType & matrixS() const </div><div class="ttdoc">Returns matrix S in the QZ decomposition. </div><div class="ttdef"><b>Definition:</b> RealQZ.h:139</div></div>
<div class="ttc" id="classEigen_1_1RealQZ_html_a1b369841b0e39a1ac80a6c32b721d242"><div class="ttname"><a href="classEigen_1_1RealQZ.html#a1b369841b0e39a1ac80a6c32b721d242">Eigen::RealQZ::setMaxIterations</a></div><div class="ttdeci">RealQZ & setMaxIterations(Index maxIters)</div><div class="ttdef"><b>Definition:</b> RealQZ.h:183</div></div>
<div class="ttc" id="classEigen_1_1RealQZ_html_a0c06d5c2034ebb329c54235369643ad2"><div class="ttname"><a href="classEigen_1_1RealQZ.html#a0c06d5c2034ebb329c54235369643ad2">Eigen::RealQZ::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> RealQZ.h:166</div></div>
<div class="ttc" id="classEigen_1_1RealQZ_html_a0d31900234ef9fea5751ce8ea693d71f"><div class="ttname"><a href="classEigen_1_1RealQZ.html#a0d31900234ef9fea5751ce8ea693d71f">Eigen::RealQZ::matrixT</a></div><div class="ttdeci">const MatrixType & matrixT() const </div><div class="ttdoc">Returns matrix S in the QZ decomposition. </div><div class="ttdef"><b>Definition:</b> RealQZ.h:148</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_1RealQZ_html_a4b2119ce39103693d003d8e434f00a3a"><div class="ttname"><a href="classEigen_1_1RealQZ.html#a4b2119ce39103693d003d8e434f00a3a">Eigen::RealQZ::RealQZ</a></div><div class="ttdeci">RealQZ(Index size=RowsAtCompileTime==Dynamic?1:RowsAtCompileTime)</div><div class="ttdoc">Default constructor. </div><div class="ttdef"><b>Definition:</b> RealQZ.h:86</div></div>
<div class="ttc" id="classEigen_1_1RealQZ_html_ae231768dac8df88971dc1871f6493571"><div class="ttname"><a href="classEigen_1_1RealQZ.html#ae231768dac8df88971dc1871f6493571">Eigen::RealQZ::iterations</a></div><div class="ttdeci">Index iterations() const </div><div class="ttdoc">Returns number of performed QR-like iterations. </div><div class="ttdef"><b>Definition:</b> RealQZ.h:174</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_1RealQZ_html_a498541357c143f345f0af5d6a6b9b3c3"><div class="ttname"><a href="classEigen_1_1RealQZ.html#a498541357c143f345f0af5d6a6b9b3c3">Eigen::RealQZ::matrixQ</a></div><div class="ttdeci">const MatrixType & matrixQ() const </div><div class="ttdoc">Returns matrix Q in the QZ decomposition. </div><div class="ttdef"><b>Definition:</b> RealQZ.h:119</div></div>
<div class="ttc" id="classEigen_1_1RealQZ_html_ab81ec305afdcbf94ed288d382710e8d7"><div class="ttname"><a href="classEigen_1_1RealQZ.html#ab81ec305afdcbf94ed288d382710e8d7">Eigen::RealQZ::RealQZ</a></div><div class="ttdeci">RealQZ(const MatrixType &A, const MatrixType &B, bool computeQZ=true)</div><div class="ttdoc">Constructor; computes real QZ decomposition of given matrices. </div><div class="ttdef"><b>Definition:</b> RealQZ.h:104</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_1RealQZ_html"><div class="ttname"><a href="classEigen_1_1RealQZ.html">Eigen::RealQZ</a></div><div class="ttdoc">Performs a real QZ decomposition of a pair of square matrices. </div><div class="ttdef"><b>Definition:</b> RealQZ.h:57</div></div>
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