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<div class="title">MatrixSquareRoot.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) 2011 Jitse Niesen <jitse@maths.leeds.ac.uk></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_MATRIX_SQUARE_ROOT</span></div>
<div class="line"><a name="l00011"></a><span class="lineno"> 11</span> <span class="preprocessor"></span><span class="preprocessor">#define EIGEN_MATRIX_SQUARE_ROOT</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="l00026"></a><span class="lineno"> 26</span> <span class="keyword">template</span> <<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00027"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixSquareRootQuasiTriangular.html"> 27</a></span> <span class="keyword">class </span><a class="code" href="classEigen_1_1MatrixSquareRootQuasiTriangular.html">MatrixSquareRootQuasiTriangular</a></div>
<div class="line"><a name="l00028"></a><span class="lineno"> 28</span> {</div>
<div class="line"><a name="l00029"></a><span class="lineno"> 29</span>  <span class="keyword">public</span>:</div>
<div class="line"><a name="l00030"></a><span class="lineno"> 30</span> </div>
<div class="line"><a name="l00039"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixSquareRootQuasiTriangular.html#a5938579430b2d461b3331d5912cfcf33"> 39</a></span>  <a class="code" href="classEigen_1_1MatrixSquareRootQuasiTriangular.html#a5938579430b2d461b3331d5912cfcf33">MatrixSquareRootQuasiTriangular</a>(<span class="keyword">const</span> MatrixType& A) </div>
<div class="line"><a name="l00040"></a><span class="lineno"> 40</span>  : m_A(A) </div>
<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>  {</div>
<div class="line"><a name="l00042"></a><span class="lineno"> 42</span>  eigen_assert(A.rows() == A.cols());</div>
<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>  }</div>
<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>  </div>
<div class="line"><a name="l00053"></a><span class="lineno"> 53</span>  <span class="keyword">template</span> <<span class="keyword">typename</span> ResultType> <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1MatrixSquareRootQuasiTriangular.html#a37407499d669c7dd9af708e7dd6f9217">compute</a>(ResultType &result); </div>
<div class="line"><a name="l00054"></a><span class="lineno"> 54</span>  </div>
<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>  <span class="keyword">private</span>:</div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::Index Index;</div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::Scalar Scalar;</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="keywordtype">void</span> computeDiagonalPartOfSqrt(MatrixType& sqrtT, <span class="keyword">const</span> MatrixType& T);</div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>  <span class="keywordtype">void</span> computeOffDiagonalPartOfSqrt(MatrixType& sqrtT, <span class="keyword">const</span> MatrixType& T);</div>
<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>  <span class="keywordtype">void</span> compute2x2diagonalBlock(MatrixType& sqrtT, <span class="keyword">const</span> MatrixType& T, <span class="keyword">typename</span> MatrixType::Index i);</div>
<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>  <span class="keywordtype">void</span> compute1x1offDiagonalBlock(MatrixType& sqrtT, <span class="keyword">const</span> MatrixType& T, </div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>  <span class="keyword">typename</span> MatrixType::Index i, <span class="keyword">typename</span> MatrixType::Index j);</div>
<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>  <span class="keywordtype">void</span> compute1x2offDiagonalBlock(MatrixType& sqrtT, <span class="keyword">const</span> MatrixType& T, </div>
<div class="line"><a name="l00065"></a><span class="lineno"> 65</span>  <span class="keyword">typename</span> MatrixType::Index i, <span class="keyword">typename</span> MatrixType::Index j);</div>
<div class="line"><a name="l00066"></a><span class="lineno"> 66</span>  <span class="keywordtype">void</span> compute2x1offDiagonalBlock(MatrixType& sqrtT, <span class="keyword">const</span> MatrixType& T, </div>
<div class="line"><a name="l00067"></a><span class="lineno"> 67</span>  <span class="keyword">typename</span> MatrixType::Index i, <span class="keyword">typename</span> MatrixType::Index j);</div>
<div class="line"><a name="l00068"></a><span class="lineno"> 68</span>  <span class="keywordtype">void</span> compute2x2offDiagonalBlock(MatrixType& sqrtT, <span class="keyword">const</span> MatrixType& T, </div>
<div class="line"><a name="l00069"></a><span class="lineno"> 69</span>  <span class="keyword">typename</span> MatrixType::Index i, <span class="keyword">typename</span> MatrixType::Index j);</div>
<div class="line"><a name="l00070"></a><span class="lineno"> 70</span>  </div>
<div class="line"><a name="l00071"></a><span class="lineno"> 71</span>  <span class="keyword">template</span> <<span class="keyword">typename</span> SmallMatrixType></div>
<div class="line"><a name="l00072"></a><span class="lineno"> 72</span>  <span class="keyword">static</span> <span class="keywordtype">void</span> solveAuxiliaryEquation(SmallMatrixType& X, <span class="keyword">const</span> SmallMatrixType& A, </div>
<div class="line"><a name="l00073"></a><span class="lineno"> 73</span>  <span class="keyword">const</span> SmallMatrixType& B, <span class="keyword">const</span> SmallMatrixType& C);</div>
<div class="line"><a name="l00074"></a><span class="lineno"> 74</span>  </div>
<div class="line"><a name="l00075"></a><span class="lineno"> 75</span>  <span class="keyword">const</span> MatrixType& m_A;</div>
<div class="line"><a name="l00076"></a><span class="lineno"> 76</span> };</div>
<div class="line"><a name="l00077"></a><span class="lineno"> 77</span> </div>
<div class="line"><a name="l00078"></a><span class="lineno"> 78</span> <span class="keyword">template</span> <<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00079"></a><span class="lineno"> 79</span> <span class="keyword">template</span> <<span class="keyword">typename</span> ResultType> </div>
<div class="line"><a name="l00080"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixSquareRootQuasiTriangular.html#a37407499d669c7dd9af708e7dd6f9217"> 80</a></span> <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1MatrixSquareRootQuasiTriangular.html#a37407499d669c7dd9af708e7dd6f9217">MatrixSquareRootQuasiTriangular<MatrixType>::compute</a>(ResultType &result)</div>
<div class="line"><a name="l00081"></a><span class="lineno"> 81</span> {</div>
<div class="line"><a name="l00082"></a><span class="lineno"> 82</span>  result.resize(m_A.rows(), m_A.cols());</div>
<div class="line"><a name="l00083"></a><span class="lineno"> 83</span>  computeDiagonalPartOfSqrt(result, m_A);</div>
<div class="line"><a name="l00084"></a><span class="lineno"> 84</span>  computeOffDiagonalPartOfSqrt(result, m_A);</div>
<div class="line"><a name="l00085"></a><span class="lineno"> 85</span> }</div>
<div class="line"><a name="l00086"></a><span class="lineno"> 86</span> </div>
<div class="line"><a name="l00087"></a><span class="lineno"> 87</span> <span class="comment">// pre: T is quasi-upper-triangular and sqrtT is a zero matrix of the same size</span></div>
<div class="line"><a name="l00088"></a><span class="lineno"> 88</span> <span class="comment">// post: the diagonal blocks of sqrtT are the square roots of the diagonal blocks of T</span></div>
<div class="line"><a name="l00089"></a><span class="lineno"> 89</span> <span class="keyword">template</span> <<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00090"></a><span class="lineno"> 90</span> <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1MatrixSquareRootQuasiTriangular.html">MatrixSquareRootQuasiTriangular<MatrixType>::computeDiagonalPartOfSqrt</a>(MatrixType& sqrtT, </div>
<div class="line"><a name="l00091"></a><span class="lineno"> 91</span>  <span class="keyword">const</span> MatrixType& T)</div>
<div class="line"><a name="l00092"></a><span class="lineno"> 92</span> {</div>
<div class="line"><a name="l00093"></a><span class="lineno"> 93</span>  <span class="keyword">using</span> std::sqrt;</div>
<div class="line"><a name="l00094"></a><span class="lineno"> 94</span>  <span class="keyword">const</span> Index size = m_A.rows();</div>
<div class="line"><a name="l00095"></a><span class="lineno"> 95</span>  <span class="keywordflow">for</span> (Index i = 0; i < size; i++) {</div>
<div class="line"><a name="l00096"></a><span class="lineno"> 96</span>  <span class="keywordflow">if</span> (i == size - 1 || T.coeff(i+1, i) == 0) {</div>
<div class="line"><a name="l00097"></a><span class="lineno"> 97</span>  eigen_assert(T(i,i) >= 0);</div>
<div class="line"><a name="l00098"></a><span class="lineno"> 98</span>  sqrtT.coeffRef(i,i) = sqrt(T.coeff(i,i));</div>
<div class="line"><a name="l00099"></a><span class="lineno"> 99</span>  }</div>
<div class="line"><a name="l00100"></a><span class="lineno"> 100</span>  <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00101"></a><span class="lineno"> 101</span>  compute2x2diagonalBlock(sqrtT, T, i);</div>
<div class="line"><a name="l00102"></a><span class="lineno"> 102</span>  ++i;</div>
<div class="line"><a name="l00103"></a><span class="lineno"> 103</span>  }</div>
<div class="line"><a name="l00104"></a><span class="lineno"> 104</span>  }</div>
<div class="line"><a name="l00105"></a><span class="lineno"> 105</span> }</div>
<div class="line"><a name="l00106"></a><span class="lineno"> 106</span> </div>
<div class="line"><a name="l00107"></a><span class="lineno"> 107</span> <span class="comment">// pre: T is quasi-upper-triangular and diagonal blocks of sqrtT are square root of diagonal blocks of T.</span></div>
<div class="line"><a name="l00108"></a><span class="lineno"> 108</span> <span class="comment">// post: sqrtT is the square root of T.</span></div>
<div class="line"><a name="l00109"></a><span class="lineno"> 109</span> <span class="keyword">template</span> <<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00110"></a><span class="lineno"> 110</span> <span class="keywordtype">void</span> MatrixSquareRootQuasiTriangular<MatrixType>::computeOffDiagonalPartOfSqrt(MatrixType& sqrtT, </div>
<div class="line"><a name="l00111"></a><span class="lineno"> 111</span>  <span class="keyword">const</span> MatrixType& T)</div>
<div class="line"><a name="l00112"></a><span class="lineno"> 112</span> {</div>
<div class="line"><a name="l00113"></a><span class="lineno"> 113</span>  <span class="keyword">const</span> Index size = m_A.rows();</div>
<div class="line"><a name="l00114"></a><span class="lineno"> 114</span>  <span class="keywordflow">for</span> (Index j = 1; j < size; j++) {</div>
<div class="line"><a name="l00115"></a><span class="lineno"> 115</span>  <span class="keywordflow">if</span> (T.coeff(j, j-1) != 0) <span class="comment">// if T(j-1:j, j-1:j) is a 2-by-2 block</span></div>
<div class="line"><a name="l00116"></a><span class="lineno"> 116</span>  <span class="keywordflow">continue</span>;</div>
<div class="line"><a name="l00117"></a><span class="lineno"> 117</span>  <span class="keywordflow">for</span> (Index i = j-1; i >= 0; i--) {</div>
<div class="line"><a name="l00118"></a><span class="lineno"> 118</span>  <span class="keywordflow">if</span> (i > 0 && T.coeff(i, i-1) != 0) <span class="comment">// if T(i-1:i, i-1:i) is a 2-by-2 block</span></div>
<div class="line"><a name="l00119"></a><span class="lineno"> 119</span>  <span class="keywordflow">continue</span>;</div>
<div class="line"><a name="l00120"></a><span class="lineno"> 120</span>  <span class="keywordtype">bool</span> iBlockIs2x2 = (i < size - 1) && (T.coeff(i+1, i) != 0);</div>
<div class="line"><a name="l00121"></a><span class="lineno"> 121</span>  <span class="keywordtype">bool</span> jBlockIs2x2 = (j < size - 1) && (T.coeff(j+1, j) != 0);</div>
<div class="line"><a name="l00122"></a><span class="lineno"> 122</span>  <span class="keywordflow">if</span> (iBlockIs2x2 && jBlockIs2x2) </div>
<div class="line"><a name="l00123"></a><span class="lineno"> 123</span>  compute2x2offDiagonalBlock(sqrtT, T, i, j);</div>
<div class="line"><a name="l00124"></a><span class="lineno"> 124</span>  <span class="keywordflow">else</span> <span class="keywordflow">if</span> (iBlockIs2x2 && !jBlockIs2x2) </div>
<div class="line"><a name="l00125"></a><span class="lineno"> 125</span>  compute2x1offDiagonalBlock(sqrtT, T, i, j);</div>
<div class="line"><a name="l00126"></a><span class="lineno"> 126</span>  <span class="keywordflow">else</span> <span class="keywordflow">if</span> (!iBlockIs2x2 && jBlockIs2x2) </div>
<div class="line"><a name="l00127"></a><span class="lineno"> 127</span>  compute1x2offDiagonalBlock(sqrtT, T, i, j);</div>
<div class="line"><a name="l00128"></a><span class="lineno"> 128</span>  <span class="keywordflow">else</span> <span class="keywordflow">if</span> (!iBlockIs2x2 && !jBlockIs2x2) </div>
<div class="line"><a name="l00129"></a><span class="lineno"> 129</span>  compute1x1offDiagonalBlock(sqrtT, T, i, j);</div>
<div class="line"><a name="l00130"></a><span class="lineno"> 130</span>  }</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="l00133"></a><span class="lineno"> 133</span> </div>
<div class="line"><a name="l00134"></a><span class="lineno"> 134</span> <span class="comment">// pre: T.block(i,i,2,2) has complex conjugate eigenvalues</span></div>
<div class="line"><a name="l00135"></a><span class="lineno"> 135</span> <span class="comment">// post: sqrtT.block(i,i,2,2) is square root of T.block(i,i,2,2)</span></div>
<div class="line"><a name="l00136"></a><span class="lineno"> 136</span> <span class="keyword">template</span> <<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00137"></a><span class="lineno"> 137</span> <span class="keywordtype">void</span> MatrixSquareRootQuasiTriangular<MatrixType></div>
<div class="line"><a name="l00138"></a><span class="lineno"> 138</span>  ::compute2x2diagonalBlock(MatrixType& sqrtT, <span class="keyword">const</span> MatrixType& T, <span class="keyword">typename</span> MatrixType::Index i)</div>
<div class="line"><a name="l00139"></a><span class="lineno"> 139</span> {</div>
<div class="line"><a name="l00140"></a><span class="lineno"> 140</span>  <span class="comment">// TODO: This case (2-by-2 blocks with complex conjugate eigenvalues) is probably hidden somewhere</span></div>
<div class="line"><a name="l00141"></a><span class="lineno"> 141</span>  <span class="comment">// in EigenSolver. If we expose it, we could call it directly from here.</span></div>
<div class="line"><a name="l00142"></a><span class="lineno"> 142</span>  Matrix<Scalar,2,2> block = T.template block<2,2>(i,i);</div>
<div class="line"><a name="l00143"></a><span class="lineno"> 143</span>  EigenSolver<Matrix<Scalar,2,2> > es(block);</div>
<div class="line"><a name="l00144"></a><span class="lineno"> 144</span>  sqrtT.template block<2,2>(i,i)</div>
<div class="line"><a name="l00145"></a><span class="lineno"> 145</span>  = (es.eigenvectors() * es.eigenvalues().cwiseSqrt().asDiagonal() * es.eigenvectors().inverse()).real();</div>
<div class="line"><a name="l00146"></a><span class="lineno"> 146</span> }</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> <span class="comment">// pre: block structure of T is such that (i,j) is a 1x1 block,</span></div>
<div class="line"><a name="l00149"></a><span class="lineno"> 149</span> <span class="comment">// all blocks of sqrtT to left of and below (i,j) are correct</span></div>
<div class="line"><a name="l00150"></a><span class="lineno"> 150</span> <span class="comment">// post: sqrtT(i,j) has the correct value</span></div>
<div class="line"><a name="l00151"></a><span class="lineno"> 151</span> <span class="keyword">template</span> <<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00152"></a><span class="lineno"> 152</span> <span class="keywordtype">void</span> MatrixSquareRootQuasiTriangular<MatrixType></div>
<div class="line"><a name="l00153"></a><span class="lineno"> 153</span>  ::compute1x1offDiagonalBlock(MatrixType& sqrtT, <span class="keyword">const</span> MatrixType& T, </div>
<div class="line"><a name="l00154"></a><span class="lineno"> 154</span>  <span class="keyword">typename</span> MatrixType::Index i, <span class="keyword">typename</span> MatrixType::Index j)</div>
<div class="line"><a name="l00155"></a><span class="lineno"> 155</span> {</div>
<div class="line"><a name="l00156"></a><span class="lineno"> 156</span>  Scalar tmp = (sqrtT.row(i).segment(i+1,j-i-1) * sqrtT.col(j).segment(i+1,j-i-1)).value();</div>
<div class="line"><a name="l00157"></a><span class="lineno"> 157</span>  sqrtT.coeffRef(i,j) = (T.coeff(i,j) - tmp) / (sqrtT.coeff(i,i) + sqrtT.coeff(j,j));</div>
<div class="line"><a name="l00158"></a><span class="lineno"> 158</span> }</div>
<div class="line"><a name="l00159"></a><span class="lineno"> 159</span> </div>
<div class="line"><a name="l00160"></a><span class="lineno"> 160</span> <span class="comment">// similar to compute1x1offDiagonalBlock()</span></div>
<div class="line"><a name="l00161"></a><span class="lineno"> 161</span> <span class="keyword">template</span> <<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00162"></a><span class="lineno"> 162</span> <span class="keywordtype">void</span> MatrixSquareRootQuasiTriangular<MatrixType></div>
<div class="line"><a name="l00163"></a><span class="lineno"> 163</span>  ::compute1x2offDiagonalBlock(MatrixType& sqrtT, <span class="keyword">const</span> MatrixType& T, </div>
<div class="line"><a name="l00164"></a><span class="lineno"> 164</span>  <span class="keyword">typename</span> MatrixType::Index i, <span class="keyword">typename</span> MatrixType::Index j)</div>
<div class="line"><a name="l00165"></a><span class="lineno"> 165</span> {</div>
<div class="line"><a name="l00166"></a><span class="lineno"> 166</span>  Matrix<Scalar,1,2> rhs = T.template block<1,2>(i,j);</div>
<div class="line"><a name="l00167"></a><span class="lineno"> 167</span>  <span class="keywordflow">if</span> (j-i > 1)</div>
<div class="line"><a name="l00168"></a><span class="lineno"> 168</span>  rhs -= sqrtT.block(i, i+1, 1, j-i-1) * sqrtT.block(i+1, j, j-i-1, 2);</div>
<div class="line"><a name="l00169"></a><span class="lineno"> 169</span>  Matrix<Scalar,2,2> A = sqrtT.coeff(i,i) * Matrix<Scalar,2,2>::Identity();</div>
<div class="line"><a name="l00170"></a><span class="lineno"> 170</span>  A += sqrtT.template block<2,2>(j,j).transpose();</div>
<div class="line"><a name="l00171"></a><span class="lineno"> 171</span>  sqrtT.template block<1,2>(i,j).transpose() = A.fullPivLu().solve(rhs.transpose());</div>
<div class="line"><a name="l00172"></a><span class="lineno"> 172</span> }</div>
<div class="line"><a name="l00173"></a><span class="lineno"> 173</span> </div>
<div class="line"><a name="l00174"></a><span class="lineno"> 174</span> <span class="comment">// similar to compute1x1offDiagonalBlock()</span></div>
<div class="line"><a name="l00175"></a><span class="lineno"> 175</span> <span class="keyword">template</span> <<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00176"></a><span class="lineno"> 176</span> <span class="keywordtype">void</span> MatrixSquareRootQuasiTriangular<MatrixType></div>
<div class="line"><a name="l00177"></a><span class="lineno"> 177</span>  ::compute2x1offDiagonalBlock(MatrixType& sqrtT, <span class="keyword">const</span> MatrixType& T, </div>
<div class="line"><a name="l00178"></a><span class="lineno"> 178</span>  <span class="keyword">typename</span> MatrixType::Index i, <span class="keyword">typename</span> MatrixType::Index j)</div>
<div class="line"><a name="l00179"></a><span class="lineno"> 179</span> {</div>
<div class="line"><a name="l00180"></a><span class="lineno"> 180</span>  Matrix<Scalar,2,1> rhs = T.template block<2,1>(i,j);</div>
<div class="line"><a name="l00181"></a><span class="lineno"> 181</span>  <span class="keywordflow">if</span> (j-i > 2)</div>
<div class="line"><a name="l00182"></a><span class="lineno"> 182</span>  rhs -= sqrtT.block(i, i+2, 2, j-i-2) * sqrtT.block(i+2, j, j-i-2, 1);</div>
<div class="line"><a name="l00183"></a><span class="lineno"> 183</span>  Matrix<Scalar,2,2> A = sqrtT.coeff(j,j) * Matrix<Scalar,2,2>::Identity();</div>
<div class="line"><a name="l00184"></a><span class="lineno"> 184</span>  A += sqrtT.template block<2,2>(i,i);</div>
<div class="line"><a name="l00185"></a><span class="lineno"> 185</span>  sqrtT.template block<2,1>(i,j) = A.fullPivLu().solve(rhs);</div>
<div class="line"><a name="l00186"></a><span class="lineno"> 186</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> <span class="comment">// similar to compute1x1offDiagonalBlock()</span></div>
<div class="line"><a name="l00189"></a><span class="lineno"> 189</span> <span class="keyword">template</span> <<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00190"></a><span class="lineno"> 190</span> <span class="keywordtype">void</span> MatrixSquareRootQuasiTriangular<MatrixType></div>
<div class="line"><a name="l00191"></a><span class="lineno"> 191</span>  ::compute2x2offDiagonalBlock(MatrixType& sqrtT, <span class="keyword">const</span> MatrixType& T, </div>
<div class="line"><a name="l00192"></a><span class="lineno"> 192</span>  <span class="keyword">typename</span> MatrixType::Index i, <span class="keyword">typename</span> MatrixType::Index j)</div>
<div class="line"><a name="l00193"></a><span class="lineno"> 193</span> {</div>
<div class="line"><a name="l00194"></a><span class="lineno"> 194</span>  Matrix<Scalar,2,2> A = sqrtT.template block<2,2>(i,i);</div>
<div class="line"><a name="l00195"></a><span class="lineno"> 195</span>  Matrix<Scalar,2,2> B = sqrtT.template block<2,2>(j,j);</div>
<div class="line"><a name="l00196"></a><span class="lineno"> 196</span>  Matrix<Scalar,2,2> C = T.template block<2,2>(i,j);</div>
<div class="line"><a name="l00197"></a><span class="lineno"> 197</span>  <span class="keywordflow">if</span> (j-i > 2)</div>
<div class="line"><a name="l00198"></a><span class="lineno"> 198</span>  C -= sqrtT.block(i, i+2, 2, j-i-2) * sqrtT.block(i+2, j, j-i-2, 2);</div>
<div class="line"><a name="l00199"></a><span class="lineno"> 199</span>  Matrix<Scalar,2,2> X;</div>
<div class="line"><a name="l00200"></a><span class="lineno"> 200</span>  solveAuxiliaryEquation(X, A, B, C);</div>
<div class="line"><a name="l00201"></a><span class="lineno"> 201</span>  sqrtT.template block<2,2>(i,j) = X;</div>
<div class="line"><a name="l00202"></a><span class="lineno"> 202</span> }</div>
<div class="line"><a name="l00203"></a><span class="lineno"> 203</span> </div>
<div class="line"><a name="l00204"></a><span class="lineno"> 204</span> <span class="comment">// solves the equation A X + X B = C where all matrices are 2-by-2</span></div>
<div class="line"><a name="l00205"></a><span class="lineno"> 205</span> <span class="keyword">template</span> <<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00206"></a><span class="lineno"> 206</span> <span class="keyword">template</span> <<span class="keyword">typename</span> SmallMatrixType></div>
<div class="line"><a name="l00207"></a><span class="lineno"> 207</span> <span class="keywordtype">void</span> MatrixSquareRootQuasiTriangular<MatrixType></div>
<div class="line"><a name="l00208"></a><span class="lineno"> 208</span>  ::solveAuxiliaryEquation(SmallMatrixType& X, <span class="keyword">const</span> SmallMatrixType& A,</div>
<div class="line"><a name="l00209"></a><span class="lineno"> 209</span>  <span class="keyword">const</span> SmallMatrixType& B, <span class="keyword">const</span> SmallMatrixType& C)</div>
<div class="line"><a name="l00210"></a><span class="lineno"> 210</span> {</div>
<div class="line"><a name="l00211"></a><span class="lineno"> 211</span>  EIGEN_STATIC_ASSERT((internal::is_same<SmallMatrixType, Matrix<Scalar,2,2> >::value),</div>
<div class="line"><a name="l00212"></a><span class="lineno"> 212</span>  EIGEN_INTERNAL_ERROR_PLEASE_FILE_A_BUG_REPORT);</div>
<div class="line"><a name="l00213"></a><span class="lineno"> 213</span> </div>
<div class="line"><a name="l00214"></a><span class="lineno"> 214</span>  Matrix<Scalar,4,4> coeffMatrix = Matrix<Scalar,4,4>::Zero();</div>
<div class="line"><a name="l00215"></a><span class="lineno"> 215</span>  coeffMatrix.coeffRef(0,0) = A.coeff(0,0) + B.coeff(0,0);</div>
<div class="line"><a name="l00216"></a><span class="lineno"> 216</span>  coeffMatrix.coeffRef(1,1) = A.coeff(0,0) + B.coeff(1,1);</div>
<div class="line"><a name="l00217"></a><span class="lineno"> 217</span>  coeffMatrix.coeffRef(2,2) = A.coeff(1,1) + B.coeff(0,0);</div>
<div class="line"><a name="l00218"></a><span class="lineno"> 218</span>  coeffMatrix.coeffRef(3,3) = A.coeff(1,1) + B.coeff(1,1);</div>
<div class="line"><a name="l00219"></a><span class="lineno"> 219</span>  coeffMatrix.coeffRef(0,1) = B.coeff(1,0);</div>
<div class="line"><a name="l00220"></a><span class="lineno"> 220</span>  coeffMatrix.coeffRef(0,2) = A.coeff(0,1);</div>
<div class="line"><a name="l00221"></a><span class="lineno"> 221</span>  coeffMatrix.coeffRef(1,0) = B.coeff(0,1);</div>
<div class="line"><a name="l00222"></a><span class="lineno"> 222</span>  coeffMatrix.coeffRef(1,3) = A.coeff(0,1);</div>
<div class="line"><a name="l00223"></a><span class="lineno"> 223</span>  coeffMatrix.coeffRef(2,0) = A.coeff(1,0);</div>
<div class="line"><a name="l00224"></a><span class="lineno"> 224</span>  coeffMatrix.coeffRef(2,3) = B.coeff(1,0);</div>
<div class="line"><a name="l00225"></a><span class="lineno"> 225</span>  coeffMatrix.coeffRef(3,1) = A.coeff(1,0);</div>
<div class="line"><a name="l00226"></a><span class="lineno"> 226</span>  coeffMatrix.coeffRef(3,2) = B.coeff(0,1);</div>
<div class="line"><a name="l00227"></a><span class="lineno"> 227</span>  </div>
<div class="line"><a name="l00228"></a><span class="lineno"> 228</span>  Matrix<Scalar,4,1> rhs;</div>
<div class="line"><a name="l00229"></a><span class="lineno"> 229</span>  rhs.coeffRef(0) = C.coeff(0,0);</div>
<div class="line"><a name="l00230"></a><span class="lineno"> 230</span>  rhs.coeffRef(1) = C.coeff(0,1);</div>
<div class="line"><a name="l00231"></a><span class="lineno"> 231</span>  rhs.coeffRef(2) = C.coeff(1,0);</div>
<div class="line"><a name="l00232"></a><span class="lineno"> 232</span>  rhs.coeffRef(3) = C.coeff(1,1);</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>  Matrix<Scalar,4,1> result;</div>
<div class="line"><a name="l00235"></a><span class="lineno"> 235</span>  result = coeffMatrix.fullPivLu().solve(rhs);</div>
<div class="line"><a name="l00236"></a><span class="lineno"> 236</span> </div>
<div class="line"><a name="l00237"></a><span class="lineno"> 237</span>  X.coeffRef(0,0) = result.coeff(0);</div>
<div class="line"><a name="l00238"></a><span class="lineno"> 238</span>  X.coeffRef(0,1) = result.coeff(1);</div>
<div class="line"><a name="l00239"></a><span class="lineno"> 239</span>  X.coeffRef(1,0) = result.coeff(2);</div>
<div class="line"><a name="l00240"></a><span class="lineno"> 240</span>  X.coeffRef(1,1) = result.coeff(3);</div>
<div class="line"><a name="l00241"></a><span class="lineno"> 241</span> }</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="l00255"></a><span class="lineno"> 255</span> <span class="keyword">template</span> <<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00256"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixSquareRootTriangular.html"> 256</a></span> <span class="keyword">class </span><a class="code" href="classEigen_1_1MatrixSquareRootTriangular.html">MatrixSquareRootTriangular</a></div>
<div class="line"><a name="l00257"></a><span class="lineno"> 257</span> {</div>
<div class="line"><a name="l00258"></a><span class="lineno"> 258</span>  <span class="keyword">public</span>:</div>
<div class="line"><a name="l00259"></a><span class="lineno"> 259</span>  <a class="code" href="classEigen_1_1MatrixSquareRootTriangular.html">MatrixSquareRootTriangular</a>(<span class="keyword">const</span> MatrixType& A) </div>
<div class="line"><a name="l00260"></a><span class="lineno"> 260</span>  : m_A(A) </div>
<div class="line"><a name="l00261"></a><span class="lineno"> 261</span>  {</div>
<div class="line"><a name="l00262"></a><span class="lineno"> 262</span>  eigen_assert(A.rows() == A.cols());</div>
<div class="line"><a name="l00263"></a><span class="lineno"> 263</span>  }</div>
<div class="line"><a name="l00264"></a><span class="lineno"> 264</span> </div>
<div class="line"><a name="l00274"></a><span class="lineno"> 274</span>  <span class="keyword">template</span> <<span class="keyword">typename</span> ResultType> <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1MatrixSquareRootTriangular.html#a37407499d669c7dd9af708e7dd6f9217">compute</a>(ResultType &result); </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>  <span class="keyword">private</span>:</div>
<div class="line"><a name="l00277"></a><span class="lineno"> 277</span>  <span class="keyword">const</span> MatrixType& m_A;</div>
<div class="line"><a name="l00278"></a><span class="lineno"> 278</span> };</div>
<div class="line"><a name="l00279"></a><span class="lineno"> 279</span> </div>
<div class="line"><a name="l00280"></a><span class="lineno"> 280</span> <span class="keyword">template</span> <<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00281"></a><span class="lineno"> 281</span> <span class="keyword">template</span> <<span class="keyword">typename</span> ResultType> </div>
<div class="line"><a name="l00282"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixSquareRootTriangular.html#a37407499d669c7dd9af708e7dd6f9217"> 282</a></span> <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1MatrixSquareRootTriangular.html#a37407499d669c7dd9af708e7dd6f9217">MatrixSquareRootTriangular<MatrixType>::compute</a>(ResultType &result)</div>
<div class="line"><a name="l00283"></a><span class="lineno"> 283</span> {</div>
<div class="line"><a name="l00284"></a><span class="lineno"> 284</span>  <span class="keyword">using</span> std::sqrt;</div>
<div class="line"><a name="l00285"></a><span class="lineno"> 285</span> </div>
<div class="line"><a name="l00286"></a><span class="lineno"> 286</span>  <span class="comment">// Compute square root of m_A and store it in upper triangular part of result</span></div>
<div class="line"><a name="l00287"></a><span class="lineno"> 287</span>  <span class="comment">// This uses that the square root of triangular matrices can be computed directly.</span></div>
<div class="line"><a name="l00288"></a><span class="lineno"> 288</span>  result.resize(m_A.rows(), m_A.cols());</div>
<div class="line"><a name="l00289"></a><span class="lineno"> 289</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::Index Index;</div>
<div class="line"><a name="l00290"></a><span class="lineno"> 290</span>  <span class="keywordflow">for</span> (Index i = 0; i < m_A.rows(); i++) {</div>
<div class="line"><a name="l00291"></a><span class="lineno"> 291</span>  result.coeffRef(i,i) = sqrt(m_A.coeff(i,i));</div>
<div class="line"><a name="l00292"></a><span class="lineno"> 292</span>  }</div>
<div class="line"><a name="l00293"></a><span class="lineno"> 293</span>  <span class="keywordflow">for</span> (Index j = 1; j < m_A.cols(); j++) {</div>
<div class="line"><a name="l00294"></a><span class="lineno"> 294</span>  <span class="keywordflow">for</span> (Index i = j-1; i >= 0; i--) {</div>
<div class="line"><a name="l00295"></a><span class="lineno"> 295</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::Scalar Scalar;</div>
<div class="line"><a name="l00296"></a><span class="lineno"> 296</span>  <span class="comment">// if i = j-1, then segment has length 0 so tmp = 0</span></div>
<div class="line"><a name="l00297"></a><span class="lineno"> 297</span>  Scalar tmp = (result.row(i).segment(i+1,j-i-1) * result.col(j).segment(i+1,j-i-1)).value();</div>
<div class="line"><a name="l00298"></a><span class="lineno"> 298</span>  <span class="comment">// denominator may be zero if original matrix is singular</span></div>
<div class="line"><a name="l00299"></a><span class="lineno"> 299</span>  result.coeffRef(i,j) = (m_A.coeff(i,j) - tmp) / (result.coeff(i,i) + result.coeff(j,j));</div>
<div class="line"><a name="l00300"></a><span class="lineno"> 300</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> }</div>
<div class="line"><a name="l00303"></a><span class="lineno"> 303</span> </div>
<div class="line"><a name="l00304"></a><span class="lineno"> 304</span> </div>
<div class="line"><a name="l00312"></a><span class="lineno"> 312</span> template <typename MatrixType, int IsComplex = NumTraits<typename internal::traits<MatrixType>::Scalar>::IsComplex></div>
<div class="line"><a name="l00313"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixSquareRoot.html"> 313</a></span> <span class="keyword">class </span><a class="code" href="classEigen_1_1MatrixSquareRoot.html">MatrixSquareRoot</a></div>
<div class="line"><a name="l00314"></a><span class="lineno"> 314</span> {</div>
<div class="line"><a name="l00315"></a><span class="lineno"> 315</span>  <span class="keyword">public</span>:</div>
<div class="line"><a name="l00316"></a><span class="lineno"> 316</span> </div>
<div class="line"><a name="l00324"></a><span class="lineno"> 324</span>  <a class="code" href="classEigen_1_1MatrixSquareRoot.html#a5233cb8199f82024af00482971ea8e58">MatrixSquareRoot</a>(<span class="keyword">const</span> MatrixType& A); </div>
<div class="line"><a name="l00325"></a><span class="lineno"> 325</span>  </div>
<div class="line"><a name="l00333"></a><span class="lineno"> 333</span>  <span class="keyword">template</span> <<span class="keyword">typename</span> ResultType> <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1MatrixSquareRoot.html#a37407499d669c7dd9af708e7dd6f9217">compute</a>(ResultType &result); </div>
<div class="line"><a name="l00334"></a><span class="lineno"> 334</span> };</div>
<div class="line"><a name="l00335"></a><span class="lineno"> 335</span> </div>
<div class="line"><a name="l00336"></a><span class="lineno"> 336</span> </div>
<div class="line"><a name="l00337"></a><span class="lineno"> 337</span> <span class="comment">// ********** Partial specialization for real matrices **********</span></div>
<div class="line"><a name="l00338"></a><span class="lineno"> 338</span> </div>
<div class="line"><a name="l00339"></a><span class="lineno"> 339</span> <span class="keyword">template</span> <<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00340"></a><span class="lineno"> 340</span> <span class="keyword">class </span><a class="code" href="classEigen_1_1MatrixSquareRoot.html">MatrixSquareRoot</a><MatrixType, 0></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>  <span class="keyword">public</span>:</div>
<div class="line"><a name="l00343"></a><span class="lineno"> 343</span> </div>
<div class="line"><a name="l00344"></a><span class="lineno"> 344</span>  <a class="code" href="classEigen_1_1MatrixSquareRoot.html#a5233cb8199f82024af00482971ea8e58">MatrixSquareRoot</a>(<span class="keyword">const</span> MatrixType& A) </div>
<div class="line"><a name="l00345"></a><span class="lineno"> 345</span>  : m_A(A) </div>
<div class="line"><a name="l00346"></a><span class="lineno"> 346</span>  { </div>
<div class="line"><a name="l00347"></a><span class="lineno"> 347</span>  eigen_assert(A.rows() == A.cols());</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>  </div>
<div class="line"><a name="l00350"></a><span class="lineno"> 350</span>  <span class="keyword">template</span> <<span class="keyword">typename</span> ResultType> <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1MatrixSquareRoot.html#a37407499d669c7dd9af708e7dd6f9217">compute</a>(ResultType &result)</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>  <span class="comment">// Compute Schur decomposition of m_A</span></div>
<div class="line"><a name="l00353"></a><span class="lineno"> 353</span>  <span class="keyword">const</span> RealSchur<MatrixType> schurOfA(m_A); </div>
<div class="line"><a name="l00354"></a><span class="lineno"> 354</span>  <span class="keyword">const</span> MatrixType& T = schurOfA.matrixT();</div>
<div class="line"><a name="l00355"></a><span class="lineno"> 355</span>  <span class="keyword">const</span> MatrixType& U = schurOfA.matrixU();</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>  <span class="comment">// Compute square root of T</span></div>
<div class="line"><a name="l00358"></a><span class="lineno"> 358</span>  MatrixType sqrtT = MatrixType::Zero(m_A.rows(), m_A.cols());</div>
<div class="line"><a name="l00359"></a><span class="lineno"> 359</span>  MatrixSquareRootQuasiTriangular<MatrixType>(T).<a class="code" href="classEigen_1_1MatrixSquareRoot.html#a37407499d669c7dd9af708e7dd6f9217">compute</a>(sqrtT);</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>  <span class="comment">// Compute square root of m_A</span></div>
<div class="line"><a name="l00362"></a><span class="lineno"> 362</span>  result = U * sqrtT * U.adjoint();</div>
<div class="line"><a name="l00363"></a><span class="lineno"> 363</span>  }</div>
<div class="line"><a name="l00364"></a><span class="lineno"> 364</span>  </div>
<div class="line"><a name="l00365"></a><span class="lineno"> 365</span>  <span class="keyword">private</span>:</div>
<div class="line"><a name="l00366"></a><span class="lineno"> 366</span>  <span class="keyword">const</span> MatrixType& m_A;</div>
<div class="line"><a name="l00367"></a><span class="lineno"> 367</span> };</div>
<div class="line"><a name="l00368"></a><span class="lineno"> 368</span> </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">// ********** Partial specialization for complex matrices **********</span></div>
<div class="line"><a name="l00371"></a><span class="lineno"> 371</span> </div>
<div class="line"><a name="l00372"></a><span class="lineno"> 372</span> <span class="keyword">template</span> <<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00373"></a><span class="lineno"> 373</span> <span class="keyword">class </span>MatrixSquareRoot<MatrixType, 1></div>
<div class="line"><a name="l00374"></a><span class="lineno"> 374</span> {</div>
<div class="line"><a name="l00375"></a><span class="lineno"> 375</span>  <span class="keyword">public</span>:</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>  <a class="code" href="classEigen_1_1MatrixSquareRoot.html#a5233cb8199f82024af00482971ea8e58">MatrixSquareRoot</a>(<span class="keyword">const</span> MatrixType& A) </div>
<div class="line"><a name="l00378"></a><span class="lineno"> 378</span>  : m_A(A) </div>
<div class="line"><a name="l00379"></a><span class="lineno"> 379</span>  { </div>
<div class="line"><a name="l00380"></a><span class="lineno"> 380</span>  eigen_assert(A.rows() == A.cols());</div>
<div class="line"><a name="l00381"></a><span class="lineno"> 381</span>  }</div>
<div class="line"><a name="l00382"></a><span class="lineno"> 382</span>  </div>
<div class="line"><a name="l00383"></a><span class="lineno"> 383</span>  <span class="keyword">template</span> <<span class="keyword">typename</span> ResultType> <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1MatrixSquareRoot.html#a37407499d669c7dd9af708e7dd6f9217">compute</a>(ResultType &result)</div>
<div class="line"><a name="l00384"></a><span class="lineno"> 384</span>  {</div>
<div class="line"><a name="l00385"></a><span class="lineno"> 385</span>  <span class="comment">// Compute Schur decomposition of m_A</span></div>
<div class="line"><a name="l00386"></a><span class="lineno"> 386</span>  <span class="keyword">const</span> ComplexSchur<MatrixType> schurOfA(m_A); </div>
<div class="line"><a name="l00387"></a><span class="lineno"> 387</span>  <span class="keyword">const</span> MatrixType& T = schurOfA.matrixT();</div>
<div class="line"><a name="l00388"></a><span class="lineno"> 388</span>  <span class="keyword">const</span> MatrixType& U = schurOfA.matrixU();</div>
<div class="line"><a name="l00389"></a><span class="lineno"> 389</span>  </div>
<div class="line"><a name="l00390"></a><span class="lineno"> 390</span>  <span class="comment">// Compute square root of T</span></div>
<div class="line"><a name="l00391"></a><span class="lineno"> 391</span>  MatrixType sqrtT;</div>
<div class="line"><a name="l00392"></a><span class="lineno"> 392</span>  MatrixSquareRootTriangular<MatrixType>(T).<a class="code" href="classEigen_1_1MatrixSquareRoot.html#a37407499d669c7dd9af708e7dd6f9217">compute</a>(sqrtT);</div>
<div class="line"><a name="l00393"></a><span class="lineno"> 393</span>  </div>
<div class="line"><a name="l00394"></a><span class="lineno"> 394</span>  <span class="comment">// Compute square root of m_A</span></div>
<div class="line"><a name="l00395"></a><span class="lineno"> 395</span>  result = U * (sqrtT.template triangularView<Upper>() * U.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="l00398"></a><span class="lineno"> 398</span>  <span class="keyword">private</span>:</div>
<div class="line"><a name="l00399"></a><span class="lineno"> 399</span>  <span class="keyword">const</span> MatrixType& m_A;</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> </div>
<div class="line"><a name="l00402"></a><span class="lineno"> 402</span> </div>
<div class="line"><a name="l00415"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixSquareRootReturnValue.html"> 415</a></span> <span class="keyword">template</span><<span class="keyword">typename</span> Derived> <span class="keyword">class </span><a class="code" href="classEigen_1_1MatrixSquareRootReturnValue.html">MatrixSquareRootReturnValue</a></div>
<div class="line"><a name="l00416"></a><span class="lineno"> 416</span> : <span class="keyword">public</span> ReturnByValue<MatrixSquareRootReturnValue<Derived> ></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>  <span class="keyword">typedef</span> <span class="keyword">typename</span> Derived::Index Index;</div>
<div class="line"><a name="l00419"></a><span class="lineno"> 419</span>  <span class="keyword">public</span>:</div>
<div class="line"><a name="l00425"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixSquareRootReturnValue.html#a04d143e094b57f9a3299a87d5166d117"> 425</a></span>  <a class="code" href="classEigen_1_1MatrixSquareRootReturnValue.html#a04d143e094b57f9a3299a87d5166d117">MatrixSquareRootReturnValue</a>(<span class="keyword">const</span> Derived& src) : m_src(src) { }</div>
<div class="line"><a name="l00426"></a><span class="lineno"> 426</span> </div>
<div class="line"><a name="l00432"></a><span class="lineno"> 432</span>  <span class="keyword">template</span> <<span class="keyword">typename</span> ResultType></div>
<div class="line"><a name="l00433"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixSquareRootReturnValue.html#a4f4ce27ebcf7fe1e0078d20d0393c766"> 433</a></span>  <span class="keyword">inline</span> <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1MatrixSquareRootReturnValue.html#a4f4ce27ebcf7fe1e0078d20d0393c766">evalTo</a>(ResultType& result)<span class="keyword"> const</span></div>
<div class="line"><a name="l00434"></a><span class="lineno"> 434</span> <span class="keyword"> </span>{</div>
<div class="line"><a name="l00435"></a><span class="lineno"> 435</span>  <span class="keyword">const</span> <span class="keyword">typename</span> Derived::PlainObject srcEvaluated = m_src.eval();</div>
<div class="line"><a name="l00436"></a><span class="lineno"> 436</span>  <a class="code" href="classEigen_1_1MatrixSquareRoot.html">MatrixSquareRoot<typename Derived::PlainObject></a> me(srcEvaluated);</div>
<div class="line"><a name="l00437"></a><span class="lineno"> 437</span>  me.compute(result);</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> </div>
<div class="line"><a name="l00440"></a><span class="lineno"> 440</span>  Index rows()<span class="keyword"> const </span>{ <span class="keywordflow">return</span> m_src.rows(); }</div>
<div class="line"><a name="l00441"></a><span class="lineno"> 441</span>  Index cols()<span class="keyword"> const </span>{ <span class="keywordflow">return</span> m_src.cols(); }</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">protected</span>:</div>
<div class="line"><a name="l00444"></a><span class="lineno"> 444</span>  <span class="keyword">const</span> Derived& m_src;</div>
<div class="line"><a name="l00445"></a><span class="lineno"> 445</span>  <span class="keyword">private</span>:</div>
<div class="line"><a name="l00446"></a><span class="lineno"> 446</span>  <a class="code" href="classEigen_1_1MatrixSquareRootReturnValue.html#a04d143e094b57f9a3299a87d5166d117">MatrixSquareRootReturnValue</a>& operator=(<span class="keyword">const</span> <a class="code" href="classEigen_1_1MatrixSquareRootReturnValue.html#a04d143e094b57f9a3299a87d5166d117">MatrixSquareRootReturnValue</a>&);</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> </div>
<div class="line"><a name="l00449"></a><span class="lineno"> 449</span> <span class="keyword">namespace </span>internal {</div>
<div class="line"><a name="l00450"></a><span class="lineno"> 450</span> <span class="keyword">template</span><<span class="keyword">typename</span> Derived></div>
<div class="line"><a name="l00451"></a><span class="lineno"> 451</span> <span class="keyword">struct </span>traits<MatrixSquareRootReturnValue<Derived> ></div>
<div class="line"><a name="l00452"></a><span class="lineno"> 452</span> {</div>
<div class="line"><a name="l00453"></a><span class="lineno"> 453</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> Derived::PlainObject ReturnType;</div>
<div class="line"><a name="l00454"></a><span class="lineno"> 454</span> };</div>
<div class="line"><a name="l00455"></a><span class="lineno"> 455</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">template</span> <<span class="keyword">typename</span> Derived></div>
<div class="line"><a name="l00458"></a><span class="lineno"> 458</span> <span class="keyword">const</span> MatrixSquareRootReturnValue<Derived> MatrixBase<Derived>::sqrt()<span class="keyword"> const</span></div>
<div class="line"><a name="l00459"></a><span class="lineno"> 459</span> <span class="keyword"></span>{</div>
<div class="line"><a name="l00460"></a><span class="lineno"> 460</span>  eigen_assert(rows() == cols());</div>
<div class="line"><a name="l00461"></a><span class="lineno"> 461</span>  <span class="keywordflow">return</span> MatrixSquareRootReturnValue<Derived>(derived());</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> } <span class="comment">// end namespace Eigen</span></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> <span class="preprocessor">#endif // EIGEN_MATRIX_FUNCTION</span></div>
<div class="ttc" id="classEigen_1_1MatrixSquareRootQuasiTriangular_html_a37407499d669c7dd9af708e7dd6f9217"><div class="ttname"><a href="classEigen_1_1MatrixSquareRootQuasiTriangular.html#a37407499d669c7dd9af708e7dd6f9217">Eigen::MatrixSquareRootQuasiTriangular::compute</a></div><div class="ttdeci">void compute(ResultType &result)</div><div class="ttdoc">Compute the matrix square root. </div><div class="ttdef"><b>Definition:</b> MatrixSquareRoot.h:80</div></div>
<div class="ttc" id="classEigen_1_1MatrixSquareRoot_html_a5233cb8199f82024af00482971ea8e58"><div class="ttname"><a href="classEigen_1_1MatrixSquareRoot.html#a5233cb8199f82024af00482971ea8e58">Eigen::MatrixSquareRoot::MatrixSquareRoot</a></div><div class="ttdeci">MatrixSquareRoot(const MatrixType &A)</div><div class="ttdoc">Constructor. </div></div>
<div class="ttc" id="classEigen_1_1MatrixSquareRootReturnValue_html"><div class="ttname"><a href="classEigen_1_1MatrixSquareRootReturnValue.html">Eigen::MatrixSquareRootReturnValue</a></div><div class="ttdoc">Proxy for the matrix square root of some matrix (expression). </div><div class="ttdef"><b>Definition:</b> MatrixSquareRoot.h:415</div></div>
<div class="ttc" id="classEigen_1_1MatrixSquareRoot_html_a37407499d669c7dd9af708e7dd6f9217"><div class="ttname"><a href="classEigen_1_1MatrixSquareRoot.html#a37407499d669c7dd9af708e7dd6f9217">Eigen::MatrixSquareRoot::compute</a></div><div class="ttdeci">void compute(ResultType &result)</div><div class="ttdoc">Compute the matrix square root. </div></div>
<div class="ttc" id="classEigen_1_1MatrixSquareRootTriangular_html_a37407499d669c7dd9af708e7dd6f9217"><div class="ttname"><a href="classEigen_1_1MatrixSquareRootTriangular.html#a37407499d669c7dd9af708e7dd6f9217">Eigen::MatrixSquareRootTriangular::compute</a></div><div class="ttdeci">void compute(ResultType &result)</div><div class="ttdoc">Compute the matrix square root. </div><div class="ttdef"><b>Definition:</b> MatrixSquareRoot.h:282</div></div>
<div class="ttc" id="classEigen_1_1MatrixSquareRootQuasiTriangular_html"><div class="ttname"><a href="classEigen_1_1MatrixSquareRootQuasiTriangular.html">Eigen::MatrixSquareRootQuasiTriangular</a></div><div class="ttdoc">Class for computing matrix square roots of upper quasi-triangular matrices. </div><div class="ttdef"><b>Definition:</b> MatrixSquareRoot.h:27</div></div>
<div class="ttc" id="classEigen_1_1MatrixSquareRootReturnValue_html_a4f4ce27ebcf7fe1e0078d20d0393c766"><div class="ttname"><a href="classEigen_1_1MatrixSquareRootReturnValue.html#a4f4ce27ebcf7fe1e0078d20d0393c766">Eigen::MatrixSquareRootReturnValue::evalTo</a></div><div class="ttdeci">void evalTo(ResultType &result) const </div><div class="ttdoc">Compute the matrix square root. </div><div class="ttdef"><b>Definition:</b> MatrixSquareRoot.h:433</div></div>
<div class="ttc" id="classEigen_1_1MatrixSquareRootTriangular_html"><div class="ttname"><a href="classEigen_1_1MatrixSquareRootTriangular.html">Eigen::MatrixSquareRootTriangular</a></div><div class="ttdoc">Class for computing matrix square roots of upper triangular matrices. </div><div class="ttdef"><b>Definition:</b> MatrixSquareRoot.h:256</div></div>
<div class="ttc" id="classEigen_1_1MatrixSquareRootQuasiTriangular_html_a5938579430b2d461b3331d5912cfcf33"><div class="ttname"><a href="classEigen_1_1MatrixSquareRootQuasiTriangular.html#a5938579430b2d461b3331d5912cfcf33">Eigen::MatrixSquareRootQuasiTriangular::MatrixSquareRootQuasiTriangular</a></div><div class="ttdeci">MatrixSquareRootQuasiTriangular(const MatrixType &A)</div><div class="ttdoc">Constructor. </div><div class="ttdef"><b>Definition:</b> MatrixSquareRoot.h:39</div></div>
<div class="ttc" id="classEigen_1_1MatrixSquareRoot_html"><div class="ttname"><a href="classEigen_1_1MatrixSquareRoot.html">Eigen::MatrixSquareRoot</a></div><div class="ttdoc">Class for computing matrix square roots of general matrices. </div><div class="ttdef"><b>Definition:</b> MatrixSquareRoot.h:313</div></div>
<div class="ttc" id="classEigen_1_1MatrixSquareRootReturnValue_html_a04d143e094b57f9a3299a87d5166d117"><div class="ttname"><a href="classEigen_1_1MatrixSquareRootReturnValue.html#a04d143e094b57f9a3299a87d5166d117">Eigen::MatrixSquareRootReturnValue::MatrixSquareRootReturnValue</a></div><div class="ttdeci">MatrixSquareRootReturnValue(const Derived &src)</div><div class="ttdoc">Constructor. </div><div class="ttdef"><b>Definition:</b> MatrixSquareRoot.h:425</div></div>
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