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<div class="title">MatrixPower.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, 2013 Chen-Pang He <jdh8@ms63.hinet.net></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_POWER</span></div>
<div class="line"><a name="l00011"></a><span class="lineno"> 11</span> <span class="preprocessor"></span><span class="preprocessor">#define EIGEN_MATRIX_POWER</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="l00015"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixPower.html"> 15</a></span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType> <span class="keyword">class </span><a class="code" href="classEigen_1_1MatrixPower.html">MatrixPower</a>;</div>
<div class="line"><a name="l00016"></a><span class="lineno"> 16</span> </div>
<div class="line"><a name="l00017"></a><span class="lineno"> 17</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00018"></a><span class="lineno"> 18</span> <span class="keyword">class </span>MatrixPowerRetval : <span class="keyword">public</span> ReturnByValue< MatrixPowerRetval<MatrixType> ></div>
<div class="line"><a name="l00019"></a><span class="lineno"> 19</span> {</div>
<div class="line"><a name="l00020"></a><span class="lineno"> 20</span>  <span class="keyword">public</span>:</div>
<div class="line"><a name="l00021"></a><span class="lineno"> 21</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::RealScalar RealScalar;</div>
<div class="line"><a name="l00022"></a><span class="lineno"> 22</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::Index Index;</div>
<div class="line"><a name="l00023"></a><span class="lineno"> 23</span> </div>
<div class="line"><a name="l00024"></a><span class="lineno"> 24</span>  MatrixPowerRetval(<a class="code" href="classEigen_1_1MatrixPower.html">MatrixPower<MatrixType></a>& pow, RealScalar p) : m_pow(pow), m_p(p)</div>
<div class="line"><a name="l00025"></a><span class="lineno"> 25</span>  { }</div>
<div class="line"><a name="l00026"></a><span class="lineno"> 26</span> </div>
<div class="line"><a name="l00027"></a><span class="lineno"> 27</span>  <span class="keyword">template</span><<span class="keyword">typename</span> ResultType></div>
<div class="line"><a name="l00028"></a><span class="lineno"> 28</span>  <span class="keyword">inline</span> <span class="keywordtype">void</span> evalTo(ResultType& res)<span class="keyword"> const</span></div>
<div class="line"><a name="l00029"></a><span class="lineno"> 29</span> <span class="keyword"> </span>{ m_pow.compute(res, m_p); }</div>
<div class="line"><a name="l00030"></a><span class="lineno"> 30</span> </div>
<div class="line"><a name="l00031"></a><span class="lineno"> 31</span>  Index rows()<span class="keyword"> const </span>{ <span class="keywordflow">return</span> m_pow.rows(); }</div>
<div class="line"><a name="l00032"></a><span class="lineno"> 32</span>  Index cols()<span class="keyword"> const </span>{ <span class="keywordflow">return</span> m_pow.cols(); }</div>
<div class="line"><a name="l00033"></a><span class="lineno"> 33</span> </div>
<div class="line"><a name="l00034"></a><span class="lineno"> 34</span>  <span class="keyword">private</span>:</div>
<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>  MatrixPower<MatrixType>& m_pow;</div>
<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>  <span class="keyword">const</span> RealScalar m_p;</div>
<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>  MatrixPowerRetval& operator=(<span class="keyword">const</span> MatrixPowerRetval&);</div>
<div class="line"><a name="l00038"></a><span class="lineno"> 38</span> };</div>
<div class="line"><a name="l00039"></a><span class="lineno"> 39</span> </div>
<div class="line"><a name="l00040"></a><span class="lineno"> 40</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00041"></a><span class="lineno"> 41</span> <span class="keyword">class </span>MatrixPowerAtomic</div>
<div class="line"><a name="l00042"></a><span class="lineno"> 42</span> {</div>
<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>  <span class="keyword">private</span>:</div>
<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>  <span class="keyword">enum</span> {</div>
<div class="line"><a name="l00045"></a><span class="lineno"> 45</span>  RowsAtCompileTime = MatrixType::RowsAtCompileTime,</div>
<div class="line"><a name="l00046"></a><span class="lineno"> 46</span>  MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime</div>
<div class="line"><a name="l00047"></a><span class="lineno"> 47</span>  };</div>
<div class="line"><a name="l00048"></a><span class="lineno"> 48</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::Scalar Scalar;</div>
<div class="line"><a name="l00049"></a><span class="lineno"> 49</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::RealScalar RealScalar;</div>
<div class="line"><a name="l00050"></a><span class="lineno"> 50</span>  <span class="keyword">typedef</span> std::complex<RealScalar> ComplexScalar;</div>
<div class="line"><a name="l00051"></a><span class="lineno"> 51</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::Index Index;</div>
<div class="line"><a name="l00052"></a><span class="lineno"> 52</span>  <span class="keyword">typedef</span> Array<Scalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> ArrayType;</div>
<div class="line"><a name="l00053"></a><span class="lineno"> 53</span> </div>
<div class="line"><a name="l00054"></a><span class="lineno"> 54</span>  <span class="keyword">const</span> MatrixType& m_A;</div>
<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>  RealScalar m_p;</div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span> </div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>  <span class="keywordtype">void</span> computePade(<span class="keywordtype">int</span> degree, <span class="keyword">const</span> MatrixType& IminusT, MatrixType& res) <span class="keyword">const</span>;</div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>  <span class="keywordtype">void</span> compute2x2(MatrixType& res, RealScalar p) <span class="keyword">const</span>;</div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>  <span class="keywordtype">void</span> computeBig(MatrixType& res) <span class="keyword">const</span>;</div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>  <span class="keyword">static</span> <span class="keywordtype">int</span> getPadeDegree(<span class="keywordtype">float</span> normIminusT);</div>
<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>  <span class="keyword">static</span> <span class="keywordtype">int</span> getPadeDegree(<span class="keywordtype">double</span> normIminusT);</div>
<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>  <span class="keyword">static</span> <span class="keywordtype">int</span> getPadeDegree(<span class="keywordtype">long</span> <span class="keywordtype">double</span> normIminusT);</div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>  <span class="keyword">static</span> ComplexScalar computeSuperDiag(<span class="keyword">const</span> ComplexScalar&, <span class="keyword">const</span> ComplexScalar&, RealScalar p);</div>
<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>  <span class="keyword">static</span> RealScalar computeSuperDiag(RealScalar, RealScalar, RealScalar p);</div>
<div class="line"><a name="l00065"></a><span class="lineno"> 65</span> </div>
<div class="line"><a name="l00066"></a><span class="lineno"> 66</span>  <span class="keyword">public</span>:</div>
<div class="line"><a name="l00067"></a><span class="lineno"> 67</span>  MatrixPowerAtomic(<span class="keyword">const</span> MatrixType& T, RealScalar p);</div>
<div class="line"><a name="l00068"></a><span class="lineno"> 68</span>  <span class="keywordtype">void</span> compute(MatrixType& res) <span class="keyword">const</span>;</div>
<div class="line"><a name="l00069"></a><span class="lineno"> 69</span> };</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> MatrixType></div>
<div class="line"><a name="l00072"></a><span class="lineno"> 72</span> MatrixPowerAtomic<MatrixType>::MatrixPowerAtomic(<span class="keyword">const</span> MatrixType& T, RealScalar p) :</div>
<div class="line"><a name="l00073"></a><span class="lineno"> 73</span>  m_A(T), m_p(p)</div>
<div class="line"><a name="l00074"></a><span class="lineno"> 74</span> { eigen_assert(T.rows() == T.cols()); }</div>
<div class="line"><a name="l00075"></a><span class="lineno"> 75</span> </div>
<div class="line"><a name="l00076"></a><span class="lineno"> 76</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00077"></a><span class="lineno"> 77</span> <span class="keywordtype">void</span> MatrixPowerAtomic<MatrixType>::compute(MatrixType& res)<span class="keyword"> const</span></div>
<div class="line"><a name="l00078"></a><span class="lineno"> 78</span> <span class="keyword"></span>{</div>
<div class="line"><a name="l00079"></a><span class="lineno"> 79</span>  res.resizeLike(m_A);</div>
<div class="line"><a name="l00080"></a><span class="lineno"> 80</span>  <span class="keywordflow">switch</span> (m_A.rows()) {</div>
<div class="line"><a name="l00081"></a><span class="lineno"> 81</span>  <span class="keywordflow">case</span> 0:</div>
<div class="line"><a name="l00082"></a><span class="lineno"> 82</span>  <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00083"></a><span class="lineno"> 83</span>  <span class="keywordflow">case</span> 1:</div>
<div class="line"><a name="l00084"></a><span class="lineno"> 84</span>  res(0,0) = std::pow(m_A(0,0), m_p);</div>
<div class="line"><a name="l00085"></a><span class="lineno"> 85</span>  <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00086"></a><span class="lineno"> 86</span>  <span class="keywordflow">case</span> 2:</div>
<div class="line"><a name="l00087"></a><span class="lineno"> 87</span>  compute2x2(res, m_p);</div>
<div class="line"><a name="l00088"></a><span class="lineno"> 88</span>  <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>  <span class="keywordflow">default</span>:</div>
<div class="line"><a name="l00090"></a><span class="lineno"> 90</span>  computeBig(res);</div>
<div class="line"><a name="l00091"></a><span class="lineno"> 91</span>  }</div>
<div class="line"><a name="l00092"></a><span class="lineno"> 92</span> }</div>
<div class="line"><a name="l00093"></a><span class="lineno"> 93</span> </div>
<div class="line"><a name="l00094"></a><span class="lineno"> 94</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00095"></a><span class="lineno"> 95</span> <span class="keywordtype">void</span> MatrixPowerAtomic<MatrixType>::computePade(<span class="keywordtype">int</span> degree, <span class="keyword">const</span> MatrixType& IminusT, MatrixType& res)<span class="keyword"> const</span></div>
<div class="line"><a name="l00096"></a><span class="lineno"> 96</span> <span class="keyword"></span>{</div>
<div class="line"><a name="l00097"></a><span class="lineno"> 97</span>  <span class="keywordtype">int</span> i = degree<<1;</div>
<div class="line"><a name="l00098"></a><span class="lineno"> 98</span>  res = (m_p-degree) / ((i-1)<<1) * IminusT;</div>
<div class="line"><a name="l00099"></a><span class="lineno"> 99</span>  <span class="keywordflow">for</span> (--i; i; --i) {</div>
<div class="line"><a name="l00100"></a><span class="lineno"> 100</span>  res = (MatrixType::Identity(IminusT.rows(), IminusT.cols()) + res).template triangularView<Upper>()</div>
<div class="line"><a name="l00101"></a><span class="lineno"> 101</span>  .solve((i==1 ? -m_p : i&1 ? (-m_p-(i>>1))/(i<<1) : (m_p-(i>>1))/((i-1)<<1)) * IminusT).eval();</div>
<div class="line"><a name="l00102"></a><span class="lineno"> 102</span>  }</div>
<div class="line"><a name="l00103"></a><span class="lineno"> 103</span>  res += MatrixType::Identity(IminusT.rows(), IminusT.cols());</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> <span class="comment">// This function assumes that res has the correct size (see bug 614)</span></div>
<div class="line"><a name="l00107"></a><span class="lineno"> 107</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00108"></a><span class="lineno"> 108</span> <span class="keywordtype">void</span> MatrixPowerAtomic<MatrixType>::compute2x2(MatrixType& res, RealScalar p)<span class="keyword"> const</span></div>
<div class="line"><a name="l00109"></a><span class="lineno"> 109</span> <span class="keyword"></span>{</div>
<div class="line"><a name="l00110"></a><span class="lineno"> 110</span>  <span class="keyword">using</span> std::abs;</div>
<div class="line"><a name="l00111"></a><span class="lineno"> 111</span>  <span class="keyword">using</span> std::pow;</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>  ArrayType logTdiag = m_A.diagonal().array().log();</div>
<div class="line"><a name="l00114"></a><span class="lineno"> 114</span>  res.coeffRef(0,0) = pow(m_A.coeff(0,0), p);</div>
<div class="line"><a name="l00115"></a><span class="lineno"> 115</span> </div>
<div class="line"><a name="l00116"></a><span class="lineno"> 116</span>  <span class="keywordflow">for</span> (Index i=1; i < m_A.cols(); ++i) {</div>
<div class="line"><a name="l00117"></a><span class="lineno"> 117</span>  res.coeffRef(i,i) = pow(m_A.coeff(i,i), p);</div>
<div class="line"><a name="l00118"></a><span class="lineno"> 118</span>  <span class="keywordflow">if</span> (m_A.coeff(i-1,i-1) == m_A.coeff(i,i))</div>
<div class="line"><a name="l00119"></a><span class="lineno"> 119</span>  res.coeffRef(i-1,i) = p * pow(m_A.coeff(i,i), p-1);</div>
<div class="line"><a name="l00120"></a><span class="lineno"> 120</span>  <span class="keywordflow">else</span> <span class="keywordflow">if</span> (2*abs(m_A.coeff(i-1,i-1)) < abs(m_A.coeff(i,i)) || 2*abs(m_A.coeff(i,i)) < abs(m_A.coeff(i-1,i-1)))</div>
<div class="line"><a name="l00121"></a><span class="lineno"> 121</span>  res.coeffRef(i-1,i) = (res.coeff(i,i)-res.coeff(i-1,i-1)) / (m_A.coeff(i,i)-m_A.coeff(i-1,i-1));</div>
<div class="line"><a name="l00122"></a><span class="lineno"> 122</span>  <span class="keywordflow">else</span></div>
<div class="line"><a name="l00123"></a><span class="lineno"> 123</span>  res.coeffRef(i-1,i) = computeSuperDiag(m_A.coeff(i,i), m_A.coeff(i-1,i-1), p);</div>
<div class="line"><a name="l00124"></a><span class="lineno"> 124</span>  res.coeffRef(i-1,i) *= m_A.coeff(i-1,i);</div>
<div class="line"><a name="l00125"></a><span class="lineno"> 125</span>  }</div>
<div class="line"><a name="l00126"></a><span class="lineno"> 126</span> }</div>
<div class="line"><a name="l00127"></a><span class="lineno"> 127</span> </div>
<div class="line"><a name="l00128"></a><span class="lineno"> 128</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00129"></a><span class="lineno"> 129</span> <span class="keywordtype">void</span> MatrixPowerAtomic<MatrixType>::computeBig(MatrixType& res)<span class="keyword"> const</span></div>
<div class="line"><a name="l00130"></a><span class="lineno"> 130</span> <span class="keyword"></span>{</div>
<div class="line"><a name="l00131"></a><span class="lineno"> 131</span>  <span class="keyword">const</span> <span class="keywordtype">int</span> digits = std::numeric_limits<RealScalar>::digits;</div>
<div class="line"><a name="l00132"></a><span class="lineno"> 132</span>  <span class="keyword">const</span> RealScalar maxNormForPade = digits <= 24? 4.3386528e-1f: <span class="comment">// sigle precision</span></div>
<div class="line"><a name="l00133"></a><span class="lineno"> 133</span>  digits <= 53? 2.789358995219730e-1: <span class="comment">// double precision</span></div>
<div class="line"><a name="l00134"></a><span class="lineno"> 134</span>  digits <= 64? 2.4471944416607995472e-1L: <span class="comment">// extended precision</span></div>
<div class="line"><a name="l00135"></a><span class="lineno"> 135</span>  digits <= 106? 1.1016843812851143391275867258512e-1L: <span class="comment">// double-double</span></div>
<div class="line"><a name="l00136"></a><span class="lineno"> 136</span>  9.134603732914548552537150753385375e-2L; <span class="comment">// quadruple precision</span></div>
<div class="line"><a name="l00137"></a><span class="lineno"> 137</span>  MatrixType IminusT, sqrtT, T = m_A.template triangularView<Upper>();</div>
<div class="line"><a name="l00138"></a><span class="lineno"> 138</span>  RealScalar normIminusT;</div>
<div class="line"><a name="l00139"></a><span class="lineno"> 139</span>  <span class="keywordtype">int</span> degree, degree2, numberOfSquareRoots = 0;</div>
<div class="line"><a name="l00140"></a><span class="lineno"> 140</span>  <span class="keywordtype">bool</span> hasExtraSquareRoot = <span class="keyword">false</span>;</div>
<div class="line"><a name="l00141"></a><span class="lineno"> 141</span> </div>
<div class="line"><a name="l00142"></a><span class="lineno"> 142</span>  <span class="comment">/* FIXME</span></div>
<div class="line"><a name="l00143"></a><span class="lineno"> 143</span> <span class="comment"> * For singular T, norm(I - T) >= 1 but maxNormForPade < 1, leads to infinite</span></div>
<div class="line"><a name="l00144"></a><span class="lineno"> 144</span> <span class="comment"> * loop. We should move 0 eigenvalues to bottom right corner. We need not</span></div>
<div class="line"><a name="l00145"></a><span class="lineno"> 145</span> <span class="comment"> * worry about tiny values (e.g. 1e-300) because they will reach 1 if</span></div>
<div class="line"><a name="l00146"></a><span class="lineno"> 146</span> <span class="comment"> * repetitively sqrt'ed.</span></div>
<div class="line"><a name="l00147"></a><span class="lineno"> 147</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00148"></a><span class="lineno"> 148</span> <span class="comment"> * If the 0 eigenvalues are semisimple, they can form a 0 matrix at the</span></div>
<div class="line"><a name="l00149"></a><span class="lineno"> 149</span> <span class="comment"> * bottom right corner.</span></div>
<div class="line"><a name="l00150"></a><span class="lineno"> 150</span> <span class="comment"> *</span></div>
<div class="line"><a name="l00151"></a><span class="lineno"> 151</span> <span class="comment"> * [ T A ]^p [ T^p (T^-1 T^p A) ]</span></div>
<div class="line"><a name="l00152"></a><span class="lineno"> 152</span> <span class="comment"> * [ ] = [ ]</span></div>
<div class="line"><a name="l00153"></a><span class="lineno"> 153</span> <span class="comment"> * [ 0 0 ] [ 0 0 ]</span></div>
<div class="line"><a name="l00154"></a><span class="lineno"> 154</span> <span class="comment"> */</span></div>
<div class="line"><a name="l00155"></a><span class="lineno"> 155</span>  <span class="keywordflow">for</span> (Index i=0; i < m_A.cols(); ++i)</div>
<div class="line"><a name="l00156"></a><span class="lineno"> 156</span>  eigen_assert(m_A(i,i) != RealScalar(0));</div>
<div class="line"><a name="l00157"></a><span class="lineno"> 157</span> </div>
<div class="line"><a name="l00158"></a><span class="lineno"> 158</span>  <span class="keywordflow">while</span> (<span class="keyword">true</span>) {</div>
<div class="line"><a name="l00159"></a><span class="lineno"> 159</span>  IminusT = MatrixType::Identity(m_A.rows(), m_A.cols()) - T;</div>
<div class="line"><a name="l00160"></a><span class="lineno"> 160</span>  normIminusT = IminusT.cwiseAbs().colwise().sum().maxCoeff();</div>
<div class="line"><a name="l00161"></a><span class="lineno"> 161</span>  <span class="keywordflow">if</span> (normIminusT < maxNormForPade) {</div>
<div class="line"><a name="l00162"></a><span class="lineno"> 162</span>  degree = getPadeDegree(normIminusT);</div>
<div class="line"><a name="l00163"></a><span class="lineno"> 163</span>  degree2 = getPadeDegree(normIminusT/2);</div>
<div class="line"><a name="l00164"></a><span class="lineno"> 164</span>  <span class="keywordflow">if</span> (degree - degree2 <= 1 || hasExtraSquareRoot)</div>
<div class="line"><a name="l00165"></a><span class="lineno"> 165</span>  <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00166"></a><span class="lineno"> 166</span>  hasExtraSquareRoot = <span class="keyword">true</span>;</div>
<div class="line"><a name="l00167"></a><span class="lineno"> 167</span>  }</div>
<div class="line"><a name="l00168"></a><span class="lineno"> 168</span>  MatrixSquareRootTriangular<MatrixType>(T).compute(sqrtT);</div>
<div class="line"><a name="l00169"></a><span class="lineno"> 169</span>  T = sqrtT.template triangularView<Upper>();</div>
<div class="line"><a name="l00170"></a><span class="lineno"> 170</span>  ++numberOfSquareRoots;</div>
<div class="line"><a name="l00171"></a><span class="lineno"> 171</span>  }</div>
<div class="line"><a name="l00172"></a><span class="lineno"> 172</span>  computePade(degree, IminusT, res);</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="keywordflow">for</span> (; numberOfSquareRoots; --numberOfSquareRoots) {</div>
<div class="line"><a name="l00175"></a><span class="lineno"> 175</span>  compute2x2(res, std::ldexp(m_p, -numberOfSquareRoots));</div>
<div class="line"><a name="l00176"></a><span class="lineno"> 176</span>  res = res.template triangularView<Upper>() * res;</div>
<div class="line"><a name="l00177"></a><span class="lineno"> 177</span>  }</div>
<div class="line"><a name="l00178"></a><span class="lineno"> 178</span>  compute2x2(res, m_p);</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>  </div>
<div class="line"><a name="l00181"></a><span class="lineno"> 181</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00182"></a><span class="lineno"> 182</span> <span class="keyword">inline</span> <span class="keywordtype">int</span> MatrixPowerAtomic<MatrixType>::getPadeDegree(<span class="keywordtype">float</span> normIminusT)</div>
<div class="line"><a name="l00183"></a><span class="lineno"> 183</span> {</div>
<div class="line"><a name="l00184"></a><span class="lineno"> 184</span>  <span class="keyword">const</span> <span class="keywordtype">float</span> maxNormForPade[] = { 2.8064004e-1f <span class="comment">/* degree = 3 */</span> , 4.3386528e-1f };</div>
<div class="line"><a name="l00185"></a><span class="lineno"> 185</span>  <span class="keywordtype">int</span> degree = 3;</div>
<div class="line"><a name="l00186"></a><span class="lineno"> 186</span>  <span class="keywordflow">for</span> (; degree <= 4; ++degree)</div>
<div class="line"><a name="l00187"></a><span class="lineno"> 187</span>  <span class="keywordflow">if</span> (normIminusT <= maxNormForPade[degree - 3])</div>
<div class="line"><a name="l00188"></a><span class="lineno"> 188</span>  <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00189"></a><span class="lineno"> 189</span>  <span class="keywordflow">return</span> degree;</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> </div>
<div class="line"><a name="l00192"></a><span class="lineno"> 192</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00193"></a><span class="lineno"> 193</span> <span class="keyword">inline</span> <span class="keywordtype">int</span> MatrixPowerAtomic<MatrixType>::getPadeDegree(<span class="keywordtype">double</span> normIminusT)</div>
<div class="line"><a name="l00194"></a><span class="lineno"> 194</span> {</div>
<div class="line"><a name="l00195"></a><span class="lineno"> 195</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> maxNormForPade[] = { 1.884160592658218e-2 <span class="comment">/* degree = 3 */</span> , 6.038881904059573e-2, 1.239917516308172e-1,</div>
<div class="line"><a name="l00196"></a><span class="lineno"> 196</span>  1.999045567181744e-1, 2.789358995219730e-1 };</div>
<div class="line"><a name="l00197"></a><span class="lineno"> 197</span>  <span class="keywordtype">int</span> degree = 3;</div>
<div class="line"><a name="l00198"></a><span class="lineno"> 198</span>  <span class="keywordflow">for</span> (; degree <= 7; ++degree)</div>
<div class="line"><a name="l00199"></a><span class="lineno"> 199</span>  <span class="keywordflow">if</span> (normIminusT <= maxNormForPade[degree - 3])</div>
<div class="line"><a name="l00200"></a><span class="lineno"> 200</span>  <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00201"></a><span class="lineno"> 201</span>  <span class="keywordflow">return</span> degree;</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="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00205"></a><span class="lineno"> 205</span> <span class="keyword">inline</span> <span class="keywordtype">int</span> MatrixPowerAtomic<MatrixType>::getPadeDegree(<span class="keywordtype">long</span> <span class="keywordtype">double</span> normIminusT)</div>
<div class="line"><a name="l00206"></a><span class="lineno"> 206</span> {</div>
<div class="line"><a name="l00207"></a><span class="lineno"> 207</span> <span class="preprocessor">#if LDBL_MANT_DIG == 53</span></div>
<div class="line"><a name="l00208"></a><span class="lineno"> 208</span> <span class="preprocessor"></span> <span class="keyword">const</span> <span class="keywordtype">int</span> maxPadeDegree = 7;</div>
<div class="line"><a name="l00209"></a><span class="lineno"> 209</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> maxNormForPade[] = { 1.884160592658218e-2L <span class="comment">/* degree = 3 */</span> , 6.038881904059573e-2L, 1.239917516308172e-1L,</div>
<div class="line"><a name="l00210"></a><span class="lineno"> 210</span>  1.999045567181744e-1L, 2.789358995219730e-1L };</div>
<div class="line"><a name="l00211"></a><span class="lineno"> 211</span> <span class="preprocessor">#elif LDBL_MANT_DIG <= 64</span></div>
<div class="line"><a name="l00212"></a><span class="lineno"> 212</span> <span class="preprocessor"></span> <span class="keyword">const</span> <span class="keywordtype">int</span> maxPadeDegree = 8;</div>
<div class="line"><a name="l00213"></a><span class="lineno"> 213</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> maxNormForPade[] = { 6.3854693117491799460e-3L <span class="comment">/* degree = 3 */</span> , 2.6394893435456973676e-2L,</div>
<div class="line"><a name="l00214"></a><span class="lineno"> 214</span>  6.4216043030404063729e-2L, 1.1701165502926694307e-1L, 1.7904284231268670284e-1L, 2.4471944416607995472e-1L };</div>
<div class="line"><a name="l00215"></a><span class="lineno"> 215</span> <span class="preprocessor">#elif LDBL_MANT_DIG <= 106</span></div>
<div class="line"><a name="l00216"></a><span class="lineno"> 216</span> <span class="preprocessor"></span> <span class="keyword">const</span> <span class="keywordtype">int</span> maxPadeDegree = 10;</div>
<div class="line"><a name="l00217"></a><span class="lineno"> 217</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> maxNormForPade[] = { 1.0007161601787493236741409687186e-4L <span class="comment">/* degree = 3 */</span> ,</div>
<div class="line"><a name="l00218"></a><span class="lineno"> 218</span>  1.0007161601787493236741409687186e-3L, 4.7069769360887572939882574746264e-3L, 1.3220386624169159689406653101695e-2L,</div>
<div class="line"><a name="l00219"></a><span class="lineno"> 219</span>  2.8063482381631737920612944054906e-2L, 4.9625993951953473052385361085058e-2L, 7.7367040706027886224557538328171e-2L,</div>
<div class="line"><a name="l00220"></a><span class="lineno"> 220</span>  1.1016843812851143391275867258512e-1L };</div>
<div class="line"><a name="l00221"></a><span class="lineno"> 221</span> <span class="preprocessor">#else</span></div>
<div class="line"><a name="l00222"></a><span class="lineno"> 222</span> <span class="preprocessor"></span> <span class="keyword">const</span> <span class="keywordtype">int</span> maxPadeDegree = 10;</div>
<div class="line"><a name="l00223"></a><span class="lineno"> 223</span>  <span class="keyword">const</span> <span class="keywordtype">double</span> maxNormForPade[] = { 5.524506147036624377378713555116378e-5L <span class="comment">/* degree = 3 */</span> ,</div>
<div class="line"><a name="l00224"></a><span class="lineno"> 224</span>  6.640600568157479679823602193345995e-4L, 3.227716520106894279249709728084626e-3L,</div>
<div class="line"><a name="l00225"></a><span class="lineno"> 225</span>  9.619593944683432960546978734646284e-3L, 2.134595382433742403911124458161147e-2L,</div>
<div class="line"><a name="l00226"></a><span class="lineno"> 226</span>  3.908166513900489428442993794761185e-2L, 6.266780814639442865832535460550138e-2L,</div>
<div class="line"><a name="l00227"></a><span class="lineno"> 227</span>  9.134603732914548552537150753385375e-2L };</div>
<div class="line"><a name="l00228"></a><span class="lineno"> 228</span> <span class="preprocessor">#endif</span></div>
<div class="line"><a name="l00229"></a><span class="lineno"> 229</span> <span class="preprocessor"></span> <span class="keywordtype">int</span> degree = 3;</div>
<div class="line"><a name="l00230"></a><span class="lineno"> 230</span>  <span class="keywordflow">for</span> (; degree <= maxPadeDegree; ++degree)</div>
<div class="line"><a name="l00231"></a><span class="lineno"> 231</span>  <span class="keywordflow">if</span> (normIminusT <= maxNormForPade[degree - 3])</div>
<div class="line"><a name="l00232"></a><span class="lineno"> 232</span>  <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00233"></a><span class="lineno"> 233</span>  <span class="keywordflow">return</span> degree;</div>
<div class="line"><a name="l00234"></a><span class="lineno"> 234</span> }</div>
<div class="line"><a name="l00235"></a><span class="lineno"> 235</span> </div>
<div class="line"><a name="l00236"></a><span class="lineno"> 236</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00237"></a><span class="lineno"> 237</span> <span class="keyword">inline</span> <span class="keyword">typename</span> MatrixPowerAtomic<MatrixType>::ComplexScalar</div>
<div class="line"><a name="l00238"></a><span class="lineno"> 238</span> MatrixPowerAtomic<MatrixType>::computeSuperDiag(<span class="keyword">const</span> ComplexScalar& curr, <span class="keyword">const</span> ComplexScalar& prev, RealScalar p)</div>
<div class="line"><a name="l00239"></a><span class="lineno"> 239</span> {</div>
<div class="line"><a name="l00240"></a><span class="lineno"> 240</span>  ComplexScalar logCurr = std::log(curr);</div>
<div class="line"><a name="l00241"></a><span class="lineno"> 241</span>  ComplexScalar logPrev = std::log(prev);</div>
<div class="line"><a name="l00242"></a><span class="lineno"> 242</span>  <span class="keywordtype">int</span> unwindingNumber = std::ceil((numext::imag(logCurr - logPrev) - M_PI) / (2*M_PI));</div>
<div class="line"><a name="l00243"></a><span class="lineno"> 243</span>  ComplexScalar w = numext::atanh2(curr - prev, curr + prev) + ComplexScalar(0, M_PI*unwindingNumber);</div>
<div class="line"><a name="l00244"></a><span class="lineno"> 244</span>  <span class="keywordflow">return</span> RealScalar(2) * std::exp(RealScalar(0.5) * p * (logCurr + logPrev)) * std::sinh(p * w) / (curr - prev);</div>
<div class="line"><a name="l00245"></a><span class="lineno"> 245</span> }</div>
<div class="line"><a name="l00246"></a><span class="lineno"> 246</span> </div>
<div class="line"><a name="l00247"></a><span class="lineno"> 247</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00248"></a><span class="lineno"> 248</span> <span class="keyword">inline</span> <span class="keyword">typename</span> MatrixPowerAtomic<MatrixType>::RealScalar</div>
<div class="line"><a name="l00249"></a><span class="lineno"> 249</span> MatrixPowerAtomic<MatrixType>::computeSuperDiag(RealScalar curr, RealScalar prev, RealScalar p)</div>
<div class="line"><a name="l00250"></a><span class="lineno"> 250</span> {</div>
<div class="line"><a name="l00251"></a><span class="lineno"> 251</span>  RealScalar w = numext::atanh2(curr - prev, curr + prev);</div>
<div class="line"><a name="l00252"></a><span class="lineno"> 252</span>  <span class="keywordflow">return</span> 2 * std::exp(p * (std::log(curr) + std::log(prev)) / 2) * std::sinh(p * w) / (curr - prev);</div>
<div class="line"><a name="l00253"></a><span class="lineno"> 253</span> }</div>
<div class="line"><a name="l00254"></a><span class="lineno"> 254</span> </div>
<div class="line"><a name="l00274"></a><span class="lineno"> 274</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00275"></a><span class="lineno"> 275</span> <span class="keyword">class </span>MatrixPower</div>
<div class="line"><a name="l00276"></a><span class="lineno"> 276</span> {</div>
<div class="line"><a name="l00277"></a><span class="lineno"> 277</span>  <span class="keyword">private</span>:</div>
<div class="line"><a name="l00278"></a><span class="lineno"> 278</span>  <span class="keyword">enum</span> {</div>
<div class="line"><a name="l00279"></a><span class="lineno"> 279</span>  RowsAtCompileTime = MatrixType::RowsAtCompileTime,</div>
<div class="line"><a name="l00280"></a><span class="lineno"> 280</span>  ColsAtCompileTime = MatrixType::ColsAtCompileTime,</div>
<div class="line"><a name="l00281"></a><span class="lineno"> 281</span>  MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,</div>
<div class="line"><a name="l00282"></a><span class="lineno"> 282</span>  MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime</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">typedef</span> <span class="keyword">typename</span> MatrixType::Scalar Scalar;</div>
<div class="line"><a name="l00285"></a><span class="lineno"> 285</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::RealScalar RealScalar;</div>
<div class="line"><a name="l00286"></a><span class="lineno"> 286</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixType::Index Index;</div>
<div class="line"><a name="l00287"></a><span class="lineno"> 287</span> </div>
<div class="line"><a name="l00288"></a><span class="lineno"> 288</span>  <span class="keyword">public</span>:</div>
<div class="line"><a name="l00297"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixPower.html#a0c03931b4fff167eeaf6bab5161ae9cd"> 297</a></span>  <span class="keyword">explicit</span> <a class="code" href="classEigen_1_1MatrixPower.html#a0c03931b4fff167eeaf6bab5161ae9cd">MatrixPower</a>(<span class="keyword">const</span> MatrixType& A) : m_A(A), m_conditionNumber(0)</div>
<div class="line"><a name="l00298"></a><span class="lineno"> 298</span>  { eigen_assert(A.rows() == A.cols()); }</div>
<div class="line"><a name="l00299"></a><span class="lineno"> 299</span> </div>
<div class="line"><a name="l00307"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixPower.html#a77c0aa664c889e4aac24406385196e8a"> 307</a></span>  <span class="keyword">const</span> MatrixPowerRetval<MatrixType> <a class="code" href="classEigen_1_1MatrixPower.html#a77c0aa664c889e4aac24406385196e8a">operator()</a>(RealScalar p)</div>
<div class="line"><a name="l00308"></a><span class="lineno"> 308</span>  { <span class="keywordflow">return</span> MatrixPowerRetval<MatrixType>(*<span class="keyword">this</span>, p); }</div>
<div class="line"><a name="l00309"></a><span class="lineno"> 309</span> </div>
<div class="line"><a name="l00317"></a><span class="lineno"> 317</span>  <span class="keyword">template</span><<span class="keyword">typename</span> ResultType></div>
<div class="line"><a name="l00318"></a><span class="lineno"> 318</span>  <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1MatrixPower.html#a95d38473ae16ef259e0d334538d050d6">compute</a>(ResultType& res, RealScalar p);</div>
<div class="line"><a name="l00319"></a><span class="lineno"> 319</span>  </div>
<div class="line"><a name="l00320"></a><span class="lineno"> 320</span>  Index rows()<span class="keyword"> const </span>{ <span class="keywordflow">return</span> m_A.rows(); }</div>
<div class="line"><a name="l00321"></a><span class="lineno"> 321</span>  Index cols()<span class="keyword"> const </span>{ <span class="keywordflow">return</span> m_A.cols(); }</div>
<div class="line"><a name="l00322"></a><span class="lineno"> 322</span> </div>
<div class="line"><a name="l00323"></a><span class="lineno"> 323</span>  <span class="keyword">private</span>:</div>
<div class="line"><a name="l00324"></a><span class="lineno"> 324</span>  <span class="keyword">typedef</span> std::complex<RealScalar> ComplexScalar;</div>
<div class="line"><a name="l00325"></a><span class="lineno"> 325</span>  <span class="keyword">typedef</span> Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, MatrixType::Options,</div>
<div class="line"><a name="l00326"></a><span class="lineno"> 326</span>  MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrix;</div>
<div class="line"><a name="l00327"></a><span class="lineno"> 327</span> </div>
<div class="line"><a name="l00328"></a><span class="lineno"> 328</span>  <span class="keyword">typename</span> MatrixType::Nested m_A;</div>
<div class="line"><a name="l00329"></a><span class="lineno"> 329</span>  MatrixType m_tmp;</div>
<div class="line"><a name="l00330"></a><span class="lineno"> 330</span>  ComplexMatrix m_T, m_U, m_fT;</div>
<div class="line"><a name="l00331"></a><span class="lineno"> 331</span>  RealScalar m_conditionNumber;</div>
<div class="line"><a name="l00332"></a><span class="lineno"> 332</span> </div>
<div class="line"><a name="l00333"></a><span class="lineno"> 333</span>  RealScalar modfAndInit(RealScalar, RealScalar*);</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>  <span class="keyword">template</span><<span class="keyword">typename</span> ResultType></div>
<div class="line"><a name="l00336"></a><span class="lineno"> 336</span>  <span class="keywordtype">void</span> computeIntPower(ResultType&, RealScalar);</div>
<div class="line"><a name="l00337"></a><span class="lineno"> 337</span> </div>
<div class="line"><a name="l00338"></a><span class="lineno"> 338</span>  <span class="keyword">template</span><<span class="keyword">typename</span> ResultType></div>
<div class="line"><a name="l00339"></a><span class="lineno"> 339</span>  <span class="keywordtype">void</span> computeFracPower(ResultType&, RealScalar);</div>
<div class="line"><a name="l00340"></a><span class="lineno"> 340</span> </div>
<div class="line"><a name="l00341"></a><span class="lineno"> 341</span>  <span class="keyword">template</span><<span class="keywordtype">int</span> Rows, <span class="keywordtype">int</span> Cols, <span class="keywordtype">int</span> Options, <span class="keywordtype">int</span> MaxRows, <span class="keywordtype">int</span> MaxCols></div>
<div class="line"><a name="l00342"></a><span class="lineno"> 342</span>  <span class="keyword">static</span> <span class="keywordtype">void</span> revertSchur(</div>
<div class="line"><a name="l00343"></a><span class="lineno"> 343</span>  Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,</div>
<div class="line"><a name="l00344"></a><span class="lineno"> 344</span>  <span class="keyword">const</span> ComplexMatrix& T,</div>
<div class="line"><a name="l00345"></a><span class="lineno"> 345</span>  <span class="keyword">const</span> ComplexMatrix& U);</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>  <span class="keyword">template</span><<span class="keywordtype">int</span> Rows, <span class="keywordtype">int</span> Cols, <span class="keywordtype">int</span> Options, <span class="keywordtype">int</span> MaxRows, <span class="keywordtype">int</span> MaxCols></div>
<div class="line"><a name="l00348"></a><span class="lineno"> 348</span>  <span class="keyword">static</span> <span class="keywordtype">void</span> revertSchur(</div>
<div class="line"><a name="l00349"></a><span class="lineno"> 349</span>  Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,</div>
<div class="line"><a name="l00350"></a><span class="lineno"> 350</span>  <span class="keyword">const</span> ComplexMatrix& T,</div>
<div class="line"><a name="l00351"></a><span class="lineno"> 351</span>  <span class="keyword">const</span> ComplexMatrix& U);</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> </div>
<div class="line"><a name="l00354"></a><span class="lineno"> 354</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00355"></a><span class="lineno"> 355</span> <span class="keyword">template</span><<span class="keyword">typename</span> ResultType></div>
<div class="line"><a name="l00356"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixPower.html#a95d38473ae16ef259e0d334538d050d6"> 356</a></span> <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1MatrixPower.html#a95d38473ae16ef259e0d334538d050d6">MatrixPower<MatrixType>::compute</a>(ResultType& res, RealScalar p)</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>  <span class="keywordflow">switch</span> (cols()) {</div>
<div class="line"><a name="l00359"></a><span class="lineno"> 359</span>  <span class="keywordflow">case</span> 0:</div>
<div class="line"><a name="l00360"></a><span class="lineno"> 360</span>  <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00361"></a><span class="lineno"> 361</span>  <span class="keywordflow">case</span> 1:</div>
<div class="line"><a name="l00362"></a><span class="lineno"> 362</span>  res(0,0) = std::pow(m_A.coeff(0,0), p);</div>
<div class="line"><a name="l00363"></a><span class="lineno"> 363</span>  <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00364"></a><span class="lineno"> 364</span>  <span class="keywordflow">default</span>:</div>
<div class="line"><a name="l00365"></a><span class="lineno"> 365</span>  RealScalar intpart, x = modfAndInit(p, &intpart);</div>
<div class="line"><a name="l00366"></a><span class="lineno"> 366</span>  computeIntPower(res, intpart);</div>
<div class="line"><a name="l00367"></a><span class="lineno"> 367</span>  computeFracPower(res, x);</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> </div>
<div class="line"><a name="l00371"></a><span class="lineno"> 371</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00372"></a><span class="lineno"> 372</span> <span class="keyword">typename</span> MatrixPower<MatrixType>::RealScalar</div>
<div class="line"><a name="l00373"></a><span class="lineno"> 373</span> <a class="code" href="classEigen_1_1MatrixPower.html">MatrixPower<MatrixType>::modfAndInit</a>(RealScalar x, RealScalar* intpart)</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">typedef</span> Array<RealScalar, RowsAtCompileTime, 1, ColMajor, MaxRowsAtCompileTime> RealArray;</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>  *intpart = std::floor(x);</div>
<div class="line"><a name="l00378"></a><span class="lineno"> 378</span>  RealScalar res = x - *intpart;</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>  <span class="keywordflow">if</span> (!m_conditionNumber && res) {</div>
<div class="line"><a name="l00381"></a><span class="lineno"> 381</span>  <span class="keyword">const</span> ComplexSchur<MatrixType> schurOfA(m_A);</div>
<div class="line"><a name="l00382"></a><span class="lineno"> 382</span>  m_T = schurOfA.matrixT();</div>
<div class="line"><a name="l00383"></a><span class="lineno"> 383</span>  m_U = schurOfA.matrixU();</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="keyword">const</span> RealArray absTdiag = m_T.diagonal().array().abs();</div>
<div class="line"><a name="l00386"></a><span class="lineno"> 386</span>  m_conditionNumber = absTdiag.maxCoeff() / absTdiag.minCoeff();</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> </div>
<div class="line"><a name="l00389"></a><span class="lineno"> 389</span>  <span class="keywordflow">if</span> (res>RealScalar(0.5) && res>(1-res)*std::pow(m_conditionNumber, res)) {</div>
<div class="line"><a name="l00390"></a><span class="lineno"> 390</span>  --res;</div>
<div class="line"><a name="l00391"></a><span class="lineno"> 391</span>  ++*intpart;</div>
<div class="line"><a name="l00392"></a><span class="lineno"> 392</span>  }</div>
<div class="line"><a name="l00393"></a><span class="lineno"> 393</span>  <span class="keywordflow">return</span> res;</div>
<div class="line"><a name="l00394"></a><span class="lineno"> 394</span> }</div>
<div class="line"><a name="l00395"></a><span class="lineno"> 395</span> </div>
<div class="line"><a name="l00396"></a><span class="lineno"> 396</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00397"></a><span class="lineno"> 397</span> <span class="keyword">template</span><<span class="keyword">typename</span> ResultType></div>
<div class="line"><a name="l00398"></a><span class="lineno"> 398</span> <span class="keywordtype">void</span> MatrixPower<MatrixType>::computeIntPower(ResultType& res, RealScalar p)</div>
<div class="line"><a name="l00399"></a><span class="lineno"> 399</span> {</div>
<div class="line"><a name="l00400"></a><span class="lineno"> 400</span>  RealScalar pp = std::abs(p);</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="keywordflow">if</span> (p<0) m_tmp = m_A.inverse();</div>
<div class="line"><a name="l00403"></a><span class="lineno"> 403</span>  <span class="keywordflow">else</span> m_tmp = m_A;</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>  res = MatrixType::Identity(rows(), cols());</div>
<div class="line"><a name="l00406"></a><span class="lineno"> 406</span>  <span class="keywordflow">while</span> (pp >= 1) {</div>
<div class="line"><a name="l00407"></a><span class="lineno"> 407</span>  <span class="keywordflow">if</span> (std::fmod(pp, 2) >= 1)</div>
<div class="line"><a name="l00408"></a><span class="lineno"> 408</span>  res = m_tmp * res;</div>
<div class="line"><a name="l00409"></a><span class="lineno"> 409</span>  m_tmp *= m_tmp;</div>
<div class="line"><a name="l00410"></a><span class="lineno"> 410</span>  pp /= 2;</div>
<div class="line"><a name="l00411"></a><span class="lineno"> 411</span>  }</div>
<div class="line"><a name="l00412"></a><span class="lineno"> 412</span> }</div>
<div class="line"><a name="l00413"></a><span class="lineno"> 413</span> </div>
<div class="line"><a name="l00414"></a><span class="lineno"> 414</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00415"></a><span class="lineno"> 415</span> <span class="keyword">template</span><<span class="keyword">typename</span> ResultType></div>
<div class="line"><a name="l00416"></a><span class="lineno"> 416</span> <span class="keywordtype">void</span> MatrixPower<MatrixType>::computeFracPower(ResultType& res, RealScalar p)</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="keywordflow">if</span> (p) {</div>
<div class="line"><a name="l00419"></a><span class="lineno"> 419</span>  eigen_assert(m_conditionNumber);</div>
<div class="line"><a name="l00420"></a><span class="lineno"> 420</span>  MatrixPowerAtomic<ComplexMatrix>(m_T, p).compute(m_fT);</div>
<div class="line"><a name="l00421"></a><span class="lineno"> 421</span>  revertSchur(m_tmp, m_fT, m_U);</div>
<div class="line"><a name="l00422"></a><span class="lineno"> 422</span>  res = m_tmp * res;</div>
<div class="line"><a name="l00423"></a><span class="lineno"> 423</span>  }</div>
<div class="line"><a name="l00424"></a><span class="lineno"> 424</span> }</div>
<div class="line"><a name="l00425"></a><span class="lineno"> 425</span> </div>
<div class="line"><a name="l00426"></a><span class="lineno"> 426</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00427"></a><span class="lineno"> 427</span> <span class="keyword">template</span><<span class="keywordtype">int</span> Rows, <span class="keywordtype">int</span> Cols, <span class="keywordtype">int</span> Options, <span class="keywordtype">int</span> MaxRows, <span class="keywordtype">int</span> MaxCols></div>
<div class="line"><a name="l00428"></a><span class="lineno"> 428</span> <span class="keyword">inline</span> <span class="keywordtype">void</span> MatrixPower<MatrixType>::revertSchur(</div>
<div class="line"><a name="l00429"></a><span class="lineno"> 429</span>  Matrix<ComplexScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,</div>
<div class="line"><a name="l00430"></a><span class="lineno"> 430</span>  <span class="keyword">const</span> ComplexMatrix& T,</div>
<div class="line"><a name="l00431"></a><span class="lineno"> 431</span>  <span class="keyword">const</span> ComplexMatrix& U)</div>
<div class="line"><a name="l00432"></a><span class="lineno"> 432</span> { res.noalias() = U * (T.template triangularView<Upper>() * U.adjoint()); }</div>
<div class="line"><a name="l00433"></a><span class="lineno"> 433</span> </div>
<div class="line"><a name="l00434"></a><span class="lineno"> 434</span> <span class="keyword">template</span><<span class="keyword">typename</span> MatrixType></div>
<div class="line"><a name="l00435"></a><span class="lineno"> 435</span> <span class="keyword">template</span><<span class="keywordtype">int</span> Rows, <span class="keywordtype">int</span> Cols, <span class="keywordtype">int</span> Options, <span class="keywordtype">int</span> MaxRows, <span class="keywordtype">int</span> MaxCols></div>
<div class="line"><a name="l00436"></a><span class="lineno"> 436</span> <span class="keyword">inline</span> <span class="keywordtype">void</span> MatrixPower<MatrixType>::revertSchur(</div>
<div class="line"><a name="l00437"></a><span class="lineno"> 437</span>  Matrix<RealScalar, Rows, Cols, Options, MaxRows, MaxCols>& res,</div>
<div class="line"><a name="l00438"></a><span class="lineno"> 438</span>  <span class="keyword">const</span> ComplexMatrix& T,</div>
<div class="line"><a name="l00439"></a><span class="lineno"> 439</span>  <span class="keyword">const</span> ComplexMatrix& U)</div>
<div class="line"><a name="l00440"></a><span class="lineno"> 440</span> { res.noalias() = (U * (T.template triangularView<Upper>() * U.adjoint())).real(); }</div>
<div class="line"><a name="l00441"></a><span class="lineno"> 441</span> </div>
<div class="line"><a name="l00455"></a><span class="lineno"> 455</span> <span class="keyword">template</span><<span class="keyword">typename</span> Derived></div>
<div class="line"><a name="l00456"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixPowerReturnValue.html"> 456</a></span> <span class="keyword">class </span><a class="code" href="classEigen_1_1MatrixPowerReturnValue.html">MatrixPowerReturnValue</a> : <span class="keyword">public</span> ReturnByValue< MatrixPowerReturnValue<Derived> ></div>
<div class="line"><a name="l00457"></a><span class="lineno"> 457</span> {</div>
<div class="line"><a name="l00458"></a><span class="lineno"> 458</span>  <span class="keyword">public</span>:</div>
<div class="line"><a name="l00459"></a><span class="lineno"> 459</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> Derived::PlainObject PlainObject;</div>
<div class="line"><a name="l00460"></a><span class="lineno"> 460</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> Derived::RealScalar RealScalar;</div>
<div class="line"><a name="l00461"></a><span class="lineno"> 461</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> Derived::Index Index;</div>
<div class="line"><a name="l00462"></a><span class="lineno"> 462</span> </div>
<div class="line"><a name="l00469"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixPowerReturnValue.html#a9d2732414d64fe27eae61ea71378b186"> 469</a></span>  <a class="code" href="classEigen_1_1MatrixPowerReturnValue.html#a9d2732414d64fe27eae61ea71378b186">MatrixPowerReturnValue</a>(<span class="keyword">const</span> Derived& A, RealScalar p) : m_A(A), m_p(p)</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="l00478"></a><span class="lineno"> 478</span>  <span class="keyword">template</span><<span class="keyword">typename</span> ResultType></div>
<div class="line"><a name="l00479"></a><span class="lineno"><a class="line" href="classEigen_1_1MatrixPowerReturnValue.html#a86a2614cdaae1d0f395c075e5060cf1a"> 479</a></span>  <span class="keyword">inline</span> <span class="keywordtype">void</span> <a class="code" href="classEigen_1_1MatrixPowerReturnValue.html#a86a2614cdaae1d0f395c075e5060cf1a">evalTo</a>(ResultType& res)<span class="keyword"> const</span></div>
<div class="line"><a name="l00480"></a><span class="lineno"> 480</span> <span class="keyword"> </span>{ <a class="code" href="classEigen_1_1MatrixPower.html">MatrixPower<PlainObject></a>(m_A.eval()).compute(res, m_p); }</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>  Index rows()<span class="keyword"> const </span>{ <span class="keywordflow">return</span> m_A.rows(); }</div>
<div class="line"><a name="l00483"></a><span class="lineno"> 483</span>  Index cols()<span class="keyword"> const </span>{ <span class="keywordflow">return</span> m_A.cols(); }</div>
<div class="line"><a name="l00484"></a><span class="lineno"> 484</span> </div>
<div class="line"><a name="l00485"></a><span class="lineno"> 485</span>  <span class="keyword">private</span>:</div>
<div class="line"><a name="l00486"></a><span class="lineno"> 486</span>  <span class="keyword">const</span> Derived& m_A;</div>
<div class="line"><a name="l00487"></a><span class="lineno"> 487</span>  <span class="keyword">const</span> RealScalar m_p;</div>
<div class="line"><a name="l00488"></a><span class="lineno"> 488</span>  <a class="code" href="classEigen_1_1MatrixPowerReturnValue.html#a9d2732414d64fe27eae61ea71378b186">MatrixPowerReturnValue</a>& operator=(<span class="keyword">const</span> <a class="code" href="classEigen_1_1MatrixPowerReturnValue.html#a9d2732414d64fe27eae61ea71378b186">MatrixPowerReturnValue</a>&);</div>
<div class="line"><a name="l00489"></a><span class="lineno"> 489</span> };</div>
<div class="line"><a name="l00490"></a><span class="lineno"> 490</span> </div>
<div class="line"><a name="l00491"></a><span class="lineno"> 491</span> <span class="keyword">namespace </span>internal {</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="keyword">template</span><<span class="keyword">typename</span> MatrixPowerType></div>
<div class="line"><a name="l00494"></a><span class="lineno"> 494</span> <span class="keyword">struct </span>traits< MatrixPowerRetval<MatrixPowerType> ></div>
<div class="line"><a name="l00495"></a><span class="lineno"> 495</span> { <span class="keyword">typedef</span> <span class="keyword">typename</span> MatrixPowerType::PlainObject ReturnType; };</div>
<div class="line"><a name="l00496"></a><span class="lineno"> 496</span> </div>
<div class="line"><a name="l00497"></a><span class="lineno"> 497</span> <span class="keyword">template</span><<span class="keyword">typename</span> Derived></div>
<div class="line"><a name="l00498"></a><span class="lineno"> 498</span> <span class="keyword">struct </span>traits< MatrixPowerReturnValue<Derived> ></div>
<div class="line"><a name="l00499"></a><span class="lineno"> 499</span> { <span class="keyword">typedef</span> <span class="keyword">typename</span> Derived::PlainObject ReturnType; };</div>
<div class="line"><a name="l00500"></a><span class="lineno"> 500</span> </div>
<div class="line"><a name="l00501"></a><span class="lineno"> 501</span> }</div>
<div class="line"><a name="l00502"></a><span class="lineno"> 502</span> </div>
<div class="line"><a name="l00503"></a><span class="lineno"> 503</span> <span class="keyword">template</span><<span class="keyword">typename</span> Derived></div>
<div class="line"><a name="l00504"></a><span class="lineno"> 504</span> <span class="keyword">const</span> MatrixPowerReturnValue<Derived> MatrixBase<Derived>::pow(<span class="keyword">const</span> RealScalar& p)<span class="keyword"> const</span></div>
<div class="line"><a name="l00505"></a><span class="lineno"> 505</span> <span class="keyword"></span>{ <span class="keywordflow">return</span> MatrixPowerReturnValue<Derived>(derived(), p); }</div>
<div class="line"><a name="l00506"></a><span class="lineno"> 506</span> </div>
<div class="line"><a name="l00507"></a><span class="lineno"> 507</span> } <span class="comment">// namespace Eigen</span></div>
<div class="line"><a name="l00508"></a><span class="lineno"> 508</span> </div>
<div class="line"><a name="l00509"></a><span class="lineno"> 509</span> <span class="preprocessor">#endif // EIGEN_MATRIX_POWER</span></div>
<div class="ttc" id="classEigen_1_1MatrixPower_html"><div class="ttname"><a href="classEigen_1_1MatrixPower.html">Eigen::MatrixPower</a></div><div class="ttdoc">Class for computing matrix powers. </div><div class="ttdef"><b>Definition:</b> MatrixPower.h:15</div></div>
<div class="ttc" id="classEigen_1_1MatrixPowerReturnValue_html_a9d2732414d64fe27eae61ea71378b186"><div class="ttname"><a href="classEigen_1_1MatrixPowerReturnValue.html#a9d2732414d64fe27eae61ea71378b186">Eigen::MatrixPowerReturnValue::MatrixPowerReturnValue</a></div><div class="ttdeci">MatrixPowerReturnValue(const Derived &A, RealScalar p)</div><div class="ttdoc">Constructor. </div><div class="ttdef"><b>Definition:</b> MatrixPower.h:469</div></div>
<div class="ttc" id="classEigen_1_1MatrixPower_html_a0c03931b4fff167eeaf6bab5161ae9cd"><div class="ttname"><a href="classEigen_1_1MatrixPower.html#a0c03931b4fff167eeaf6bab5161ae9cd">Eigen::MatrixPower::MatrixPower</a></div><div class="ttdeci">MatrixPower(const MatrixType &A)</div><div class="ttdoc">Constructor. </div><div class="ttdef"><b>Definition:</b> MatrixPower.h:297</div></div>
<div class="ttc" id="classEigen_1_1MatrixPower_html_a95d38473ae16ef259e0d334538d050d6"><div class="ttname"><a href="classEigen_1_1MatrixPower.html#a95d38473ae16ef259e0d334538d050d6">Eigen::MatrixPower::compute</a></div><div class="ttdeci">void compute(ResultType &res, RealScalar p)</div><div class="ttdoc">Compute the matrix power. </div><div class="ttdef"><b>Definition:</b> MatrixPower.h:356</div></div>
<div class="ttc" id="classEigen_1_1MatrixPower_html_a77c0aa664c889e4aac24406385196e8a"><div class="ttname"><a href="classEigen_1_1MatrixPower.html#a77c0aa664c889e4aac24406385196e8a">Eigen::MatrixPower::operator()</a></div><div class="ttdeci">const MatrixPowerRetval< MatrixType > operator()(RealScalar p)</div><div class="ttdoc">Returns the matrix power. </div><div class="ttdef"><b>Definition:</b> MatrixPower.h:307</div></div>
<div class="ttc" id="classEigen_1_1MatrixPowerReturnValue_html_a86a2614cdaae1d0f395c075e5060cf1a"><div class="ttname"><a href="classEigen_1_1MatrixPowerReturnValue.html#a86a2614cdaae1d0f395c075e5060cf1a">Eigen::MatrixPowerReturnValue::evalTo</a></div><div class="ttdeci">void evalTo(ResultType &res) const </div><div class="ttdoc">Compute the matrix power. </div><div class="ttdef"><b>Definition:</b> MatrixPower.h:479</div></div>
<div class="ttc" id="classEigen_1_1MatrixPowerReturnValue_html"><div class="ttname"><a href="classEigen_1_1MatrixPowerReturnValue.html">Eigen::MatrixPowerReturnValue</a></div><div class="ttdoc">Proxy for the matrix power of some matrix (expression). </div><div class="ttdef"><b>Definition:</b> MatrixPower.h:456</div></div>
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