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<div class="title">tql2.hpp</div> </div>
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<a href="tql2_8hpp.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno"> 1</span> <span class="preprocessor">#ifndef VIENNACL_LINALG_TQL2_HPP</span></div>
<div class="line"><a name="l00002"></a><span class="lineno"> 2</span> <span class="preprocessor">#define VIENNACL_LINALG_TQL2_HPP</span></div>
<div class="line"><a name="l00003"></a><span class="lineno"> 3</span> </div>
<div class="line"><a name="l00004"></a><span class="lineno"> 4</span> <span class="comment">/* =========================================================================</span></div>
<div class="line"><a name="l00005"></a><span class="lineno"> 5</span> <span class="comment"> Copyright (c) 2010-2016, Institute for Microelectronics,</span></div>
<div class="line"><a name="l00006"></a><span class="lineno"> 6</span> <span class="comment"> Institute for Analysis and Scientific Computing,</span></div>
<div class="line"><a name="l00007"></a><span class="lineno"> 7</span> <span class="comment"> TU Wien.</span></div>
<div class="line"><a name="l00008"></a><span class="lineno"> 8</span> <span class="comment"> Portions of this software are copyright by UChicago Argonne, LLC.</span></div>
<div class="line"><a name="l00009"></a><span class="lineno"> 9</span> <span class="comment"></span></div>
<div class="line"><a name="l00010"></a><span class="lineno"> 10</span> <span class="comment"> -----------------</span></div>
<div class="line"><a name="l00011"></a><span class="lineno"> 11</span> <span class="comment"> ViennaCL - The Vienna Computing Library</span></div>
<div class="line"><a name="l00012"></a><span class="lineno"> 12</span> <span class="comment"> -----------------</span></div>
<div class="line"><a name="l00013"></a><span class="lineno"> 13</span> <span class="comment"></span></div>
<div class="line"><a name="l00014"></a><span class="lineno"> 14</span> <span class="comment"> Project Head: Karl Rupp rupp@iue.tuwien.ac.at</span></div>
<div class="line"><a name="l00015"></a><span class="lineno"> 15</span> <span class="comment"></span></div>
<div class="line"><a name="l00016"></a><span class="lineno"> 16</span> <span class="comment"> (A list of authors and contributors can be found in the manual)</span></div>
<div class="line"><a name="l00017"></a><span class="lineno"> 17</span> <span class="comment"></span></div>
<div class="line"><a name="l00018"></a><span class="lineno"> 18</span> <span class="comment"> License: MIT (X11), see file LICENSE in the base directory</span></div>
<div class="line"><a name="l00019"></a><span class="lineno"> 19</span> <span class="comment">============================================================================= */</span></div>
<div class="line"><a name="l00020"></a><span class="lineno"> 20</span> </div>
<div class="line"><a name="l00026"></a><span class="lineno"> 26</span> <span class="preprocessor">#include "<a class="code" href="vector_8hpp.html">viennacl/vector.hpp</a>"</span></div>
<div class="line"><a name="l00027"></a><span class="lineno"> 27</span> <span class="preprocessor">#include "<a class="code" href="matrix_8hpp.html">viennacl/matrix.hpp</a>"</span></div>
<div class="line"><a name="l00028"></a><span class="lineno"> 28</span> <span class="preprocessor">#include <iomanip></span></div>
<div class="line"><a name="l00029"></a><span class="lineno"> 29</span> </div>
<div class="line"><a name="l00030"></a><span class="lineno"> 30</span> <span class="preprocessor">#include "<a class="code" href="qr-method-common_8hpp.html">viennacl/linalg/qr-method-common.hpp</a>"</span></div>
<div class="line"><a name="l00031"></a><span class="lineno"> 31</span> <span class="preprocessor">#include "<a class="code" href="prod_8hpp.html">viennacl/linalg/prod.hpp</a>"</span></div>
<div class="line"><a name="l00032"></a><span class="lineno"> 32</span> </div>
<div class="line"><a name="l00033"></a><span class="lineno"> 33</span> <span class="keyword">namespace </span>viennacl</div>
<div class="line"><a name="l00034"></a><span class="lineno"> 34</span> {</div>
<div class="line"><a name="l00035"></a><span class="lineno"> 35</span> <span class="keyword">namespace </span>linalg</div>
<div class="line"><a name="l00036"></a><span class="lineno"> 36</span> {</div>
<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>  <span class="comment">// Symmetric tridiagonal QL algorithm.</span></div>
<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>  <span class="comment">// This is derived from the Algol procedures tql1, by Bowdler, Martin, Reinsch, and Wilkinson,</span></div>
<div class="line"><a name="l00039"></a><span class="lineno"> 39</span>  <span class="comment">// Handbook for Auto. Comp., Vol.ii-Linear Algebra, and the corresponding Fortran subroutine in EISPACK.</span></div>
<div class="line"><a name="l00040"></a><span class="lineno"> 40</span>  <span class="keyword">template</span> <<span class="keyword">typename</span> SCALARTYPE, <span class="keyword">typename</span> VectorType></div>
<div class="line"><a name="l00041"></a><span class="lineno"><a class="line" href="namespaceviennacl_1_1linalg.html#a528690f82b77f9fc12ee90038ee404ea"> 41</a></span>  <span class="keywordtype">void</span> <a class="code" href="namespaceviennacl_1_1linalg.html#a528690f82b77f9fc12ee90038ee404ea">tql1</a>(<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a> n,</div>
<div class="line"><a name="l00042"></a><span class="lineno"> 42</span>  VectorType & d,</div>
<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>  VectorType & e)</div>
<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>  {</div>
<div class="line"><a name="l00045"></a><span class="lineno"> 45</span>  <span class="keywordflow">for</span> (<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a> i = 1; i < n; i++)</div>
<div class="line"><a name="l00046"></a><span class="lineno"> 46</span>  e[i - 1] = e[i];</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> </div>
<div class="line"><a name="l00049"></a><span class="lineno"> 49</span>  e[n - 1] = 0;</div>
<div class="line"><a name="l00050"></a><span class="lineno"> 50</span> </div>
<div class="line"><a name="l00051"></a><span class="lineno"> 51</span>  SCALARTYPE f = 0.;</div>
<div class="line"><a name="l00052"></a><span class="lineno"> 52</span>  SCALARTYPE tst1 = 0.;</div>
<div class="line"><a name="l00053"></a><span class="lineno"> 53</span>  SCALARTYPE eps = <span class="keyword">static_cast<</span>SCALARTYPE<span class="keyword">></span>(1e-6);</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> </div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>  <span class="keywordflow">for</span> (<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a> l = 0; l < n; l++)</div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>  {</div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>  <span class="comment">// Find small subdiagonal element.</span></div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>  tst1 = std::max<SCALARTYPE>(tst1, std::fabs(d[l]) + std::fabs(e[l]));</div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>  <a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a> m = l;</div>
<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>  <span class="keywordflow">while</span> (m < n)</div>
<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>  {</div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>  <span class="keywordflow">if</span> (std::fabs(e[m]) <= eps * tst1)</div>
<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>  <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00065"></a><span class="lineno"> 65</span>  m++;</div>
<div class="line"><a name="l00066"></a><span class="lineno"> 66</span>  }</div>
<div class="line"><a name="l00067"></a><span class="lineno"> 67</span> </div>
<div class="line"><a name="l00068"></a><span class="lineno"> 68</span>  <span class="comment">// If m == l, d[l) is an eigenvalue, otherwise, iterate.</span></div>
<div class="line"><a name="l00069"></a><span class="lineno"> 69</span>  <span class="keywordflow">if</span> (m > l)</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>  <a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a> iter = 0;</div>
<div class="line"><a name="l00072"></a><span class="lineno"> 72</span>  <span class="keywordflow">do</span></div>
<div class="line"><a name="l00073"></a><span class="lineno"> 73</span>  {</div>
<div class="line"><a name="l00074"></a><span class="lineno"> 74</span>  iter = iter + 1; <span class="comment">// (Could check iteration count here.)</span></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="comment">// Compute implicit shift</span></div>
<div class="line"><a name="l00077"></a><span class="lineno"> 77</span>  SCALARTYPE g = d[l];</div>
<div class="line"><a name="l00078"></a><span class="lineno"> 78</span>  SCALARTYPE p = (d[l + 1] - g) / (2 * e[l]);</div>
<div class="line"><a name="l00079"></a><span class="lineno"> 79</span>  SCALARTYPE r = viennacl::linalg::detail::pythag<SCALARTYPE>(p, 1);</div>
<div class="line"><a name="l00080"></a><span class="lineno"> 80</span>  <span class="keywordflow">if</span> (p < 0)</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>  r = -r;</div>
<div class="line"><a name="l00083"></a><span class="lineno"> 83</span>  }</div>
<div class="line"><a name="l00084"></a><span class="lineno"> 84</span> </div>
<div class="line"><a name="l00085"></a><span class="lineno"> 85</span>  d[l] = e[l] / (p + r);</div>
<div class="line"><a name="l00086"></a><span class="lineno"> 86</span>  d[l + 1] = e[l] * (p + r);</div>
<div class="line"><a name="l00087"></a><span class="lineno"> 87</span>  SCALARTYPE h = g - d[l];</div>
<div class="line"><a name="l00088"></a><span class="lineno"> 88</span>  <span class="keywordflow">for</span> (<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a> i = l + 2; i < n; i++)</div>
<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>  {</div>
<div class="line"><a name="l00090"></a><span class="lineno"> 90</span>  d[i] -= h;</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>  f = f + h;</div>
<div class="line"><a name="l00094"></a><span class="lineno"> 94</span> </div>
<div class="line"><a name="l00095"></a><span class="lineno"> 95</span>  <span class="comment">// Implicit QL transformation.</span></div>
<div class="line"><a name="l00096"></a><span class="lineno"> 96</span>  p = d[m];</div>
<div class="line"><a name="l00097"></a><span class="lineno"> 97</span>  SCALARTYPE c = 1;</div>
<div class="line"><a name="l00098"></a><span class="lineno"> 98</span>  SCALARTYPE s = 0;</div>
<div class="line"><a name="l00099"></a><span class="lineno"> 99</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = <span class="keywordtype">int</span>(m - 1); i >= int(l); i--)</div>
<div class="line"><a name="l00100"></a><span class="lineno"> 100</span>  {</div>
<div class="line"><a name="l00101"></a><span class="lineno"> 101</span>  g = c * e[<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a>(i)];</div>
<div class="line"><a name="l00102"></a><span class="lineno"> 102</span>  h = c * p;</div>
<div class="line"><a name="l00103"></a><span class="lineno"> 103</span>  r = viennacl::linalg::detail::pythag<SCALARTYPE>(p, e[<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a>(i)]);</div>
<div class="line"><a name="l00104"></a><span class="lineno"> 104</span>  e[<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a>(i) + 1] = s * r;</div>
<div class="line"><a name="l00105"></a><span class="lineno"> 105</span>  s = e[<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a>(i)] / r;</div>
<div class="line"><a name="l00106"></a><span class="lineno"> 106</span>  c = p / r;</div>
<div class="line"><a name="l00107"></a><span class="lineno"> 107</span>  p = c * d[<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a>(i)] - s * g;</div>
<div class="line"><a name="l00108"></a><span class="lineno"> 108</span>  d[<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a>(i) + 1] = h + s * (c * g + s * d[<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a>(i)]);</div>
<div class="line"><a name="l00109"></a><span class="lineno"> 109</span>  }</div>
<div class="line"><a name="l00110"></a><span class="lineno"> 110</span>  e[l] = s * p;</div>
<div class="line"><a name="l00111"></a><span class="lineno"> 111</span>  d[l] = c * p;</div>
<div class="line"><a name="l00112"></a><span class="lineno"> 112</span>  <span class="comment">// Check for convergence.</span></div>
<div class="line"><a name="l00113"></a><span class="lineno"> 113</span>  }</div>
<div class="line"><a name="l00114"></a><span class="lineno"> 114</span>  <span class="keywordflow">while</span> (std::fabs(e[l]) > eps * tst1);</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>  d[l] = d[l] + f;</div>
<div class="line"><a name="l00117"></a><span class="lineno"> 117</span>  e[l] = 0;</div>
<div class="line"><a name="l00118"></a><span class="lineno"> 118</span>  }</div>
<div class="line"><a name="l00119"></a><span class="lineno"> 119</span>  }</div>
<div class="line"><a name="l00120"></a><span class="lineno"> 120</span> </div>
<div class="line"><a name="l00121"></a><span class="lineno"> 121</span> </div>
<div class="line"><a name="l00122"></a><span class="lineno"> 122</span> </div>
<div class="line"><a name="l00123"></a><span class="lineno"> 123</span> </div>
<div class="line"><a name="l00124"></a><span class="lineno"> 124</span> </div>
<div class="line"><a name="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> <span class="comment">// Symmetric tridiagonal QL algorithm.</span></div>
<div class="line"><a name="l00128"></a><span class="lineno"> 128</span> <span class="comment">// This is derived from the Algol procedures tql2, by Bowdler, Martin, Reinsch, and Wilkinson,</span></div>
<div class="line"><a name="l00129"></a><span class="lineno"> 129</span> <span class="comment">// Handbook for Auto. Comp., Vol.ii-Linear Algebra, and the corresponding Fortran subroutine in EISPACK.</span></div>
<div class="line"><a name="l00130"></a><span class="lineno"> 130</span> <span class="keyword">template</span> <<span class="keyword">typename</span> SCALARTYPE, <span class="keyword">typename</span> VectorType, <span class="keyword">typename</span> F></div>
<div class="line"><a name="l00131"></a><span class="lineno"><a class="line" href="namespaceviennacl_1_1linalg.html#a3fe3eb20d4dcda794259fd6a6d30afb6"> 131</a></span> <span class="keywordtype">void</span> <a class="code" href="namespaceviennacl_1_1linalg.html#a3fe3eb20d4dcda794259fd6a6d30afb6">tql2</a>(<a class="code" href="classviennacl_1_1matrix__base.html">matrix_base<SCALARTYPE, F></a> & Q,</div>
<div class="line"><a name="l00132"></a><span class="lineno"> 132</span>  VectorType & d,</div>
<div class="line"><a name="l00133"></a><span class="lineno"> 133</span>  VectorType & e)</div>
<div class="line"><a name="l00134"></a><span class="lineno"> 134</span> {</div>
<div class="line"><a name="l00135"></a><span class="lineno"> 135</span>  <a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a> n = <span class="keyword">static_cast<</span><a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a><span class="keyword">></span>(<a class="code" href="namespaceviennacl_1_1traits.html#aa756f5d6820722094cae0d8b9bb6d5e2">viennacl::traits::size1</a>(Q));</div>
<div class="line"><a name="l00136"></a><span class="lineno"> 136</span> </div>
<div class="line"><a name="l00137"></a><span class="lineno"> 137</span>  <span class="comment">//boost::numeric::ublas::vector<SCALARTYPE> cs(n), ss(n);</span></div>
<div class="line"><a name="l00138"></a><span class="lineno"> 138</span>  std::vector<SCALARTYPE> cs(n), ss(n);</div>
<div class="line"><a name="l00139"></a><span class="lineno"> 139</span>  <a class="code" href="classviennacl_1_1vector.html">viennacl::vector<SCALARTYPE></a> tmp1(n), tmp2(n);</div>
<div class="line"><a name="l00140"></a><span class="lineno"> 140</span> </div>
<div class="line"><a name="l00141"></a><span class="lineno"> 141</span>  <span class="keywordflow">for</span> (<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a> i = 1; i < n; i++)</div>
<div class="line"><a name="l00142"></a><span class="lineno"> 142</span>  e[i - 1] = e[i];</div>
<div class="line"><a name="l00143"></a><span class="lineno"> 143</span> </div>
<div class="line"><a name="l00144"></a><span class="lineno"> 144</span>  e[n - 1] = 0;</div>
<div class="line"><a name="l00145"></a><span class="lineno"> 145</span> </div>
<div class="line"><a name="l00146"></a><span class="lineno"> 146</span>  SCALARTYPE f = 0;</div>
<div class="line"><a name="l00147"></a><span class="lineno"> 147</span>  SCALARTYPE tst1 = 0;</div>
<div class="line"><a name="l00148"></a><span class="lineno"> 148</span>  SCALARTYPE eps = <span class="keyword">static_cast<</span>SCALARTYPE<span class="keyword">></span>(viennacl::linalg::detail::EPS);</div>
<div class="line"><a name="l00149"></a><span class="lineno"> 149</span> </div>
<div class="line"><a name="l00150"></a><span class="lineno"> 150</span>  <span class="keywordflow">for</span> (<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a> l = 0; l < n; l++)</div>
<div class="line"><a name="l00151"></a><span class="lineno"> 151</span>  {</div>
<div class="line"><a name="l00152"></a><span class="lineno"> 152</span>  <span class="comment">// Find small subdiagonal element.</span></div>
<div class="line"><a name="l00153"></a><span class="lineno"> 153</span>  tst1 = std::max<SCALARTYPE>(tst1, std::fabs(d[l]) + std::fabs(e[l]));</div>
<div class="line"><a name="l00154"></a><span class="lineno"> 154</span>  <a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a> m = l;</div>
<div class="line"><a name="l00155"></a><span class="lineno"> 155</span>  <span class="keywordflow">while</span> (m < n)</div>
<div class="line"><a name="l00156"></a><span class="lineno"> 156</span>  {</div>
<div class="line"><a name="l00157"></a><span class="lineno"> 157</span>  <span class="keywordflow">if</span> (std::fabs(e[m]) <= eps * tst1)</div>
<div class="line"><a name="l00158"></a><span class="lineno"> 158</span>  <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00159"></a><span class="lineno"> 159</span>  m++;</div>
<div class="line"><a name="l00160"></a><span class="lineno"> 160</span>  }</div>
<div class="line"><a name="l00161"></a><span class="lineno"> 161</span> </div>
<div class="line"><a name="l00162"></a><span class="lineno"> 162</span>  <span class="comment">// If m == l, d[l) is an eigenvalue, otherwise, iterate.</span></div>
<div class="line"><a name="l00163"></a><span class="lineno"> 163</span>  <span class="keywordflow">if</span> (m > l)</div>
<div class="line"><a name="l00164"></a><span class="lineno"> 164</span>  {</div>
<div class="line"><a name="l00165"></a><span class="lineno"> 165</span>  <a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a> iter = 0;</div>
<div class="line"><a name="l00166"></a><span class="lineno"> 166</span>  <span class="keywordflow">do</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>  iter = iter + 1; <span class="comment">// (Could check iteration count here.)</span></div>
<div class="line"><a name="l00169"></a><span class="lineno"> 169</span> </div>
<div class="line"><a name="l00170"></a><span class="lineno"> 170</span>  <span class="comment">// Compute implicit shift</span></div>
<div class="line"><a name="l00171"></a><span class="lineno"> 171</span>  SCALARTYPE g = d[l];</div>
<div class="line"><a name="l00172"></a><span class="lineno"> 172</span>  SCALARTYPE p = (d[l + 1] - g) / (2 * e[l]);</div>
<div class="line"><a name="l00173"></a><span class="lineno"> 173</span>  SCALARTYPE r = viennacl::linalg::detail::pythag<SCALARTYPE>(p, 1);</div>
<div class="line"><a name="l00174"></a><span class="lineno"> 174</span>  <span class="keywordflow">if</span> (p < 0)</div>
<div class="line"><a name="l00175"></a><span class="lineno"> 175</span>  {</div>
<div class="line"><a name="l00176"></a><span class="lineno"> 176</span>  r = -r;</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> </div>
<div class="line"><a name="l00179"></a><span class="lineno"> 179</span>  d[l] = e[l] / (p + r);</div>
<div class="line"><a name="l00180"></a><span class="lineno"> 180</span>  d[l + 1] = e[l] * (p + r);</div>
<div class="line"><a name="l00181"></a><span class="lineno"> 181</span>  SCALARTYPE dl1 = d[l + 1];</div>
<div class="line"><a name="l00182"></a><span class="lineno"> 182</span>  SCALARTYPE h = g - d[l];</div>
<div class="line"><a name="l00183"></a><span class="lineno"> 183</span>  <span class="keywordflow">for</span> (<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a> i = l + 2; i < n; i++)</div>
<div class="line"><a name="l00184"></a><span class="lineno"> 184</span>  {</div>
<div class="line"><a name="l00185"></a><span class="lineno"> 185</span>  d[i] -= h;</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>  f = f + h;</div>
<div class="line"><a name="l00189"></a><span class="lineno"> 189</span> </div>
<div class="line"><a name="l00190"></a><span class="lineno"> 190</span>  <span class="comment">// Implicit QL transformation.</span></div>
<div class="line"><a name="l00191"></a><span class="lineno"> 191</span>  p = d[m];</div>
<div class="line"><a name="l00192"></a><span class="lineno"> 192</span>  SCALARTYPE c = 1;</div>
<div class="line"><a name="l00193"></a><span class="lineno"> 193</span>  SCALARTYPE c2 = c;</div>
<div class="line"><a name="l00194"></a><span class="lineno"> 194</span>  SCALARTYPE c3 = c;</div>
<div class="line"><a name="l00195"></a><span class="lineno"> 195</span>  SCALARTYPE el1 = e[l + 1];</div>
<div class="line"><a name="l00196"></a><span class="lineno"> 196</span>  SCALARTYPE s = 0;</div>
<div class="line"><a name="l00197"></a><span class="lineno"> 197</span>  SCALARTYPE <a class="code" href="global__variables_8cpp.html#a8163feae57063e5c0bb2e640ff4e8c7a">s2</a> = 0;</div>
<div class="line"><a name="l00198"></a><span class="lineno"> 198</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = <span class="keywordtype">int</span>(m - 1); i >= int(l); i--)</div>
<div class="line"><a name="l00199"></a><span class="lineno"> 199</span>  {</div>
<div class="line"><a name="l00200"></a><span class="lineno"> 200</span>  c3 = c2;</div>
<div class="line"><a name="l00201"></a><span class="lineno"> 201</span>  c2 = c;</div>
<div class="line"><a name="l00202"></a><span class="lineno"> 202</span>  s2 = s;</div>
<div class="line"><a name="l00203"></a><span class="lineno"> 203</span>  g = c * e[<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a>(i)];</div>
<div class="line"><a name="l00204"></a><span class="lineno"> 204</span>  h = c * p;</div>
<div class="line"><a name="l00205"></a><span class="lineno"> 205</span>  r = <a class="code" href="namespaceviennacl_1_1linalg_1_1detail.html#a0737c94fac817b0d0c8444463be7ceb7">viennacl::linalg::detail::pythag</a>(p, e[<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a>(i)]);</div>
<div class="line"><a name="l00206"></a><span class="lineno"> 206</span>  e[<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a>(i) + 1] = s * r;</div>
<div class="line"><a name="l00207"></a><span class="lineno"> 207</span>  s = e[<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a>(i)] / r;</div>
<div class="line"><a name="l00208"></a><span class="lineno"> 208</span>  c = p / r;</div>
<div class="line"><a name="l00209"></a><span class="lineno"> 209</span>  p = c * d[<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a>(i)] - s * g;</div>
<div class="line"><a name="l00210"></a><span class="lineno"> 210</span>  d[<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a>(i) + 1] = h + s * (c * g + s * d[<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a>(i)]);</div>
<div class="line"><a name="l00211"></a><span class="lineno"> 211</span> </div>
<div class="line"><a name="l00212"></a><span class="lineno"> 212</span> </div>
<div class="line"><a name="l00213"></a><span class="lineno"> 213</span>  cs[<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a>(i)] = c;</div>
<div class="line"><a name="l00214"></a><span class="lineno"> 214</span>  ss[<a class="code" href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">vcl_size_t</a>(i)] = s;</div>
<div class="line"><a name="l00215"></a><span class="lineno"> 215</span>  }</div>
<div class="line"><a name="l00216"></a><span class="lineno"> 216</span> </div>
<div class="line"><a name="l00217"></a><span class="lineno"> 217</span> </div>
<div class="line"><a name="l00218"></a><span class="lineno"> 218</span>  p = -s * s2 * c3 * el1 * e[l] / dl1;</div>
<div class="line"><a name="l00219"></a><span class="lineno"> 219</span>  e[l] = s * p;</div>
<div class="line"><a name="l00220"></a><span class="lineno"> 220</span>  d[l] = c * p;</div>
<div class="line"><a name="l00221"></a><span class="lineno"> 221</span> </div>
<div class="line"><a name="l00222"></a><span class="lineno"> 222</span>  <a class="code" href="namespaceviennacl.html#a10b7f8cf6b8864a7aa196d670481a453">viennacl::copy</a>(cs, tmp1);</div>
<div class="line"><a name="l00223"></a><span class="lineno"> 223</span>  <a class="code" href="namespaceviennacl.html#a10b7f8cf6b8864a7aa196d670481a453">viennacl::copy</a>(ss, tmp2);</div>
<div class="line"><a name="l00224"></a><span class="lineno"> 224</span> </div>
<div class="line"><a name="l00225"></a><span class="lineno"> 225</span>  <a class="code" href="namespaceviennacl_1_1linalg.html#a22c5bc49dfdc3c85b101694cca8d23cb">viennacl::linalg::givens_next</a>(Q, tmp1, tmp2, <span class="keywordtype">int</span>(l), <span class="keywordtype">int</span>(m));</div>
<div class="line"><a name="l00226"></a><span class="lineno"> 226</span> </div>
<div class="line"><a name="l00227"></a><span class="lineno"> 227</span>  <span class="comment">// Check for convergence.</span></div>
<div class="line"><a name="l00228"></a><span class="lineno"> 228</span>  }</div>
<div class="line"><a name="l00229"></a><span class="lineno"> 229</span>  <span class="keywordflow">while</span> (std::fabs(e[l]) > eps * tst1);</div>
<div class="line"><a name="l00230"></a><span class="lineno"> 230</span>  }</div>
<div class="line"><a name="l00231"></a><span class="lineno"> 231</span>  d[l] = d[l] + f;</div>
<div class="line"><a name="l00232"></a><span class="lineno"> 232</span>  e[l] = 0;</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> </div>
<div class="line"><a name="l00235"></a><span class="lineno"> 235</span>  <span class="comment">// Sort eigenvalues and corresponding vectors.</span></div>
<div class="line"><a name="l00236"></a><span class="lineno"> 236</span> <span class="comment">/*</span></div>
<div class="line"><a name="l00237"></a><span class="lineno"> 237</span> <span class="comment"> for (int i = 0; i < n-1; i++) {</span></div>
<div class="line"><a name="l00238"></a><span class="lineno"> 238</span> <span class="comment"> int k = i;</span></div>
<div class="line"><a name="l00239"></a><span class="lineno"> 239</span> <span class="comment"> SCALARTYPE p = d[i];</span></div>
<div class="line"><a name="l00240"></a><span class="lineno"> 240</span> <span class="comment"> for (int j = i+1; j < n; j++) {</span></div>
<div class="line"><a name="l00241"></a><span class="lineno"> 241</span> <span class="comment"> if (d[j] > p) {</span></div>
<div class="line"><a name="l00242"></a><span class="lineno"> 242</span> <span class="comment"> k = j;</span></div>
<div class="line"><a name="l00243"></a><span class="lineno"> 243</span> <span class="comment"> p = d[j);</span></div>
<div class="line"><a name="l00244"></a><span class="lineno"> 244</span> <span class="comment"> }</span></div>
<div class="line"><a name="l00245"></a><span class="lineno"> 245</span> <span class="comment"> }</span></div>
<div class="line"><a name="l00246"></a><span class="lineno"> 246</span> <span class="comment"> if (k != i) {</span></div>
<div class="line"><a name="l00247"></a><span class="lineno"> 247</span> <span class="comment"> d[k] = d[i];</span></div>
<div class="line"><a name="l00248"></a><span class="lineno"> 248</span> <span class="comment"> d[i] = p;</span></div>
<div class="line"><a name="l00249"></a><span class="lineno"> 249</span> <span class="comment"> for (int j = 0; j < n; j++) {</span></div>
<div class="line"><a name="l00250"></a><span class="lineno"> 250</span> <span class="comment"> p = Q(j, i);</span></div>
<div class="line"><a name="l00251"></a><span class="lineno"> 251</span> <span class="comment"> Q(j, i) = Q(j, k);</span></div>
<div class="line"><a name="l00252"></a><span class="lineno"> 252</span> <span class="comment"> Q(j, k) = p;</span></div>
<div class="line"><a name="l00253"></a><span class="lineno"> 253</span> <span class="comment"> }</span></div>
<div class="line"><a name="l00254"></a><span class="lineno"> 254</span> <span class="comment"> }</span></div>
<div class="line"><a name="l00255"></a><span class="lineno"> 255</span> <span class="comment"> }</span></div>
<div class="line"><a name="l00256"></a><span class="lineno"> 256</span> <span class="comment"></span></div>
<div class="line"><a name="l00257"></a><span class="lineno"> 257</span> <span class="comment">*/</span></div>
<div class="line"><a name="l00258"></a><span class="lineno"> 258</span> </div>
<div class="line"><a name="l00259"></a><span class="lineno"> 259</span> }</div>
<div class="line"><a name="l00260"></a><span class="lineno"> 260</span> } <span class="comment">// namespace linalg</span></div>
<div class="line"><a name="l00261"></a><span class="lineno"> 261</span> } <span class="comment">// namespace viennacl</span></div>
<div class="line"><a name="l00262"></a><span class="lineno"> 262</span> <span class="preprocessor">#endif</span></div>
<div class="ttc" id="prod_8hpp_html"><div class="ttname"><a href="prod_8hpp.html">prod.hpp</a></div><div class="ttdoc">Generic interface for matrix-vector and matrix-matrix products. See viennacl/linalg/vector_operations...</div></div>
<div class="ttc" id="matrix_8hpp_html"><div class="ttname"><a href="matrix_8hpp.html">matrix.hpp</a></div><div class="ttdoc">Implementation of the dense matrix class. </div></div>
<div class="ttc" id="namespaceviennacl_1_1traits_html_aa756f5d6820722094cae0d8b9bb6d5e2"><div class="ttname"><a href="namespaceviennacl_1_1traits.html#aa756f5d6820722094cae0d8b9bb6d5e2">viennacl::traits::size1</a></div><div class="ttdeci">vcl_size_t size1(MatrixType const &mat)</div><div class="ttdoc">Generic routine for obtaining the number of rows of a matrix (ViennaCL, uBLAS, etc.) </div><div class="ttdef"><b>Definition:</b> <a href="size_8hpp_source.html#l00163">size.hpp:163</a></div></div>
<div class="ttc" id="namespaceviennacl_1_1linalg_html_a3fe3eb20d4dcda794259fd6a6d30afb6"><div class="ttname"><a href="namespaceviennacl_1_1linalg.html#a3fe3eb20d4dcda794259fd6a6d30afb6">viennacl::linalg::tql2</a></div><div class="ttdeci">void tql2(matrix_base< SCALARTYPE, F > &Q, VectorType &d, VectorType &e)</div><div class="ttdef"><b>Definition:</b> <a href="tql2_8hpp_source.html#l00131">tql2.hpp:131</a></div></div>
<div class="ttc" id="classviennacl_1_1matrix__base_html"><div class="ttname"><a href="classviennacl_1_1matrix__base.html">viennacl::matrix_base</a></div><div class="ttdef"><b>Definition:</b> <a href="matrix__def_8hpp_source.html#l00103">matrix_def.hpp:103</a></div></div>
<div class="ttc" id="global__variables_8cpp_html_a8163feae57063e5c0bb2e640ff4e8c7a"><div class="ttname"><a href="global__variables_8cpp.html#a8163feae57063e5c0bb2e640ff4e8c7a">s2</a></div><div class="ttdeci">viennacl::scalar< int > s2</div><div class="ttdef"><b>Definition:</b> <a href="global__variables_8cpp_source.html#l00058">global_variables.cpp:58</a></div></div>
<div class="ttc" id="qr-method-common_8hpp_html"><div class="ttname"><a href="qr-method-common_8hpp.html">qr-method-common.hpp</a></div><div class="ttdoc">Common routines used for the QR method and SVD. Experimental. </div></div>
<div class="ttc" id="namespaceviennacl_html_a98a0afcc513111ffa0bd138f891930df"><div class="ttname"><a href="namespaceviennacl.html#a98a0afcc513111ffa0bd138f891930df">viennacl::vcl_size_t</a></div><div class="ttdeci">std::size_t vcl_size_t</div><div class="ttdef"><b>Definition:</b> <a href="forwards_8h_source.html#l00075">forwards.h:75</a></div></div>
<div class="ttc" id="classviennacl_1_1vector_html"><div class="ttname"><a href="classviennacl_1_1vector.html">viennacl::vector</a></div><div class="ttdef"><b>Definition:</b> <a href="forwards_8h_source.html#l00266">forwards.h:266</a></div></div>
<div class="ttc" id="namespaceviennacl_1_1linalg_1_1detail_html_a0737c94fac817b0d0c8444463be7ceb7"><div class="ttname"><a href="namespaceviennacl_1_1linalg_1_1detail.html#a0737c94fac817b0d0c8444463be7ceb7">viennacl::linalg::detail::pythag</a></div><div class="ttdeci">SCALARTYPE pythag(SCALARTYPE a, SCALARTYPE b)</div><div class="ttdef"><b>Definition:</b> <a href="qr-method-common_8hpp_source.html#l00065">qr-method-common.hpp:65</a></div></div>
<div class="ttc" id="vector_8hpp_html"><div class="ttname"><a href="vector_8hpp.html">vector.hpp</a></div><div class="ttdoc">The vector type with operator-overloads and proxy classes is defined here. Linear algebra operations ...</div></div>
<div class="ttc" id="namespaceviennacl_html_a10b7f8cf6b8864a7aa196d670481a453"><div class="ttname"><a href="namespaceviennacl.html#a10b7f8cf6b8864a7aa196d670481a453">viennacl::copy</a></div><div class="ttdeci">void copy(std::vector< NumericT > &cpu_vec, circulant_matrix< NumericT, AlignmentV > &gpu_mat)</div><div class="ttdoc">Copies a circulant matrix from the std::vector to the OpenCL device (either GPU or multi-core CPU) ...</div><div class="ttdef"><b>Definition:</b> <a href="circulant__matrix_8hpp_source.html#l00150">circulant_matrix.hpp:150</a></div></div>
<div class="ttc" id="namespaceviennacl_1_1linalg_html_a528690f82b77f9fc12ee90038ee404ea"><div class="ttname"><a href="namespaceviennacl_1_1linalg.html#a528690f82b77f9fc12ee90038ee404ea">viennacl::linalg::tql1</a></div><div class="ttdeci">void tql1(vcl_size_t n, VectorType &d, VectorType &e)</div><div class="ttdef"><b>Definition:</b> <a href="tql2_8hpp_source.html#l00041">tql2.hpp:41</a></div></div>
<div class="ttc" id="namespaceviennacl_1_1linalg_html_a22c5bc49dfdc3c85b101694cca8d23cb"><div class="ttname"><a href="namespaceviennacl_1_1linalg.html#a22c5bc49dfdc3c85b101694cca8d23cb">viennacl::linalg::givens_next</a></div><div class="ttdeci">void givens_next(matrix_base< NumericT > &Q, vector_base< NumericT > &tmp1, vector_base< NumericT > &tmp2, int l, int m)</div><div class="ttdoc">This function updates the matrix Q. It is part of the tql2 algorithm. </div><div class="ttdef"><b>Definition:</b> <a href="matrix__operations_8hpp_source.html#l01092">matrix_operations.hpp:1092</a></div></div>
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