<?xml version="1.0" encoding="UTF-8" ?>
<!DOCTYPE Runge-Kutta PUBLIC "-//spaceroots.org//DTD Runge-Kutta V1.2//EN"
                             "http://www.spaceroots.org/Runge-Kutta.dtd" >

<!-- $Id: DormandPrince853.xml,v 1.3 2004/05/23 17:41:52 luc Exp $ -->

<Runge-Kutta name="Dormand-Prince 8(5,3)" >

  <time-steps>
    <zero/>
    <quadratic-surd> <p1>12</p1> <p2>-2</p2> <d>6</d> <q>135</q></quadratic-surd>
    <quadratic-surd> <p1> 6</p1> <p2>-1</p2> <d>6</d> <q> 45</q></quadratic-surd>
    <quadratic-surd> <p1> 6</p1> <p2>-1</p2> <d>6</d> <q> 30</q></quadratic-surd>
    <quadratic-surd> <p1> 6</p1> <p2> 1</p2> <d>6</d> <q> 30</q></quadratic-surd>
    <rational> <p>  1</p> <q>  3</q> </rational>
    <rational> <p>  1</p> <q>  4</q> </rational>
    <rational> <p>  4</p> <q> 13</q> </rational>
    <rational> <p>127</p> <q>195</q> </rational>
    <rational> <p>  3</p> <q>  5</q> </rational>
    <rational> <p>  6</p> <q>  7</q> </rational>
    <one/>
  </time-steps>

  <internal-weights>
    <row>
      <!-- empty first row -->
    </row>
    <row>
      <quadratic-surd> <p1>  12</p1> <p2>  -2</p2> <d>6</d> <q> 135</q> </quadratic-surd>
    </row>
    <row>
      <quadratic-surd> <p1>   6</p1> <p2>  -1</p2> <d>6</d> <q> 180</q> </quadratic-surd>
      <quadratic-surd> <p1>   6</p1> <p2>  -1</p2> <d>6</d> <q>  60</q> </quadratic-surd>
    </row>
    <row>
      <quadratic-surd> <p1>   6</p1> <p2>  -1</p2> <d>6</d> <q> 120</q> </quadratic-surd>
      <zero/>
      <quadratic-surd> <p1>   6</p1> <p2>  -1</p2> <d>6</d> <q>  40</q> </quadratic-surd>
    </row>
    <row>
      <quadratic-surd> <p1> 462</p1> <p2> 107</p2> <d>6</d> <q>3000</q> </quadratic-surd>
      <zero/>
      <quadratic-surd> <p1>-402</p1> <p2>-197</p2> <d>6</d> <q>1000</q> </quadratic-surd>
      <quadratic-surd> <p1> 168</p1> <p2>  73</p2> <d>6</d> <q> 375</q> </quadratic-surd>
    </row>
    <row>
      <rational> <p> 1</p> <q> 27</q> </rational>
      <zero/>
      <zero/>
      <quadratic-surd> <p1>  16</p1> <p2>   1</p2> <d>6</d> <q> 108</q> </quadratic-surd>
      <quadratic-surd> <p1>  16</p1> <p2>  -1</p2> <d>6</d> <q> 108</q> </quadratic-surd>
    </row>
    <row>
      <rational> <p>19</p> <q>512</q> </rational>
      <zero/>
      <zero/>
      <quadratic-surd> <p1> 118</p1> <p2>  23</p2> <d>6</d> <q>1024</q> </quadratic-surd>
      <quadratic-surd> <p1> 118</p1> <p2> -23</p2> <d>6</d> <q>1024</q> </quadratic-surd>
      <rational> <p>-9</p> <q>512</q> </rational>
    </row>
    <row>
      <rational> <p>13772</p> <q>371293</q></rational>
      <zero/>
      <zero/>
      <quadratic-surd> <p1>51544</p1> <p2> 4784</p2> <d>6</d> <q> 371293</q> </quadratic-surd>
      <quadratic-surd> <p1>51544</p1> <p2>-4784</p2> <d>6</d> <q> 371293</q> </quadratic-surd>
      <rational> <p>-5688</p> <q>371293</q> </rational>
      <rational> <p> 3072</p> <q>371293</q> </rational>
    </row>
    <row>
      <rational> <p>58656157643</p> <q>93983540625</q> </rational>
      <zero/>
      <zero/>
      <quadratic-surd> <p1>-1324889724104</p1> <p2>-318801444819</p2> <d>6</d> <q>626556937500</q> </quadratic-surd>
      <quadratic-surd> <p1>-1324889724104</p1> <p2> 318801444819</p2> <d>6</d> <q>626556937500</q> </quadratic-surd>
      <rational> <p>  96044563816</p> <q>  3480871875</q> </rational>
      <rational> <p>5682451879168</p> <q>281950621875</q> </rational>
      <rational> <p>  -165125654 </p> <q>     3796875</q> </rational>
    </row>
    <row>
      <rational> <p>    8909899</p> <q>  18653125</q> </rational>
      <zero/>
      <zero/>
      <quadratic-surd> <p1>-4521408</p1> <p2>-1137963</p2> <d>6</d> <q>2937500</q> </quadratic-surd>
      <quadratic-surd> <p1>-4521408</p1> <p2> 1137963</p2> <d>6</d> <q>2937500</q> </quadratic-surd>
      <rational> <p>   96663078</p> <q>   4553125</q> </rational>
      <rational> <p> 2107245056</p> <q> 137915625</q> </rational>
      <rational> <p>-4913652016</p> <q> 147609375</q> </rational>
      <rational> <p>  -78894270</p> <q>3880452869</q> </rational>
    </row>
    <row>
      <rational> <p>     -20401265806</p> <q>      21769653311</q> </rational>
      <zero/>
      <zero/>
      <quadratic-surd> <p1>354216</p1> <p2> 94326</p2> <d>6</d> <q>112847</q> </quadratic-surd>
      <quadratic-surd> <p1>354216</p1> <p2>-94326</p2> <d>6</d> <q>112847</q> </quadratic-surd>
      <rational> <p>     -43306765128</p> <q>       5313852383</q> </rational>
      <rational> <p>  -20866708358144</p> <q>    1126708119789</q> </rational>
      <rational> <p>   14886003438020</p> <q>     654632330667</q> </rational>
      <rational> <p>35290686222309375</p> <q>14152473387134411</q> </rational>
      <rational> <p>      -1477884375</p> <q>        485066827</q> </rational>
    </row>
    <row>
      <rational> <p>          39815761</p> <q>        17514443</q> </rational>
      <zero/>
      <zero/>
      <quadratic-surd> <p1>-3457480</p1> <p2>-960905</p2> <d>6</d> <q>551636</q> </quadratic-surd>
      <quadratic-surd> <p1>-3457480</p1> <p2> 960905</p2> <d>6</d> <q>551636</q> </quadratic-surd>
      <rational> <p>       -844554132 </p> <q>        47026969</q> </rational>
      <rational> <p>        8444996352</p> <q>       302158619</q> </rational>
      <rational> <p>       -2509602342</p> <q>       877790785</q> </rational>
      <rational> <p>-28388795297996250</p> <q>3199510091356783</q> </rational>
      <rational> <p>         226716250</p> <q>        18341897</q> </rational>
      <rational> <p>        1371316744</p> <q>      2131383595</q> </rational>
    </row>
  </internal-weights>

  <estimation-weights>
    <rational> <p>        104257</p> <q>        1920240</q> </rational>
    <zero/>
    <zero/>
    <zero/>
    <zero/>
    <rational> <p>       3399327</p> <q>         763840</q> </rational>
    <rational> <p>      66578432</p> <q>       35198415</q> </rational>
    <rational> <p>   -1674902723</p> <q>      288716400</q> </rational>
    <rational> <p>54980371265625</p> <q>176692375811392</q> </rational>
    <rational> <p>       -734375</p> <q>        4826304</q> </rational>
    <rational> <p>     171414593</p> <q>      851261400</q> </rational>
    <rational> <p>        137909</p> <q>        3084480</q> </rational>
  </estimation-weights>

  <!-- this method uses two error estimations -->
  <!-- one should be commented out while checking the other one -->

  <!-- this error estimation is of order 5 -->
  <error-weights>
    <rational> <p>        116092271</p> <q>      8848465920</q> </rational>
    <zero/>
    <zero/>
    <zero/>
    <zero/>
    <rational> <p>         -1871647</p> <q>         1527680</q> </rational>
    <rational> <p>        -69799717</p> <q>       140793660</q> </rational>
    <rational> <p>    1230164450203</p> <q>    739113984000</q> </rational>
    <rational> <p>-1980813971228885</p> <q>5654156025964544</q> </rational>
    <rational> <p>        464500805</p> <q>      1389975552</q> </rational>
    <rational> <p>    1606764981773</p> <q>  19613062656000</q> </rational>
    <rational> <p>          -137909</p> <q>         6168960</q> </rational>
  </error-weights>

  <!-- this error estimation is of order 3 -->
  <!-- <error-weights>
    <rational> <p>        -364463</p> <q>        1920240</q> </rational>
    <zero/>
    <zero/>
    <zero/>
    <zero/>
    <rational> <p>        3399327</p> <q>         763840</q> </rational>
    <rational> <p>       66578432</p> <q>       35198415</q> </rational>
    <rational> <p>    -1674902723</p> <q>      288716400</q> </rational>
    <rational> <p>-74684743568175</p> <q>176692375811392</q> </rational>
    <rational> <p>        -734375</p> <q>        4826304</q> </rational>
    <rational> <p>      171414593</p> <q>      851261400</q> </rational>
    <rational> <p>          69869</p> <q>        3084480</q> </rational>
  </error-weights> -->

</Runge-Kutta>
