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      <revision>67</revision>
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    <calculation>
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      <header alignment="Center" color="#a9a9a9">&amp;[DATE] &amp;[TIME] - &amp;[FILENAME]</header>
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    <math>
      <input>
        <e type="operand">M</e>
        <e type="function" preserve="true" args="1">rank</e>
      </input>
    </math>
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  <region id="1" left="18" top="54" width="709" height="56" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="eng">
      <p>Returns the description text of a function or variable with the given name. This text
is set using the description context menu option in the math region where the name is
defined.</p>
    </text>
    <text lang="ger">
      <p>Rang einer Matrix M</p>
    </text>
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    <math>
      <input>
        <e type="operand">A</e>
        <e type="operand">0</e>
        <e type="operand">1</e>
        <e type="operand">1</e>
        <e type="operand">0</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operator" args="2">:</e>
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    </math>
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  <region id="3" left="126" top="117" width="68" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">det</e>
      </input>
      <result action="numeric">
        <e type="operand">1</e>
        <e type="operator" args="1">-</e>
      </result>
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    <math>
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">rank</e>
      </input>
      <result action="numeric">
        <e type="operand">2</e>
      </result>
    </math>
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  <region id="5" left="18" top="162" width="68" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">C</e>
        <e type="operand">a</e>
        <e type="operand">b</e>
        <e type="operand">c</e>
        <e type="operand">d</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
  <region id="6" left="126" top="171" width="110" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">C</e>
        <e type="function" preserve="true" args="1">det</e>
      </input>
      <result action="symbolic">
        <e type="operand">a</e>
        <e type="operand">d</e>
        <e type="operator" args="2">*</e>
        <e type="operand">c</e>
        <e type="operand">b</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">-</e>
      </result>
    </math>
  </region>
  <region id="7" left="234" top="171" width="98" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">C</e>
        <e type="function" preserve="true" args="1">rank</e>
      </input>
      <result action="symbolic">
        <e type="operand">2</e>
      </result>
    </math>
  </region>
  <region id="8" left="18" top="216" width="98" height="44" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">D</e>
        <e type="operand">a</e>
        <e type="operand">b</e>
        <e type="operand">2</e>
        <e type="operand">a</e>
        <e type="operator" args="2">*</e>
        <e type="operand">2</e>
        <e type="operand">b</e>
        <e type="operator" args="2">*</e>
        <e type="operand">2</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="6">mat</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
  <region id="9" left="126" top="225" width="59" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">D</e>
        <e type="function" preserve="true" args="1">det</e>
      </input>
      <result action="symbolic">
        <e type="operand">0</e>
      </result>
    </math>
  </region>
  <region id="10" left="234" top="225" width="98" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">D</e>
        <e type="function" preserve="true" args="1">rank</e>
      </input>
      <result action="symbolic">
        <e type="operand">1</e>
      </result>
    </math>
  </region>
  <region id="11" left="18" top="270" width="689" height="72" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p>Die numerische Berechnung von Determinante und Rang ist nicht robust, da die 
Zwischenergebnisse für die interne Zahlendarstellung zu groß werden. Im Einzelfall 
kann die symbolische Auswertung noch sinnvolle Werte liefern, stößt aber auch an 
Grenzen der Zahlendarstellung. </p>
    </text>
  </region>
  <region id="12" left="18" top="342" width="196" height="188" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">A</e>
        <e type="operand">8</e>
        <e type="operand">0</e>
        <e type="operand">4</e>
        <e type="operand">8</e>
        <e type="operand">8</e>
        <e type="operand">2</e>
        <e type="operand">8</e>
        <e type="operand">3</e>
        <e type="operand">3</e>
        <e type="operand">2</e>
        <e type="operand">5</e>
        <e type="operand">3</e>
        <e type="operand">0</e>
        <e type="operand">7</e>
        <e type="operand">4</e>
        <e type="operand">7</e>
        <e type="operand">5</e>
        <e type="operand">4</e>
        <e type="operand">6</e>
        <e type="operand">7</e>
        <e type="operand">5</e>
        <e type="operand">8</e>
        <e type="operand">4</e>
        <e type="operand">1</e>
        <e type="operand">8</e>
        <e type="operand">3</e>
        <e type="operand">1</e>
        <e type="operand">8</e>
        <e type="operand">6</e>
        <e type="operand">9</e>
        <e type="operand">6</e>
        <e type="operand">3</e>
        <e type="operand">5</e>
        <e type="operand">0</e>
        <e type="operand">0</e>
        <e type="operand">3</e>
        <e type="operand">8</e>
        <e type="operand">9</e>
        <e type="operand">8</e>
        <e type="operand">0</e>
        <e type="operand">5</e>
        <e type="operand">8</e>
        <e type="operand">3</e>
        <e type="operand">5</e>
        <e type="operand">8</e>
        <e type="operand">3</e>
        <e type="operand">4</e>
        <e type="operand">6</e>
        <e type="operand">6</e>
        <e type="operand">5</e>
        <e type="operand">9</e>
        <e type="operand">6</e>
        <e type="operand">7</e>
        <e type="operand">2</e>
        <e type="operand">8</e>
        <e type="operand">9</e>
        <e type="operand">6</e>
        <e type="operand">3</e>
        <e type="operand">9</e>
        <e type="operand">5</e>
        <e type="operand">1</e>
        <e type="operand">9</e>
        <e type="operand">6</e>
        <e type="operand">4</e>
        <e type="operand">5</e>
        <e type="operand">1</e>
        <e type="operand">8</e>
        <e type="operand">0</e>
        <e type="operand">6</e>
        <e type="operand">1</e>
        <e type="operand">9</e>
        <e type="operand">5</e>
        <e type="operand">5</e>
        <e type="operand">5</e>
        <e type="operand">1</e>
        <e type="operand">7</e>
        <e type="operand">5</e>
        <e type="operand">8</e>
        <e type="operand">2</e>
        <e type="operand">4</e>
        <e type="operand">8</e>
        <e type="operand">4</e>
        <e type="operand">2</e>
        <e type="operand">1</e>
        <e type="operand">9</e>
        <e type="operand">9</e>
        <e type="operand">5</e>
        <e type="operand">2</e>
        <e type="operand">1</e>
        <e type="operand">1</e>
        <e type="operand">6</e>
        <e type="operand">5</e>
        <e type="operand">9</e>
        <e type="operand">3</e>
        <e type="operand">9</e>
        <e type="operand">5</e>
        <e type="operand">1</e>
        <e type="operand">8</e>
        <e type="operand">0</e>
        <e type="operand">7</e>
        <e type="operand">10</e>
        <e type="operand">10</e>
        <e type="function" preserve="true" args="102">mat</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
  <region id="13" left="225" top="342" width="98" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math error="43">
      <description active="true" position="Top" lang="eng">
        <p>numeric optimization</p>
      </description>
      <description active="true" position="Top" lang="ger">
        <p>Numerische Optimierung</p>
      </description>
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">rank</e>
      </input>
      <result action="numeric">
        <e type="operand">#</e>
      </result>
    </math>
  </region>
  <region id="14" left="225" top="396" width="59" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math error="43">
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">det</e>
      </input>
      <result action="numeric">
        <e type="operand">#</e>
      </result>
    </math>
  </region>
  <region id="15" left="225" top="441" width="106" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <description active="true" position="Top" lang="eng">
        <p>symbolic optimization</p>
      </description>
      <description active="true" position="Top" lang="ger">
        <p>symbolische Optimierung</p>
      </description>
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">rank</e>
      </input>
      <result action="symbolic">
        <e type="operand">10</e>
      </result>
    </math>
  </region>
  <region id="16" left="225" top="477" width="3070" height="50" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">det</e>
      </input>
      <result action="symbolic">
        <e type="operand">2006726199281670</e>
        <e type="operand">32418090381882800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000</e>
        <e type="operand">2</e>
        <e type="operator" args="2">^</e>
        <e type="operator" args="2">*</e>
        <e type="operand">16213093592340300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000</e>
        <e type="operand">165724166765075000000000000000000000000000000000000000000000000000000000</e>
        <e type="operator" args="2">*</e>
        <e type="operator" args="2">/</e>
      </result>
    </math>
  </region>
  <region id="17" left="18" top="540" width="504" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="eng">
      <p>Maple and Matlab have no problems computing the determinant.</p>
    </text>
    <text lang="ger">
      <p>Maple und Matlab finden den Wert problemlos.</p>
    </text>
  </region>
  <region id="18" left="27" top="576" width="178" height="28" color="#000000" bgColor="#ffffff" fontSize="10">
    <math>
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">det</e>
        <e type="function" preserve="true" args="1">maple</e>
      </input>
      <result action="symbolic">
        <e type="operand">784894425</e>
      </result>
    </math>
  </region>
  <region id="19" left="27" top="603" width="172" height="26" color="#000000" bgColor="#ffffff" fontSize="10">
    <math decimalPlaces="10" exponentialThreshold="14">
      <input>
        <e type="operand">A</e>
        <e type="function" preserve="true" args="1">mwdet</e>
      </input>
      <result action="numeric">
        <e type="operand">784894425</e>
      </result>
    </math>
  </region>
  <region id="20" left="18" top="648" width="53" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p href="det.sm">det()</p>
    </text>
  </region>
  <region id="21" left="18" top="684" width="127" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="eng">
      <p href="Function index.sm">Function index</p>
    </text>
    <text lang="ger">
      <p href="Function index.sm">Funktionsindex</p>
    </text>
  </region>
  <region id="22" left="18" top="702" width="78" height="24" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="eng">
      <p href="Handbook.sm">Contents</p>
    </text>
    <text lang="ger">
      <p href="Handbook.sm">Inhalt</p>
    </text>
  </region>
  <region id="23" left="18" top="738" width="450" height="40" color="#000000" bgColor="#ffffff" fontSize="10">
    <text lang="ger">
      <p italic="true">$Author: mkraska $
$Date: 2013-08-25 22:05:19 +0200 (So, 25. Aug 2013) $ </p>
    </text>
  </region>
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