Quellcode-Bibliothek chap2_mj.html   Sprache: HTML

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<h3>2 <span class="Heading">Extending Gauss Functionality</span></h3>

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<h4>2.1 <span class="

p< classpkg</strong a lot functionality for echelon of. These be by < class">pkg">GAP matrix type over fields. However, these algorithms are not capable of computing a reduced row echelon form (RREF) of a matrix,thereisnowayto Gaussupwards".Whileisneccessarythings   computationsthis  anumberfeatures     < =pkg<strong< =">homalg/strong M. < =".#biBhomalg-package]/>/>

<p>Parallel to this development I worked on <strong class="pkg">SCO</strong> <a href</code)./p>

<p>It should

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<ulBy the ( how " " is)< class"impleMath>(A) reduced with">(B\)/>. However, the left side of thematrix justservesthesingle purpose of tricking the Gaussian into doing what wewant. Therefore,itwasa step implement < class=func>ReduceMat<span="RefLink>.-/pan
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<liReducing with (code=""DecideZeroRows)<>

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<li><p>Compute the relations between module elements (<code class="code">SyzygiesGeneratorsOfRows

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<> additionto tasks relatively tools matrixmanipulation needed rangingfrom and to the rows a matrix, to the forcommunication might helpful supply class=pkg"homalg/> with some moreadvanced .

<p> the tasks can quitedifficult when,for example, working noncommutativepolynomial ringsinthe < class"> case they can all be done as long as you can compute a Reduced Row Echelon Form. This is clear for BasisOfRowModule,astherowsof RREF already abasis themodule.< =func"EchelonMat/ode< ="chap4_mj.html#X8499C9FD7AD9908F"< class">21span<> ""GAP internal method ><table class="GAPDocTable">
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<td class="tdcenter<>Approachingthis led tothe method< class=func"EchelonMatTransformation( ="hap4_mj.tml#">< class"efLink"422/></>)  additionally the matrix="(\/span>, such that < class=">\= T cdot\</panSimilar <ode="code"SemiEchelonMatTransformation> < class\T)/>  split the needed create the vectors  the, and the  that to rows on computations fields itwas  step write class"KernelMat/>(ahref=chap4_mjhtmlX78E97A0E7F1ED8AA">span=RefLink4 after REF returns kernel./>
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<p>By computing the RREF (notice how important "Gaussing upwards" is here) <span class="SimpleMath">\(A\)</span> is reduced with <span class="SimpleMath">\(B\)</span>. However, the left

<p>Note: When, much later, it became clear that it was importantjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 0

<p>The third procedure, <code class="code">SygygiesGeneratorsOfRows

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<p>and computing its Row Echelon Form. Then the row relations are generated by the rows to<r>

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<The  over <=">(ℤ/\ p^n\\)"(=(,ldots0h*,\dots) neq0\>gneq)/>  thatspan class="SimpleMath">\(g \cdot h = 0\)</pan thevectorspan="SimpleMath>\g\ )/> as rowvectorhastobereducedandcan reducededuced with work allowed implementation ></code(ahref.#"spanclass="4-<spana  overclass">( / langle ^rangle)/>.

<p>This concludes>

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