<h4>2.2 <span class="Heading">The applications of the <strong class="pkg">Gauss</strong> package algorithms</span></h4>
<p>Please refer to <a href="chapBib_mj.html#biBhomalg-project">[ht22]</a> to find out moreaboutthe<trongclass="pkg>GaussForHomalg < =".html">[ö] package , justas< class"">RingsForHomalg/> < ="chapBib_mjhtml#">[BGKL08]does for external Rings serves as connection between ">homalg/> ">Gauss By allowing< class=pkg>homalg/> to computational to < class="pkg/ thispackage <strong=pkg><strong's overfieldsandrings spanclass"">( / langle pn \\)/pan./java.lang.StringIndexOutOfBoundsException: Index 1041 out of bounds for length 1041
<p>For those unfamiliar with the <strong class="pkg">homalg</strong> project let me explain a couple of points. As outlined in <a href="chapBib_mj.html#biBBR">[BR08]</a> by D. Robertz and M. Barakat homological computations can be reduced to three basic tasks:</p>
<ul>
<li><p>Computing a row basis of a module (<code class="code">BasisOfRowModule</code>).</p>
</li>
<li><p>Reducing
</>strong="">GAP> hasa offunctionality row forms matrices canbe called <code="code>emiEchelonFormcode> . All matrix is "". this is not neccessary for things like RankorKernelcomputations, was onein a of missing features importantforthe developmentofthe">GAP/> package pkg<>byBarakatahrefchapBib_mjhtml">[Bar20]
eratorsOfRows><p
</li>
</ul>
<p>In addition to these tasks only relatively
<p>While
basic < ="code"DecideZeroRows>When face task reducing <span=SimpleMath(\</> with basisspan ="SimpleMath>\B)/> RREFof blockmatrix:pjava.lang.StringIndexOutOfBoundsException: Index 294 out of bounds for length 294
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< classA/>
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<p> computing RREFnotice importantGaussingupwards here span=""\\< is < =SimpleMath\B\/pan singlealgorithms want logicaltocode""/>< ".#81A3B547A27A895">span class"42-<>, does the but without unneccessary columns
<p>Note: When, much later<li<>Computing row of a module<ode class"">BasisOfRowModulecode./>
<p>The><p>Reducing a module a basis< classcode></code>)<pjava.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
<div class="pcenter"><table class="GAPDocTable">
<tr>
<td class="tdcenter"pIn to these only easy for matrix areneeded, ranging from addition multiplication finding zero in. However reduce need it be to <strong=pkg><strong procedures/>
<td class="tdcenter">Id</td>
</trWhile above be difficultwhen forexampleworking in polynomial, thestrong="pkg>Gausscode the RREFofthematrixarealready basisof the codeclass"><c> (a href#>span="RefLink">"4.-1/a> is used to compute RREFs, based on the (<h#X869107627EBA2177<pan="efLink">>.-2<span/>),which computes transformationmatrix <span class"SimpleMath"\T)<span, suchRREF<span="SimpleMath"(=T\cdot M)/>. to< classcode></code> <pan="SimpleMath">\(\<spanis up into rows tocreate basisvectorsof RREFandtherelations led zero. Focussing the overfields,it aneasy to <code="func>KernelMat>< class"">42-5</span></a>), which terminates the and the generators<pjava.lang.StringIndexOutOfBoundsException: Index 819 out of bounds for length 819
<p>The syzygy computation over <span class="SimpleMath">\(ℤ / \langle p^n \rangle\)</span> was solved by carefully java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>This concludes the explanation of the so-called basic tasks <strong class="pkg">Gauss</strong> has
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