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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X84F709227E5EEB55">2.1 <span class="Heading">The need for extended functionality</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7F21EDDF81C27747">2.2 <span class="Heading">The applications of the <strong class="pkg">Gauss</strong> package algorithms</span></a>
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<h3

<>< idX84F709227E5EEB55name"

<h4>2.1 <span class="Heading

<p>strong="pkg>

<p>Parallel to this development I worked on <strong class="pkg">SCO</strong> <a href="chapBib.html#biBSCO">[

<p>It should be quite clear

<> am  totellyouthat  to the for over (2),  waspossible  compute GF2-Rank  the mentioned  inless than0 minutes amemory about.</p

<>22span classHeading applications  < class>auss> <span>

<h4>2.2 <span class

<pPlease to =chapBib#"ht22homalg/> .Most the for strongclass""Gausspackage there. project, strongclass"">GaussForHomalgchapBib>Gr08a/>package which,just < class""><strong<="chapBibhtmlbiBRingsForHomalg"[]/a   externalRingsserves    strong=pkg<strongandclasspkg<strong  strong=pkghomalg delegatetasks< =pkgGauss/>thispackage strong""homalg'scapabilities todense overfields ringsof the form ℤ⟨ ^

<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.html#biBBR">[BR08]</a> by D. Robertz and M. Barakat homological computations can be reduced to three basic tasks:</p>

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<li><p>Compute the relations between

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<pIn to tasks easy for  are, ranging addition multiplicationto the rows  . Howevertoreduce need communication might helpfulto supply strong=pkg</strong> some advanced./>

<p>While the above tasks can be quite difficult when><pCompute relations module (code="">yzygiesGeneratorsOfRows>)./>

<p/

<div>Inaddition to tasksonly easy  matrix  needed  from  multiplication finding  rows a . However reduce  for it be to < class"homalg/> advancedprocedures

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<td <p>While  tasks  quite when, example in  rings  the classpkg</> case can be  long  can aReduced Echelon. This  for =code>asisOfRowModulecode,as rows  RREF  matrix  a  of . code=""EchelonMat>(a =chap4html>span=RefLink.-<span> is to ,  on < classpkg</stronginternal <code="">SemiEchelonMatcode  Row .</p
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<tdpLets at pointbasic < class="code>ecideZeroRows/code:When youfacethetaskreducing a spanc=SimpleMath">A</span> with a given basis <span class="SimpleMath">B</span>, you can compute the RREF of the following block matrix:</p>
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<p>By computing the RREF <div ="">table="GAPDocTable">

<p>Note: When<r>

<pThe third procedure code="">SygygiesGeneratorsOfRows/code,is with relationsbetween rows amatrix rowrepresenting   element  a field these are exactlythe ofthe. One be bytaking amatrix</>

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</<>By   RREF(notice important "Gaussing upwards" is here) <span class="SimpleMath">A</span> is reduced with<span class"SimpleMath>B/span.However, left side of the matrix just serves the oftricking the Gaussian algorithms doingwhat want., was a logical steptoimplementcodeclass"">ReduceMat/code < ="chap4#">span class="RefLink">4.2-3), which does the same thing but without needing unneccessary columns.

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

<p>Approaching this problem led to>

<pThesyzygy computation over<span="SimpleMath"ℤ p />was     basiswithzero-divisingheadIffor < class""> =0,.,,,,..*,h≠,/spanthere <span class=""> ≠ 0</span  thatspanclass"">⋅ =0span  < classSimpleMath>g⋅v/span>isregardedasan  rowvector has be and be . some  thisallowed theimplementationofcodeclass"func">KernelMat<code <a href"chap4.htmlX78E97A0E7F1ED8AA"><span class="RefLink">4.2-5</span></a>) for matrices over <span class="SimpleMath">ℤ / ⟨ p^n ⟩</span>.</p>

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<p>As you can see, the development of hermite algorithms was not continued for dense matrices. There are two reasons for that: <strong class="pkg">GAP</strong> already has very good algorithms for ℤ, and for small matrices the disadvantage of computing over ℤ, potentially leading to coefficient explosion, is marginal.</p>

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