Quellcodebibliothek Statistik Leitseite products/sources/formale Sprachen/GAP/pkg/gradedringforhomalg/doc/   (Algebra von RWTH Aachen Version 4.15.1©)  Datei vom 9.6.2024 mit Größe 12 kB image not shown  

Quelle  chapA_mj.html   Sprache: HTML

 
 products/sources/formale Sprachen/GAP/pkg/gradedringforhomalg/doc/chapA_mj.html


<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<script type="text/javascript"
  src="https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<title>GAP (GradedRingForHomalg) - Appendix A: The Matrix Tool Operations</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chapA"  onload="jscontent()">


<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chapA_mj.html">A</a>  <a href="chapB_mj.html">B</a>  <a href="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap4_mj.html">[Previous Chapter]</a>    <a href="chapB_mj.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chapA.html">[MathJax off]</a></p>
<p><a id="X7B2993CB7B012115" name="X7B2993CB7B012115"></a></p>
<div class="ChapSects"><a href="chapA_mj.html#X7B2993CB7B012115">A <span class="Heading">The Matrix Tool Operations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA_mj.html#X7988F0AF7D87FD23">A.1 <span class="Heading">The Tool Operations <em>without</em> a Fallback Method</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA_mj.html#X7912E42C81296637">A.2 <span class="Heading">The Tool Operations with a Fallback Method</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapA_mj.html#X84C78C6B7C0890BF">A.2-1 MonomialMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapA_mj.html#X7F9F9C978703E871">A.2-2 RandomMatrixBetweenGradedFreeLeftModules</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapA_mj.html#X82E813D982331C0B">A.2-3 RandomMatrixBetweenGradedFreeRightModules</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapA_mj.html#X7EA1DC07822DE322">A.2-4 Diff</a></span>
</div></div>
</div>

<h3>A <span class="Heading">The Matrix Tool Operations</span></h3>

<p>The functions listed below are components of the <code class="code">homalgTable</code> object stored in the ring. They are only indirectly accessible through standard methods that invoke them.</p>

<p><a id="X7988F0AF7D87FD23" name="X7988F0AF7D87FD23"></a></p>

<h4>A.1 <span class="Heading">The Tool Operations <em>without</em> a Fallback Method</span></h4>

<p>There are matrix methods for which <strong class="pkg">homalg</strong> needs a <code class="code">homalgTable</code> entry for non-internal rings, as it cannot provide a suitable fallback. Below is the list of these <code class="code">homalgTable</code> entries.</p>

<p><a id="X7912E42C81296637" name="X7912E42C81296637"></a></p>

<h4>A.2 <span class="Heading">The Tool Operations with a Fallback Method</span></h4>

<p>These are the methods for which it is recommended for performance reasons to have a <code class="code">homalgTable</code> entry for non-internal rings. <strong class="pkg">homalg</strong> only provides a generic fallback method.</p>

<p><a id="X84C78C6B7C0890BF" name="X84C78C6B7C0890BF"></a></p>

<h5>A.2-1 MonomialMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MonomialMatrix</code>( <var class="Arg">d</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> matrix</p>

<p>The column matrix of <var class="Arg">d</var>-th monomials of the <strong class="pkg">homalg</strong> graded ring <var class="Arg">R</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m := MonomialMatrix( 2, S );</span>
<A ? x 1 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">NumberRows( m );</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">m;</span>
<A 6 x 1 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( m );</span>
x^2,
x*y,
x*z,
y^2,
y*z,
z^2
(over a graded ring) 
</pre></div>

<p><a id="X7F9F9C978703E871" name="X7F9F9C978703E871"></a></p>

<h5>A.2-2 RandomMatrixBetweenGradedFreeLeftModules</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomMatrixBetweenGradedFreeLeftModules</code>( <var class="Arg">degreesS</var>, <var class="Arg">degreesT</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> matrix</p>

<p>A random <span class="SimpleMath">\(r \times c \)</span>-matrix between the graded free <em>left</em> modules <span class="SimpleMath">\(\textit{R}^{(-\textit{degreesS})} \to \textit{R}^{(-\textit{degreesT})}\)</span>, where <span class="SimpleMath">\(r = \)</span><code class="code">Length</code><span class="SimpleMath">\((\)</span><var class="Arg">degreesS</var><span class="SimpleMath">\()\)</span> and <span class="SimpleMath">\(c = \)</span><code class="code">Length</code><span class="SimpleMath">\((\)</span><var class="Arg">degreesT</var><span class="SimpleMath">\()\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rand := RandomMatrixBetweenGradedFreeLeftModules( [ 2, 3, 4 ], [ 1, 2 ], S );</span>
<A 3 x 2 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">#Display( rand );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">#a-2*b+2*c,                                                2,                 </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">#a^2-a*b+b^2-2*b*c+5*c^2,                                  3*c,               </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">#2*a^3-3*a^2*b+2*a*b^2+3*a^2*c+a*b*c-2*b^2*c-3*b*c^2-2*c^3,a^2-4*a*b-3*a*c-c^2</span>
</pre></div>

<p><a id="X82E813D982331C0B" name="X82E813D982331C0B"></a></p>

<h5>A.2-3 RandomMatrixBetweenGradedFreeRightModules</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomMatrixBetweenGradedFreeRightModules</code>( <var class="Arg">degreesT</var>, <var class="Arg">degreesS</var>, <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> matrix</p>

<p>A random <span class="SimpleMath">\(r \times c \)</span>-matrix between the graded free <em>right</em> modules <span class="SimpleMath">\(\textit{R}^{(-\textit{degreesS})} \to \textit{R}^{(-\textit{degreesT})}\)</span>, where <span class="SimpleMath">\(r = \)</span><code class="code">Length</code><span class="SimpleMath">\((\)</span><var class="Arg">degreesT</var><span class="SimpleMath">\()\)</span> and <span class="SimpleMath">\(c = \)</span><code class="code">Length</code><span class="SimpleMath">\((\)</span><var class="Arg">degreesS</var><span class="SimpleMath">\()\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GradedRing( R );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rand := RandomMatrixBetweenGradedFreeRightModules( [ 1, 2 ], [ 2, 3, 4 ], S );</span>
<A 2 x 3 matrix over a graded ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">#Display( rand );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">#a-2*b-c,a*b+b^2-b*c,2*a^3-a*b^2-4*b^3+4*a^2*c-3*a*b*c-b^2*c+a*c^2+5*b*c^2-2*c^3,</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">#-5,     -2*a+c,     -2*a^2-a*b-2*b^2-3*a*c                                      </span>
</pre></div>

<p><a id="X7EA1DC07822DE322" name="X7EA1DC07822DE322"></a></p>

<h5>A.2-4 Diff</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Diff</code>( <var class="Arg">D</var>, <var class="Arg">N</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a <strong class="pkg">homalg</strong> matrix</p>

<p>If <var class="Arg">D</var> is a <span class="SimpleMath">\(f \times p\)</span>-matrix and <var class="Arg">N</var> is a <span class="SimpleMath">\(g \times q\)</span>-matrix then <span class="SimpleMath">\(H=Diff(\)</span><var class="Arg">D</var>,<var class="Arg">N</var><span class="SimpleMath">\()\)</span> is an <span class="SimpleMath">\(fg \times pq\)</span>-matrix whose entry <span class="SimpleMath">\(H[g*(i-1)+j,q*(k-1)+l]\)</span> is the result of differentiating <var class="Arg">N</var><span class="SimpleMath">\([j,l]\)</span> by the differential operator corresponding to <var class="Arg">D</var><span class="SimpleMath">\([i,k]\)</span>. (Here we follow the Macaulay2 convention.)</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := HomalgFieldOfRationalsInDefaultCAS( ) * "a,b,c" * "x,y,z";;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D := HomalgMatrix( "[ \
<span class="GAPprompt">></span> <span class="GAPinput">x,2*y,   \</span>
<span class="GAPprompt">></span> <span class="GAPinput">y,a-b^2, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">z,y-b    \</span>
<span class="GAPprompt">></span> <span class="GAPinput">]", 3, 2, S );
<A 3 x 2 matrix over an external ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := HomalgMatrix( "[ \
<span class="GAPprompt">></span> <span class="GAPinput">x^2-a*y^3,x^3-z^2*y,x*y-b,x*z-c, \</span>
<span class="GAPprompt">></span> <span class="GAPinput">x,        x*y,      a-b,  x*a*b  \</span>
<span class="GAPprompt">></span> <span class="GAPinput">]", 2, 4, S );
<A 2 x 4 matrix over an external ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">H := Diff( D, N );</span>
<A 6 x 8 matrix over an external ring>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( H );</span>
2*x,     3*x^2, y,z,  -6*a*y^2,-2*z^2,2*x,0,  
1,       y,     0,a*b,0,       2*x,   0,  0,  
-3*a*y^2,-z^2,  x,0,  -y^3,    0,     0,  0,  
0,       x,     0,0,  0,       0,     1,  b*x,
0,       -2*y*z,0,x,  -3*a*y^2,-z^2,  x+1,0,  
0,       0,     0,0,  0,       x,     1,  -a*x
</pre></div>


<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap4_mj.html">[Previous Chapter]</a>    <a href="chapB_mj.html">[Next Chapter]</a>   </div>


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chapA_mj.html">A</a>  <a href="chapB_mj.html">B</a>  <a href="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>

100%


¤ Dauer der Verarbeitung: 0.17 Sekunden  (vorverarbeitet)  ¤

*© Formatika GbR, Deutschland




Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.