http://apcocoa.uni-passau.de/wiki/api.php?action=feedcontributions&user=132.231.10.62&feedformat=atomApCoCoAWiki - User contributions [en]2024-03-29T07:28:12ZUser contributionsMediaWiki 1.35.0http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.RatPoints&diff=12316ApCoCoA-1:Num.RatPoints2011-11-01T12:16:42Z<p>132.231.10.62: Changed AppBB to BB</p>
<hr />
<div> <command><br />
<title>Num.RatPoints</title><br />
<short_description>Computes the zero set of an exact zero dimensional border basis. The zeros are computed approximately using the eigenvalues of the transposed multiplication matrices.</short_description><br />
<syntax><br />
Num.RatPoints(BB:LIST, OrderIdeal:LIST)):LIST of MAT<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This command computes a set of points, which are the zeros of an exact border basis. This border basis is close to the approximate border basis <tt>AppBB</tt>. The set of (complex) points is represented as two matrices. The <tt>j</tt>-th column of the first matrix gives the real part of a point and the <tt>j</tt>-th column of the second matrix gives the imaginary part. For computation the function is using the <ref>Num.EigenValues</ref> command.<br />
<br />
<itemize><br />
<item>@param <em>AppBB</em> An approximate border basis.</item><br />
<item>@param <em>OrderIdeal</em> The associated order ideal</item><br />
<item>@return A set of points in matrix form described above.</item><br />
</itemize><br />
<br />
<example><br />
Use P::=QQ[x,y,z];<br />
<br />
Points := Mat([[2/3,0,0],[0,10,0],[0,0,1/3]]);<br />
R:=Num.ABM(Points, 0);<br />
Dec(Num.RatPoints(R[1],R[2]),2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
[Mat([<br />
[<quotes>0.66</quotes>, <quotes>0.00</quotes>, <quotes>0</quotes>],<br />
[<quotes>0</quotes>, <quotes>0</quotes>, <quotes>10</quotes>],<br />
[<quotes>0</quotes>, <quotes>0.33</quotes>, <quotes>0</quotes>]<br />
]), Mat([<br />
[<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0</quotes>],<br />
[<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0</quotes>],<br />
[<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0</quotes>]<br />
])]<br />
-------------------------------<br />
<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>polynomial</type><br />
<type>points</type><br />
</types><br />
<key>Num.RatPoints</key><br />
<key>RatPoints</key><br />
<key>numerical.RatPoints</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=Downloads&diff=12180Downloads2011-05-06T09:14:08Z<p>132.231.10.62: </p>
<hr />
<div>The following releases provide the functionality of CoCoA 4.7 together with the additional capabilities of the ApCoCoA library. They are equipped with a graphical user interface. For the official CoCoA releases, visit the [http://cocoa.dima.unige.it CoCoA home page].<br />
<br />
<br />
=ApCoCoA=<br />
<br />
<br />
==ApCoCoA-1.7==<br />
<br />
The current version of ApCoCoA is ApCoCoA-1.7!<br />
<br />
===Eclipse GUI - Standalone Version===<br />
<br />
'''Short installation guide:'''<br />
# Download and unpack the Eclipse GUI package.<br />
# Go into the folder apcocoa and start apcocoa.exe<br />
Here you can download a short [[Media:quick_start.pdf|quick-start User Guide]] and here you find a [[ApCoCoA:tut_eclipse_gui_de|Tutorial]] in German.<br />
<br />
* Windows x86-Installer: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.7.0.exe ApCoCoA-1.7 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.7.0.md5 md5sum] (2011/05/06)<br />
<br />
* Windows x86-Zip: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.7.0-win32.zip ApCoCoA-1.7 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.7.0-win32.zip.md5 md5sum] (2011/05/06)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.6.0-linux-x86.tar.gz ApCoCoA-1.6 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.6.0-linux-x86.tar.gz.md5 md5sum] (2010/12/17)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.6.0-linux-x86_64.tar.gz ApCoCoA-1.6 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.6.0-linux-x86_64.tar.gz.md5 md5sum] (2010/12/17)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.6.0-macosx.tar.gz ApCoCoA-1.6 Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.6.0-macosx.tar.gz.md5 md5sum] (2010/12/17)<br/><b>Please note:</b> ApCoCoAServer for Mac OS X requires an Intel 64-bit based platform!<br />
<br />
===ApCoCoA as Eclipse plug-in===<br />
<br />
For using ApCoCoA-1.7 as Eclipse plug-in please go to [[HowTo:Install_and_Work_with_the_Eclipse_GUI|eclipse-GUI installation guide]].<br />
<br />
<br />
===Well-known Qt-GUI===<br />
<br />
* Windows x86-Installer: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.7.0QT.exe ApCoCoA-1.7QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.7.0QT.md5 md5sum] (2011/05/06)<br />
<br />
* Windows x86-Tgz: [http://www.apcocoa.org/download/apcocoa/win32/apcocoa-1.7.0qt-win.tgz ApCoCoA-1.7QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/apcocoa-1.7.0qt-win.tgz.md5 md5sum] (2011/05/06)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.6.0-QT-linux-x86.tgz ApCoCoA-1.6QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.6.0-QT-linux-x86.tgz.md5 md5sum] (2010/12/17)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.6.0-QT-linux-x86_64.tgz ApCoCoA-1.6QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.6.0-QT-linux-x86_64.tgz.md5 md5sum] (2010/12/17)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/apcocoa-1.6.0-QT-macosx.tgz ApCoCoA-1.6QT Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/apcocoa-1.6.0-QT-macosx.tgz.md5 md5sum] (2010/12/17)<br />
<br />
<br />
==ApCoCoA-1.6==<br />
<br />
The current version of ApCoCoA is ApCoCoA-1.6!<br />
<br />
===Eclipse GUI - Standalone Version===<br />
<br />
'''Short installation guide:'''<br />
# Download and unpack the Eclipse GUI package.<br />
# Go into the folder apcocoa and start apcocoa.exe<br />
Here you can download a short [[Media:quick_start.pdf|quick-start User Guide]] and here you find a [[ApCoCoA:tut_eclipse_gui_de|Tutorial]] in German.<br />
<br />
* Windows x86-Installer: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.6.0.exe ApCoCoA-1.6 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.6.0.md5 md5sum] (2010/12/17)<br />
<br />
* Windows x86-Zip: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.6.0-win32.zip ApCoCoA-1.6 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.6.0-win32.zip.md5 md5sum] (2010/12/17)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.6.0-linux-x86.tar.gz ApCoCoA-1.6 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.6.0-linux-x86.tar.gz.md5 md5sum] (2010/12/17)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.6.0-linux-x86_64.tar.gz ApCoCoA-1.6 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.6.0-linux-x86_64.tar.gz.md5 md5sum] (2010/12/17)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.6.0-macosx.tar.gz ApCoCoA-1.6 Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.6.0-macosx.tar.gz.md5 md5sum] (2010/12/17)<br/><b>Please note:</b> ApCoCoAServer for Mac OS X requires an Intel 64-bit based platform!<br />
<br />
===ApCoCoA as Eclipse plug-in===<br />
<br />
For using ApCoCoA-1.6 as Eclipse plug-in please go to [[HowTo:Install_and_Work_with_the_Eclipse_GUI|eclipse-GUI installation guide]].<br />
<br />
<br />
===Well-known Qt-GUI===<br />
<br />
* Windows x86-Installer: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.6.0QT.exe ApCoCoA-1.6QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.6.0QT.md5 md5sum] (2010/12/17)<br />
<br />
* Windows x86-Tgz: [http://www.apcocoa.org/download/apcocoa/win32/apcocoa-1.6.0qt-win.tgz ApCoCoA-1.6QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/apcocoa-1.6.0qt-win.tgz.md5 md5sum] (2010/12/17)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.6.0-QT-linux-x86.tgz ApCoCoA-1.6QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.6.0-QT-linux-x86.tgz.md5 md5sum] (2010/12/17)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.6.0-QT-linux-x86_64.tgz ApCoCoA-1.6QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.6.0-QT-linux-x86_64.tgz.md5 md5sum] (2010/12/17)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/apcocoa-1.6.0-QT-macosx.tgz ApCoCoA-1.6QT Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/apcocoa-1.6.0-QT-macosx.tgz.md5 md5sum] (2010/12/17)<br />
<br />
==ApCoCoA-1.5==<br />
<br />
===Eclipse GUI - Standalone Version===<br />
<br />
'''Short installation guide:'''<br />
# Download and unpack the Eclipse GUI package.<br />
# Go into the folder apcocoa and start apcocoa.exe<br />
Here you can download a short [[Media:quick_start.pdf|quick-start User Guide]] and here you find a [[ApCoCoA:tut_eclipse_gui_de|Tutorial]] in German.<br />
<br />
* Windows x86-Installer: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1-SR2.exe ApCoCoA-1.5.1 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1-SR2.md5 md5sum] (2010/10/15)<br />
<br />
* Windows x86-Zip: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1-SR2-win32.zip ApCoCoA-1.5.1 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1-SR2-win32.zip.md5 md5sum] (2010/10/15)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.5.1-linux-x86.tar.gz ApCoCoA-1.5.1 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.5.1-linux-x86.tar.gz.md5 md5sum] (2010/10/15)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.5.1-linux-x86_64.tar.gz ApCoCoA-1.5.1 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.5.1-linux-x86_64.tar.gz.md5 md5sum] (2010/10/15)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.5.1-macosx.tar.gz ApCoCoA-1.5.1 Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.5.1-macosx.tar.gz.md5 md5sum] (2010/10/15)<br/><b>Please note:</b> ApCoCoAServer for Mac OS X requires an Intel 64-bit based platform!<br />
<br />
===ApCoCoA as Eclipse plug-in===<br />
<br />
For using ApCoCoA-1.5.1 as Eclipse plug-in please go to [[HowTo:Install_and_Work_with_the_Eclipse_GUI|eclipse-GUI installation guide]].<br />
<br />
<br />
===Well-known Qt-GUI===<br />
<br />
* Windows x86-Installer: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1QT.exe ApCoCoA-1.5.1QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1QT.md5 md5sum] (2010/07/29)<br />
<br />
* Windows x86-Tgz: [http://www.apcocoa.org/download/apcocoa/win32/apcocoa-1.5.1qt-win.tgz ApCoCoA-1.5.1QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/apcocoa-1.5.1qt-win.tgz.md5 md5sum] (2010/10/15)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.5.1qt-linux-x86.tgz ApCoCoA-1.5.1QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.5.1qt-linux-x86.tgz.md5 md5sum] (2010/10/15)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.5.1qt-linux-x86_64.tgz ApCoCoA-1.5.1QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.5.1qt-linux-x86_64.tgz.md5 md5sum] (2010/10/15)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/apcocoa-1.5.1-QT-macosx.tgz ApCoCoA-1.5QT Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/apcocoa-1.5.1-QT-macosx.tgz.md5 md5sum] (2010/07/29)<br />
<br />
<br />
==Older versions==<br />
<br />
For using an earlier version of ApCoCoA please go to [[ApCoCoA:Older Versions]]!<br />
<br />
<br />
=ApCoCoALib=<br />
To build ApCoCoALib from source, please follow the [[ApCoCoALib:CompilationInstructions|compilation instructions]].<br />
<br />
==Repository==<br />
The [[ApCoCoA:SourceCodeManagement|subversion repository]] contains the bleeding-edge developmental versions of ApCoCoALib.<br />
<br />
==Older Versions==<br />
For using an earlier version of ApCoCoALib please go to [[ApCoCoA:Older Versions]]!<br />
<br />
<br />
=ApCoCoA-Matlab-Interface=<br />
Only available for MS Windows. For details see [[ApCoCoA:MatlabToolbox]].<br />
*[http://www.apcocoa.org/~mmachnik/interface/Setup1-02.zip Version 1.02] - Based on CoCoALib 0.99.25 and ApCoCoALib as of 30.01.2009.<br />
<br />
[[Category:ApCoCoA|{{PAGENAME}}]]<br />
[[Category:ApCoCoA|Lib{{PAGENAME}}]]</div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.ABM&diff=11952ApCoCoA-1:Num.ABM2011-03-30T18:15:43Z<p>132.231.10.62: Added new parameter</p>
<hr />
<div><command><br />
<title>Num.ABM</title><br />
<br />
<short_description>Computes the border basis of an almost vanishing ideal for a set of points using the ABM algorithm.</short_description><br />
<syntax><br />
Num.ABM(Points:MAT, Epsilon:RAT):Object<br />
Num.ABM(Points:MAT, Epsilon:RAT, Delta:RAT, ForbiddenTerms:LIST, NormalizeType:INT):Object<br />
</syntax><br />
<br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This command computes a border basis of an almost vanishing ideal for a set of points. <br />
<par/><br />
The current ring has to be a ring over the rational numbers with a standard-degree<br />
compatible term-ordering. The matrix <tt>Points</tt> contains the points: each<br />
point is a row in the matrix, so the number of columns must equal the<br />
number of indeterminates in the current ring. <br />
<br />
<itemize><br />
<item>@param <em>Points</em> The points for which a border basis is computed.</item><br />
<br />
<item>@param <em>Epsilon</em> A positive rational number describing the maximal admissible least squares error for a polynomial. (Bigger values for <tt>Epsilon</tt> lead to bigger errors of the polynomials evaluated at the point set). <tt>Epsilon</tt> should be in the interval (0,1). As a rule of thumb, <tt>Epsilon</tt> is the expected percentage of error on the input points. </item><br />
<br />
<item>@return A list of two results. First the border basis as a list of polynomials, second the vector space basis of <tt>P/I</tt> as a list of terms.</item><br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positiv rational number. <tt>Delta</tt> describes the computing precision. In different steps, it is crucial, if a value is 0 or not. The algorithm assumes every value in <tt>[-Delta, Delta]</tt> to be 0. The default value for <tt>Delta</tt> is 0.00000000001.</item><br />
<item>@param <em>ForbiddenTerms</em> A list containing the terms which are not allowed to show up in the order ideal.</item><br />
<br />
<item>@param <em>NormalizeType</em> A integer of the range 1..4. The default value is 2. This parameter describes, if/how the input points are normalized. If <tt>NormalizeType</tt> equals 1, each coordinate is divided by the maximal absolute value of the corresponding column of the matrix. This ensures that all coordinates of points are in [-1,1]. With <tt>NormalizeType=2</tt> no normalization is done at all. <tt>NormalizeType=3</tt> shifts each coordinate to [-1,1]. So it's minimum is mapped to -1 and the maximum to one, describing a unique affine mapping. The last option is <tt>NormalizeType=4</tt>. In this case, each coordinate is normalized, using the column's euclidian norm.</item><br />
</itemize><br />
<br />
<example><br />
Use P::=QQ[x,y,z];<br />
<br />
Points := Mat([[1,0,0],[0,0,1],[0,0.99,0]]);<br />
Res := Num.ABM(Points,0.1);<br />
<br />
Dec(Res[1],2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
[<quotes>1 x +1.01 y +0.99 z -0.99 </quotes>, <quotes>1 z^2 -0.99 z +0.00 </quotes>, <quotes>1 yz </quotes>, <quotes>1 xz </quotes>, <quotes>1 y^2 -0.98 y -0.00 </quotes>, <quotes>1 xy </quotes>]<br />
-------------------------------<br />
</example><br />
</description><br />
<br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.SubABM</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>points</type><br />
</types><br />
<key>ABM</key><br />
<key>Num.ABM</key><br />
<key>numerical.ABM</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.SVD&diff=11946ApCoCoA-1:Num.SVD2011-02-27T19:41:01Z<p>132.231.10.62: Added reference to SingularValues</p>
<hr />
<div> <command><br />
<title>Num.SVD</title><br />
<short_description>Computes the singular value decomposition of a matrix.</short_description><br />
<syntax><br />
Num.SVD(A:MAT):[U:MAT,S:MAT,VT:MAT]<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This command computes the singular value decomposition of the given matrix <tt>A</tt>. Let <tt>A</tt> be a <tt>(m x n)</tt> matrix. Then <tt>A</tt> is decomposed into the product of an orthogonal <tt>(m x m)</tt> matrix <tt>U</tt>, a transposed matrix <tt>VT</tt> of an orthogonal <tt>(n x n)</tt> matrix <tt>V</tt> and a real <tt>(m x n)</tt> matrix <tt>S</tt>, which contains the singular values of the matrix <tt>A</tt>.<br />
<br />
<itemize><br />
<item>@param <em>A</em> The matrix we want to decompose.</item><br />
<item>@return A list of three matrices <tt>[U, S, VT]</tt> such that <tt>A=U*S*VT</tt>.</item><br />
</itemize><br />
<br />
<example><br />
D:=[[1,2,7,18],[2,4,9,12],[23,8,9,10]];<br />
Dec(Num.SVD(D),3);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[Mat([<br />
[<quotes>-0.473</quotes>, <quotes>-0.666</quotes>, <quotes>-0.575</quotes>],<br />
[<quotes>-0.415</quotes>, <quotes>-0.407</quotes>, <quotes>0.813</quotes>],<br />
[<quotes>-0.776</quotes>, <quotes>0.624</quotes>, <quotes>-0.084</quotes>]<br />
]), Mat([<br />
[<quotes>33.091</quotes>, <quotes>17.047</quotes>, <quotes>3.365</quotes>]<br />
]), Mat([<br />
[<quotes>-0.579</quotes>, <quotes>-0.266</quotes>, <quotes>-0.424</quotes>, <quotes>-0.642</quotes>],<br />
[<quotes>0.755</quotes>, <quotes>0.119</quotes>, <quotes>-0.159</quotes>, <quotes>-0.624</quotes>],<br />
[<quotes>-0.265</quotes>, <quotes>0.423</quotes>, <quotes>0.750</quotes>, <quotes>-0.431</quotes>],<br />
[<quotes>-0.153</quotes>, <quotes>0.857</quotes>, <quotes>-0.480</quotes>, <quotes>0.100</quotes>]<br />
])]<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.QR</see><br />
<see>Num.SingularValues</see><br />
<see>Num.EigenValues</see><br />
<see>Num.EigenValuesAndVectors</see><br />
<see>Num.EigenValuesAndAllVectors</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>matrix</type><br />
</types><br />
<key>numerical.svd</key><br />
<key>svd</key><br />
<key>num.svd</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.SingularValues&diff=11945ApCoCoA-1:Num.SingularValues2011-02-27T19:38:09Z<p>132.231.10.62: Added SingularValues command</p>
<hr />
<div><command><br />
<title>Num.SingularValues</title><br />
<short_description>Computes the singular values of a matrix.</short_description><br />
<syntax><br />
Num.SVD(A:MAT):[S:LIST]<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This command computes the singular values of the given matrix <tt>A</tt>.<br />
<br />
<itemize><br />
<item>@param <em>A</em> The matrix we want to analyze.</item><br />
<item>@return A list containing the singular values of <tt>A</tt>.</item><br />
</itemize><br />
<br />
<example><br />
D:=[[1,2,7,18],[2,4,9,12],[23,8,9,10]];<br />
Dec(Num.SingularValues(D),3);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[<quotes>33.091</quotes>, <quotes>17.047</quotes>, <quotes>3.365</quotes>]<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.QR</see><br />
<see>Num.SVD</see><br />
<see>Num.EigenValues</see><br />
<see>Num.EigenValuesAndVectors</see><br />
<see>Num.EigenValuesAndAllVectors</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>matrix</type><br />
</types><br />
<key>numerical.singularvalues</key><br />
<key>singularvalues</key><br />
<key>num.singularvalues</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:HowTo:Testing_For_Releases&diff=11756ApCoCoA-1:HowTo:Testing For Releases2010-12-13T15:03:04Z<p>132.231.10.62: </p>
<hr />
<div>=General Remarks=<br />
* Take the following remarks as a guidance to get started. <br />
* It's mainly an incomplete list of features that have to be tested, if you miss something, add it.<br />
<br />
=What to test=<br />
<br />
On Win, Linux 32 and 64, MacOS X:<br />
* Moccha (Standalone, Plugin, Update, WINstaller) + ApCoCoA (Server + BBFServer) + Documentation<br />
* Qt + ApCoCoA (Server + BBFServer) + Documentation<br />
<br />
=Test plans=<br />
<br />
==Moccha - Standalone==<br />
* Download and unzip latest from [ftp://ftp.apcocoa.org/builds/apcocoa/nightly/moccha/ here].<br />
* Test GUI features using a new workspace.<br />
* Test using an already existing workspace (one that was created with an older version). Check, that everything there is still there and works.<br />
* GUI functionalities to test:<br />
** Create new Project and files in the project (package, cocoa, interactive).<br />
** Work with the different editor windows: Send cocoa commands using Alt+Enter, the corresponding button and the menu entry.<br />
** Start the ApCoCoAServer and execute the ApCoCoA Test Suite:<pre>$apcocoa/ts/aunit.RegisterAllTests();&#13;$apcocoa/ts/aunit.RunTests();</pre> You'll find the tests under ''<library>/apcocoa/ts''. See if there are tests covering your functionality. If not, [[Test-Suite_Template | add some tests]].<br />
** History functionality in interactive editor window.<br />
** Code completion: see, if your own functions are all present.<br />
** Call the help system and look for some functions, see if your own functions are present, do full text search for your own functions.<br />
** Change the preferences and see if they are applied correctly (server port, userinit, colors,...).<br />
* Start the ApCoCoABBFServer and execute some commands (please note that there is not button in the GUI to start the ApCoCoABBFServer).<br />
<br />
==Moccha - Plugin==<br />
<br />
* Download and install some eclipse version from [http://www.eclipse.org/ eclipse.org].<br />
* Install the ApCoCoA Plugin from http://www.apcocoa.org/testupdatesite.<br />
* Test some of the GUI functionalities, at least, create a new project, an interactive file, start the server, and execute the Test Suite, check the documentation.<br />
<br />
==Moccha - Update==<br />
<br />
* Download and install an old version of Moccha from [http://www.apcocoa.org/wiki?title=ApCoCoA:Downloads here].<br />
* Create a project and some files, change some preferences.<br />
* Change the preconfigured update site to http://www.apcocoa.org/testupdatesite<br />
* Update Moccha.<br />
* See if everything works as expected.<br />
* Test some of the GUI functionalities; at least create a new project, an interactive file, start the server, and execute the Test Suite.<br />
<br />
==Qt==<br />
<br />
* Download and install the Qt distribution.<br />
* Start the server and run the Test Suite (compare Moccha - Standalone for further instructions).<br />
<br />
=Reporting Bugs=<br />
<br />
Please report bugs [[LWBugTrackerForTesting| here]].</div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=Downloads&diff=11346Downloads2010-11-22T10:27:59Z<p>132.231.10.62: </p>
<hr />
<div>The following releases provide the functionality of CoCoA 4.7 together with the additional capabilities of the ApCoCoA library. They are equipped with a graphical user interface. For the official CoCoA releases, visit the [http://cocoa.dima.unige.it CoCoA home page].<br />
<br />
<br />
=ApCoCoA=<br />
<br />
<br />
==ApCoCoA-1.5==<br />
<br />
The current version of ApCoCoA is ApCoCoA-1.5.1!<br />
<br />
===Eclipse GUI - Standalone Version===<br />
<br />
'''Short installation guide:'''<br />
# Download and unpack the Eclipse GUI package.<br />
# Go into the folder apcocoa and start apcocoa.exe<br />
Here you can download a short [[Media:quick_start.pdf|quick-start User Guide]] and here you find a [[ApCoCoA:tut_eclipse_gui_de|Tutorial]] in German.<br />
<br />
* Windows x86-Installer: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1-SR2.exe ApCoCoA-1.5.1 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1-SR2.md5 md5sum] (2010/10/15)<br />
<br />
* Windows x86-Zip: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1-SR2-win32.zip ApCoCoA-1.5.1 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1-SR2-win32.zip.md5 md5sum] (2010/10/15)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.5.1-linux-x86.tar.gz ApCoCoA-1.5.1 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.5.1-linux-x86.tar.gz.md5 md5sum] (2010/10/15)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.5.1-linux-x86_64.tar.gz ApCoCoA-1.5.1 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.5.1-linux-x86_64.tar.gz.md5 md5sum] (2010/10/15)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.5.1-macosx.tar.gz ApCoCoA-1.5.1 Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.5.1-macosx.tar.gz.md5 md5sum] (2010/10/15)<br/><b>Please note:</b> ApCoCoAServer for Mac OS X requires an Intel 64-bit based platform!<br />
<br />
===ApCoCoA as Eclipse plug-in===<br />
<br />
For using ApCoCoA-1.5.1 as Eclipse plug-in please go to [[HowTo:Install_and_Work_with_the_Eclipse_GUI|eclipse-GUI installation guide]].<br />
<br />
<br />
===Well-known Qt-GUI===<br />
<br />
* Windows x86-Installer: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1QT.exe ApCoCoA-1.5.1QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1QT.md5 md5sum] (2010/07/29)<br />
<br />
* Windows x86-Tgz: [http://www.apcocoa.org/download/apcocoa/win32/apcocoa-1.5.1qt-win.tgz ApCoCoA-1.5.1QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/apcocoa-1.5.1qt-win.tgz.md5 md5sum] (2010/10/15)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.5.1qt-linux-x86.tgz ApCoCoA-1.5.1QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.5.1qt-linux-x86.tgz.md5 md5sum] (2010/10/15)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.5.1qt-linux-x86_64.tgz ApCoCoA-1.5.1QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.5.1qt-linux-x86_64.tgz.md5 md5sum] (2010/10/15)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/apcocoa-1.5qt-macosx.tgz ApCoCoA-1.5QT Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/apcocoa-1.5qt-macosx.tgz.md5 md5sum] (2010/07/29)<br />
<br />
==ApCoCoA-1.4==<br />
<br />
===Eclipse GUI - Standalone Version===<br />
<br />
'''Short installation guide:'''<br />
# Download and unpack the Eclipse GUI package.<br />
# Go into the folder apcocoa and start apcocoa.exe<br />
Here you can download a short [[Media:quick_start.pdf|quick-start User Guide]] and here you find a [[ApCoCoA:tut_eclipse_gui_de|Tutorial]] in German.<br />
<br />
* Windows x86-Installer: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.4.exe ApCoCoA-1.4.0 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.4.md5 md5sum] (2010/05/30)<br />
<br />
* Windows x86-Zip: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.4.0-win32.zip ApCoCoA-1.4.0 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.4.0-win32.zip.md5 md5sum] (2010/05/28)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.4.0-linux-x86.tar.gz ApCoCoA-1.4.0 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.4.0-linux-x86.tar.gz.md5 md5sum] (2010/05/28)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.4.0-linux-x86_64.tar.gz ApCoCoA-1.4.0 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.4.0-linux-x86_64.tar.gz.md5 md5sum] (2010/05/28)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.4.0-macosx.tar.gz ApCoCoA-1.4.0 Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.4.0-macosx.tar.gz.md5 md5sum] (2010/05/28)<br/><b>Please note:</b> ApCoCoAServer for Mac OS X requires an Intel 64-bit based platform!<br />
<br />
===ApCoCoA as Eclipse plug-in===<br />
<br />
For using ApCoCoA-1.4 as Eclipse plug-in please go to [[HowTo:Install_and_Work_with_the_Eclipse_GUI|eclipse-GUI installation guide]].<br />
<br />
<br />
===Well-known Qt-GUI===<br />
<br />
* Windows x86-Installer: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.4QT.exe ApCoCoA-1.4QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.4QT.md5 md5sum] (2010/05/30)<br />
<br />
* Windows x86-Tgz: [http://www.apcocoa.org/download/apcocoa/win32/apcocoa-1.4qt-win.tgz ApCoCoA-1.4QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/apcocoa-1.4qt-win.tgz.md5 md5sum] (2010/05/28)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.4qt-linux-x86.tgz ApCoCoA-1.4QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.4qt-linux-x86.tgz.md5 md5sum] (2010/05/28)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.4qt-linux-x86_64.tgz ApCoCoA-1.4QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.4qt-linux-x86_64.tgz.md5 md5sum] (2010/05/28)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/apcocoa-1.4qt-macosx.tgz ApCoCoA-1.4QT Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/apcocoa-1.4qt-macosx.tgz.md5 md5sum] (2010/05/28)<br />
<br />
==ApCoCoA-1.3==<br />
<br />
===Eclipse GUI - Standalone Version===<br />
<br />
'''Short installation guide:'''<br />
# Download and unpack the Eclipse GUI package.<br />
# Go into the folder apcocoa and start apcocoa.exe<br />
Here you can download a short [[Media:quick_start.pdf|quick-start User Guide]] and here you find a [[ApCoCoA:tut_eclipse_gui_de|Tutorial]] in German.<br />
<br />
* Windows x86: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.3.exe ApCoCoA-1.3 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.3.md5 md5sum] (2009/10/16)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.3-linux-x86.tgz ApCoCoA-1.3 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.3-linux-x86.tgz.md5 md5sum] (2009/10/16)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.3-linux-x86_64.tgz ApCoCoA-1.3 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.3-linux-x86_64.tgz.md5 md5sum] (2009/10/16)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.3.1-osx-intel.tgz ApCoCoA-1.3.1 Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.3.1-osx-intel.md5 md5sum] (2009/11/11)<br/><b>Please note:</b> ApCoCoAServer for Mac OS X requires an Intel 64-bit based platform!<br />
<br />
===ApCoCoA as Eclipse plug-in===<br />
<br />
For using ApCoCoA-1.3 as Eclipse plug-in please go to [[HowTo:Install_and_Work_with_the_Eclipse_GUI|eclipse-GUI installation guide]].<br />
<br />
<br />
===Well-known Qt-GUI===<br />
<br />
* Windows x86: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.3QT.exe ApCoCoA-1.3QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.3QT.md5 md5sum] (2009/10/16)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.3qt-linux-x86.tgz ApCoCoA-1.3QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.3qt-linux-x86.tgz.md5 md5sum] (2009/10/16)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.3qt-linux-x86_64.tgz ApCoCoA-1.3QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.3qt-linux-x86_64.tgz.md5 md5sum] (2009/10/16)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.3QT.tgz ApCoCoA-1.3QT Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.3QT.md5 md5sum] (2009/10/16)<br />
<br />
==Older versions==<br />
<br />
For using an earlier version of ApCoCoA please go to [[ApCoCoA:Older Versions]]!<br />
<br />
<br />
=ApCoCoALib=<br />
To build ApCoCoALib from source, please follow the [[ApCoCoALib:CompilationInstructions|compilation instructions]].<br />
<br />
==Repository==<br />
The [[ApCoCoA:SourceCodeManagement|subversion repository]] contains the bleeding-edge developmental versions of ApCoCoALib.<br />
<br />
==Older Versions==<br />
For using an earlier version of ApCoCoALib please go to [[ApCoCoA:Older Versions]]!<br />
<br />
<br />
=ApCoCoA-Matlab-Interface=<br />
Only available for MS Windows. For details see [[ApCoCoA:MatlabToolbox]].<br />
*[http://www.apcocoa.org/~mmachnik/interface/Setup1-02.zip Version 1.02] - Based on CoCoALib 0.99.25 and ApCoCoALib as of 30.01.2009.<br />
<br />
[[Category:ApCoCoA|{{PAGENAME}}]]<br />
[[Category:ApCoCoA|Lib{{PAGENAME}}]]</div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=Downloads&diff=11328Downloads2010-10-18T08:40:34Z<p>132.231.10.62: Updated installer links for 1.5.1</p>
<hr />
<div>The following releases provide the functionality of CoCoA 4.7 together with the additional capabilities of the ApCoCoA library. They are equipped with a graphical user interface. For the official CoCoA releases, visit the [http://cocoa.dima.unige.it CoCoA home page].<br />
<br />
<br />
=ApCoCoA=<br />
<br />
<br />
==ApCoCoA-1.5==<br />
<br />
The current version of ApCoCoA is ApCoCoA-1.5.1!<br />
<br />
===Eclipse GUI - Standalone Version===<br />
<br />
'''Short installation guide:'''<br />
# Download and unpack the Eclipse GUI package.<br />
# Go into the folder apcocoa and start apcocoa.exe<br />
Here you can download a short [[Media:quick_start.pdf|quick-start User Guide]] and here you find a [[ApCoCoA:tut_eclipse_gui_de|Tutorial]] in German.<br />
<br />
* Windows x86-Installer: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1.exe ApCoCoA-1.5.1 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1.md5 md5sum] (2010/10/15)<br />
<br />
* Windows x86-Zip: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1-win32.zip ApCoCoA-1.5.1 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1-win32.zip.md5 md5sum] (2010/10/15)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.5.1-linux-x86.tar.gz ApCoCoA-1.5.1 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.5.1-linux-x86.tar.gz.md5 md5sum] (2010/10/15)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.5.1-linux-x86_64.tar.gz ApCoCoA-1.5.1 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.5.1-linux-x86_64.tar.gz.md5 md5sum] (2010/10/15)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.5.1-macosx.tar.gz ApCoCoA-1.5.1 Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.5.1-macosx.tar.gz.md5 md5sum] (2010/10/15)<br/><b>Please note:</b> ApCoCoAServer for Mac OS X requires an Intel 64-bit based platform!<br />
<br />
===ApCoCoA as Eclipse plug-in===<br />
<br />
For using ApCoCoA-1.5.1 as Eclipse plug-in please go to [[HowTo:Install_and_Work_with_the_Eclipse_GUI|eclipse-GUI installation guide]].<br />
<br />
<br />
===Well-known Qt-GUI===<br />
<br />
* Windows x86-Installer: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1QT.exe ApCoCoA-1.5.1QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.5.1QT.md5 md5sum] (2010/07/29)<br />
<br />
* Windows x86-Tgz: [http://www.apcocoa.org/download/apcocoa/win32/apcocoa-1.5.1qt-win.tgz ApCoCoA-1.5.1QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/apcocoa-1.5.1qt-win.tgz.md5 md5sum] (2010/10/15)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.5.1qt-linux-x86.tgz ApCoCoA-1.5.1QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.5.1qt-linux-x86.tgz.md5 md5sum] (2010/10/15)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.5.1qt-linux-x86_64.tgz ApCoCoA-1.5.1QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.5.1qt-linux-x86_64.tgz.md5 md5sum] (2010/10/15)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/apcocoa-1.5qt-macosx.tgz ApCoCoA-1.5QT Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/apcocoa-1.5qt-macosx.tgz.md5 md5sum] (2010/07/29)<br />
<br />
==ApCoCoA-1.4==<br />
<br />
===Eclipse GUI - Standalone Version===<br />
<br />
'''Short installation guide:'''<br />
# Download and unpack the Eclipse GUI package.<br />
# Go into the folder apcocoa and start apcocoa.exe<br />
Here you can download a short [[Media:quick_start.pdf|quick-start User Guide]] and here you find a [[ApCoCoA:tut_eclipse_gui_de|Tutorial]] in German.<br />
<br />
* Windows x86-Installer: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.4.exe ApCoCoA-1.4.0 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.4.md5 md5sum] (2010/05/30)<br />
<br />
* Windows x86-Zip: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.4.0-win32.zip ApCoCoA-1.4.0 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.4.0-win32.zip.md5 md5sum] (2010/05/28)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.4.0-linux-x86.tar.gz ApCoCoA-1.4.0 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.4.0-linux-x86.tar.gz.md5 md5sum] (2010/05/28)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.4.0-linux-x86_64.tar.gz ApCoCoA-1.4.0 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.4.0-linux-x86_64.tar.gz.md5 md5sum] (2010/05/28)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.4.0-macosx.tar.gz ApCoCoA-1.4.0 Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.4.0-macosx.tar.gz.md5 md5sum] (2010/05/28)<br/><b>Please note:</b> ApCoCoAServer for Mac OS X requires an Intel 64-bit based platform!<br />
<br />
===ApCoCoA as Eclipse plug-in===<br />
<br />
For using ApCoCoA-1.4 as Eclipse plug-in please go to [[HowTo:Install_and_Work_with_the_Eclipse_GUI|eclipse-GUI installation guide]].<br />
<br />
<br />
===Well-known Qt-GUI===<br />
<br />
* Windows x86-Installer: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.4QT.exe ApCoCoA-1.4QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.4QT.md5 md5sum] (2010/05/30)<br />
<br />
* Windows x86-Tgz: [http://www.apcocoa.org/download/apcocoa/win32/apcocoa-1.4qt-win.tgz ApCoCoA-1.4QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/apcocoa-1.4qt-win.tgz.md5 md5sum] (2010/05/28)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.4qt-linux-x86.tgz ApCoCoA-1.4QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.4qt-linux-x86.tgz.md5 md5sum] (2010/05/28)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.4qt-linux-x86_64.tgz ApCoCoA-1.4QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.4qt-linux-x86_64.tgz.md5 md5sum] (2010/05/28)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/apcocoa-1.4qt-macosx.tgz ApCoCoA-1.4QT Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/apcocoa-1.4qt-macosx.tgz.md5 md5sum] (2010/05/28)<br />
<br />
==ApCoCoA-1.3==<br />
<br />
===Eclipse GUI - Standalone Version===<br />
<br />
'''Short installation guide:'''<br />
# Download and unpack the Eclipse GUI package.<br />
# Go into the folder apcocoa and start apcocoa.exe<br />
Here you can download a short [[Media:quick_start.pdf|quick-start User Guide]] and here you find a [[ApCoCoA:tut_eclipse_gui_de|Tutorial]] in German.<br />
<br />
* Windows x86: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.3.exe ApCoCoA-1.3 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.3.md5 md5sum] (2009/10/16)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.3-linux-x86.tgz ApCoCoA-1.3 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-1.3-linux-x86.tgz.md5 md5sum] (2009/10/16)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.3-linux-x86_64.tgz ApCoCoA-1.3 Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/ApCoCoA-1.3-linux-x86_64.tgz.md5 md5sum] (2009/10/16)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.3.1-osx-intel.tgz ApCoCoA-1.3.1 Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.3.1-osx-intel.md5 md5sum] (2009/11/11)<br/><b>Please note:</b> ApCoCoAServer for Mac OS X requires an Intel 64-bit based platform!<br />
<br />
===ApCoCoA as Eclipse plug-in===<br />
<br />
For using ApCoCoA-1.3 as Eclipse plug-in please go to [[HowTo:Install_and_Work_with_the_Eclipse_GUI|eclipse-GUI installation guide]].<br />
<br />
<br />
===Well-known Qt-GUI===<br />
<br />
* Windows x86: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.3QT.exe ApCoCoA-1.3QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.3QT.md5 md5sum] (2009/10/16)<br />
<br />
* Linux x86: [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.3qt-linux-x86.tgz ApCoCoA-1.3QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.3qt-linux-x86.tgz.md5 md5sum] (2009/10/16)<br />
<br />
* Linux x86_64: [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.3qt-linux-x86_64.tgz ApCoCoA-1.3QT Release] with [http://www.apcocoa.org/download/apcocoa/linux-x86_64/apcocoa-1.3qt-linux-x86_64.tgz.md5 md5sum] (2009/10/16)<br />
<br />
* MacOSx: [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.3QT.tgz ApCoCoA-1.3QT Release] with [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.3QT.md5 md5sum] (2009/10/16)<br />
<br />
==Older versions==<br />
<br />
For using an earlier version of ApCoCoA please go to [[ApCoCoA:Older Versions]]!<br />
<br />
<br />
=ApCoCoALib=<br />
To build ApCoCoALib from source, please follow the [[ApCoCoALib:CompilationInstructions|compilation instructions]].<br />
<br />
==Repository==<br />
The [[ApCoCoA:SourceCodeManagement|subversion repository]] contains the bleeding-edge developmental versions of ApCoCoALib.<br />
<br />
==Older Versions==<br />
For using an earlier version of ApCoCoALib please go to [[ApCoCoA:Older Versions]]!<br />
<br />
<br />
=ApCoCoA-Matlab-Interface=<br />
Only available for MS Windows. For details see [[ApCoCoA:MatlabToolbox]].<br />
*[http://www.apcocoa.org/~mmachnik/interface/Setup1-02.zip Version 1.02] - Based on CoCoALib 0.99.25 and ApCoCoALib as of 30.01.2009.<br />
<br />
[[Category:ApCoCoA|{{PAGENAME}}]]<br />
[[Category:ApCoCoA|Lib{{PAGENAME}}]]</div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.SimDiag&diff=11206ApCoCoA-1:Num.SimDiag2010-10-07T13:54:12Z<p>132.231.10.62: </p>
<hr />
<div> <command><br />
<title>Num.SimDiag</title><br />
<short_description>Computes an approximate diagonalization of a set of matrices.</short_description><br />
<syntax><br />
Num.SimDiag(A:LIST, MaxIt:INT):[B:MAT, C:MAT]<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function returns a list of two matrices, containing the approximate (almost) eigenvectors of the matrices in <tt>A</tt> and its inverse. <br />
<br />
<itemize><br />
<item>@param <em>A</em> A list of quadratic matrices with rational entries.</item><br />
<item>@param <em>MaxIt</em> The maximum number of iterations.</item><br />
<item>@return The output is a list of two matrices <tt>[B:MAT, C:MAT]</tt>. The first matrix <tt>B</tt> contains the real almost eigenvectors of the matrices in <tt>A</tt>. The matrix <tt>C</tt> is the inverse of <tt>A</tt>.</item><br />
</itemize><br />
<br />
<br />
<example><br />
M1 := Mat([[0, 0, -0.079, -0.018],[0, 0, 0.032, -0.012], [1, 0, 1.056, -0.012],[0, 1, -0.060, 1.025]]);<br />
M2 := Mat([[0, -0.063, 0, -0.018],[1, 1.026, 0, -0.012], [0, 0, 0, -0.012], [0, 0, 1, 1.025]]);<br />
M1 := Transposed(M1);<br />
M2 := Transposed(M2);<br />
Result := Num.SimDiag([M1,M2],10);<br />
<br />
Dec(Result[2]*M1*Result[1],3);<br />
Dec(Result[2]*M2*Result[1],3);<br />
<br />
<br />
Mat([<br />
[<quotes>0.062</quotes>, <quotes>0.016</quotes>, <quotes>0.000</quotes>, <quotes>0.006</quotes>],<br />
[<quotes>0.021</quotes>, <quotes>0.030</quotes>, <quotes>-0.002</quotes>, <quotes>-0.000</quotes>],<br />
[<quotes>0.000</quotes>, <quotes>0.005</quotes>, <quotes>1.006</quotes>, <quotes>-0.035</quotes>],<br />
[<quotes>-0.000</quotes>, <quotes>-0.000</quotes>, <quotes>-0.031</quotes>, <quotes>0.982</quotes>]<br />
])<br />
-------------------------------<br />
Mat([<br />
[<quotes>0.048</quotes>, <quotes>0.000</quotes>, <quotes>0.030</quotes>, <quotes>-0.005</quotes>],<br />
[<quotes>0.000</quotes>, <quotes>0.991</quotes>, <quotes>-0.002</quotes>, <quotes>-0.021</quotes>],<br />
[<quotes>0.020</quotes>, <quotes>0.005</quotes>, <quotes>0.029</quotes>, <quotes>-0.000</quotes>],<br />
[<quotes>0.000</quotes>, <quotes>-0.030</quotes>, <quotes>-0.000</quotes>, <quotes>0.982</quotes>]<br />
])<br />
-----------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.QR</see><br />
<see>Num.SVD</see><br />
<see>Num.EigenValues</see><br />
<see>Num.EigenValuesAndVectors</see><br />
<see>Num.EigenValuesAndAllVectors</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>matrix</type><br />
</types><br />
<key>simdiag</key><br />
<key>num.simdiag</key><br />
<key>numerical.simdiag</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.SimDiag&diff=11205ApCoCoA-1:Num.SimDiag2010-10-07T13:49:06Z<p>132.231.10.62: </p>
<hr />
<div> <command><br />
<title>Num.SimDiag</title><br />
<short_description>Computes an approximate diagonalization of a set of matrices.</short_description><br />
<syntax><br />
Num.SimDiag(A:LSIT):[B:MAT, C:MAT]<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function returns a list of two matrices, containing the approximate (almost) eigenvectors of the matrices in <tt>A</tt> and its inverse. <br />
<br />
<itemize><br />
<item>@param <em>A</em> A list of quadratic matrices with rational entries.</item><br />
<item>@return The output is a list of two matrices <tt>[B:MAT, C:MAT]</tt>. The first matrix <tt>B</tt> contains the real almost eigenvectors of the matrices in <tt>A</tt>. The matrix <tt>C</tt> is the inverse of <tt>A</tt>.</item><br />
</itemize><br />
<br />
<br />
<example><br />
M1 := Mat([[0, 0, -0.079, -0.018],[0, 0, 0.032, -0.012], [1, 0, 1.056, -0.012],[0, 1, -0.060, 1.025]]);<br />
M2 := Mat([[0, -0.063, 0, -0.018],[1, 1.026, 0, -0.012], [0, 0, 0, -0.012], [0, 0, 1, 1.025]]);<br />
M1 := Transposed(M1);<br />
M2 := Transposed(M2);<br />
Result := Num.SimDiag([M1,M2],10);<br />
<br />
Dec(Result[2]*M1*Result[1],3);<br />
Dec(Result[2]*M2*Result[1],3);<br />
<br />
<br />
Mat([<br />
[<quotes>0.062</quotes>, <quotes>0.016</quotes>, <quotes>0.000</quotes>, <quotes>0.006</quotes>],<br />
[<quotes>0.021</quotes>, <quotes>0.030</quotes>, <quotes>-0.002</quotes>, <quotes>-0.000</quotes>],<br />
[<quotes>0.000</quotes>, <quotes>0.005</quotes>, <quotes>1.006</quotes>, <quotes>-0.035</quotes>],<br />
[<quotes>-0.000</quotes>, <quotes>-0.000</quotes>, <quotes>-0.031</quotes>, <quotes>0.982</quotes>]<br />
])<br />
-------------------------------<br />
Mat([<br />
[<quotes>0.048</quotes>, <quotes>0.000</quotes>, <quotes>0.030</quotes>, <quotes>-0.005</quotes>],<br />
[<quotes>0.000</quotes>, <quotes>0.991</quotes>, <quotes>-0.002</quotes>, <quotes>-0.021</quotes>],<br />
[<quotes>0.020</quotes>, <quotes>0.005</quotes>, <quotes>0.029</quotes>, <quotes>-0.000</quotes>],<br />
[<quotes>0.000</quotes>, <quotes>-0.030</quotes>, <quotes>-0.000</quotes>, <quotes>0.982</quotes>]<br />
])<br />
-----------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.QR</see><br />
<see>Num.SVD</see><br />
<see>Num.EigenValues</see><br />
<see>Num.EigenValuesAndVectors</see><br />
<see>Num.EigenValuesAndAllVectors</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>matrix</type><br />
</types><br />
<key>simdiag</key><br />
<key>num.simdiag</key><br />
<key>numerical.simdiag</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.SimDiag&diff=11204ApCoCoA-1:Num.SimDiag2010-10-07T13:46:19Z<p>132.231.10.62: </p>
<hr />
<div> <command><br />
<title>Num.SimDiag</title><br />
<short_description>Computes an approximate diagonalization of a set of matrices.</short_description><br />
<syntax><br />
Num.SimDiag(A:LSIT):[B:MAT, C:MAT]<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function returns a list of two matrices, containing the approximate (almost) eigenvectors of the matrices in <tt>A</tt> and its inverse. <br />
<br />
<itemize><br />
<item>@param <em>A</em> A list of quadratic matrices with rational entries.</item><br />
<item>@return The output is a list of two matrices <tt>[B:MAT, C:MAT]</tt>. The first matrix <tt>B</tt> contains the real almost eigenvectors of the matrices in <tt>A</tt>. The matrix <tt>C</tt> is the inverse of <tt>A</tt>.</item><br />
</itemize><br />
<br />
<br />
<example><br />
M1 := Mat([[0, 0, -0.079, -0.018],[0, 0, 0.032, -0.012], [1, 0, 1.056, -0.012],[0, 1, -0.060, 1.025]]);<br />
M2 := Mat([[0, -0.063, 0, -0.018],[1, 1.026, 0, -0.012], [0, 0, 0, -0.012], [0, 0, 1, 1.025]]);<br />
M1 := Transposed(M1);<br />
M2 := Transposed(M2);<br />
Result := Num.SimDiag([M1,M2],10);<br />
<br />
Dec(Result[2]*M1*Result[1],3);<br />
Dec(Result[2]*M2*Result[1],3);<br />
<br />
Mat([<br />
[<quote>0.062</quote>, <quote>0.016</quote>, <quote>0.000</quote>, <quote>0.006</quote>],<br />
[<quote>0.021</quote>, <quote>0.030</quote>, <quote>-0.002</quote>, <quote>-0.000</quote>],<br />
[<quote>0.000</quote>, <quote>0.005</quote>, <quote>1.006</quote>, <quote>-0.035</quote>],<br />
[<quote>-0.000</quote>, <quote>-0.000</quote>, <quote>-0.031</quote>, <quote>0.982</quote>]<br />
])<br />
-------------------------------<br />
Mat([<br />
[<quote>0.048</quote>, <quote>0.000</quote>, <quote>0.030</quote>, <quote>-0.005</quote>],<br />
[<quote>0.000</quote>, <quote>0.991</quote>, <quote>-0.002</quote>, <quote>-0.021</quote>],<br />
[<quote>0.020</quote>, <quote>0.005</quote>, <quote>0.029</quote>, <quote>-0.000</quote>],<br />
[<quote>0.000</quote>, <quote>-0.030</quote>, <quote>-0.000</quote>, <quote>0.982</quote>]<br />
])<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.QR</see><br />
<see>Num.SVD</see><br />
<see>Num.EigenValues</see><br />
<see>Num.EigenValuesAndVectors</see><br />
<see>Num.EigenValuesAndAllVectors</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>matrix</type><br />
</types><br />
<key>simdiag</key><br />
<key>num.simdiag</key><br />
<key>numerical.simdiag</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.SimDiag&diff=11203ApCoCoA-1:Num.SimDiag2010-10-07T13:37:54Z<p>132.231.10.62: Added SimDiag command</p>
<hr />
<div> <command><br />
<title>Num.SimDiag</title><br />
<short_description>Computes an approximate diagonalization of a set of matrices.</short_description><br />
<syntax><br />
Num.SimDiag(A:LSIT):[B:MAT, C:MAT]<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This function returns a list of two matrices, containing the approximate (almost) eigenvectors of the matrices in <tt>A</tt> and its inverse. <br />
<br />
<itemize><br />
<item>@param <em>A</em> A list of quadratic matrices with rational entries.</item><br />
<item>@return The output is a list of two matrices <tt>[B:MAT, C:MAT]</tt>. The first matrix <tt>B</tt> contains the real almost eigenvectors of the matrices in <tt>A</tt>. The matrix <tt>C</tt> is the inverse of <tt>A</tt>.</item><br />
</itemize><br />
<br />
<br />
<example><br />
M1 := Mat([[0, 0, -0.079, -0.018],[0, 0, 0.032, -0.012], [1, 0, 1.056, -0.012],[0, 1, -0.060, 1.025]]);<br />
M2 := Mat([[0, -0.063, 0, -0.018],[1, 1.026, 0, -0.012], [0, 0, 0, -0.012], [0, 0, 1, 1.025]]);<br />
M1 := Transposed(M1);<br />
M2 := Transposed(M2);<br />
Result := Num.SimDiag([M1],10);<br />
<br />
Dec(Result[2]*M1*Result[1],3);<br />
Dec(Result[2]*M2*Result[1],3);<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.QR</see><br />
<see>Num.SVD</see><br />
<see>Num.EigenValues</see><br />
<see>Num.EigenValuesAndVectors</see><br />
<see>Num.EigenValuesAndAllVectors</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>matrix</type><br />
</types><br />
<key>simdiag</key><br />
<key>num.simdiag</key><br />
<key>numerical.simdiag</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.CABM&diff=10563ApCoCoA-1:Num.CABM2010-05-26T16:04:24Z<p>132.231.10.62: Minor update</p>
<hr />
<div><command><br />
<title>Num.CABM</title><br />
<br />
<short_description>Computes the border basis of an almost vanishing ideal for a set of complex points.</short_description><br />
<syntax><br />
Num.CABM(PointsReal:MAT, PointsComp:MAT, Epsilon:RAT):Object<br />
Num.CABM(PointsReal:MAT, PointsComp:MAT, Epsilon:RAT, Delta:RAT, NormalizeType:INT):Object<br />
</syntax><br />
<br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This command computes a border basis of an almost vanishing ideal for a set of complex points. <br />
<par/><br />
The current ring has to be a ring over the rational numbers with a standard-degree<br />
compatible term-ordering. The matrix <tt>PointsReal</tt> and <tt>PointsComp</tt> contain the points: each<br />
point is a row in the matrix, so the number of columns must equal the<br />
number of indeterminates in the current ring. <br />
<br />
<itemize><br />
<item>@param <em>PointsReal</em> The real part of the points for which a border basis is computed.</item><br />
<br />
<item>@param <em>PointsComp</em> The imaginary part of the points for which a border basis is computed.</item><br />
<br />
<item>@param <em>Epsilon</em> A positive rational number describing the maximal admissible least squares error for a polynomial. (Bigger values for <tt>Epsilon</tt> lead to bigger errors of the polynomials evaluated at the point set). <tt>Epsilon</tt> should be in the interval (0,1). As a rule of thumb, <tt>Epsilon</tt> is the expected percentage of error on the input points. </item><br />
<br />
<item>@return A list of two lists. First the border basis as a list of polynomials. Two polynomials always belong together containing the real and the complex part. Second the vector space basis of <tt>P/I</tt> as a list of terms.</item><br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positiv rational number. <tt>Delta</tt> describes the computing precision. In different steps, it is crucial, if a value is 0 or not. The algorithm assumes every value in <tt>[-Delta, Delta]</tt> to be 0. The default value for <tt>Delta</tt> is 0.00000000001.</item><br />
<br />
<item>@param <em>NormalizeType</em> An integer of the range 1..4. The default value is 2. This parameter describes, if/how the input points are normalized. If <tt>NormalizeType</tt> equals 1, each coordinate is divided by the maximal absolute value of the corresponding column of the matrix. This ensures that all coordinates of points are in [-1,1]. With <tt>NormalizeType=2</tt> no normalization is done at all. <tt>NormalizeType=3</tt> shifts each coordinate to [-1,1]. So it's minimum is mapped to -1 and the maximum to one, describing a unique affine mapping. The last option is <tt>NormalizeType=4</tt>. In this case, each coordinate is normalized, using the euclidian norm of the column.</item><br />
</itemize><br />
<br />
<example><br />
Use P ::= Q[x];<br />
<br />
PointsReal := Mat([[0],[0]]);<br />
PointsComp := Mat([[1],[-1]]);<br />
<br />
Res := Num.CABM(PointsReal, PointsComp, 0.1);<br />
<br />
Dec(Res,3);<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[[<quotes>-0.707 x^2 -0.707 </quotes>, <quotes>0</quotes>], [<quotes>1</quotes>, <quotes> x </quotes>]]<br />
-------------------------------<br />
</example><br />
</description><br />
<br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.SubABM</see><br />
<see>Num.ABM</see><br />
<see>Num.BBABM</see><br />
<see>Num.DABM</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>points</type><br />
</types><br />
<key>CABM</key><br />
<key>Num.CABM</key><br />
<key>numerical.CABM</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.ABM&diff=10378ApCoCoA-1:Num.ABM2010-05-12T16:36:58Z<p>132.231.10.62: Small correction</p>
<hr />
<div><command><br />
<title>Num.ABM</title><br />
<br />
<short_description>Computes the border basis of an almost vanishing ideal for a set of points using the ABM algorithm.</short_description><br />
<syntax><br />
Num.ABM(Points:MAT, Epsilon:RAT):Object<br />
Num.ABM(Points:MAT, Epsilon:RAT, Delta:RAT, NormalizeType:INT):Object<br />
</syntax><br />
<br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This command computes a border basis of an almost vanishing ideal for a set of points. <br />
<par/><br />
The current ring has to be a ring over the rational numbers with a standard-degree<br />
compatible term-ordering. The matrix <tt>Points</tt> contains the points: each<br />
point is a row in the matrix, so the number of columns must equal the<br />
number of indeterminates in the current ring. <br />
<br />
<itemize><br />
<item>@param <em>Points</em> The points for which a border basis is computed.</item><br />
<br />
<item>@param <em>Epsilon</em> A positive rational number describing the maximal admissible least squares error for a polynomial. (Bigger values for <tt>Epsilon</tt> lead to bigger errors of the polynomials evaluated at the point set). <tt>Epsilon</tt> should be in the interval (0,1). As a rule of thumb, <tt>Epsilon</tt> is the expected percentage of error on the input points. </item><br />
<br />
<item>@return A list of two results. First the border basis as a list of polynomials, second the vector space basis of <tt>P/I</tt> as a list of terms.</item><br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positiv rational number. <tt>Delta</tt> describes the computing precision. In different steps, it is crucial, if a value is 0 or not. The algorithm assumes every value in <tt>[-Delta, Delta]</tt> to be 0. The default value for <tt>Delta</tt> is 0.00000000001.</item><br />
<br />
<item>@param <em>NormalizeType</em> A integer of the range 1..4. The default value is 2. This parameter describes, if/how the input points are normalized. If <tt>NormalizeType</tt> equals 1, each coordinate is divided by the maximal absolute value of the corresponding column of the matrix. This ensures that all coordinates of points are in [-1,1]. With <tt>NormalizeType=2</tt> no normalization is done at all. <tt>NormalizeType=3</tt> shifts each coordinate to [-1,1]. So it's minimum is mapped to -1 and the maximum to one, describing a unique affine mapping. The last option is <tt>NormalizeType=4</tt>. In this case, each coordinate is normalized, using the column's euclidian norm.</item><br />
</itemize><br />
<br />
<example><br />
Use P::=QQ[x,y,z];<br />
<br />
Points := Mat([[1,0,0],[0,0,1],[0,0.99,0]]);<br />
Res := Num.ABM(Points,0.1);<br />
<br />
Dec(Res[1],2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
[<quotes>1 x +1.01 y +0.99 z -0.99 </quotes>, <quotes>1 z^2 -0.99 z +0.00 </quotes>, <quotes>1 yz </quotes>, <quotes>1 xz </quotes>, <quotes>1 y^2 -0.98 y -0.00 </quotes>, <quotes>1 xy </quotes>]<br />
-------------------------------<br />
</example><br />
</description><br />
<br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.SubABM</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>points</type><br />
</types><br />
<key>ABM</key><br />
<key>Num.ABM</key><br />
<key>numerical.ABM</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.BBABM&diff=10377ApCoCoA-1:Num.BBABM2010-05-12T16:34:50Z<p>132.231.10.62: </p>
<hr />
<div><command><br />
<title>Num.BBABM</title><br />
<br />
<short_description>Computes the border basis of an almost vanishing ideal for a set of points using the BB ABM algorithm.</short_description><br />
<syntax><br />
Num.BBABM(Points:MAT, Epsilon:RAT):Object<br />
Num.BBABM(Points:MAT, Epsilon:RAT, Delta:RAT, NormalizeType:INT):Object<br />
</syntax><br />
<br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This command computes a border basis of an almost vanishing ideal for a set of points. In contrast to the ABM algorithm the border polynomials are computed in such a way, that the residual is calculated on polynomials that have leading borderterm coefficient one.<br />
<par/><br />
The current ring has to be a ring over the rational numbers with a standard-degree<br />
compatible term-ordering. The matrix <tt>Points</tt> contains the points: each<br />
point is a row in the matrix, so the number of columns must equal the<br />
number of indeterminates in the current ring. <br />
<br />
<itemize><br />
<item>@param <em>Points</em> The points for which a border basis is computed.</item><br />
<item>@param <em>Epsilon</em> A positive rational number describing the maximal admissible least squares error for a polynomial. (Bigger values for <tt>Epsilon</tt> lead to bigger errors of the polynomials evaluated at the point set). <tt>Epsilon</tt> should be in the interval (0,1). As a rule of thumb, <tt>Epsilon</tt> is the expected percentage of error on the input points. </item><br />
<item>@return A list of two results. First the border basis as a list of polynomials, second the vector space basis of <tt>P/I</tt> as a list of terms.</item><br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positiv rational number. <tt>Delta</tt> describes the computing precision. In different steps, it is crucial, if a value is 0 or not. The algorithm assumes every value in <tt>[-Delta, Delta]</tt> to be 0. The default value for <tt>Delta</tt> is 0.00000000001.</item><br />
<br />
<item>@param <em>NormalizeType</em> A integer of the range 1..4. The default value is 2. This parameter describes, if/how the input points are normalized. If <tt>NormalizeType</tt> equals 1, each coordinate is divided by the maximal absolute value of the corresponding column of the matrix. This ensures that all coordinates of points are in [-1,1]. With <tt>NormalizeType=2</tt> no normalization is done at all. <tt>NormalizeType=3</tt> shifts each coordinate to [-1,1]. So it's minimum is mapped to -1 and the maximum to one, describing a unique affine mapping. The last option is <tt>NormalizeType=4</tt>. In this case, each coordinate is normalized, using the column's euclidian norm. Due to backward compatibility, the default is 1, although 3 is in most cases a better choice.</item><br />
</itemize><br />
<br />
<example><br />
Use P::=QQ[x,y,z];<br />
<br />
Points := Mat([[1,0,0],[0,0,1],[0,0.99,0]]);<br />
Res := Num.BBABM(Points,0.1);<br />
<br />
Dec(Res[1],2);<br />
-- CoCoAServer: computing Cpu Time = 0.015<br />
-------------------------------<br />
[<quotes>1 x +1.01 y +1 z -0.99 </quotes>, <quotes>1 z^2 -1 z </quotes>, <quotes>1 yz </quotes>, <quotes>1 xz </quotes>, <quotes>1 y^2 -0.98 y </quotes>, <quotes>1 xy </quotes>]<br />
--------------------------------------------------------------<br />
</example><br />
</description><br />
<br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.SubBBABM</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>points</type><br />
</types><br />
<key>BBABM</key><br />
<key>Num.BBABM</key><br />
<key>numerical.BBABM</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.CABM&diff=10366ApCoCoA-1:Num.CABM2010-05-12T14:42:00Z<p>132.231.10.62: </p>
<hr />
<div><command><br />
<title>Num.CABM</title><br />
<br />
<short_description>Computes the border basis of an almost vanishing ideal for a set of points.</short_description><br />
<syntax><br />
Num.CABM(PointsReal:MAT, PointsComp:MAT, Epsilon:RAT):Object<br />
Num.CABM(PointsReal:MAT, PointsComp:MAT, Epsilon:RAT, Delta:RAT, NormalizeType:INT):Object<br />
</syntax><br />
<br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This command computes a border basis of an almost vanishing ideal for a set of complex points. <br />
<par/><br />
The current ring has to be a ring over the rational numbers with a standard-degree<br />
compatible term-ordering. The matrix <tt>PointsReal</tt> and <tt>PointsComp</tt> contain the points: each<br />
point is a row in the matrix, so the number of columns must equal the<br />
number of indeterminates in the current ring. <br />
<br />
<itemize><br />
<item>@param <em>PointsReal</em> The real component of the points for which a border basis is computed.</item><br />
<br />
<item>@param <em>PointsComp</em> The complex component of the points for which a border basis is computed.</item><br />
<br />
<item>@param <em>Epsilon</em> A positive rational number describing the maximal admissible least squares error for a polynomial. (Bigger values for <tt>Epsilon</tt> lead to bigger errors of the polynomials evaluated at the point set). <tt>Epsilon</tt> should be in the interval (0,1). As a rule of thumb, <tt>Epsilon</tt> is the expected percentage of error on the input points. </item><br />
<br />
<item>@return A list of two results. First the border basis as a list of polynomials, second the vector space basis of <tt>P/I</tt> as a list of terms.</item><br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positiv rational number. <tt>Delta</tt> describes the computing precision. In different steps, it is crucial, if a value is 0 or not. The algorithm assumes every value in <tt>[-Delta, Delta]</tt> to be 0. The default value for <tt>Delta</tt> is 0.00000000001.</item><br />
<br />
<item>@param <em>NormalizeType</em> A integer of the range 1..4. The default value is 2. This parameter describes, if/how the input points are normalized. If <tt>NormalizeType</tt> equals 1, each coordinate is divided by the maximal absolute value of the corresponding column of the matrix. This ensures that all coordinates of points are in [-1,1]. With <tt>NormalizeType=2</tt> no normalization is done at all. <tt>NormalizeType=3</tt> shifts each coordinate to [-1,1]. So it's minimum is mapped to -1 and the maximum to one, describing a unique affine mapping. The last option is <tt>NormalizeType=4</tt>. In this case, each coordinate is normalized, using the column's euclidian norm.</item><br />
</itemize><br />
<br />
<example><br />
Use P ::= Q[x];<br />
<br />
PointsReal := Mat([[0],[0]]);<br />
PointsComp := Mat([[1],[-1]]);<br />
<br />
Res := Num.CABM(PointsReal, PointsComp, 0.1);<br />
<br />
Dec(Res,3);<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[[<quotes>-0.707 x^2 -0.707 </quotes>, <quotes>0</quotes>], [<quotes>1</quotes>, <quotes> x </quotes>]]<br />
-------------------------------<br />
</example><br />
</description><br />
<br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.SubABM</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>points</type><br />
</types><br />
<key>ABM</key><br />
<key>Num.ABM</key><br />
<key>numerical.ABM</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.CABM&diff=10365ApCoCoA-1:Num.CABM2010-05-12T14:27:26Z<p>132.231.10.62: </p>
<hr />
<div><command><br />
<title>Num.CABM</title><br />
<br />
<short_description>Computes the border basis of an almost vanishing ideal for a set of points.</short_description><br />
<syntax><br />
Num.CABM(PointsReal:MAT, PointsComp:MAT, Epsilon:RAT):Object<br />
Num.CABM(PointsReal:MAT, PointsComp:MAT, Epsilon:RAT, Delta:RAT, NormalizeType:INT):Object<br />
</syntax><br />
<br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This command computes a border basis of an almost vanishing ideal for a set of complex points. <br />
<par/><br />
The current ring has to be a ring over the rational numbers with a standard-degree<br />
compatible term-ordering. The matrix <tt>PointsReal</tt> and <tt>PointsComp</tt> contain the points: each<br />
point is a row in the matrix, so the number of columns must equal the<br />
number of indeterminates in the current ring. <br />
<br />
<itemize><br />
<item>@param <em>PointsReal</em> The real component of the points for which a border basis is computed.</item><br />
<br />
<item>@param <em>PointsComp</em> The complex component of the points for which a border basis is computed.</item><br />
<br />
<item>@param <em>Epsilon</em> A positive rational number describing the maximal admissible least squares error for a polynomial. (Bigger values for <tt>Epsilon</tt> lead to bigger errors of the polynomials evaluated at the point set). <tt>Epsilon</tt> should be in the interval (0,1). As a rule of thumb, <tt>Epsilon</tt> is the expected percentage of error on the input points. </item><br />
<br />
<item>@return A list of two results. First the border basis as a list of polynomials, second the vector space basis of <tt>P/I</tt> as a list of terms.</item><br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positiv rational number. <tt>Delta</tt> describes the computing precision. In different steps, it is crucial, if a value is 0 or not. The algorithm assumes every value in <tt>[-Delta, Delta]</tt> to be 0. The default value for <tt>Delta</tt> is 0.00000000001.</item><br />
<br />
<item>@param <em>NormalizeType</em> A integer of the range 1..4. The default value is 2. This parameter describes, if/how the input points are normalized. If <tt>NormalizeType</tt> equals 1, each coordinate is divided by the maximal absolute value of the corresponding column of the matrix. This ensures that all coordinates of points are in [-1,1]. With <tt>NormalizeType=2</tt> no normalization is done at all. <tt>NormalizeType=3</tt> shifts each coordinate to [-1,1]. So it's minimum is mapped to -1 and the maximum to one, describing a unique affine mapping. The last option is <tt>NormalizeType=4</tt>. In this case, each coordinate is normalized, using the column's euclidian norm.</item><br />
</itemize><br />
<br />
<example><br />
Use P::=QQ[x,y,z];<br />
<br />
Points := Mat([[1,0,0],[0,0,1],[0,0.99,0]]);<br />
Res := Num.ABM(Points,0.1);<br />
<br />
Dec(Res[1],2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
[<quotes>1 x +1.01 y +0.99 z -0.99 </quotes>, <quotes>1 z^2 -0.99 z +0.00 </quotes>, <quotes>1 yz </quotes>, <quotes>1 xz </quotes>, <quotes>1 y^2 -0.98 y -0.00 </quotes>, <quotes>1 xy </quotes>]<br />
-------------------------------<br />
</example><br />
</description><br />
<br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.SubABM</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>points</type><br />
</types><br />
<key>ABM</key><br />
<key>Num.ABM</key><br />
<key>numerical.ABM</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.CABM&diff=10364ApCoCoA-1:Num.CABM2010-05-12T14:18:16Z<p>132.231.10.62: Added CABM</p>
<hr />
<div><command><br />
<title>Num.CABM</title><br />
<br />
<short_description>Computes the border basis of an almost vanishing ideal for a set of points.</short_description><br />
<syntax><br />
Num.CABM(PointsReal:MAT, PointsComp:MAT, Epsilon:RAT):Object<br />
Num.CABM(PointsReal:MAT, PointsComp:MAT, Epsilon:RAT, Delta:RAT, NormalizeType:INT):Object<br />
</syntax><br />
<br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This command computes a border basis of an almost vanishing ideal for a set of points. <br />
<par/><br />
The current ring has to be a ring over the rational numbers with a standard-degree<br />
compatible term-ordering. The matrix <tt>Points</tt> contains the points: each<br />
point is a row in the matrix, so the number of columns must equal the<br />
number of indeterminates in the current ring. <br />
<br />
<itemize><br />
<item>@param <em>Points</em> The points for which a border basis is computed.</item><br />
<br />
<item>@param <em>Epsilon</em> A positive rational number describing the maximal admissible least squares error for a polynomial. (Bigger values for <tt>Epsilon</tt> lead to bigger errors of the polynomials evaluated at the point set). <tt>Epsilon</tt> should be in the interval (0,1). As a rule of thumb, <tt>Epsilon</tt> is the expected percentage of error on the input points. </item><br />
<br />
<item>@return A list of two results. First the border basis as a list of polynomials, second the vector space basis of <tt>P/I</tt> as a list of terms.</item><br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positiv rational number. <tt>Delta</tt> describes the computing precision. In different steps, it is crucial, if a value is 0 or not. The algorithm assumes every value in <tt>[-Delta, Delta]</tt> to be 0. The default value for <tt>Delta</tt> is 0.00000000001.</item><br />
<br />
<item>@param <em>NormalizeType</em> A integer of the range 1..4. The default value is 2. This parameter describes, if/how the input points are normalized. If <tt>NormalizeType</tt> equals 1, each coordinate is divided by the maximal absolute value of the corresponding column of the matrix. This ensures that all coordinates of points are in [-1,1]. With <tt>NormalizeType=2</tt> no normalization is done at all. <tt>NormalizeType=3</tt> shifts each coordinate to [-1,1]. So it's minimum is mapped to -1 and the maximum to one, describing a unique affine mapping. The last option is <tt>NormalizeType=4</tt>. In this case, each coordinate is normalized, using the column's euclidian norm.</item><br />
</itemize><br />
<br />
<example><br />
Use P::=QQ[x,y,z];<br />
<br />
Points := Mat([[1,0,0],[0,0,1],[0,0.99,0]]);<br />
Res := Num.ABM(Points,0.1);<br />
<br />
Dec(Res[1],2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
[<quotes>1 x +1.01 y +0.99 z -0.99 </quotes>, <quotes>1 z^2 -0.99 z +0.00 </quotes>, <quotes>1 yz </quotes>, <quotes>1 xz </quotes>, <quotes>1 y^2 -0.98 y -0.00 </quotes>, <quotes>1 xy </quotes>]<br />
-------------------------------<br />
</example><br />
</description><br />
<br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.SubABM</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>points</type><br />
</types><br />
<key>ABM</key><br />
<key>Num.ABM</key><br />
<key>numerical.ABM</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.ABM&diff=10315ApCoCoA-1:Num.ABM2010-04-19T13:55:33Z<p>132.231.10.62: </p>
<hr />
<div><command><br />
<title>Num.ABM</title><br />
<br />
<short_description>Computes the border basis of an almost vanishing ideal for a set of points.</short_description><br />
<syntax><br />
Num.ABM(Points:MAT, Epsilon:RAT):Object<br />
Num.ABM(Points:MAT, Epsilon:RAT, Delta:RAT, NormalizeType:INT):Object<br />
</syntax><br />
<br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This command computes a border basis of an almost vanishing ideal for a set of points. <br />
<par/><br />
The current ring has to be a ring over the rational numbers with a standard-degree<br />
compatible term-ordering. The matrix <tt>Points</tt> contains the points: each<br />
point is a row in the matrix, so the number of columns must equal the<br />
number of indeterminates in the current ring. <br />
<br />
<itemize><br />
<item>@param <em>Points</em> The points for which a border basis is computed.</item><br />
<br />
<item>@param <em>Epsilon</em> A positive rational number describing the maximal admissible least squares error for a polynomial. (Bigger values for <tt>Epsilon</tt> lead to bigger errors of the polynomials evaluated at the point set). <tt>Epsilon</tt> should be in the interval (0,1). As a rule of thumb, <tt>Epsilon</tt> is the expected percentage of error on the input points. </item><br />
<br />
<item>@return A list of two results. First the border basis as a list of polynomials, second the vector space basis of <tt>P/I</tt> as a list of terms.</item><br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positiv rational number. <tt>Delta</tt> describes the computing precision. In different steps, it is crucial, if a value is 0 or not. The algorithm assumes every value in <tt>[-Delta, Delta]</tt> to be 0. The default value for <tt>Delta</tt> is 0.00000000001.</item><br />
<br />
<item>@param <em>NormalizeType</em> A integer of the range 1..4. The default value is 2. This parameter describes, if/how the input points are normalized. If <tt>NormalizeType</tt> equals 1, each coordinate is divided by the maximal absolute value of the corresponding column of the matrix. This ensures that all coordinates of points are in [-1,1]. With <tt>NormalizeType=2</tt> no normalization is done at all. <tt>NormalizeType=3</tt> shifts each coordinate to [-1,1]. So it's minimum is mapped to -1 and the maximum to one, describing a unique affine mapping. The last option is <tt>NormalizeType=4</tt>. In this case, each coordinate is normalized, using the column's euclidian norm.</item><br />
</itemize><br />
<br />
<example><br />
Use P::=QQ[x,y,z];<br />
<br />
Points := Mat([[1,0,0],[0,0,1],[0,0.99,0]]);<br />
Res := Num.ABM(Points,0.1);<br />
<br />
Dec(Res[1],2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
[<quotes>1 x +1.01 y +0.99 z -0.99 </quotes>, <quotes>1 z^2 -0.99 z +0.00 </quotes>, <quotes>1 yz </quotes>, <quotes>1 xz </quotes>, <quotes>1 y^2 -0.98 y -0.00 </quotes>, <quotes>1 xy </quotes>]<br />
-------------------------------<br />
</example><br />
</description><br />
<br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.SubABM</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>points</type><br />
</types><br />
<key>ABM</key><br />
<key>Num.ABM</key><br />
<key>numerical.ABM</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=Downloads&diff=10248Downloads2009-10-16T10:58:08Z<p>132.231.10.62: Links for Windows versions of ApCoCoA 1.3</p>
<hr />
<div>The following releases provide the functionality of CoCoA 4.7 together with the additional capabilities of the ApCoCoA library. They are equipped with a graphical user interface. For the official CoCoA releases, visit the [http://cocoa.dima.unige.it CoCoA home page].<br />
<br />
=ApCoCoA Eclipse GUI - Standalone Version=<br />
<br />
==ApCoCoA-GUI-0.2-RC1 with ApCoCoA 1.2==<br />
<br />
'''Short installation guide:'''<br />
# Download and unpack the Eclipse GUI package.<br />
# Go into the folder apcocoa and start apcocoa.exe<br />
<br />
* Windows x86<br />
** [http://www.apcocoa.org/download/moccha/win32/ApCoCoABuild-win32.win32.x86-RC1.zip ApCoCoA GUI Release Candidate 1] with [http://www.apcocoa.org/download/moccha/win32/ApCoCoABuild-win32.win32.x86-RC1.zip.md5 md5sum] (2009/09/07)<br />
<br />
=ApCoCoA=<br />
<br />
==ApCoCoA-1.3==<br />
<br />
===Eclipse GUI - Standalone Version===<br />
<br />
'''Short installation guide:'''<br />
# Download and unpack the Eclipse GUI package.<br />
# Go into the folder apcocoa and start apcocoa.exe<br />
<br />
* Windows x86: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.3.exe ApCoCoA-1.3 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.3.md5 md5sum] (2009/10/16)<br />
<br />
* Linux x86:<br />
<br />
* Linux x86-64:<br />
<br />
* MacOSx:<br />
<br />
<br />
===ApCoCoA as eclipse-plugin===<br />
<br />
For using ApCoCoA-1.3 as eclipse-plugin please go to [[HowTo:Install_and_Work_with_the_Eclipse_GUI|eclipse-GUI installation guide]].<br />
<br />
<br />
===Well-known Qt-GUI===<br />
<br />
* Windows x86: [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.3QT.exe ApCoCoA-1.3QT Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.3QT.md5 md5sum] (2009/10/16)<br />
<br />
* Linux x86:<br />
<br />
* Linux x86-64:<br />
<br />
* MacOSx:<br />
<br />
<br />
==ApCoCoA-1.2==<br />
<br />
<font color=red>'''Please note:''' It is recommended to use ApCoCoA-1.2 with the beta version of the new eclipse-GUI. For Installation go to [[HowTo:Install_and_Work_with_the_Eclipse_GUI|eclipse-GUI installation guide]]. As soon as possible there will be a standalone version of the eclipse-GUI, too. Below you find the executables for ApCoCoA-1.2 using the old, well-known Qt-GUI.</font><br />
<br />
* Windows x86:<br />
** [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.2.exe ApCoCoA-1.2 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.2.exe.md5 md5sum] (2009/07/15)<br />
<br />
* Linux x86:<br />
**[http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.2-linux32-x86.tar.gz ApCoCoA-1.2-Release-linux32-x86] with [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.2-linux32-x86.tar.gz.md5 md5sum] and [[ApCoCoA:Build_specifications#ApCoCoA-1.2 Linux 32bit|build specifications]] (2009/07/15)<br />
<br />
* Linux x86-64:<br />
**[http://www.apcocoa.org/download/apcocoa/linux-x86-64/apcocoa-1.2-linux64-x86-64.tar.gz ApCoCoA-1.2-Release-linux64-x86] with [http://www.apcocoa.org/download/apcocoa/linux-x86-64/apcocoa-1.2-linux64-x86-64.tar.gz.md5 md5sum] and [[ApCoCoA:Build_specifications#ApCoCoA-1.2 Linux 64bit|build specifications]] (2009/07/15)<br />
<br />
* MacOSx:<br />
** [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.2-osx-intel.tgz ApCoCoA-1.2-osx-intel] with [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.2-osx-intel.tgz.md5 md5sum] and [[ApCoCoA:Build_specifications#ApCoCoA-1.2 Mac_OS_X|build specifications]] (2009/07/15)<br />
<br />
==ApCoCoA-1.1==<br />
<br />
* Windows x86:<br />
** [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.1.exe ApCoCoA-1.1 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.1.exe.md5 md5sum] (2009/04/30)<br />
<br />
* Linux x86:<br />
**[http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.1-linux-x86.tar.gz ApCoCoA-1.1-Release-linux32-x86] with [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.1-linux-x86.tar.gz.md5 md5sum] and [[ApCoCoA:Build_specifications#ApCoCoA-1.1 Linux 32bit|build specifications]] (2009/04/30)<br />
<br />
* Linux x86-64:<br />
**[http://www.apcocoa.org/download/apcocoa/linux-x86-64/apcocoa-1.1-linux-x86-64.tar.gz ApCoCoA-1.1-Release-linux64-x86] with [http://www.apcocoa.org/download/apcocoa/linux-x86-64/apcocoa-1.1-linux-x86-64.tar.gz.md5 md5sum] and [[ApCoCoA:Build_specifications#ApCoCoA-1.1 Linux 64bit|build specifications]] (2009/04/30)<br />
<br />
* MacOSx:<br />
** [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.1-osx-intel.dmg ApCoCoA-1.1-osx-intel] with [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.1-osx-intel.dmg.md5 md5sum] and [[ApCoCoA:Build_specifications#ApCoCoA-1.1 Mac_OS_X|build specifications]] (2009/04/30)<br />
<br />
==ApCoCoA-1.0==<br />
* Windows x86:<br />
**[http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.0.exe ApCoCoA-1.0 Release] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.0.exe.md5 md5sum] (2008/12/22)<br />
<br />
* Linux x86:<br />
**[http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.0-linux-x86.tar.gz ApCoCoA-1.0-Release-linux32-x86] with [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.0-linux-x86.tar.gz.md5 md5sum] (2008/12/22)<br />
<br />
* Linux x86-64:<br />
**[http://www.apcocoa.org/download/apcocoa/linux-x86-64/apcocoa-1.0-text-linux-x86-64.tar.gz ApCoCoA-1.0-Release-linux64-x86] with [http://www.apcocoa.org/download/apcocoa/linux-x86-64/apcocoa-1.0-text-linux-x86-64.tar.gz.md5 md5sum] (2008/12/22)<br />
<br />
* MacOSx:<br />
** [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.0.0-osx-intel.pkg.tgz ApCoCoA-1.0.0-osx-intel] with [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.0.0-osx-intel.pkg.tgz.md5 md5sum] (2008/12/22)<br />
<br />
<br />
<br />
==Older versions==<br />
* Windows x86:<br />
**[http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.0.0-rc7.exe ApCoCoA-1.0.0-rc7] with [http://www.apcocoa.org/download/apcocoa/win32/ApCoCoA-1.0.0-rc7.exe.md5sum md5sum] (2007/11/19)<br />
**[http://apcocoa.org/download/apcocoa/win32/ApCoCoA-4.7.1-0.99.09-win-x86.exe ApCoCoA-4.7.1-0.99.09-win-32] with [http://apcocoa.org/download/apcocoa/win32/ApCoCoA-4.7.1-0.99.09-win-x86.exe.md5 md5sum] (2007/10/19)<br />
**[http://apcocoa.org/download/apcocoa/win32/ApCoCoA-4.7.1-0.99.04-win-x86.exe ApCoCoA-4.7.1-0.99.04-win-32] with [http://apcocoa.org/download/apcocoa/win32/ApCoCoA-4.7.1-0.99.04-win-x86.exe.md5 md5sum]<br />
<br />
* Linux x86:<br />
**[http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.0-rc7-linux-x86.tar.gz ApCoCoA-1.0-rc7-linux-x86] with [http://www.apcocoa.org/download/apcocoa/linux-x86/apcocoa-1.0-rc7-linux-x86.tar.gz.md5 md5sum] (2007/11/19)<br />
**[http://apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-4.7.1-0.99.04-linux-x86.tar.gz ApCoCoA-4.7.1-0.99.04-linux-x86] with [http://apcocoa.org/download/apcocoa/linux-x86/ApCoCoA-4.7.1-0.99.04-linux-x86.tar.gz.md5 md5sum]<br />
<br />
* Linux x86-64:<br />
**[http://apcocoa.org/download/apcocoa/linux-x86-64/ApCoCoA-4.7.1-0.99.04-linux-x86-64.tar.gz ApCoCoA-4.7.1-0.99.04-linux-x86-64] with [http://apcocoa.org/download/apcocoa/linux-x86-64/ApCoCoA-4.7.1-0.99.04-linux-x86-64.tar.gz.md5 md5sum]<br />
<br />
* MacOSX:<br />
**[http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.0.0-rc7-ppc-only.dmg ApCoCoA-1.0.0-rc7-ppc-only] with [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-1.0.0-rc7-ppc-only.dmg.md5 md5sum] (2007/11/19)<br />
**[http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-4.7.1-0.99.04.dmg ApCoCoA-4.7.1-0.99.04-osx-universal] with [http://www.apcocoa.org/download/apcocoa/osx-universal/ApCoCoA-4.7.1-0.99.04.dmg.md5 md5sum]<br />
<br />
<br />
<br />
=ApCoCoALib=<br />
To build ApCoCoALib from source, please follow the [[ApCoCoALib:CompilationInstructions|compilation instructions]].<br />
<br />
==Repository==<br />
The [[ApCoCoA:SourceCodeManagement|subversion repository]] contains the bleeding-edge developmental versions of ApCoCoALib.<br />
<br />
==ApCoCoALib-1.02==<br />
<br />
[http://www.apcocoa.org/download/apcocoa/sources/ApCoCoALib-1.02.tar.gz ApCoCoALib-1.02.tar.gz] (1.4MB) with these [[ApCoCoA:Build specifications#ApCoCoA-1.2_build_specifications|build specifications]].<br />
<br />
==ApCoCoALib-1.01==<br />
<br />
[http://www.apcocoa.org/download/apcocoa/sources/ApCoCoALib-1.01.tar.gz ApCoCoALib-1.01.tar.gz] (1.3MB) with these [[ApCoCoA:Build specifications#ApCoCoA-1.1_build_specifications|build specifications]].<br />
<br />
==ApCoCoALib-1.00==<br />
[http://www.apcocoa.org/hg/ApCoCoALib-1.00/archive/tip.tar.gz ApCoCoALib.tar.gz] compiles against [http://www.apcocoa.org/download/cocoa/sources/CoCoALib-0.9914.tgz CoCoALib-0.9914.tgz] (1.1MB)<br />
<br />
<br />
==Deprecated sources==<br />
*[http://www.apcocoa.org/download/apcocoa/sources/ApCoCoALib-0.99.07.tar.gz ApCoCoALib-0.99.07.tar.gz] (246KB) - backup of the mercurial repository from v0.99.07, compiles against [http://www.apcocoa.org/download/cocoa/sources/CoCoALib-0.9907.tgz CoCoALib-0.9907.tgz] (1.1MB)<br />
*[http://www.apcocoa.org/download/apcocoa/sources/ApCoCoALib-0.99.05.tar.gz ApCoCoALib-0.99.05.tar.gz] (175KB) - backup of the mercurial repository from v0.99.05, compiles against [http://www.apcocoa.org/download/cocoa/sources/CoCoALib-0.9905.tgz CoCoALib-0.9905.tgz] (1.4MB)<br />
<br />
=ApCoCoA-Matlab-Interface=<br />
Only available for MS Windows. For details see [[ApCoCoA:MatlabToolbox]].<br />
*[http://www.apcocoa.org/~mmachnik/interface/Setup1-02.zip Version 1.02] - Based on CoCoALib 0.99.25 and ApCoCoALib as of 30.01.2009.<br />
<br />
[[Category:ApCoCoA|{{PAGENAME}}]]<br />
[[Category:ApCoCoA|Lib{{PAGENAME}}]]</div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.EXTABM&diff=10123ApCoCoA-1:Num.EXTABM2009-10-08T09:10:36Z<p>132.231.10.62: </p>
<hr />
<div><command><br />
<title>Num.EXTABM</title><br />
<br />
<short_description>Computes the border basis of an almost vanishing ideal for a set of points.</short_description><br />
<syntax><br />
Num.EXTABM(Points:MAT, Val:MAT, Epsilon:RAT):Object<br />
Num.EXTABM(Points:MAT, Val:MAT, Epsilon:RAT, Delta:RAT, NormalizeType:INT):Object<br />
</syntax><br />
<br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This command computes a border basis of an almost vanishing ideal for a set of points. A special property of the polynomials is, that when evaluated at the original set of points, one obtains approximately <tt>Val</tt>. To obtain an approximate ideal an additional indeterminate has to be added to each equation, which represents the coordinates of Val.<br />
<par/><br />
The current ring has to be a ring over the rational numbers with a standard-degree<br />
compatible term-ordering. The matrix <tt>Points</tt> contains the points: each<br />
point is a row in the matrix, so the number of columns must equal the<br />
number of indeterminates in the current ring. <br />
<br />
<itemize><br />
<item>@param <em>Points</em> The points for which a border basis is computed.</item><br />
<br />
<item>@param <em>Tau</em> A positive rational number describing which singular values should be treated as 0 (smaller values for <tt>Tau</tt> lead to bigger errors of the polynomials evaluated at the point set). <tt>Tau</tt> should be in the interval (0,1). As a rule of thumb, <tt>Tau</tt> is the expected percentage of error on the input points. </item><br />
<br />
<item>@return A list of two results. First the border basis as a list of polynomials, second the vector space basis of <tt>P/I</tt> as a list of terms.</item><br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positiv rational number. <tt>Delta</tt> describes the computing precision. In different steps, it is crucial, if a value is 0 or not. The algorithm assumes every value in <tt>[-Delta, Delta]</tt> to be 0. The default value for <tt>Delta</tt> is 0.00000000001.</item><br />
<br />
<item>@param <em>NormalizeType</em> A integer of the range 1..4. The default value is 2. This parameter describes, if/how the input points are normalized. If <tt>NormalizeType</tt> equals 1, each coordinate is divided by the maximal absolute value of the corresponding column of the matrix. This ensures that all coordinates of points are in [-1,1]. With <tt>NormalizeType=2</tt> no normalization is done at all. <tt>NormalizeType=3</tt> shifts each coordinate to [-1,1]. So it's minimum is mapped to -1 and the maximum to one, describing a unique affine mapping. The last option is <tt>NormalizeType=4</tt>. In this case, each coordinate is normalized, using the column's euclidian norm. Due to backward compatibility, the default is 1, although 3 is in most cases a better choice.</item><br />
</itemize><br />
<br />
<example><br />
Use P::=QQ[x,y,z];<br />
<br />
Points := Mat([[1,2,3],[4,5,6],[7,11,12]]);<br />
Val := Mat([[1],[0.1],[0.2]]);<br />
R:=Num.EXTABM(Points,Val, 0.1);<br />
<br />
Dec(-Eval(R[1],Points[1]),3);<br />
Dec(-Eval(R[1],Points[2]),3);<br />
Dec(-Eval(R[1],Points[3]),3);<br />
<br />
<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[<quotes>1.000</quotes> <quotes>0.999</quotes> <quotes>0.999</quotes> <quotes>0.999</quotes> <quotes>0.999</quotes> <quotes>1.000</quotes>]<br />
-------------------------------<br />
[<quotes>0.099</quotes> <quotes>0.099</quotes> <quotes>0.099</quotes> <quotes>0.099</quotes> <quotes>0.099</quotes> <quotes>0.099</quotes>]<br />
-------------------------------<br />
[<quotes>0.199</quotes> <quotes>0.200</quotes> <quotes>0.199</quotes> <quotes>0.200</quotes> <quotes>0.199</quotes> <quotes>0.199</quotes>]<br />
-------------------------------<br />
</example><br />
</description><br />
<br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.SubEXTABM</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>points</type><br />
</types><br />
<key>EXTABM</key><br />
<key>Num.EXTABM</key><br />
<key>numerical.EXTABM</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.Floor&diff=10101ApCoCoA-1:Num.Floor2009-10-07T12:38:15Z<p>132.231.10.62: Added Floor command</p>
<hr />
<div><command><br />
<title>Floor</title><br />
<short_description>Maps a rational number to the next smallest integer.</short_description><br />
<br />
<syntax><br />
Num.Floor(Num:RAT):INT<br />
</syntax><br />
<br />
<description><br />
Maps a rational number <tt>Num</tt> to the next smallest integer.<br />
<br />
<itemize><br />
<item>@param <em>Num</em> A rational number.</item><br />
<item>@return The next smallest integer.</item><br />
</itemize><br />
<br />
<example><br />
<br />
Num.Floor(8.1);<br />
Num.Floor(-8.1);<br />
Num.Floor(8);<br />
8<br />
-------------------------------<br />
-9<br />
-------------------------------<br />
8<br />
-------------------------------<br />
<br />
</example><br />
<br />
</description><br />
<br />
<types><br />
<type>integer</type><br />
<type>rat</type><br />
</types><br />
<br />
<seealso><br />
<see>Ceil</see><br />
<see>FPart</see><br />
<see>PrintLn</see><br />
</seealso><br />
<br />
<key>Floor</key><br />
<br />
<br />
[[Category:ApCoCoA Manual|{{PAGENAME}}]]<br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.RatPoints&diff=10075ApCoCoA-1:Num.RatPoints2009-10-06T11:23:48Z<p>132.231.10.62: Updated example</p>
<hr />
<div> <command><br />
<title>Num.RatPoints</title><br />
<short_description>Computes the zero set of an exact zero dimensional border basis. The zeros are computed approximately using the eigenvalues of the transposed multiplication matrices.</short_description><br />
<syntax><br />
Num.RatPoints(AppBB:LIST, OrderIdeal:LIST)):LIST of MAT<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This command computes a set of points, which are the zeros of an exact border basis. This border basis is close to the approximate border basis <tt>AppBB</tt>. The set of (complex) points is represented as two matrices. The <tt>j</tt>-th column of the first matrix gives the real part of a point and the <tt>j</tt>-th column of the second matrix gives the imaginary part. For computation the function is using the <ref>Num.EigenValues</ref> command.<br />
<br />
<itemize><br />
<item>@param <em>AppBB</em> An approximate border basis.</item><br />
<item>@param <em>OrderIdeal</em> The associated order ideal</item><br />
<item>@return A set of points in matrix form described above.</item><br />
</itemize><br />
<br />
<example><br />
Use P::=QQ[x,y,z];<br />
<br />
Points := Mat([[2/3,0,0],[0,10,0],[0,0,1/3]]);<br />
R:=Num.ABM(Points, 0);<br />
Dec(Num.RatPoints(R[1],R[2]),2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
[Mat([<br />
[<quotes>0.66</quotes>, <quotes>0.00</quotes>, <quotes>0</quotes>],<br />
[<quotes>0</quotes>, <quotes>0</quotes>, <quotes>10</quotes>],<br />
[<quotes>0</quotes>, <quotes>0.33</quotes>, <quotes>0</quotes>]<br />
]), Mat([<br />
[<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0</quotes>],<br />
[<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0</quotes>],<br />
[<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0</quotes>]<br />
])]<br />
-------------------------------<br />
<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>polynomial</type><br />
<type>points</type><br />
</types><br />
<key>Num.RatPoints</key><br />
<key>RatPoints</key><br />
<key>numerical.RatPoints</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.RatPoints&diff=10067ApCoCoA-1:Num.RatPoints2009-10-06T09:47:32Z<p>132.231.10.62: Corrected description</p>
<hr />
<div> <command><br />
<title>Num.RatPoints</title><br />
<short_description>Computes the zero set of an exact zero dimensional border basis. The zeros are computed approximately using the eigenvalues of the transposed multiplication matrices.</short_description><br />
<syntax><br />
Num.RatPoints(AppBB:LIST, OrderIdeal:LIST)):LIST of MAT<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
This command computes a set of points, which are the zeros of an exact border basis. This border basis is close to the approximate border basis <tt>AppBB</tt>. The set of (complex) points is represented as two matrices. The <tt>j</tt>-th column of the first matrix gives the real part of a point and the <tt>j</tt>-th column of the second matrix gives the imaginary part. For computation the function is using the <ref>Num.EigenValues</ref> command.<br />
<br />
<itemize><br />
<item>@param <em>AppBB</em> An approximate border basis.</item><br />
<item>@param <em>OrderIdeal</em> The associated order ideal</item><br />
<item>@return A set of points in matrix form described above.</item><br />
</itemize><br />
<br />
<example><br />
Use P::=QQ[x,y,z];<br />
<br />
Points := Mat([[2/3,0,0],[0,1,0],[0,0,1/3]]);<br />
R:=Num.SubAVI(Points, 0.001, [1]);<br />
Dec(Num.RatPoints(R[1],R[2]),2);<br />
<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[Mat([<br />
[<quotes>0.66</quotes>, <quotes>0.00</quotes>, <quotes>-0.00</quotes>],<br />
[<quotes>0</quotes>, <quotes>0</quotes>, <quotes>1.00</quotes>],<br />
[<quotes>0</quotes>, <quotes>0.33</quotes>, <quotes>0</quotes>]<br />
]), Mat([<br />
[<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0</quotes>],<br />
[<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0</quotes>],<br />
[<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0</quotes>]<br />
])]<br />
-------------------------------<br />
<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>polynomial</type><br />
<type>points</type><br />
</types><br />
<key>Num.RatPoints</key><br />
<key>RatPoints</key><br />
<key>numerical.RatPoints</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>132.231.10.62