http://apcocoa.uni-passau.de/wiki/api.php?action=feedcontributions&user=XMLBot&feedformat=atomApCoCoAWiki - User contributions [en]2024-03-29T04:48:25ZUser contributionsMediaWiki 1.35.0http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.NDneighbors&diff=7213ApCoCoA-1:BB.NDneighbors2007-11-08T20:35:16Z<p>XMLBot: ApCoCoA:Borderbasis.NDneighbors moved to ApCoCoA:BB.NDneighbors: new alias</p>
<hr />
<div><command><br />
<title>BB.NDneighbors</title><br />
<short_description>list of next-door neighbors</short_description><br />
<syntax><br />
BB.NDneighbors(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the list of next-door neighbors in the border of the order ideal OO. The input is a list OO of terms that specify an order ideal. The output is a list of triples <formula>[i,j,k]</formula> such that <formula>b_i = x_k * b_j</formula>.<br />
<example><br />
Use Q[x,y,z];<br />
BB.NDneighbors([1,x]);<br />
[[3, 1, 1], [4, 2, 1]]<br />
-------------------------------<br />
</example><br />
</description><br />
<see>BB.ASneighbors</see><br />
<key>kreuzer</key><br />
<key>bb.ndneighbors</key><br />
<key>borderbasis.ndneighbors</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.NDgens&diff=7211ApCoCoA-1:BB.NDgens2007-11-08T20:35:13Z<p>XMLBot: ApCoCoA:Borderbasis.NDgens moved to ApCoCoA:BB.NDgens: new alias</p>
<hr />
<div><command><br />
<title>BB.NDgens</title><br />
<short_description>generators from vanishing ideal of a border basis scheme</short_description><br />
<syntax><br />
BB.NDgens(K:INT,OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the generators of the vanishing ideal of the border basis scheme corresponding to the lifting of the K-th element of NDneighbors(OO). The inputs are an integer K in the range 1..Len(NDneighbors(OO)) and a list OO of terms that specify an order ideal. The output is a list of polynomials in the ring <formula>BBS=K[c_{ij}]</formula>.<br />
<example><br />
Use Q[x,y,z];<br />
BB.NDgens(1, [1,x]);<br />
[BBS :: c[1,5]c[2,1] - c[1,3], BBS :: c[2,1]c[2,5] + c[1,1] - c[2,3]]<br />
-------------------------------<br />
</example><br />
</description><br />
<see>BB.ASgens</see><br />
<see>BB.HomASgens</see><br />
<see>BB.HomNDgens</see><br />
<key>kreuzer</key><br />
<key>bb.ndgens</key><br />
<key>borderbasis.ndgens</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.LiftND&diff=7209ApCoCoA-1:BB.LiftND2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.LiftND moved to ApCoCoA:BB.LiftND: new alias</p>
<hr />
<div><command><br />
<title>BB.LiftND</title><br />
<short_description>border basis scheme for the lifting of next-door neighbors</short_description><br />
<syntax><br />
BB.LiftND(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Compute the equations defining the border basis scheme and coming from the lifting of next-door neighbors. The input is a list of terms OO (2nd element of type POLY). The output is a list of poly in the ring <formula>BBS=K[c_{ij}]</formula>.<br />
</description><br />
<key>kreuzer</key><br />
<key>bb.liftnd</key><br />
<key>borderbasis.liftnd</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.LiftHomND&diff=7207ApCoCoA-1:BB.LiftHomND2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.LiftHomND moved to ApCoCoA:BB.LiftHomND: new alias</p>
<hr />
<div><command><br />
<title>BB.LiftHomND</title><br />
<short_description>homogeneous border basis scheme for the lifting of next-door neighbors</short_description><br />
<syntax><br />
BB.LiftHomND(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Compute the equations defining the homogeneous border basis scheme and coming from the lifting of next-door neighbors. The input is a list of terms OO (2nd element of type POLY). The output is a list of poly in the ring <formula>BBS=K[c_{ij}]</formula>.<br />
</description><br />
<key>kreuzer</key><br />
<key>bb.lifthomnd</key><br />
<key>borderbasis.lifthomnd</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.LiftHomAS&diff=7205ApCoCoA-1:BB.LiftHomAS2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.LiftHomAS moved to ApCoCoA:BB.LiftHomAS: new alias</p>
<hr />
<div><command><br />
<title>BB.LiftHomAS</title><br />
<short_description>homogeneous border basis scheme for the lifting of across-the-street neighbors</short_description><br />
<syntax><br />
BB.LiftHomAS(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Compute the equations defining the homogeneous border basis scheme and coming from the lifting of across-the-street neighbors. The input is a list of terms OO (2nd element of type POLY). The output is a list of poly in the ring <formula>BBS=K[c_{ij}]</formula>.<br />
</description><br />
<key>kreuzer</key><br />
<key>bb.lifthomas</key><br />
<key>borderbasis.lifthomas</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.LiftAS&diff=7203ApCoCoA-1:BB.LiftAS2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.LiftAS moved to ApCoCoA:BB.LiftAS: new alias</p>
<hr />
<div><command><br />
<title>BB.LiftAS</title><br />
<short_description>border basis scheme for the lifting of across-the-street neighbors</short_description><br />
<syntax><br />
BB.LiftAS(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Compute the equations defining the border basis scheme and coming from the lifting of across-the-street neighbors. The input is a list of terms OO (2nd element of type POLY). The output is a list of poly in the ring <formula>BBS=K[c_{ij}]</formula>.<br />
</description><br />
<key>kreuzer</key><br />
<key>bb.liftas</key><br />
<key>borderbasis.liftas</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.HomNDgens&diff=7201ApCoCoA-1:BB.HomNDgens2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.HomNDgens moved to ApCoCoA:BB.HomNDgens: new alias</p>
<hr />
<div><command><br />
<title>BB.HomNDgens</title><br />
<short_description>generators of vanishing ideal of homogeneous border basis scheme</short_description><br />
<syntax><br />
BB.HomNDgens(K:INT,OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the generators of the vanishing ideal of the homogenous border basis scheme corresponding to the lifting of the K-th element of NDneighbors(OO). The inputs are an integer K in the range 1..Len(NDneighbors(OO)) and a list OO of terms that specify an order ideal. The output is a list of polynomials in the ring <formula>BBS=K[c_{ij}]</formula>.<br />
</description><br />
<see>BB.ASgens</see><br />
<see>BB.HomASgens</see><br />
<see>BB.NDgens</see><br />
<key>kreuzer</key><br />
<key>bb.homndgens</key><br />
<key>borderbasis.homndgens</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.HomBBscheme&diff=7199ApCoCoA-1:BB.HomBBscheme2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.HomBBscheme moved to ApCoCoA:BB.HomBBscheme: new alias</p>
<hr />
<div><command><br />
<title>BB.HomBBscheme</title><br />
<short_description>defining equations of homogeneous border basis scheme</short_description><br />
<syntax><br />
BB.HomBBscheme(OO:LIST):IDEAL<br />
</syntax><br />
<description><br />
Computes the defining equations of the homogeneous border basis scheme using the commutators of the generic homogeneous multiplication matrices. The input is a list OO of terms that specify an order ideal. The second element of OO must be a non-constant polynomial. The output is an ideal in the ring <formula>BBS = K[c_{ij}]</formula>.<br />
</description><br />
<see>BB.BBscheme</see><br />
<key>kreuzer</key><br />
<key>bb.hombbscheme</key><br />
<key>borderbasis.hombbscheme</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.HomASgens&diff=7197ApCoCoA-1:BB.HomASgens2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.HomASgens moved to ApCoCoA:BB.HomASgens: new alias</p>
<hr />
<div><command><br />
<title>BB.HomASgens</title><br />
<short_description>generators of vanishing ideal of homogeneous border basis scheme</short_description><br />
<syntax><br />
BB.HomASgens(K:INT,OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the generators of the vanishing ideal of the homogeneous border basis scheme corresponding to the lifting of the K-th element of ASneighbors(OO). The inputs are an integer K in the range 1..Len(ASneighbors(OO)) and a list OO of terms that specify an order ideal. The output is a list of polynomials in the ring <formula>BBS=K[c_{ij}]</formula>.<br />
</description><br />
<see>BB.ASgens</see><br />
<see>BB.HomNDgens</see><br />
<see>BB.NDgens</see><br />
<key>kreuzer</key><br />
<key>bb.homasgens</key><br />
<key>borderbasis.homasgens</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.GenericHomBB&diff=7195ApCoCoA-1:BB.GenericHomBB2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.GenericHomBB moved to ApCoCoA:BB.GenericHomBB: new alias</p>
<hr />
<div><command><br />
<title>BB.GenericHomBB</title><br />
<short_description>generic homogeneous border basis</short_description><br />
<syntax><br />
BB.GenericHomBB(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the <quotes>generic</quotes> homogeneous border basis w.r.t. an order ideal OO. The input is a list of terms OO (2nd element of type POLY). The output is a list of POLY in a <quotes>universal family ring</quotes> UF where <formula>UF = K[x_1,..,x_n,c_{ij}]</formula>.<br />
</description><br />
<key>kreuzer</key><br />
<key>bb.generichombb</key><br />
<key>borderbasis.generichombb</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.GenericBB&diff=7193ApCoCoA-1:BB.GenericBB2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.GenericBB moved to ApCoCoA:BB.GenericBB: new alias</p>
<hr />
<div><command><br />
<title>BB.GenericBB</title><br />
<short_description>generic border basis</short_description><br />
<syntax><br />
BB.GenericBB(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the <quotes>generic</quotes> border basis w.r.t. an order ideal OO i.e. the polys <formula>g_j = b_j - \sum_i c_{ij} t_i</formula>. The input is a list of terms OO (2nd element of type POLY). The output is a list of POLY in a <quotes>universal family ring</quotes> UF where <formula>UF = K[x_1,..,x_n,c_{ij}]</formula>.<br />
</description><br />
<key>kreuzer</key><br />
<key>bb.genericbb</key><br />
<key>borderbasis.genericbb</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.GenMultMat&diff=7191ApCoCoA-1:BB.GenMultMat2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.GenMultMat moved to ApCoCoA:BB.GenMultMat: new alias</p>
<hr />
<div><command><br />
<title>BB.GenMultMat</title><br />
<short_description>generic multiplication matrix</short_description><br />
<syntax><br />
BB.GenMultMat(I:INT,OO:LIST):MAT<br />
</syntax><br />
<description><br />
Computes the generic multiplication matrix for <formula>x[I]</formula> with respect to an order ideal. The inputs are an integer I and a list OO of terms that specify an order ideal. The second element of OO must be a non-constant polynomial. The output is a matrix of size <formula>Mu<times/>Mu</formula> over the ring <formula>BBS=K[c_{ij}]</formula>.<br />
</description><br />
<key>kreuzer</key><br />
<key>bb.genmultmat</key><br />
<key>borderbasis.genmultmat</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.GenHomMultMat&diff=7189ApCoCoA-1:BB.GenHomMultMat2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.GenHomMultMat moved to ApCoCoA:BB.GenHomMultMat: new alias</p>
<hr />
<div><command><br />
<title>BB.GenHomMultMat</title><br />
<short_description>generic homogeneous multiplication matrix</short_description><br />
<syntax><br />
BB.GenHomMultMat(I:INT,OO:LIST):MAT<br />
</syntax><br />
<description><br />
Computes the generic homogeneous multiplication matrix for <formula>x[I]</formula> with respect to an order ideal. The inputs are an integer I and a list OO of terms that specify an order ideal. The second element of OO must be a non-constant polynomial. The output is a matrix of size <formula>Mu<times/>Mu</formula> over the ring <formula>BBS=K[c_{ij}]</formula>.<br />
</description><br />
<key>kreuzer</key><br />
<key>bb.genhommultmat</key><br />
<key>borderbasis.genhommultmat</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.Box&diff=7187ApCoCoA-1:BB.Box2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.Box moved to ApCoCoA:BB.Box: new alias</p>
<hr />
<div><command><br />
<title>BB.Box</title><br />
<short_description>box order ideal</short_description><br />
<syntax><br />
BB.Box(D:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the box order ideal of type <formula>D=[D_1,..,D_N]</formula>. The input is a list of integers D of length NumIndets(). The output is a list of terms sorted with respect to the current term ordering.<br />
<example><br />
Use Q[x,y,z];<br />
BB.Box([2,1,1]);<br />
[1, x]<br />
-------------------------------<br />
</example><br />
</description><br />
<see>BB.Border</see><br />
<key>kreuzer</key><br />
<key>bb.box</key><br />
<key>borderbasis.box</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.BBasisForOI&diff=7185ApCoCoA-1:BB.BBasisForOI2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.BorderBasis moved to ApCoCoA:BB.BorderBasis: new alias</p>
<hr />
<div><command><br />
<title>BB.BorderBasis</title><br />
<short_description>border basis of an ideal</short_description><br />
<syntax><br />
BB.BorderBasis(F:LIST,OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the border basis of the ideal <formula>I=&lt;F&gt;</formula> with respect to the order ideal OO. Gives an error messages if no border basis exists. Uses the <formula>O_{\sigma}(I)</formula> border basis and the border basis transformation algorithm. The inputs are a list of polynomials F and a list OO of terms that specify an order ideal. The output is a list of polynomials.<br />
<example><br />
Use Q[x,y];<br />
BB.BorderBasis([x^2, xy + y^2], [1,x,y,y^2]);<br />
<br />
[xy + y^2, x^2, y^3, xy^2]<br />
-------------------------------<br />
</example><br />
</description><br />
<see>BBasis</see><br />
<key>kreuzer</key><br />
<key>bb.borderbasis</key><br />
<key>borderbasis.borderbasis</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.Border&diff=7183ApCoCoA-1:BB.Border2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.Border moved to ApCoCoA:BB.Border: new alias</p>
<hr />
<div><command><br />
<title>BB.Border</title><br />
<short_description>border of an order ideal</short_description><br />
<syntax><br />
BB.Border(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the border of an order ideal OO. The input/output is a list of terms.<br />
<example><br />
Use Q[x,y,z];<br />
BB.Border([1,x]);<br />
[z, y, xz, xy, x^2]<br />
-------------------------------<br />
</example><br />
</description><br />
<see>BBasis</see><br />
<see>BB.BorderBasis</see><br />
<see>BB.Box</see><br />
<key>kreuzer</key><br />
<key>bb.border</key><br />
<key>borderbasis.border</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.BBscheme&diff=7181ApCoCoA-1:BB.BBscheme2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.BBscheme moved to ApCoCoA:BB.BBscheme: new alias</p>
<hr />
<div><command><br />
<title>BB.BBscheme</title><br />
<short_description>defining equations of border basis scheme</short_description><br />
<syntax><br />
BB.BBscheme(OO:LIST):IDEAL<br />
</syntax><br />
<description><br />
Computes the defining equations of the border basis scheme using the commutators of the multiplication matrices. The input is a list OO of terms that specify an order ideal. The second element of OO must be a non-constant polynomial. The output is an ideal in the ring <formula>BBS = K[c_{ij}]</formula>.<br />
<example><br />
Use Q[x,y,z];<br />
BB.BBscheme([1,x]);<br />
BBS :: Ideal(c[1,5]c[2,2] - c[1,4], c[1,2]c[1,5] - c[1,5]c[2,4] + c[1,4]c[2,5],<br />
c[2,2]c[2,5] + c[1,2] - c[2,4], c[1,5]c[2,2] - c[1,4], c[1,5]c[2,1] - c[1,3],<br />
c[1,1]c[1,5] - c[1,5]c[2,3] + c[1,3]c[2,5], c[2,1]c[2,5] + c[1,1] - c[2,3],<br />
c[1,5]c[2,1] - c[1,3], c[1,4]c[2,1] - c[1,3]c[2,2],<br />
c[1,2]c[1,3] - c[1,1]c[1,4] + c[1,4]c[2,3] - c[1,3]c[2,4],<br />
c[1,2]c[2,1] - c[1,1]c[2,2] + c[2,2]c[2,3] - c[2,1]c[2,4], c[1,4]c[2,1] - c[1,3]c[2,2])<br />
-------------------------------<br />
</example><br />
</description><br />
<see>BB.HomBBscheme</see><br />
<key>kreuzer</key><br />
<key>bb.bbscheme</key><br />
<key>borderbasis.bbscheme</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.ASneighbors&diff=7179ApCoCoA-1:BB.ASneighbors2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.ASneighbors moved to ApCoCoA:BB.ASneighbors: new alias</p>
<hr />
<div><command><br />
<title>BB.ASneighbors</title><br />
<short_description>list of across-the-street neighbors</short_description><br />
<syntax><br />
BB.ASneighbors(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the list of across-the-street neighbors in the border of the order ideal OO. The input is a list OO of terms that specify an order ideal. The output is a list of quadruples <formula>[i,j,k,l]</formula> such that <formula>x_k * b_i = x_l * b_j</formula>.<br />
<example><br />
Use Q[x,y,z];<br />
BB.ASneighbors([1,x]);<br />
[[1, 2, 2, 3], [3, 4, 2, 3], [3, 5, 1, 3], [4, 5, 1, 2]]<br />
-------------------------------<br />
</example><br />
</description><br />
<see>BB.NDneighbors</see><br />
<key>kreuzer</key><br />
<key>bb.asneighbors</key><br />
<key>borderbasis.asneighbors</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.ASgens&diff=7177ApCoCoA-1:BB.ASgens2007-11-08T20:35:12Z<p>XMLBot: ApCoCoA:Borderbasis.ASgens moved to ApCoCoA:BB.ASgens: new alias</p>
<hr />
<div><command><br />
<title>BB.ASgens</title><br />
<short_description>generators from vanishing ideal of a border basis scheme</short_description><br />
<syntax><br />
BB.ASgens(K:INT,OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the generators of the vanishing ideal of the border basis scheme corresponding to the lifting of the K-th element of the list returned by ASneighbors(OO). The inputs are an integer K in the range 1..Len(ASneighbors(OO)) and a list OO of terms that specify an order ideal. The output is a list of polynomials in the ring <formula>BBS=K[c_{ij}]</formula>.<br />
<example><br />
Use Q[x,y,z];<br />
BB.ASgens(1, [1,x,y,z]);<br />
[BBS :: c[1,5]c[2,1] - c[1,3]c[2,2] + c[1,4]c[3,1] - c[1,2]c[3,2] + c[1,2]c[4,1] - c[1,1]c[4,2],<br />
BBS :: c[2,2]c[2,3] - c[2,1]c[2,5] - c[2,4]c[3,1] + c[2,2]c[3,2] - c[2,2]c[4,1] + c[2,1]c[4,2],<br />
BBS :: c[3,2]^2 + c[2,2]c[3,3] - c[3,1]c[3,4] - c[2,1]c[3,5] - c[3,2]c[4,1] + c[3,1]c[4,2] - c[1,1],<br />
BBS :: c[3,2]c[4,2] + c[2,2]c[4,3] - c[3,1]c[4,4] - c[2,1]c[4,5] + c[1,2]]<br />
-------------------------------<br />
</example><br />
</description><br />
<see>BB.HomASgens</see><br />
<see>BB.HomNDgens</see><br />
<see>BB.NDgens</see><br />
<key>kreuzer</key><br />
<key>bb.asgens</key><br />
<key>borderbasis.asgens</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Weyl.WeylMul&diff=6871ApCoCoA-1:Weyl.WeylMul2007-10-24T19:25:54Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div> <command><br />
<title>Weyl.Mul</title><br />
<short_description>multiplying two WeylPolynoms</short_description><br />
<syntax><br />
Weyl.Mul(P,Q):WeylPolynom<br />
</syntax><br />
<description><br />
{{Beta}}<br />
<br />
This method multiplies P and Q and returns P*Q as a WeylPolynom.<br />
<br />
{{Stub}}<br />
</description><br />
<seealso><br />
<see>Weyl.WeylPolynom</see><br />
<see>Weyl.NewWeylPolynom</see> <br />
<see>Weyl.IsWeylPolynom</see><br />
<see>Weyl.Add</see><br />
<see>Weyl.Sub</see><br />
<see>Weyl.Equals</see><br />
<see>Weyl.RollOver</see><br />
</seealso><br />
<wiki-category>Package_Weyl</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Weyl.WMul&diff=6865ApCoCoA-1:Weyl.WMul2007-10-24T19:25:54Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div> <command><br />
<title>Weyl.GBasis</title><br />
<short_description>computing a Groebner basis.</short_description><br />
<syntax><br />
Weyl.GBasis(P):LIST<br />
</syntax><br />
<description><br />
{{Beta}}<br />
<br />
This method computes an both sided ideal's Groebner basis and returns it as a list of WeylPolynoms. Currently, it is not yet implemented, due to some missing link in the server/client communication. <br />
<br />
{{Stub}}<br />
</description><br />
<seealso><br />
<see>Weyl.WeylIdeal</see><br />
<see>Weyl.WeylPolynom</see><br />
<see>Weyl.NewWeylIdeal</see><br />
</seealso><br />
<wiki-category>Package_Weyl</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.SVD&diff=6861ApCoCoA-1:Num.SVD2007-10-24T19:25:54Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div> <command><br />
<title>Numerical.SVD</title><br />
<short_description>Singular value decomposition of a matrix</short_description><br />
<syntax><br />
$numerical.SVD(A:Matrix):List<br />
</syntax><br />
<description><br />
This function returns a list of three matrices which form the singular<br />
value decomposition of the input matrix. The list produced is [U, S, VT].<br />
Warning: internally floating point values are used, so the result is only<br />
approximate.<br />
<br />
<example><br />
D:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10]]);<br />
$numerical.SVD(D);<br />
-------------------------------<br />
[Mat([<br />
[-2608957845014309/4503599627370496, 3400715993947695/4503599627370496, -1196230415249177/4503599627370496, -5542055005031021/36028797018963968],<br />
[-4803191187447087/18014398509481984, 4289880920686871/36028797018963968, 3813211715037953/9007199254740992, 7724713654272699/9007199254740992],<br />
[-7645273287337725/18014398509481984, -5741692259075309/36028797018963968, 3381220959856661/4503599627370496, -540919752203371/1125899906842624],<br />
[-5789886178591733/9007199254740992, -2813340077166513/4503599627370496, -7780633724302695/18014398509481984, 3606131681355807/36028797018963968]<br />
]), Mat([<br />
[1164315100749939/35184372088832, 4798366071344577/281474976710656, 3788674137264815/1125899906842624]<br />
]), Mat([<br />
[-8521591816535737/18014398509481984, -3744869794805223/9007199254740992, -6996513907843673/9007199254740992],<br />
[-3002889242741505/4503599627370496, -7337996657000815/18014398509481984, 2810636692253967/4503599627370496],<br />
[-5187087952406809/9007199254740992, 915526145687749/1125899906842624, -6091132379868651/72057594037927936]<br />
])]<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Numerical.QR</see><br />
<see>Numerical.EigenValues</see><br />
<see>Numerical.EigenValuesAndVectors</see><br />
<see>Numerical.EigenValuesAndAllVectors</see><br />
</seealso><br />
<types><br />
<type>cocoaserver</type><br />
</types><br />
<key>Heldt</key><br />
<key>numerical.svd</key><br />
<key>numericalsvd</key><br />
<wiki-category>Package_Numerical</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.QR&diff=6859ApCoCoA-1:Num.QR2007-10-24T19:25:54Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div> <command><br />
<title>Numerical.QR</title><br />
<short_description>QR-decomposition of a matrix</short_description><br />
<syntax><br />
$numerical.QR(A:Matrix):Matrix;<br />
</syntax><br />
<description><br />
This function returns a matrix, containing the <em>IMPI</em>. The matrix consists<br />
of the upper-right triangular matrix and the lower left triangular matrix,<br />
describing the input matrix's QR-decomposition.<br />
<br />
<example><br />
Points:=Mat([[1,2,3],[2,3,4],[3,4,5]]);<br />
$numerical.QR(Points);<br />
-------------------------------<br />
Mat([<br />
[-8425463406411593/2251799813685248, 7598355191047809/18014398509481984, 5698766393285857/9007199254740992],<br />
[-1504547036859213/281474976710656, 737074506864293/1125899906842624, 7744099468837749/9007199254740992],<br />
[-7823644591667907/1125899906842624, 5896596054914343/4503599627370496, -809/4611686018427387904]<br />
])<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Numerical.SVD</see><br />
<see>Numerical.EigenValues</see><br />
<see>Numerical.EigenValuesAndVectors</see><br />
<see>Numerical.EigenValuesAndAllVectors</see><br />
</seealso><br />
<types><br />
<type>cocoaserver</type><br />
</types><br />
<key>Heldt</key><br />
<key>numerical.qr</key><br />
<key>numericalqr</key><br />
<wiki-category>Package_Numerical</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.EigenValuesAndVectors&diff=6846ApCoCoA-1:Num.EigenValuesAndVectors2007-10-24T19:25:53Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div> <command><br />
<title>Numerical.EigenValuesAndVectors</title><br />
<short_description>Eigenvalues of a matrix</short_description><br />
<syntax><br />
$numerical.EigenValuesAndVectors(A:Matrix):List<br />
</syntax><br />
<description><br />
This function returns a List of two matrices, containing numerical approximation to A's eigenvalues and (right hand) eigenvectors. <br />
Therefore the input matrix A has to be rectangular!<br />
It is implemented in the ApCoCoA server, so you need a running server. It was not implemented in version 0.99.4 or previous. Also please keep in mind this method is based on blas/Lapack's eigenvalue solver and uses floating point arithmetic. This is not an exact, algebraic method!<br />
The output contains first of a matrix B, where the number of rows contains one of A's eigenvalues. The first column contains the eigenvalue's real part, the second the imaginary.<br />
The second part of the list is a matrix of the size of A, containing A's (right hand) eigenvectors. <br />
To compute only the left hand's eigenvectors apply this method to Transposed(A).<br />
<example><br />
A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]); <br />
Numerical.EigenValuesAndVectors(A); <br />
-- CoCoAServer: computing Cpu Time = 0.0038<br />
-------------------------------<br />
[Mat([<br />
[2038617447977453/70368744177664, 1593056728295919/4503599627370496, 0, 1717983664400761/562949953421312],<br />
[-3850002255576293/281474976710656, 1593056728295919/4503599627370496, 0, -1717983664400761/562949953421312]<br />
]), Mat([<br />
[-7110239176083849/18014398509481984, -5241040126502889/9007199254740992, -569232410323621/18014398509481984, 4695168387448581/18014398509481984],<br />
[-7846388397589843/18014398509481984, -3981313256671163/9007199254740992, -2719422585742633/9007199254740992, -4930385173711605/9007199254740992],<br />
[-3437594604471165/4503599627370496, 2800381393796867/4503599627370496, 6128985174171139/9007199254740992, 0],<br />
[-1207381852306067/4503599627370496, 634514467740541/2251799813685248, -2469130937097749/9007199254740992, 6644460631770309/144115188075855872]<br />
])]<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Numerical.QR</see><br />
<see>Numerical.SVD</see><br />
<see>Numerical.EigenValues</see><br />
<see>Numerical.EigenValuesAndAllVectors</see><br />
</seealso><br />
<wiki-category>Package_Numerical</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.EigenValuesAndAllVectors&diff=6845ApCoCoA-1:Num.EigenValuesAndAllVectors2007-10-24T19:25:53Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div> <command><br />
<title>Numerical.EigenValuesAndAllVectors</title><br />
<short_description>Eigenvalues and left and right eigenvectors of a matrix</short_description><br />
<syntax><br />
$numerical.EigenValues(A:Matrix):List<br />
</syntax><br />
<description><br />
This function returns a List of three matrices, containing numerical approximation to A's eigenvalues and right and left eigenvectors. <br />
Therefore the input matrix A has to be rectangular!<br />
It is implemented in the ApCoCoA server, so you need a running server. It was not implemented in version 0.99.4 or previous. Also please keep in mind this method is based on blas/Lapack's eigenvalue solver and uses floating point arithmetic. This is not an exact, algebraic method!<br />
The output contains first of a matrix B, where the number of rows contains one of A's eigenvalues. The first column contains the eigenvalue's real part, the second the imaginary.<br />
The second element of the list is a matrix of the size of A, containing A's left hand eigenvectors, while the third element in the list is a matrix containing the right hand eigenvectors.<br />
<example><br />
A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]); <br />
Numerical.EigenValuesAndAllVectors(A);<br />
-- CoCoAServer: computing Cpu Time = 0.0031<br />
-------------------------------<br />
[Mat([<br />
[2038617447977453/70368744177664, 1593056728295919/4503599627370496, 0, 1717983664400761/562949953421312],<br />
[-3850002255576293/281474976710656, 1593056728295919/4503599627370496, 0, -1717983664400761/562949953421312]<br />
]), Mat([<br />
[-4846625556027553/9007199254740992, -675715895173401/1125899906842624, 6285574018989927/36028797018963968, -7024364631742823/18014398509481984],<br />
[-5611119929071853/18014398509481984, -8025389267782659/36028797018963968, -630161806301403/4503599627370496, 7963794620848619/18014398509481984],<br />
[-3851121972702563/9007199254740992, 6293666352540409/36028797018963968, -2394868378529203/9007199254740992, -1824257157284653/36028797018963968],<br />
[-5910799605047357/9007199254740992, 6738448111784605/9007199254740992, 6552680769135833/9007199254740992, 0]<br />
]), Mat([<br />
[-7110239176083849/18014398509481984, -5241040126502889/9007199254740992, -569232410323621/18014398509481984, 4695168387448581/18014398509481984],<br />
[-7846388397589843/18014398509481984, -3981313256671163/9007199254740992, -2719422585742633/9007199254740992, -4930385173711605/9007199254740992],<br />
[-3437594604471165/4503599627370496, 2800381393796867/4503599627370496, 6128985174171139/9007199254740992, 0],<br />
[-1207381852306067/4503599627370496, 634514467740541/2251799813685248, -2469130937097749/9007199254740992, 6644460631770309/144115188075855872]<br />
])]<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Numerical.QR</see><br />
<see>Numerical.SVD</see><br />
<see>Numerical.EigenValues</see><br />
<see>Numerical.EigenValuesAndVectors</see><br />
</seealso><br />
<wiki-category>Package_Numerical</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.EigenValues&diff=6844ApCoCoA-1:Num.EigenValues2007-10-24T19:25:53Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div> <command><br />
<title>Numerical.EigenValues</title><br />
<short_description>Eigenvalues of a matrix</short_description><br />
<syntax><br />
$numerical.EigenValues(A:Matrix):List<br />
</syntax><br />
<description><br />
This function returns a matrix, containing numerical approximation to A's eigenvalues. <br />
Therefore the input matrix A has to be rectangular!<br />
It is implemented in the ApCoCoA server, so you need a running server. It was not implemented in version 0.99.4 or previous. Also please keep in mind this method is based on blas/Lapack's eigenvalue solver and uses floating point arithmetic. This is not an exact, algebraic method!<br />
The output contains of a matrix B, where the number of rows contains one of A's eigenvalues. The first column contains the eigenvalue's real part, the second the imaginary.<br />
<example><br />
A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]);<br />
Numerical.EigenValues(A);<br />
-- CoCoAServer: computing Cpu Time = 0.0049<br />
-------------------------------<br />
Mat([<br />
[2038617447977453/70368744177664, 1593056728295919/4503599627370496, 0, 1717983664400761/562949953421312],<br />
[-3850002255576293/281474976710656, 1593056728295919/4503599627370496, 0, -1717983664400761/562949953421312]<br />
])<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Numerical.QR</see><br />
<see>Numerical.SVD</see><br />
<see>Numerical.EigenValuesAndVectors</see><br />
<see>Numerical.EigenValuesAndAllVectors</see><br />
</seealso><br />
<wiki-category>Package_Numerical</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.SubAVI&diff=6842ApCoCoA-1:Num.SubAVI2007-10-24T19:25:53Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div> <command><br />
<title>Numerical.BBasisOfPointsInIdeal</title><br />
<short_description>border basis of an almost vanishing sub-ideal for a set of points and ideal</short_description><br />
<syntax><br />
$numerical.BBasisOfPointsInIdeal(Points, Epsilon, GetO, GBasis):Object<br />
</syntax><br />
<description><br />
This command computes a border basis of an almost vanishing sub-ideal for a set of points and ideal using the algorithm described in the paper<br />
<par/><br />
D. Heldt, M. Kreuzer, H. Poulisse: <em>Computing Approximate <br />
Vanishing Ideals</em> (Work in progress)<br />
<par/><br />
The current ring has to be a ring over the rationals with a standard-degree<br />
compatible term-ordering. The matrix Points 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. Epsilon is a rational &gt;0,<br />
describing which singular values should be treated as 0 (smaller values for<br />
epsilon lead to bigger errors of the polynomials evaluated at the point<br />
set). Epsilon should be in the interval (0,1). As a rule of thumb,<br />
Epsilon is the expected percentage of error on the input points. <br />
GetO must be either True or False. If it is true, the command<br />
returns a list of two values: the first contains the border basis, the<br />
second one a vector space basis of P/I comprising those power products<br />
lying outside the leading term ideal of I. If GetO is false, the function<br />
returns only the border basis (not in a list). GBasis must be a<br />
homogeneous Groebner Basis in the current ring. This basis defines the<br />
ideal we compute the approximate vanishing ideal's basis in. Warning: for<br />
reasons of efficiency the function does not check that the validity of<br />
GBasis.<br />
<br />
<example><br />
Points := Mat([[2,0,0],[0,3,0],[0,0,1]]);<br />
$numerical.BBasisOfPointsInIdeal(Points, 0.001, False,[z,y]);<br />
-------------------------------<br />
[z^2 - z, 1/3yz, 1/2xz, 1/9y^2 - 9007199254740991/27021597764222976y, 1/6xy]<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Numerical.GBasisOfPoints</see><br />
<see>Numerical.BBasisOfPoints</see><br />
<see>Numerical.HBasisOfPoints</see><br />
<see>Numerical.GBasisOfPointsInIdeal</see><br />
<see>Numerical.HBasisOfPointsInIdeal</see><br />
<see>Numerical.FirstVanishingRelations</see><br />
<see>Numerical.FirstVanishingRelationsInIdeal</see><br />
</seealso><br />
<types><br />
<type>cocoaserver</type><br />
</types><br />
<key>Heldt</key><br />
<key>numerical.bbasisofpointsinideal</key><br />
<key>numericalbbasisofpointsinideal</key><br />
<wiki-category>Package_Numerical</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.ABM&diff=6840ApCoCoA-1:Num.ABM2007-10-24T19:25:53Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div> <command><br />
<title>Numerical.BBasisOfPoints</title><br />
<short_description>Border basis of almost vanishing ideal for a set of points</short_description><br />
<syntax><br />
$numerical.BBasisOfPoints(Points, Epsilon, GetO):Object<br />
</syntax><br />
<description><br />
This command computes a border basis of an almost vanishing ideal for a set of points using the algorithm described in the paper<br />
<par/><br />
D. Heldt, M. Kreuzer, H. Poulisse, S.Pokutta: <em>Approximate Computation<br />
of Zero-Dimensional Ideals</em> Submitted: August 2006<br />
<par/><br />
The current ring has to be a ring over the rationals with a standard-degree<br />
compatible term-ordering. The matrix Points 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. Epsilon is a rational &gt;0<br />
describing which singular values should be treated as 0 (smaller values for<br />
Epsilon lead to bigger errors of the polynomials evaluated at the point<br />
set). Epsilon should be in the interval (0,1). As a rule of thumb, <br />
Epsilon is the expected percentage of error on the input points. GetO must <br />
be either True or False. If it is true, the command returns a list <br />
of two values: the first contains the border basis, the<br />
second one a vector space basis of P/I comprising those power products<br />
lying outside the leading term ideal of I. If GetO is false, the function<br />
returns only the border basis (not in a list).<br />
<example><br />
Points := Mat([[1,0,0],[0,0,1],[0,2,0]]);<br />
$numerical.BBasisOfPoints(Points,0.001,True);<br />
-------------------------------<br />
[[x + 9007199254740991/18014398509481984y + z - 1, z^2 - 9007199254740991/9007199254740992z, 1/2yz, xz, 1/4y^2 - 9007199254740991/18014398509481984y, 1/2xy], [y, z, 1]]<br />
<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Numerical.GBasisOfPoints</see><br />
<see>Numerical.HBasisOfPoints</see><br />
<see>Numerical.GBasisOfPointsInIdeal</see><br />
<see>Numerical.BBasisOfPointsInIdeal</see><br />
<see>Numerical.HBasisOfPointsInIdeal</see><br />
<see>Numerical.FirstVanishingRelations</see><br />
<see>Numerical.FirstVanishingRelationsInIdeal</see><br />
</seealso><br />
<types><br />
<type>cocoaserver</type><br />
</types><br />
<key>Heldt</key><br />
<key>numerical.bbasisofpoints</key><br />
<key>numericalbbasisofpoints</key><br />
<wiki-category>Package_Numerical</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:CharP.GBasisModSquares&diff=6839ApCoCoA-1:CharP.GBasisModSquares2007-10-24T19:25:53Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div> <command><br />
<title>Char2.GBasisModSquares</title><br />
<short_description>computing a gbasis of a given ideal, intersected with x^2-x for all indeterminates x</short_description><br />
<syntax><br />
$char2.GBasisModSquares(Ideal):List<br />
</syntax><br />
<description><br />
This function returns reduced Groebner basis for the ideal, intersected with the ideal, created by x^2-x for all indeterminates. If x^2-x for <br />
all indeterminates is in the ideal (e.g. the set of zeros is a subset of {0,1}^n) this method should produce the GBasis much faster!<br />
Please be aware, that this is much more efficient if the term ordering is Lex, DegLex or DegRevLex. Otherwise, first a DegRevLex GBasis is computed and then<br />
transformed with the FGLM-algorithm. <br />
</description><br />
<seealso><br />
<see>FGLM</see><br />
<see>GBasis</see><br />
</seealso><br />
<wiki-category>Package_char2</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.NDneighbors&diff=6838ApCoCoA-1:BB.NDneighbors2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.NDneighbors</title><br />
<short_description>Computes the list of next-door neighbors in the border of OO</short_description><br />
<syntax><br />
$borderbasis.NDneighbors(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the list of next-door neighbors in the border of OO. The input is a list of terms OO (2nd element of type POLY). The output is a list of triples [i,j,k] s.t. b_i = x_k * b_j.<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.ndneighbors</key><br />
<key>borderbasisndneighbors</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.NDgens&diff=6837ApCoCoA-1:BB.NDgens2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.NDgens</title><br />
<short_description>Computes the generators of the vanishing ideal of the border basis scheme corresponding to the lifting of the K-th element of NDneighbors(OO)</short_description><br />
<syntax><br />
$borderbasis.NDgens(K:INT,OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the generators of the vanishing ideal of the border basis scheme corresponding to the lifting of the K-th element of NDneighbors(OO). The input is an integer K In 1..<ttref>Len</ttref>(NDneighbors(OO)) and a list of terms OO (2nd element of type POLY). The output is a list of poly in the ring BBS=K[c_{ij}].<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.ndgens</key><br />
<key>borderbasisndgens</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.LiftND&diff=6836ApCoCoA-1:BB.LiftND2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.LiftND</title><br />
<short_description>Compute the equations defining the border basis scheme and coming from the lifting of next-door neighbors</short_description><br />
<syntax><br />
$borderbasis.LiftND(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Compute the equations defining the border basis scheme and coming from the lifting of next-door neighbors. The input is a list of terms OO (2nd element of type POLY). The output is a list of poly in the ring BBS=K[c_{ij}].<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.liftnd</key><br />
<key>borderbasisliftnd</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.LiftHomND&diff=6835ApCoCoA-1:BB.LiftHomND2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.LiftHomND</title><br />
<short_description>Compute the equations defining the hom. border basis scheme and coming from the lifting of next-door neighbors</short_description><br />
<syntax><br />
$borderbasis.LiftHomND(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Compute the equations defining the homogeneous border basis scheme and coming from the lifting of next-door neighbors. The input is a list of terms OO (2nd element of type POLY). The output is a list of poly in the ring BBS=K[c_{ij}].<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.lifthomnd</key><br />
<key>borderbasislifthomnd</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.LiftHomAS&diff=6834ApCoCoA-1:BB.LiftHomAS2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.LiftHomAS</title><br />
<short_description>Compute the equations defining the hom. border basis scheme and coming from the lifting of across-the-street neighbors</short_description><br />
<syntax><br />
$borderbasis.LiftHomAS(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Compute the equations defining the homogeneous border basis scheme and coming from the lifting of across-the-street neighbors. The input is a list of terms OO (2nd element of type POLY). The output is a list of poly in the ring BBS=K[c_{ij}].<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.lifthomas</key><br />
<key>borderbasislifthomas</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.LiftAS&diff=6833ApCoCoA-1:BB.LiftAS2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.LiftAS</title><br />
<short_description>Compute the equations defining the border basis scheme and coming from the lifting of across-the-street neighbors</short_description><br />
<syntax><br />
$borderbasis.LiftAS(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Compute the equations defining the border basis scheme and coming from the lifting of across-the-street neighbors. The input is a list of terms OO (2nd element of type POLY). The output is a list of poly in the ring BBS=K[c_{ij}].<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.liftas</key><br />
<key>borderbasisliftas</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.HomNDgens&diff=6832ApCoCoA-1:BB.HomNDgens2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.HomNDgens</title><br />
<short_description>Computes the generators of the vanishing ideal of the hom. border basis scheme corresp. to the lifting of the K-th element of NDneighbors(OO)</short_description><br />
<syntax><br />
$borderbasis.HomNDgens(K:INT,OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the generators of the vanishing ideal of the homogenous border basis scheme corresponding to the lifting of the K-th element of NDneighbors(OO). The input is an integer K In 1..<ttref>Len</ttref>(NDneighbors(OO)) and a list of terms OO (2nd element of type POLY). The output is a list of poly in the ring BBS=K[c_{ij}].<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.homndgens</key><br />
<key>borderbasishomndgens</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.HomBBscheme&diff=6831ApCoCoA-1:BB.HomBBscheme2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.HomBBscheme</title><br />
<short_description>Computes the defining equations of the homogeneous border basis scheme using the commutators of the generic hom. mult. matrices</short_description><br />
<syntax><br />
$borderbasis.HomBBscheme(OO:LIST):IDEAL<br />
</syntax><br />
<description><br />
Computes the defining equations of the homogeneous border basis scheme using the commutators of the generic homogeneous multiplication matrices. The input is an order ideal OO (2nd element of type POLY). The output is an ideal in the ring BBS = K[c_{ij}].<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.hombbscheme</key><br />
<key>borderbasishombbscheme</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.HomASgens&diff=6830ApCoCoA-1:BB.HomASgens2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.HomASgens</title><br />
<short_description>Computes the generators of the vanishing ideal of the hom. border basis scheme corresponding to the lifting of the K-th element of ASneighbors(OO)</short_description><br />
<syntax><br />
$borderbasis.HomASgens(K:INT,OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the generators of the vanishing ideal of the homogeneous border basis scheme corresponding to the lifting of the K-th element of ASneighbors(OO). The input is an integer K In 1..<ttref>Len</ttref>(ASneighbors(OO)) and a list of terms OO (2nd element of type POLY). The output is a list of poly in the ring BBS=K[c_{ij}].<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.homasgens</key><br />
<key>borderbasishomasgens</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.GenericHomBB&diff=6829ApCoCoA-1:BB.GenericHomBB2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.GenericHomBB</title><br />
<short_description>Computes the "generic" homogeneous border basis w.r.t. an order ideal OO</short_description><br />
<syntax><br />
$borderbasis.GenericHomBB(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the "generic" homogeneous border basis w.r.t. an order ideal OO. The input is a list of terms OO (2nd element of type POLY). The output is a list of POLY in a "universal family ring" UF where UF = K[x_1,..,x_n,c{ij}].<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.generichombb</key><br />
<key>borderbasisgenerichombb</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.GenericBB&diff=6828ApCoCoA-1:BB.GenericBB2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.GenericBB</title><br />
<short_description>Computes the "generic" border basis w.r.t. an order ideal OO</short_description><br />
<syntax><br />
$borderbasis.GenericBB(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the "generic" border basis w.r.t. an order ideal OO i.e. the polys g_j = b_j - \sum_i c_{ij} t_i. The input is a list of terms OO (2nd element of type POLY). The output is a list of POLY in a "universal family ring" UF where UF = K[x_1,..,x_n,c{ij}].<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.genericbb</key><br />
<key>borderbasisgenericbb</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.GenMultMat&diff=6827ApCoCoA-1:BB.GenMultMat2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.GenMultMat</title><br />
<short_description>Computes the generic multiplication matrix for x[I] with respect to the order ideal OO</short_description><br />
<syntax><br />
$borderbasis.GenMultMat(I:INT,OO:LIST):MAT<br />
</syntax><br />
<description><br />
Computes the generic multiplication matrix for x[I] with respect to the order ideal OO. The input is a positive integer I and an order ideal OO (2nd element of type POLY). The output is a matrix of size Mu x Mu over the ring BBS=K[c_{ij}]<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.genmultmat</key><br />
<key>borderbasisgenmultmat</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.GenHomMultMat&diff=6826ApCoCoA-1:BB.GenHomMultMat2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.GenHomMultMat</title><br />
<short_description>Computes the generic homogeneous mult. matrix for x[I] with respect to the order ideal OO</short_description><br />
<syntax><br />
$borderbasis.GenHomMultMat(I:INT,OO:LIST):MAT<br />
</syntax><br />
<description><br />
Computes the generic homogeneous multiplication matrix for x[I] with respect to the order ideal OO. The input is a positive integer I and an order ideal OO (2nd element of type POLY). The output is a matrix of size Mu x Mu over the ring BBS=K[c_{ij}]<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.genhommultmat</key><br />
<key>borderbasisgenhommultmat</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.Box&diff=6825ApCoCoA-1:BB.Box2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.Box</title><br />
<short_description>Computes the "box" order ideal of type D=[D1,..,DN]</short_description><br />
<syntax><br />
$borderbasis.Box(D:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the "box" order ideal of type D=[D1,..,DN]. The input is a list of integers D of length <ttref>NumIndets</ttref>(). The output is a list of terms (sorted w.r.t. current TO).<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.box</key><br />
<key>borderbasisbox</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.BBasisForOI&diff=6824ApCoCoA-1:BB.BBasisForOI2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.BorderBasis</title><br />
<short_description>Computes the border basis of the ideal I=&lt;F&gt; with respect to the order ideal OO</short_description><br />
<syntax><br />
$borderbasis.BorderBasis(F:LIST,OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the border basis of the ideal I=&lt;F&gt; with respect to the order ideal OO. Gives an error messages if no border basis exists. Uses the O_sigma(I) border basis and the BB transformation. The input is a list of poly F and a list of terms OO. The output is a list of poly.<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.borderbasis</key><br />
<key>borderbasisborderbasis</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.Border&diff=6823ApCoCoA-1:BB.Border2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.Border</title><br />
<short_description>Computes the border of an order ideal</short_description><br />
<syntax><br />
$borderbasis.Border(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the border of an order ideal OO. The input/output is a list of terms.<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.border</key><br />
<key>borderbasisborder</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.BBscheme&diff=6822ApCoCoA-1:BB.BBscheme2007-10-24T19:25:52Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.BBscheme</title><br />
<short_description>Computes the defining equations of the border basis scheme using the commutators of the multiplication matrices</short_description><br />
<syntax><br />
$borderbasis.BBscheme(OO:LIST):IDEAL<br />
</syntax><br />
<description><br />
Computes the defining equations of the border basis scheme using the commutators of the multiplication matrices. The input is an order ideal OO (2nd element of type POLY). The output is an ideal in the ring BBS = K[c_{ij}].<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.bbscheme</key><br />
<key>borderbasisbbscheme</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.ASneighbors&diff=6821ApCoCoA-1:BB.ASneighbors2007-10-24T19:25:51Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.ASneighbors</title><br />
<short_description>Computes the list of across-the-street neighbors in the border of OO</short_description><br />
<syntax><br />
$borderbasis.ASneighbors(OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the list of across-the-street neighbors in the border of OO. The input is a list of terms OO (2nd element of type POLY). The output is a list of quadruples [i,j,k,l] s.t. x_k * b_i = x_l * b_j.<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.asneighbors</key><br />
<key>borderbasisasneighbors</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.ASgens&diff=6820ApCoCoA-1:BB.ASgens2007-10-24T19:25:51Z<p>XMLBot: we no longer need manual sortkeys here</p>
<hr />
<div><command><br />
<title>borderbasis.ASgens</title><br />
<short_description>Computes the generators of the vanishing ideal of the border basis scheme corresponding to the lifting of the K-th element of ASneighbors(OO)</short_description><br />
<syntax><br />
$borderbasis.ASgens(K:INT,OO:LIST):LIST<br />
</syntax><br />
<description><br />
Computes the generators of the vanishing ideal of the border basis scheme corresponding to the lifting of the K-th element of ASneighbors(OO). The input is an integer K In 1..<ttref>Len</ttref>(ASneighbors(OO)) and a list of terms OO (2nd element of type POLY). The output is a list of poly in the ring BBS=K[c_{ij}].<br />
</description><br />
<key>Kreuzer</key><br />
<key>borderbasis.asgens</key><br />
<key>borderbasisasgens</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Weyl.WMul&diff=6759ApCoCoA-1:Weyl.WMul2007-10-24T10:17:58Z<p>XMLBot: ApCoCoA:WeylGBasis moved to ApCoCoA:Weyl.GBasis</p>
<hr />
<div><command><br />
<title>Weyl.GBasis</title><br />
<short_description>computing a Groebner basis.</short_description><br />
<syntax><br />
Weyl.GBasis(P):LIST<br />
</syntax><br />
<description><br />
{{Beta}}<br />
<br />
This method computes an both sided ideal's Groebner basis and returns it as a list of WeylPolynoms. Currently, it is not yet implemented, due to some missing link in the server/client communication. <br />
<br />
{{Stub}}<br />
</description><br />
<seealso><br />
<see>WeylIdeal</see><br />
<see>WeylPolynom</see><br />
<see>NewWeylIdeal</see><br />
<see>Package_Weyl</see><br />
</seealso><br />
<wiki-category>Package_Weyl</wiki-category><br />
</command></div>XMLBothttp://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Weyl.WeylMul&diff=6741ApCoCoA-1:Weyl.WeylMul2007-10-24T10:17:58Z<p>XMLBot: ApCoCoA:Mul moved to ApCoCoA:Weyl.Mul</p>
<hr />
<div><command><br />
<title>Weyl.Mul</title><br />
<short_description>multiplying two WeylPolynoms</short_description><br />
<syntax><br />
Weyl.Mul(P,Q):WeylPolynom<br />
</syntax><br />
<description><br />
{{Beta}}<br />
<br />
This method multiplies P and Q and returns P*Q as a WeylPolynom.<br />
<br />
{{Stub}}<br />
</description><br />
<seealso><br />
<see>NewWeylPolynom</see> <br />
<see>WeylPolynom</see><br />
<see>IsWeylPolynom</see><br />
<see>Add</see><br />
<see>Sub</see><br />
<see>Equals</see><br />
<see>RollOver</see><br />
<see>Package_Weyl</see><br />
</seealso><br />
<wiki-category>Package_Weyl</wiki-category><br />
</command></div>XMLBot