http://apcocoa.uni-passau.de/wiki/api.php?action=feedcontributions&user=193.79.225.178&feedformat=atomApCoCoAWiki - User contributions [en]2024-03-29T08:27:50ZUser contributionsMediaWiki 1.35.0http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:BB.TransformBBIntoGB&diff=9938ApCoCoA-1:BB.TransformBBIntoGB2009-07-15T08:54:32Z<p>193.79.225.178: Corrected example</p>
<hr />
<div><command><br />
<title>BB.TransformBBIntoGB</title><br />
<short_description>Transforms a border basis into a Groebner basis.</short_description><br />
<br />
<syntax><br />
BB.TransformBBIntoGB(BB:LIST of POLY):LIST of POLY<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 />
Let <tt>BB</tt> be a list of polynomials that form a <tt>O_sigma(I)</tt>-border basis of a zero-dimensional ideal <tt>I</tt>. This function extracts the reduced <tt>sigma</tt>-Groebner basis contained in the <tt>O_sigma(I)</tt>-border basis <tt>BB</tt> and returns it as a list of polynomials.<br />
<itemize><br />
<item>@param <em>BB</em> A border basis of an ideal.</item><br />
<item>@return A list of polynomials that represents the reduced Groebner basis of the ideal generated by the input polynomials in <tt>BB</tt>.</item><br />
</itemize><br />
<example><br />
Use ZZ/(32003)[x,y,z],DegLex;<br />
I := Ideal(<br />
4*x+5*y+6,<br />
2*x^2*z+4*y^2*z+4*y*z^2+3*x*y+25*y^2+7*x*z+2*y-3*z,<br />
x^2*y+3*x*y*z+x*z^2+15*x^2+x*y+9*y*z+7<br />
);<br />
BB := BB.BBasis(I); -- compute a border basis of I<br />
GB := BB.TransformBBIntoGB(BB);<br />
GB;<br />
<br />
-------------------------------<br />
[x + 8002y - 16000, y^2z - 5614yz^2 + 6179y^2 - 2246yz - 4492y - 3370z,<br />
y^3 + 12128yz^2 + 2045y^2 - 10508yz + 10240z^2 + 3337y - 8088z - 11495,<br />
z^4 - 928yz^2 + 15802z^3 - 8546y^2 - 13286yz - 15491z^2 - 13314y + 5553z - 11227,<br />
yz^3 - 9667yz^2 + 11342z^3 + 6752y^2 + 8104yz - 15091z^2 - 950y - 15081z + 885]<br />
-------------------------------<br />
</example><br />
</description><br />
<types><br />
<type>polynomial</type><br />
<type>borderbasis</type><br />
<type>groebner</type><br />
<type>apcocoaserver</type><br />
</types><br />
<br />
<see>Introduction to CoCoAServer</see><br />
<see>Introduction to Groebner Basis in CoCoA</see><br />
<see>BB.BBasis</see><br />
<see>GBasis</see><br />
<br />
<key>TransformBBIntoGB</key><br />
<key>BB.TransformBBIntoGB</key><br />
<key>borderbasis.TransformBBIntoGB</key><br />
<wiki-category>Package_borderbasis</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.SubAVI&diff=9937ApCoCoA-1:Num.SubAVI2009-07-15T08:44:42Z<p>193.79.225.178: correct info about default value</p>
<hr />
<div> <command><br />
<title>Num.SubAVI</title><br />
<short_description>Computes a border basis of an almost vanishing sub-ideal for a set of points and an ideal using the <ref>Num.AVI</ref> algorithm.</short_description><br />
<syntax><br />
Num.SubAVI(Points:MAT, Tau:RAT, GBasis:LIST):Object<br />
Num.SubAVI(Points:MAT, Tau:RAT, GBasis:LIST, Delta:RAT, NormalizeType:INT, RREFNormalizeType:INT, RREFUseEps:BOOL, RREFType:INT):Object<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 border basis of an almost vanishing sub-ideal for a set of points and an ideal using the <ref>Num.ABM</ref> algorithm.<br />
<par/><br />
The current ring has to be a ring over the rational numbers with a standard-degree<br />
compatible term-ordering. Each row in the matrix <tt>Points</tt> represents one point, 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 tau lead to bigger errors of the polynomials evaluated at the point set). Tau should be in the interval (0,1). As a rule of thumb, Tau is the expected percentage of error on the input points. </item><br />
<br />
<item>@param <em>GBasis</em> A homogeneous Groebner Basis in the current ring. This basis defines the ideal in which we compute the basis of the approximate vanishing ideal. <em>Warning:</em> For reasons of efficiency the function does not check the validity of <tt>GBasis</tt>.</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 P/I as a list of terms.</item><br />
<br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positive rational number which 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 set <tt>{1,2,3,4}</tt>. The default value is 2. This parameter describes, if and where required the input points are normalized. If <tt>NormalizeType</tt> equals 1, each coordinate of a point is divided by the maximal absolute value of all coordinates of this point. This ensures that all coordinates of the points are within <tt>[-1,1]</tt>. With <tt>NormalizeType=2</tt> no normalization is done at all. <tt>NormalizeType=3</tt> shifts each coordinate to <tt>[-1,1]</tt>, i.e. the minimal coordinate of a point is mapped to -1 and the maximal coordinate to 1, which describes a unique affine mapping. The last option is <tt>NormalizeType=4</tt>. In this case, each point is normalized by its euclidean norm. Although <tt>NormalizeType=3</tt> is in most cases a better choice, the default value is due to backward compatibility 1.</item><br />
<br />
<item>@param <em>RREFNormalizeType</em> Describes, how in each RREF step the columns are normalized. The options correspond to the ones for <tt>NormalizeType</tt> and the default is 1 again.</item><br />
<br />
<item>@param <em>RREFUseEps</em> A boolean value. If <tt>RREFUseEps=TRUE</tt>, the given <tt>Delta</tt> is used within the RREF to decide if a value equals 0 or not. If <tt>RREFUseEps=FALSE</tt>, a replacement value for <tt>Delta</tt> is used, which is based on the norm of the matrix.</item><br />
<br />
<item>@param <em>RREFType</em> A integer of the set <tt>{1,2}</tt>. If <tt>RREFType=1</tt>, the RREF operates column-wise, otherwise it works row-wise. The default value is 1.</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, 0.1, [x]);<br />
Dec(R[1],2);<br />
R[2];<br />
<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[<quotes>0.83 x^2 -0.55 x </quotes>, <quotes>1 xy </quotes>, <quotes>1 xz </quotes>]<br />
-------------------------------<br />
[x]<br />
-------------------------------<br />
<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.AVI</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>points</type><br />
<type>ideal</type><br />
</types><br />
<key>SubAVI</key><br />
<key>num.SubAVI</key><br />
<key>numerical.subavi</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.SubABM&diff=9936ApCoCoA-1:Num.SubABM2009-07-15T08:43:13Z<p>193.79.225.178: Added a more reasonable example</p>
<hr />
<div> <command><br />
<title>Num.SubABM</title><br />
<short_description>Computes a border basis of an almost vanishing sub-ideal for a set of points and an ideal using the <ref>Num.ABM</ref> algorithm.</short_description><br />
<syntax><br />
Num.SubABM(Points:MAT, Tau:RAT, GBasis:LIST):Object<br />
Num.SubABM(Points:MAT, Tau:RAT, GBasis:LIST, Delta:RAT, NormalizeType:INT):Object<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 border basis of an almost vanishing sub-ideal for a set of points and an ideal using the <ref>Num.ABM</ref> algorithm.<br />
<par/><br />
The current ring has to be a ring over the rational numbers with a standard-degree<br />
compatible term-ordering. Each row in the matrix <tt>Points</tt> represents one point, 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 tau lead to bigger errors of the polynomials evaluated at the point set). Tau should be in the interval (0,1). As a rule of thumb, Tau is the expected percentage of error on the input points. </item><br />
<br />
<item>@param <em>GBasis</em> A homogeneous Groebner Basis in the current ring. This basis defines the ideal in which we compute the basis of the approximate vanishing ideal. <em>Warning:</em> For reasons of efficiency the function does not check the validity of <tt>GBasis</tt>.</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 P/I as a list of terms.</item><br />
<br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positive rational number which 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 set <tt>{1,2,3,4}</tt>. The default value is 2. This parameter describes, if and where required the input points are normalized. If <tt>NormalizeType</tt> equals 1, each coordinate of a point is divided by the maximal absolute value of all coordinates of this point. This ensures that all coordinates of the points are within <tt>[-1,1]</tt>. With <tt>NormalizeType=2</tt> no normalization is done at all. <tt>NormalizeType=3</tt> shifts each coordinate to <tt>[-1,1]</tt>, i.e. the minimal coordinate of a point is mapped to -1 and the maximal coordinate to 1, which describes a unique affine mapping. The last option is <tt>NormalizeType=4</tt>. In this case, each point is normalized by its euclidean norm. Although <tt>NormalizeType=3</tt> is in most cases a better choice, the default value is due to backward compatibility 1.</item><br />
<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.SubABM(Points, 0.1, [x]);<br />
Dec(R[1],2);<br />
R[2];<br />
<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[<quotes>1 xz </quotes>, <quotes>1 xy </quotes>, <quotes>-0.83 x^2 +0.55 x </quotes>]<br />
-------------------------------<br />
[x]<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.ABM</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>points</type><br />
<type>ideal</type><br />
</types><br />
<key>SubABM</key><br />
<key>num.SubABM</key><br />
<key>numerical.subabm</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.SubABM&diff=9935ApCoCoA-1:Num.SubABM2009-07-15T07:58:02Z<p>193.79.225.178: correct info about default value</p>
<hr />
<div> <command><br />
<title>Num.SubABM</title><br />
<short_description>Computes a border basis of an almost vanishing sub-ideal for a set of points and an ideal using the <ref>Num.ABM</ref> algorithm.</short_description><br />
<syntax><br />
Num.SubABM(Points:MAT, Tau:RAT, GBasis:LIST):Object<br />
Num.SubABM(Points:MAT, Tau:RAT, GBasis:LIST, Delta:RAT, NormalizeType:INT):Object<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 border basis of an almost vanishing sub-ideal for a set of points and an ideal using the <ref>Num.ABM</ref> algorithm.<br />
<par/><br />
The current ring has to be a ring over the rational numbers with a standard-degree<br />
compatible term-ordering. Each row in the matrix <tt>Points</tt> represents one point, 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 tau lead to bigger errors of the polynomials evaluated at the point set). Tau should be in the interval (0,1). As a rule of thumb, Tau is the expected percentage of error on the input points. </item><br />
<br />
<item>@param <em>GBasis</em> A homogeneous Groebner Basis in the current ring. This basis defines the ideal in which we compute the basis of the approximate vanishing ideal. <em>Warning:</em> For reasons of efficiency the function does not check the validity of <tt>GBasis</tt>.</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 P/I as a list of terms.</item><br />
<br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positive rational number which 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 set <tt>{1,2,3,4}</tt>. The default value is 2. This parameter describes, if and where required the input points are normalized. If <tt>NormalizeType</tt> equals 1, each coordinate of a point is divided by the maximal absolute value of all coordinates of this point. This ensures that all coordinates of the points are within <tt>[-1,1]</tt>. With <tt>NormalizeType=2</tt> no normalization is done at all. <tt>NormalizeType=3</tt> shifts each coordinate to <tt>[-1,1]</tt>, i.e. the minimal coordinate of a point is mapped to -1 and the maximal coordinate to 1, which describes a unique affine mapping. The last option is <tt>NormalizeType=4</tt>. In this case, each point is normalized by its euclidean norm. Although <tt>NormalizeType=3</tt> is in most cases a better choice, the default value is due to backward compatibility 1.</item><br />
<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.SubABM(Points, 0.1, [1,x]);<br />
Dec(R[1],2);<br />
R[2];<br />
<br />
-- CoCoAServer: computing Cpu Time = 0.015<br />
-------------------------------<br />
[<quotes>0.41 x +0.27 y +0.82 z -0.27 </quotes>, <quotes>0.94 z^2 -0.31 z -0.00 </quotes>, <quotes>1 yz </quotes>, <quotes>1 xz </quotes>, <quotes>0.70 y^2 -0.70 y -0.00 </quotes>, <quotes>1 xy </quotes>]<br />
-------------------------------<br />
[1, z, y]<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.ABM</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>points</type><br />
<type>ideal</type><br />
</types><br />
<key>SubABM</key><br />
<key>num.SubABM</key><br />
<key>numerical.subabm</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:GLPK.MIPSolve&diff=9934ApCoCoA-1:GLPK.MIPSolve2009-07-15T07:53:36Z<p>193.79.225.178: Corrected example result</p>
<hr />
<div><command><br />
<title>GLPK.MIPSolve</title><br />
<short_description>Solving linear programmes.</short_description><br />
<syntax><br />
GLPK.MIPSolve(Objective_f:POLY, EQ_Poly:LIST, LE_Poly:LIST, GE_Poly:LIST, Bounds:LIST, IntNum:LIST, Binaries:LIST, MinMax:STRING)<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 />
<br />
<itemize><br />
<item>@param <em>Objective_f</em>: A linear polynomial which is equivalent to the linear objective function.</item><br />
<item>@param <em>EQ_Poly</em>: List of linear polynomials, which are equivalent to the equality-part in the list of conditions.</item><br />
<item>@param <em>LE_Poly</em>: List of linear polynomials, which are equivalent to the lower or equal-part in the list of conditions.</item><br />
<item>@param <em>GE_Poly</em>: List of linear polynomials, which are equivalent to the greater or equal-part in the list of conditions.</item><br />
<item>@param <em>Bounds</em>: List of lists with two elements. Each List contains the lower and upper bounds for each variable. You can choose between INT or RAT for the type of each bound, if you type in a (empty) string, then it means minus infinity (first place) or plus infinity (second place).</item><br />
<item>@param <em>IntNum</em>: List of variables, which should be integer. <em>Note</em>: For each variable in this list, the borders get rounded (lower bound: up and upper bound: down). In the case that the lower rounded bound becomes greater then the upper rounded bound, glpk returns: Solution Status: INTEGER UNDEFINED - Value of objective function: 0.</item><br />
<item>@param <em>Binaries</em>: List of variables, which should be binaries (0 or 1).</item><br />
<item>@param <em>MinMax</em>: Minimization (<quotes>Min</quotes>) or maximization (<quotes>Max</quotes>), that's the question.</item><br />
<item>@return List of linear polynomials, the zeros of the polynomials are the points where the optimal value of the objective function is achieved</item><br />
</itemize><br />
<br />
<example><br />
-- We want to maximize the Function y = - 1/2x, <br />
-- with the two conditions y ≤ 6 - 3/4x and y ≥ 1 - x and the bounds 0 ≤ x ≤ 6 and 1/3 ≤ y ≤ 4.<br />
<br />
-- We prename the input of GLPK.MIPSolve-function.<br />
Use S::=QQ[x,y];<br />
OF := 1/2x + y;<br />
LE := [3/4x + y - 6];<br />
GE := [x + y - 1];<br />
Bounds:=[[0,6], [1/3,4]];<br />
IntNum:=[x,y];<br />
<br />
-- Then we compute the solution with<br />
GLPK.MIPSolve(OF, [], LE, GE, Bounds, IntNum, [], <quotes>Max</quotes>);<br />
<br />
<br />
-- And we achieve:<br />
Solution Status: INTEGER OPTIMAL<br />
Value of objective function: 5<br />
[x - 2, y - 4]<br />
</example><br />
<br />
</description><br />
<types><br />
<type>apcocoaserver</type><br />
<type>linear_programs</type><br />
</types><br />
<key>mipsolve</key><br />
<key>solve linear programm</key><br />
<key>solve lp</key><br />
<key>GLPK.MIPSolve</key><br />
<wiki-category>Package_glpk</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:GLPK.LPSolve&diff=9931ApCoCoA-1:GLPK.LPSolve2009-07-14T11:49:53Z<p>193.79.225.178: Updated example</p>
<hr />
<div><command><br />
<title>GLPK.LPSolve</title><br />
<short_description>Solving linear programmes.</short_description><br />
<syntax><br />
GLPK.LPSolve(Objective_f:POLY, EQ_Poly:LIST, LE_Poly:LIST, GE_Poly:LIST, Bounds:LIST, Method:STRING, MinMax:STRING):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 />
<br />
<itemize><br />
<item>@param <em>Objective_f</em>: A linear polynomial which is equivalent to the linear objective function.</item><br />
<item>@param <em>EQ_Poly</em>: List of linear polynomials, which are equivalent to the equality-part in the list of conditions.</item><br />
<item>@param <em>LE_Poly</em>: List of linear polynomials, which are equivalent to the lower or equal-part in the list of conditions.</item><br />
<item>@param <em>GE_Poly</em>: List of linear polynomials, which are equivalent to the greater or equal-part in the list of conditions.</item><br />
<item>@param <em>Bounds</em>: List of lists with two elements. Each List contains the lower and upper bounds for each variable. You can choose between INT or RAT for the type of each bound, if you type in a (empty) string, then it means minus infinity (first place) or plus infinity (second place).</item><br />
<item>@param <em>Method</em>: You can choose between the interior-point-method (<quotes>InterP</quotes>) or the simplex-algorithm (<quotes>Simplex</quotes>). Usually you should use the simplex-algorithm.</item><br />
<item>@param <em>MinMax</em>: Minimization (<quotes>Min</quotes>) or maximization (<quotes>Max</quotes>), that's the question.</item><br />
<item>@return List of linear polynomials, the zeros of the polynomials are the points where the optimal value of the objective function is achieved</item><br />
</itemize><br />
<br />
<example><br />
-- We want to maximize the Function y = - 1/2x, <br />
-- with the two conditions y ≤ 6 - 3/4x and y ≥ 1 - x and the bounds 0 ≤ x ≤ 6 and 1/3 ≤ y ≤ 4.<br />
<br />
-- We prename the input of GLPK.LPSolve-function.<br />
Use S::=QQ[x,y];<br />
OF := 1/2x + y;<br />
LE := [3/4x + y - 6];<br />
GE := [x + y - 1];<br />
Bounds:=[[0,6], [1/3,4]];<br />
<br />
-- Then we compute the solution with<br />
GLPK.LPSolve(OF, [], LE, GE, Bounds, <quotes>Simplex</quotes>, <quotes>Max</quotes>);<br />
<br />
-- And we achieve:<br />
------------------------------------- <br />
Solution Status: OPTIMAL<br />
Value of objective function: 5333333333/1000000000<br />
[x - 266667/100000, y - 4]<br />
</example><br />
<br />
</description><br />
<types><br />
<type>apcocoaserver</type><br />
<type>linear_programs</type><br />
</types><br />
<see>Latte.Minimize</see><br />
<see>Latte.Maximize</see><br />
<br />
<key>lpsolve</key><br />
<key>solve linear program</key><br />
<key>solve lp</key><br />
<key>GLPK.LPSolve</key><br />
<wiki-category>Package_glpk</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.SubAVI&diff=9924ApCoCoA-1:Num.SubAVI2009-07-14T09:39:09Z<p>193.79.225.178: </p>
<hr />
<div> <command><br />
<title>Num.SubAVI</title><br />
<short_description>Computes a border basis of an almost vanishing sub-ideal for a set of points and an ideal using the <ref>Num.AVI</ref> algorithm.</short_description><br />
<syntax><br />
Num.SubAVI(Points:MAT, Tau:RAT, GBasis:LIST):Object<br />
Num.SubAVI(Points:MAT, Tau:RAT, GBasis:LIST, Delta:RAT, NormalizeType:INT, RREFNormalizeType:INT, RREFUseEps:BOOL, RREFType:INT):Object<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 border basis of an almost vanishing sub-ideal for a set of points and an ideal using the <ref>Num.ABM</ref> algorithm.<br />
<par/><br />
The current ring has to be a ring over the rational numbers with a standard-degree<br />
compatible term-ordering. Each row in the matrix <tt>Points</tt> represents one point, 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 tau lead to bigger errors of the polynomials evaluated at the point set). Tau should be in the interval (0,1). As a rule of thumb, Tau is the expected percentage of error on the input points. </item><br />
<br />
<item>@param <em>GBasis</em> A homogeneous Groebner Basis in the current ring. This basis defines the ideal in which we compute the basis of the approximate vanishing ideal. <em>Warning:</em> For reasons of efficiency the function does not check the validity of <tt>GBasis</tt>.</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 P/I as a list of terms.</item><br />
<br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positive rational number which 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 set <tt>{1,2,3,4}</tt>. The default value is 1. This parameter describes, if and where required the input points are normalized. If <tt>NormalizeType</tt> equals 1, each coordinate of a point is divided by the maximal absolute value of all coordinates of this point. This ensures that all coordinates of the points are within <tt>[-1,1]</tt>. With <tt>NormalizeType=2</tt> no normalization is done at all. <tt>NormalizeType=3</tt> shifts each coordinate to <tt>[-1,1]</tt>, i.e. the minimal coordinate of a point is mapped to -1 and the maximal coordinate to 1, which describes a unique affine mapping. The last option is <tt>NormalizeType=4</tt>. In this case, each point is normalized by its euclidean norm. Although <tt>NormalizeType=3</tt> is in most cases a better choice, the default value is due to backward compatibility 1.</item><br />
<br />
<item>@param <em>RREFNormalizeType</em> Describes, how in each RREF step the columns are normalized. The options correspond to the ones for <tt>NormalizeType</tt> and the default is 1 again.</item><br />
<br />
<item>@param <em>RREFUseEps</em> A boolean value. If <tt>RREFUseEps=TRUE</tt>, the given <tt>Delta</tt> is used within the RREF to decide if a value equals 0 or not. If <tt>RREFUseEps=FALSE</tt>, a replacement value for <tt>Delta</tt> is used, which is based on the norm of the matrix.</item><br />
<br />
<item>@param <em>RREFType</em> A integer of the set <tt>{1,2}</tt>. If <tt>RREFType=1</tt>, the RREF operates column-wise, otherwise it works row-wise. The default value is 1.</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, 0.1, [x]);<br />
Dec(R[1],2);<br />
R[2];<br />
<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[<quotes>0.83 x^2 -0.55 x </quotes>, <quotes>1 xy </quotes>, <quotes>1 xz </quotes>]<br />
-------------------------------<br />
[x]<br />
-------------------------------<br />
<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.AVI</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>points</type><br />
<type>ideal</type><br />
</types><br />
<key>SubAVI</key><br />
<key>num.SubAVI</key><br />
<key>numerical.subavi</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.AVI&diff=9673ApCoCoA-1:Num.AVI2009-07-06T13:18:57Z<p>193.79.225.178: </p>
<hr />
<div> <command><br />
<title>Num.AVI</title><br />
<short_description>Computes a border basis of an almost vanishing ideal for a set of points.</short_description><br />
<syntax><br />
Num.AVI(Points:MAT, Tau:RAT):Object<br />
Num.AVI(Points:MAT, Tau:RAT, Delta:RAT, NormalizeType:INT, RREFNormalizeType:INT, RREFUseEps:BOOL, RREFType: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 function computes an approximate border basis of an almost vanishing ideal for a set of points using the AVI algorithm.<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. <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 tau lead to bigger errors of the polynomials evaluated at the point set). Tau should be in the interval (0,1). As a rule of thumb, Tau 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 P/I as a list of terms.</item><br />
<br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positiv rational number. Delta describes the computing precision. In different steps, it is crucial, if a value is 0 or not. The algorithm assumes every value in [-Delta, Delta] to be 0. The default value for Delta 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 NormalizeType equals 1, each coordinate is divided by the maximal absolut value of the matrix's corresponding column. This ensures that all point's coordinates are in [-1,1]. With NormalizeType=2 no normalization is done at all. NormalizeType=3 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 NormalizeType=4. 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 />
<br />
<item>@param <em>RREFNormalizeType</em> Describes, how in each RREF steps the columns are normalized. The options correspond to the ones for NormalizeType and the default is 1 again.</item><br />
<br />
<item>@param <em>RREFUseEps</em> must be either true or false! If RREFUseEps is true, the given Delta is used within the RREF to decide if a value equals 0 or not. If this parameter is false, a replacement value for Delta is used, which is based on the matrix's norm. </item><br />
<br />
<item>@param <em>RREFType</em> This must be 1 or 2. If RREFType=1, the rref operates column-wise. Otherwise it works row-wise. The default is 1.</item><br />
</itemize><br />
<br />
<example><br />
Use P::=QQ[x,y,z];<br />
<br />
Points := Mat([[1,0,0],[0,0,1],[0,1,0]]);<br />
Num.AVI(Points,0.001);<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[[1/2x + 1/2y + 4503599627370495/9007199254740992z - 1/2, xy, 1592262918131443/2251799813685248y^2 - 1592262918131443/2251799813685248y, <br />
xz, yz, 1592262918131443/2251799813685248z^2 - 1592262918131443/2251799813685248z], [1, z, y]]<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.SubAVI</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>points</type><br />
</types><br />
<key>numerical.AVI</key><br />
<key>AVI</key><br />
<key>num.avi</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.ABM&diff=9672ApCoCoA-1:Num.ABM2009-07-06T13:16:50Z<p>193.79.225.178: </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, Tau:RAT):Object<br />
Num.ABM(Points:MAT, Tau: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 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. <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 tau lead to bigger errors of the polynomials evaluated at the point set). Tau should be in the interval (0,1). As a rule of thumb, Tau 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 P/I as a list of terms.</item><br />
<br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positiv rational number. Delta describes the computing precision. In different steps, it is crucial, if a value is 0 or not. The algorithm assumes every value in [-Delta, Delta] to be 0. The default value for Delta 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 NormalizeType equals 1, each coordinate is divided by the maximal absolut value of the matrix's corresponding column. This ensures that all point's coordinates are in [-1,1]. With NormalizeType=2 no normalization is done at all. NormalizeType=3 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 NormalizeType=4. 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 />
<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<br />
-------------------------------<br />
[<quotes>-0.49 x -0.50 y -0.49 z +0.49 </quotes>, <quotes>-0.70 z^2 +0.70 z -0.00 </quotes>, <quotes>1 yz </quotes>, <quotes>1 xz </quotes>, <quotes>0.71 y^2 -0.70 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>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.SubAVI&diff=9540ApCoCoA-1:Num.SubAVI2009-04-29T09:12:45Z<p>193.79.225.178: Corrected a copy and past error</p>
<hr />
<div> <command><br />
<title>Num.SubAVI</title><br />
<short_description>Computes a border basis of an almost vanishing sub-ideal for a set of points and ideal.</short_description><br />
<syntax><br />
Num.SubAVI(Points:MAT, Tau:RAT, GBasis:LIST):Object<br />
Num.SubAVI(Points:MAT, Tau:RAT, GBasis:LIST, Delta:RAT, NormalizeType:INT, RREFNormalizeType:INT, RREFUseEps:BOOL, RREFType:INT):Object<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 border basis of an almost vanishing sub-ideal for a set of points and ideal.<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. <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 tau lead to bigger errors of the polynomials evaluated at the point set). Tau should be in the interval (0,1). As a rule of thumb, Tau is the expected percentage of error on the input points. </item><br />
<br />
<item>@param <em>GBasis</em> A homogeneous Groebner Basis in the current ring. This basis defines the ideal we compute the approximate vanishing ideal's basis in. Warning: for reasons of efficiency the function does not check the validity of GBasis.</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 P/I as a list of terms.</item><br />
<br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positiv rational number. Delta describes the computing precision. In different steps, it is crucial, if a value is 0 or not. The algorithm assumes every value in [-Delta, Delta] to be 0. The default value for Delta is 0.00000000001.</item><br />
<br />
<item>@param <em>NormalizeType</em> A integer of the range 1..4. The default value is 1. This parameter describes, if / how the input points are normalized. If NormalizeType equals 1, each coordinate is divided by the maximal absolut value of the matrix's corresponding column. This ensures that all point's coordinates are in [-1,1]. With NormalizeType=2 no normalization is done at all. NormalizeType=3 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 NormalizeType=4. 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 />
<br />
<item>@param <em>RREFNormalizeType</em> Describes, how in each RREF steps the columns are normalized. The options correspond to the ones for NormalizeType and the default is 1 again.</item><br />
<br />
<item>@param <em>RREFUseEps</em> must be either true or false! If RREFUseEps is true, the given Delta is used within the RREF to decide if a value equals 0 or not. If this parameter is false, a replacement value for Delta is used, which is based on the matrix's norm. </item><br />
<br />
<item>@param <em>RREFType</em> This must be 1 or 2. If RREFType=1, the rref operates column-wise. Otherwise it works row-wise. The default is 1.</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, 0.1, [1,x]);<br />
Dec(R[1],2);<br />
R[2];<br />
<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[<quotes>0.83 x^2 -0.55 x </quotes>, <quotes>1 xy </quotes>, <quotes>1 xz </quotes>, <quotes>0.00 x </quotes>, <quotes>0.41 x +0.27 y +0.82 z -0.27 </quotes>, <quotes>1 xz </quotes>, <quotes>1 yz </quotes>, <quotes>0.94 z^2 -0.31 z </quotes>]<br />
-------------------------------<br />
[1, x, z]<br />
-------------------------------<br />
<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.AVI</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>points</type><br />
<type>ideal</type><br />
</types><br />
<key>SubAVI</key><br />
<key>num.SubAVI</key><br />
<key>numerical.subavi</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.SubAVI&diff=9539ApCoCoA-1:Num.SubAVI2009-04-29T09:11:46Z<p>193.79.225.178: Corrected a copy and past error</p>
<hr />
<div> <command><br />
<title>Num.SubAVI</title><br />
<short_description>Computes a border basis of an almost vanishing sub-ideal for a set of points and ideal.</short_description><br />
<syntax><br />
Num.SubAVI(Points:MAT, Tau:RAT, GBasis:LIST):Object<br />
Num.SubAVI(Points:MAT, Tau:RAT, GBasis:LIST, Delta:RAT, NormalizeType:INT, RREFNormalizeType:INT, RREFUseEps:BOOL, RREFType:INT):Object<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 border basis of an almost vanishing sub-ideal for a set of points and ideal.<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. <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 tau lead to bigger errors of the polynomials evaluated at the point set). Tau should be in the interval (0,1). As a rule of thumb, Tau is the expected percentage of error on the input points. </item><br />
<br />
<item>@param <em>GBasis</em> A homogeneous Groebner Basis in the current ring. This basis defines the ideal we compute the approximate vanishing ideal's basis in. Warning: for reasons of efficiency the function does not check the validity of GBasis.</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 P/I as a list of terms.</item><br />
<br />
</itemize><br />
<br />
The following parameters are optional:<br />
<itemize><br />
<item>@param <em>Delta</em> A positiv rational number. Delta describes the computing precision. In different steps, it is crucial, if a value is 0 or not. The algorithm assumes every value in [-Delta, Delta] to be 0. The default value for Delta is 0.00000000001.</item><br />
<br />
<item>@param <em>NormalizeType</em> A integer of the range 1..4. The default value is 1. This parameter describes, if / how the input points are normalized. If NormalizeType equals 1, each coordinate is divided by the maximal absolut value of the matrix's corresponding column. This ensures that all point's coordinates are in [-1,1]. With NormalizeType=2 no normalization is done at all. NormalizeType=3 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 NormalizeType=4. 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 />
<br />
<item>@param <em>RREFNormalizeType</em> Describes, how in each RREF steps the columns are normalized. The options correspond to the ones for NormalizeType and the default is 1 again.</item><br />
<br />
<item>@param <em>RREFUseEps</em> must be either true or false! If RREFUseEps is true, the given Delta is used within the RREF to decide if a value equals 0 or not. If this parameter is false, a replacement value for Delta is used, which is based on the matrix's norm. </item><br />
<br />
<item>@param <em>RREFType</em> This must be 1 or 2. If RREFType=1, the rref operates column-wise. Otherwise it works row-wise. The default is 1.</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, 0.1, [1,x]);<br />
Dec(R[1],2);<br />
R[2];<br />
<br />
-- CoCoAServer: computing Cpu Time = 0<br />
-------------------------------<br />
[<quotes>0.83 x^2 -0.55 x <quotes>, <quotes>1 xy <quotes>, <quotes>1 xz <quotes>, <quotes>0.00 x <quotes>, <quotes>0.41 x +0.27 y +0.82 z -0.27 <quotes>, <quotes>1 xz <quotes>, <quotes>1 yz <quotes>, <quotes>0.94 z^2 -0.31 z <quotes>]<br />
-------------------------------<br />
[1, x, z]<br />
-------------------------------<br />
<br />
</example><br />
</description><br />
<seealso><br />
<see>Introduction to CoCoAServer</see><br />
<see>Num.AVI</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>points</type><br />
<type>ideal</type><br />
</types><br />
<key>SubAVI</key><br />
<key>num.SubAVI</key><br />
<key>numerical.subavi</key><br />
<wiki-category>Package_numerical</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.RatPoints&diff=9358ApCoCoA-1:Num.RatPoints2009-04-27T07:51:41Z<p>193.79.225.178: </p>
<hr />
<div> <command><br />
<title>Num.RatPoints</title><br />
<short_description>Calculates a set of points which are the zeros of an exact border basis, which is close to AppBB.</short_description><br />
<syntax><br />
Num.RatPoints(AppBB:LIST, OrderIdeal:LIST)):[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 />
Calculates a set of points which are the zeros of an exact border basis which is close to AppBB. Uses the eigenvalue methode.<br />
<br />
<itemize><br />
<item>@param <em>AppBB</em> An approximate border basis.</item><br />
<item>@param <em>Points</em> The asscoiated order ideal</item><br />
<item>@return A set of points in matrix form.</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 />
["0.66", "0.00", "-0.00"],<br />
["0", "0", "1.00"],<br />
["0", "0.33", "0"]<br />
]), Mat([<br />
["0", "0", "0"],<br />
["0", "0", "0"],<br />
["0", "0", "0"]<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>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.EigenValuesAndVectors&diff=8261ApCoCoA-1:Num.EigenValuesAndVectors2009-03-30T13:58:03Z<p>193.79.225.178: </p>
<hr />
<div> <command><br />
<title>Numerical.EigenValuesAndVectors</title><br />
<short_description>Computes the eigenvalues and eigenvectors of a matrix</short_description><br />
<syntax><br />
Num.EigenValuesAndVectors(A:Matrix):[B:Matrix, C:Matrix, D:Matrix]<br />
</syntax><br />
<description><br />
{{ApCoCoAServer}} Please also note that you will have to use an ApCoCoAServer with enabled BLAS/LAPACK support.<br />
<br />
This function returns a list of three matrices, containing numerical approximation to A's eigenvalues and (right hand) eigenvectors. <br />
The input matrix A has to be a square matrix!<br />
The output [B:Matrix, C:Matrix, D:Matrix] contains a matrix B, where each column contains one of A's eigenvalues. The first row contains the eigenvalue's real part, the second row the imaginary.<br />
The matrices C and D both have the same dimensions as A. Column j of matrix C contains the real part of the eigenvector corresponding to eigenvalue j and column j of matrix D contains the imaginary part of the eigenvector correspsonding to eigenvalue j.<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 />
Dec(Num.EigenValuesAndVectors(A),3); <br />
<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
[Mat([<br />
["28.970", "-13.677", "0.353", "0.353"],<br />
["0", "0", "3.051", "-3.051"]<br />
]), Mat([<br />
["0.394", "-0.581", "0.260", "0.260"],<br />
["0.435", "-0.442", "-0.547", "-0.547"],<br />
["0.763", "0.621", "0", "0"],<br />
["0.268", "0.281", "0.046", "0.046"]<br />
]), Mat([<br />
["0", "0", "-0.031", "0.031"],<br />
["0", "0", "-0.301", "0.301"],<br />
["0", "0", "0.680", "-0.680"],<br />
["0", "0", "-0.274", "0.274"]<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 />
<types><br />
<type>cocoaserver</type><br />
</types><br />
<key>Num.EigenValuesAndVectors</key><br />
<key>EigenValuesAndVectors</key><br />
<wiki-category>Package_Numerical</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.EigenValues&diff=8260ApCoCoA-1:Num.EigenValues2009-03-30T13:56:07Z<p>193.79.225.178: </p>
<hr />
<div> <command><br />
<title>Num.EigenValues</title><br />
<short_description>Computes the eigenvalues of a matrix</short_description><br />
<syntax><br />
Num.EigenValues(A:Matrix):B:Matrix<br />
</syntax><br />
<description><br />
{{ApCoCoAServer}} Please also note that you will have to use an ApCoCoAServer with enabled BLAS/LAPACK support.<br />
<br />
This function returns a matrix, containing numerical approximation to A's eigenvalues. <br />
The input matrix A has to be quadratic!<br />
The output contains a matrix B. Each of B's columns describes one of the eigenvalues of A. The first row of B contains the real part of the eigenvalues, the second row the imaginary ones. <br />
<br />
<example><br />
A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]);<br />
Dec(Num.EigenValues(A),3);<br />
-- CoCoAServer: computing Cpu Time = 0.015<br />
-------------------------------<br />
Mat([<br />
["28.970", "-13.677", "0.353", "0.353"],<br />
["0", "0", "3.051", "-3.051"]<br />
])<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 />
<types><br />
<type>cocoaserver</type><br />
</types><br />
<key>Num.EigenValues</key><br />
<key>EigenValues</key><br />
<wiki-category>Package_Numerical</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.EigenValuesAndAllVectors&diff=8259ApCoCoA-1:Num.EigenValuesAndAllVectors2009-03-30T13:54:38Z<p>193.79.225.178: </p>
<hr />
<div> <command><br />
<title>Num.EigenValuesAndAllVectors</title><br />
<short_description>Computes eigenvalues and left and right eigenvectors of a matrix</short_description><br />
<syntax><br />
Num.EigenValuesAndAllVectors(A:Matrix):[B:Matrix, C:Matrix, D:Matrix, E:Matrix, F:Matrix]<br />
</syntax><br />
<description><br />
{{ApCoCoAServer}} Please also note that you will have to use an ApCoCoAServer with enabled BLAS/LAPACK support.<br />
<br />
This function returns a list of five matrices, containing numerical approximation to A's eigenvalues and right and left eigenvectors. <br />
The input matrix A has to be a square matrix!<br />
The output [B:Matrix, C:Matrix, D:Matrix, E:Matrix, F:Matrix] contains a matrix B, where each column contains one of A's eigenvalues. The first row contains the eigenvalue's real part, the second row the imaginary.<br />
The matrices C, D, E and F all have the same dimensions as A. Column j of matrix C contains the real part of the right eigenvector corresponding to eigenvalue j and column j of matrix D contains the imaginary part of the right eigenvector correspsonding to eigenvalue j. The matrices E and F store the left eigenvectors in the same way as C and D.<br />
<example><br />
A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]); <br />
Dec(Num.EigenValuesAndAllVectors(A),3);<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
[Mat([<br />
["28.970", "-13.677", "0.353", "0.353"],<br />
["0", "0", "3.051", "-3.051"]<br />
]), Mat([<br />
["0.538", "-0.600", "0.389", "0.389"],<br />
["0.311", "-0.222", "-0.442", "-0.442"],<br />
["0.427", "0.174", "0.050", "0.050"],<br />
["0.656", "0.748", "0", "0"]<br />
]), Mat([<br />
["0", "0", "-0.174", "0.174"],<br />
["0", "0", "0.139", "-0.139"],<br />
["0", "0", "0.265", "-0.265"],<br />
["0", "0", "-0.727", "0.727"]<br />
]), Mat([<br />
["0.394", "-0.581", "0.260", "0.260"],<br />
["0.435", "-0.442", "-0.547", "-0.547"],<br />
["0.763", "0.621", "0", "0"],<br />
["0.268", "0.281", "0.046", "0.046"]<br />
]), Mat([<br />
["0", "0", "-0.031", "0.031"],<br />
["0", "0", "-0.301", "0.301"],<br />
["0", "0", "0.680", "-0.680"],<br />
["0", "0", "-0.274", "0.274"]<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 />
<types><br />
<type>cocoaserver</type><br />
</types><br />
<key>Num.EigenValuesAndAllVectors</key><br />
<key>EigenValuesAndAllVectors</key><br />
<wiki-category>Package_Numerical</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.EigenValuesAndAllVectors&diff=8258ApCoCoA-1:Num.EigenValuesAndAllVectors2009-03-30T13:54:01Z<p>193.79.225.178: </p>
<hr />
<div> <command><br />
<title>Num.EigenValuesAndAllVectors</title><br />
<short_description>Computes eigenvalues and left and right eigenvectors of a matrix</short_description><br />
<syntax><br />
Num.EigenValuesAndAllVectors(A:Matrix):[B:Matrix, C:Matrix, D:Matrix, E:Matrix, F:Matrix]<br />
</syntax><br />
<description><br />
{{ApCoCoAServer}} Please also note that you will have to use an ApCoCoAServer with enabled BLAS/LAPACK support.<br />
<br />
This function returns a List of five matrices, containing numerical approximation to A's eigenvalues and right and left eigenvectors. <br />
The input matrix A has to be a square matrix!<br />
The output [B:Matrix, C:Matrix, D:Matrix, E:Matrix, F:Matrix] contains a matrix B, where each column contains one of A's eigenvalues. The first row contains the eigenvalue's real part, the second row the imaginary.<br />
The matrices C, D, E and F all have the same dimensions as A. Column j of matrix C contains the real part of the right eigenvector corresponding to eigenvalue j and column j of matrix D contains the imaginary part of the right eigenvector correspsonding to eigenvalue j. The matrices E and F store the left eigenvectors in the same way as C and D.<br />
<example><br />
A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]); <br />
Dec(Num.EigenValuesAndAllVectors(A),3);<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
[Mat([<br />
["28.970", "-13.677", "0.353", "0.353"],<br />
["0", "0", "3.051", "-3.051"]<br />
]), Mat([<br />
["0.538", "-0.600", "0.389", "0.389"],<br />
["0.311", "-0.222", "-0.442", "-0.442"],<br />
["0.427", "0.174", "0.050", "0.050"],<br />
["0.656", "0.748", "0", "0"]<br />
]), Mat([<br />
["0", "0", "-0.174", "0.174"],<br />
["0", "0", "0.139", "-0.139"],<br />
["0", "0", "0.265", "-0.265"],<br />
["0", "0", "-0.727", "0.727"]<br />
]), Mat([<br />
["0.394", "-0.581", "0.260", "0.260"],<br />
["0.435", "-0.442", "-0.547", "-0.547"],<br />
["0.763", "0.621", "0", "0"],<br />
["0.268", "0.281", "0.046", "0.046"]<br />
]), Mat([<br />
["0", "0", "-0.031", "0.031"],<br />
["0", "0", "-0.301", "0.301"],<br />
["0", "0", "0.680", "-0.680"],<br />
["0", "0", "-0.274", "0.274"]<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 />
<types><br />
<type>cocoaserver</type><br />
</types><br />
<key>Num.EigenValuesAndAllVectors</key><br />
<key>EigenValuesAndAllVectors</key><br />
<wiki-category>Package_Numerical</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.EigenValues&diff=8255ApCoCoA-1:Num.EigenValues2009-03-30T13:39:17Z<p>193.79.225.178: Updated the docu</p>
<hr />
<div> <command><br />
<title>Num.EigenValues</title><br />
<short_description>Computes the eigenvalues of a matrix</short_description><br />
<syntax><br />
Num.EigenValues(A:Matrix):[B:Matrix]<br />
</syntax><br />
<description><br />
{{ApCoCoAServer}} Please also note that you will have to use an ApCoCoAServer with enabled BLAS/LAPACK support.<br />
<br />
This function returns a matrix, containing numerical approximation to A's eigenvalues. <br />
The input matrix A has to be quadratic!<br />
The output contains a matrix B. Each B's columns describe one of the eigenvalues of A. The first row of B contains the real part of the eigenvalues, the second row the imaginary ones. <br />
<br />
<example><br />
A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]);<br />
Dec(Num.EigenValues(A),3);<br />
-- CoCoAServer: computing Cpu Time = 0.015<br />
-------------------------------<br />
Mat([<br />
["28.970", "-13.677", "0.353", "0.353"],<br />
["0", "0", "3.051", "-3.051"]<br />
])<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 />
<types><br />
<type>cocoaserver</type><br />
</types><br />
<key>Num.EigenValues</key><br />
<key>EigenValues</key><br />
<wiki-category>Package_Numerical</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.EigenValuesAndVectors&diff=8254ApCoCoA-1:Num.EigenValuesAndVectors2009-03-30T13:33:04Z<p>193.79.225.178: Updated the Doku</p>
<hr />
<div> <command><br />
<title>Numerical.EigenValuesAndVectors</title><br />
<short_description>Computes the eigenvalues and eigenvectors of a matrix</short_description><br />
<syntax><br />
Num.EigenValuesAndVectors(A:Matrix):[B:Matrix, C:Matrix, D:Matrix]<br />
</syntax><br />
<description><br />
{{ApCoCoAServer}} Please also note that you will have to use an ApCoCoAServer with enabled BLAS/LAPACK support.<br />
<br />
This function returns a list of three matrices, containing numerical approximation to A's eigenvalues and (right hand) eigenvectors. <br />
The input matrix A has to be a square matrix!<br />
The output [B:Matrix, C:Matrix, D:Matrix] contains a matrix B, where each column contains one of A's eigenvalues. The first row contains the eigenvalue's real part, the second row the imaginary.<br />
The matrices C and B have both the same dimensions as A. Column j of matrix C contains the real part of the eigenvector corresponding to eigenvalue j and column j of matrix D contains the imaginary part of the eigenvector correspsonding to eigenvalue j.<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 />
Dec(Num.EigenValuesAndVectors(A),3); <br />
<br />
-- CoCoAServer: computing Cpu Time = 0.016<br />
-------------------------------<br />
[Mat([<br />
["28.970", "-13.677", "0.353", "0.353"],<br />
["0", "0", "3.051", "-3.051"]<br />
]), Mat([<br />
["0.394", "-0.581", "0.260", "0.260"],<br />
["0.435", "-0.442", "-0.547", "-0.547"],<br />
["0.763", "0.621", "0", "0"],<br />
["0.268", "0.281", "0.046", "0.046"]<br />
]), Mat([<br />
["0", "0", "-0.031", "0.031"],<br />
["0", "0", "-0.301", "0.301"],<br />
["0", "0", "0.680", "-0.680"],<br />
["0", "0", "-0.274", "0.274"]<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 />
<types><br />
<type>cocoaserver</type><br />
</types><br />
<key>Num.EigenValuesAndVectors</key><br />
<key>EigenValuesAndVectors</key><br />
<wiki-category>Package_Numerical</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.EigenValuesAndVectors&diff=8252ApCoCoA-1:Num.EigenValuesAndVectors2009-03-30T10:51:43Z<p>193.79.225.178: </p>
<hr />
<div> <command><br />
<title>Numerical.EigenValuesAndVectors</title><br />
<short_description>Computes the eigenvalues and eigenvectors of a matrix</short_description><br />
<syntax><br />
$numerical.EigenValuesAndVectors(A:Matrix):List<br />
</syntax><br />
<description><br />
{{ApCoCoAServer}} Please also note that you will have to use an ApCoCoAServer with enabled BLAS/LAPACK support.<br />
<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 a matrix B, where each column contains one of A's eigenvalues. The first row contains the eigenvalue's real part, the second the imaginary.<br />
The second element C of the returned list is a matrix of the size of A, containing the (right hand) eigenvectors of A. Column j contains the eigenvector corresponding to eigenvalue j if the imaginary part of j is zero. If eigenvalue j had also an imaginary part, then eigenvalue j+1 is the complex conjugate of j and the eigenvector of j is composed of the real part stored in column j and the imaginary part stored in column j+1 of matrix C. The eigenvector of j+1 is the complex conjugate of eigenvector j.<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 />
[4077234895954899/140737488355328, -3850002255576291/281474976710656, 3186113456591853/9007199254740992, 3186113456591853/9007199254740992],<br />
[0, 0, 6871934657603045/2251799813685248, -6871934657603045/2251799813685248]<br />
]), Mat([<br />
[1777559794020963/4503599627370496, 5241040126502889/9007199254740992, -4553859282588877/144115188075855872, 4695168387448585/18014398509481984],<br />
[7846388397589841/18014398509481984, 3981313256671163/9007199254740992, -5438845171485265/18014398509481984, -4930385173711607/9007199254740992],<br />
[6875189208942329/9007199254740992, -5600762787593733/9007199254740992, 11970674168303/17592186044416, 0],<br />
[2414763704612135/9007199254740992, -5076115741924331/18014398509481984, -2469130937097749/9007199254740992, 3322230315885151/72057594037927936]<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 />
<types><br />
<type>cocoaserver</type><br />
</types><br />
<key>heldt</key><br />
<key>numerical.eigenvaluesandvectors</key><br />
<wiki-category>Package_Numerical</wiki-category><br />
</command></div>193.79.225.178http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:Num.EigenValuesAndAllVectors&diff=8203ApCoCoA-1:Num.EigenValuesAndAllVectors2009-03-17T10:04:39Z<p>193.79.225.178: </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.EigenValuesAndAllVectors(A:Matrix):List<br />
</syntax><br />
<description><br />
{{ApCoCoAServer}} Please also note that you will have to use an ApCoCoAServer with enabled BLAS/LAPACK support.<br />
<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 list contains first a matrix B. Each row of B describe 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. Column j contains the eigenvector corresponding to eigenvalue j if the imaginary part of j is zero. If eigenvalue j had also an imaginary part, then eigenvalue j+1 is the complex conjugate of j and the eigenvector of j is composed of the real part stored in column j and the imaginary part stored in column j+1. The eigenvector of j+1 is the complex conjugate of eigenvector j.<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 />
[4077234895954899/140737488355328, -3850002255576291/281474976710656, 3186113456591853/9007199254740992, 3186113456591853/9007199254740992],<br />
[0, 0, 6871934657603045/2251799813685248, -6871934657603045/2251799813685248]<br />
]), Mat([<br />
[1211656389006889/2251799813685248, 5405727161387211/9007199254740992, 1571393504747479/9007199254740992, -7024364631742825/18014398509481984],<br />
[350694995566991/1125899906842624, 4012694633891333/18014398509481984, -5041294450411215/36028797018963968, 7963794620848619/18014398509481984],<br />
[7702243945405117/18014398509481984, -6293666352540407/36028797018963968, -4789736757058405/18014398509481984, -3648514314569325/72057594037927936],<br />
[5910799605047357/9007199254740992, -6738448111784603/9007199254740992, 3276340384567917/4503599627370496, 0]<br />
]), Mat([<br />
[1777559794020963/4503599627370496, 5241040126502889/9007199254740992, -4553859282588877/144115188075855872, 4695168387448585/18014398509481984],<br />
[7846388397589841/18014398509481984, 3981313256671163/9007199254740992, -5438845171485265/18014398509481984, -4930385173711607/9007199254740992],<br />
[6875189208942329/9007199254740992, -5600762787593733/9007199254740992, 11970674168303/17592186044416, 0],<br />
[2414763704612135/9007199254740992, -5076115741924331/18014398509481984, -2469130937097749/9007199254740992, 3322230315885151/72057594037927936]<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 />
<types><br />
<type>cocoaserver</type><br />
</types><br />
<key>heldt</key><br />
<key>numerical.eigenvaluesandallvectors</key><br />
<wiki-category>Package_Numerical</wiki-category><br />
</command></div>193.79.225.178