ApCoCoA-1:MatlabCoCoAIdeal: Difference between revisions
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<table border="1"> | <table border="1"> | ||
<tr> | <tr> | ||
<td> basering | <td> basering </td><td> A CoCoARing | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> gens | <td> gens </td><td> an array of polynomials | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> gBasis | <td> gBasis </td><td> the Groebner-Basis of an Ideal | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> numBasisElements | <td> numBasisElements </td><td> The number of basis elements. If gBasis is not yet computed the number of generators is stored. | ||
</tr> | </tr> | ||
</table> | </table> | ||
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<table border="1"> | <table border="1"> | ||
<tr> | <tr> | ||
<td><b>Method</b> | <td><b>Method</b></td><td><b>Operator</b></td><td><b>Description</b></td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> computeGBasis | <td> computeGBasis </td><td> </td><td> computes the GBasis of the given Ideal and returns a new ideal with the same set of generators and the computed GBasis.</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> computeLT | <td> computeLT </td><td> </td><td> computes the leading term of the ideal. Returns a CoCoAPoly.</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> display | <td> display </td><td> </td><td> Is used by Matlab to print a CoCoAIdeal to the command window. E.g. when a command is not terminated by a ';'.</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> eq | <td> eq </td><td>==</td><td> Compares two ideals. Returns 1 if all generators are equal.</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> ge | <td> ge </td><td>>=</td><td> Compares two ideals according to their leading term. Returns 1 if first ideal is greater equal the second ideal.</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> get | <td> get </td><td> </td><td> Returns the attributes of an ideal. Syntax: <br> | ||
'BaseRing': Returns the ring over which the polynomials in the generator list are defined. | get(idealVar, 'Keyword'), where idealVar is a CoCoAIdeal and 'Keyword' is one of the keywords:<br> | ||
'Gens': returns the generator array of CoCoAPoly's | 'BaseRing': Returns the ring over which the polynomials in the generator list are defined.<br> | ||
'GBasis': returns the GBasis of the Ideal if it has been computed. Otherwise [] is returned. | 'Gens': returns the generator array of CoCoAPoly's<br> | ||
'NumElements': Returns the number of basis elements. If gBasis is not yet computed the number of generators is returned. | 'GBasis': returns the GBasis of the Ideal if it has been computed. Otherwise [] is returned.<br> | ||
'String': Returns the polynomial as a string. | 'NumElements': Returns the number of basis elements. If gBasis is not yet computed the number of generators is returned.<br> | ||
'String': Returns the polynomial as a string.<br> | |||
'Latex' will return a string using Latex syntax.</td> | 'Latex' will return a string using Latex syntax.</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> gt | <td> gt </td><td>></td><td> Compares two ideals according to their leading term. Returns 1 if first ideal is greater than the second ideal.</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> isContained | <td> isContained </td><td> </td><td> Returns 1 if the first ideal is contained in the second ideal.</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> isElem | <td> isElem </td><td> </td><td> Returns 1 if the polynomial (2. argument) is in the ideal (1. argument)</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> le | <td> le </td><td><=</td><td> Compares two ideals according to their leading term. Returns 1 if first ideal is less equal the second ideal.</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> lt | <td> lt </td><td><</td><td> Compares two ideals according to their leading term. Returns 1 if first ideal is less than the second ideal.</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> ne | <td> ne </td><td>~=</td><td> returns NOT eq.</td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
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* Ideal generated by two polynomials. | * Ideal generated by two polynomials. | ||
<matlab> | <matlab> | ||
p1 = CoCoAPoly( | p1 = CoCoAPoly('x^2-1'); | ||
p2 = CoCoAPoly( | p2 = CoCoAPoly('y+x'); | ||
myIdeal = CoCoAIdeal([p1 p2]); | myIdeal = CoCoAIdeal([p1 p2]); | ||
</matlab> | </matlab> | ||
[[Category:MatlabToolbox]] | |||
Latest revision as of 16:54, 19 June 2008
Introduction
The CoCoAIdeal class represents an ideal of polynomials over a given polynomial ring (See <a href="MatlabCoCoAPoly.html">Poly class</a>).
Constructor
The constructor for a CoCoAIdeal can have two arguments:
- Generators: Array of polynomials
- Optional: 1/0 : 1 = Compute Groebner-Basis of Ideal
If no input is given the default ideal (1) over Q[x[1]] is returned Generators should be a 1xn CoCoAPoly array. A nx1 array is transposed.
Attributes
| basering | A CoCoARing |
| gens | an array of polynomials |
| gBasis | the Groebner-Basis of an Ideal |
| numBasisElements | The number of basis elements. If gBasis is not yet computed the number of generators is stored. |
Methods
| Method | Operator | Description |
| computeGBasis | computes the GBasis of the given Ideal and returns a new ideal with the same set of generators and the computed GBasis. | |
| computeLT | computes the leading term of the ideal. Returns a CoCoAPoly. | |
| display | Is used by Matlab to print a CoCoAIdeal to the command window. E.g. when a command is not terminated by a ';'. | |
| eq | == | Compares two ideals. Returns 1 if all generators are equal. |
| ge | >= | Compares two ideals according to their leading term. Returns 1 if first ideal is greater equal the second ideal. |
| get | Returns the attributes of an ideal. Syntax: get(idealVar, 'Keyword'), where idealVar is a CoCoAIdeal and 'Keyword' is one of the keywords: |
|
| gt | > | Compares two ideals according to their leading term. Returns 1 if first ideal is greater than the second ideal. |
| isContained | Returns 1 if the first ideal is contained in the second ideal. | |
| isElem | Returns 1 if the polynomial (2. argument) is in the ideal (1. argument) | |
| le | <= | Compares two ideals according to their leading term. Returns 1 if first ideal is less equal the second ideal. |
| lt | < | Compares two ideals according to their leading term. Returns 1 if first ideal is less than the second ideal. |
| ne | ~= | returns NOT eq. |
Examples
A collection of examples is provided in the file TestCoCoAIdeal.m. This file defines the function TestCoCoAIdeal which will run and display many examples. A few common examples are presented below.
- Ideal generated by two polynomials.
<matlab> p1 = CoCoAPoly('x^2-1'); p2 = CoCoAPoly('y+x'); myIdeal = CoCoAIdeal([p1 p2]); </matlab>
