# Difference between revisions of "ApCoCoA-1:Num.DABM"

S schuster (talk | contribs) (New page: <command> <title>Num.DABM</title> <short_description>In a differential setting, computes the border basis of an almost vanishing ideal for a set of points using the ABM algorithm.</shor...) |
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The matrix <tt>Points</tt> contains the data points: each | The matrix <tt>Points</tt> contains the data points: each | ||

point is a row in the matrix, so the number of columns must equal the | point is a row in the matrix, so the number of columns must equal the | ||

− | number of indeterminates, namely NumOfIndets, in the current ring. The number of rows must be greater than Nl+Nr (again, because of the Savitzky-Golay filter). | + | number of indeterminates, namely NumOfIndets, in the current ring. The number of rows must be greater than <tt>Nl+Nr</tt> (again, because of the Savitzky-Golay filter). |

<itemize> | <itemize> | ||

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<example> | <example> | ||

Use Q[x[1..2,0..3]]; | Use Q[x[1..2,0..3]]; | ||

− | Use Q[x[1..2,0..3]], Ord(DA.DiffTO( | + | Use Q[x[1..2,0..3]], Ord(DA.DiffTO(<quotes>Ord</quotes>)); |

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<see>Introduction to CoCoAServer</see> | <see>Introduction to CoCoAServer</see> | ||

<see>Num.ABM</see> | <see>Num.ABM</see> | ||

+ | <see>Num.CABM</see> | ||

+ | <see>Num.BBABM</see> | ||

</seealso> | </seealso> | ||

<types> | <types> | ||

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<type>points</type> | <type>points</type> | ||

</types> | </types> | ||

− | <key> | + | <key>DABM</key> |

<key>Num.DABM</key> | <key>Num.DABM</key> | ||

<key>numerical.DABM</key> | <key>numerical.DABM</key> | ||

<wiki-category>Package_numerical</wiki-category> | <wiki-category>Package_numerical</wiki-category> | ||

</command> | </command> |

## Revision as of 08:36, 25 May 2010

## Num.DABM

In a differential setting, computes the border basis of an almost vanishing ideal for a set of points using the ABM algorithm.

### Syntax

DABM(Points:MAT, GoUpToOrder:INT, Epsilon:RAT, Nl:INT, Nr:INT, Deg:INT)

### Description

*Please note:* The function(s) explained on this page is/are using the *ApCoCoAServer*. You will have to start the ApCoCoAServer in order to use it/them.

This command computes a differential border basis of an almost vanishing ideal for a set of points.

The current ring has to be a polynomial ring over the rational numbers with a standard-degree

compatible term-ordering. The indeterminates have to be given as x[1..NumOfIndets,0..MaxOrd], where MaxOrd is the maximal order which should be considered by the algorithm, i.e., the order up to which derivative values are computed by the Savitzky-Golay filter involved.

The matrix `Points` contains the data points: each
point is a row in the matrix, so the number of columns must equal the
number of indeterminates, namely NumOfIndets, in the current ring. The number of rows must be greater than `Nl+Nr` (again, because of the Savitzky-Golay filter).

@param

*Points*The points for which a border basis is computed.@param

*GoUpToOrder*The order up to which derivatives are taken to construct the model polynomials. It must be between 0 and the maximal differential order MaxOrd in the definition of the current ring.@param

*Epsilon*A positive rational number describing the maximal admissible least squares error for a polynomial. (Bigger values for`Epsilon`lead to bigger errors of the polynomials evaluated at the point set).`Epsilon`should be in the interval (0,1). As a rule of thumb,`Epsilon`is the expected percentage of error on the input points.@param

*Nl*Number of left data points to be considered by the Savitzky-Golay filter.@param

*Nr*Number of right data points to be considered by the Savitzky-Golay filter.@param

*Deg*Maximum degree of the interpolation polynomial used by Savitzky-Golay filter. Must be greater or equal MaxOrd.@return The border basis as a list of polynomials.

#### Example

Use Q[x[1..2,0..3]]; Use Q[x[1..2,0..3]], Ord(DA.DiffTO(<quotes>Ord</quotes>)); Nl := 2; Nr := 2; Deg := 4; GoUpToOrder:=2; Epsilon:=0.1; -- Data is given by [[cos(I), sin(I)] | I In 1..35]. Points := Mat([[0.540302, 0.841471], [-0.416147, 0.909297], [-0.989992, 0.14112], [-0.653644, -0.756802], [0.283662, -0.958924], [0.96017, -0.279415], [0.753902, 0.656987], [-0.1455, 0.989358], [-0.91113, 0.412118], [-0.839072, -0.544021], [0.0044257, -0.99999], [0.843854, -0.536573], [0.907447, 0.420167], [0.136737, 0.990607], [-0.759688, 0.650288], [-0.957659, -0.287903], [-0.275163, -0.961397], [0.660317, -0.750987], [0.988705, 0.149877], [0.408082, 0.912945], [-0.547729, 0.836656], [-0.999961, -0.00885131], [-0.532833, -0.84622], [0.424179, -0.905578], [0.991203, -0.132352], [0.646919, 0.762558], [-0.292139, 0.956376], [-0.962606, 0.270906], [-0.748058, -0.663634], [0.154251, -0.988032], [0.914742, -0.404038], [0.834223, 0.551427], [-0.0132767, 0.999912], [-0.84857, 0.529083], [-0.903692, -0.428183] ]); Result:=Num.DABM(Points, GoUpToOrder, Epsilon, Nl, Nr, Deg); Foreach X In Result Do PrintLn Dec(X,2); EndForeach; ------------------------------- -- CoCoAServer: computing Cpu Time = 0.1423 ------------------------------- 0.70 x[1,1] +0.70 x[2,0] -0.00 x[1,0] +0.00 0.70 x[2,1] +0.00 x[2,0] -0.70 x[1,0] +0.00 [...]

### See also