border basis of an almost vanishing sub-ideal for a set of points and ideal
$numerical.BBasisOfPointsInIdeal(Points, Tau, GetO, GBasis, Delta, NormalizeType, RREFNormalizeType, RREFUseEps, RREFType):Object
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
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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
D. Heldt, M. Kreuzer, H. Poulisse: Computing Approximate Vanishing Ideals (Work in progress)
The current ring has to be a ring over the rationals with a standard-degree compatible term-ordering. The matrix Points contains the points: each point is a row in the matrix, so the number of columns must equal the number of indeterminates in the current ring.
Tau is a rational <formula>>0</formula>,
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 <formula>(0,1)</formula>. As a rule of thumb, tau is the expected percentage of error on the input points.
GetO must be either True or False. If it is true, the command
returns a list of two values: the first contains the border basis, the second one a vector space basis of <formula>P/I</formula> comprising those power products lying outside the leading term ideal of I. If GetO is false, the function returns only the border basis (not in a list).
GBasis must be 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.
The last 5 parameters are optional.
Delta must be a positiv rational. 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.
NormalizeType must be one of 1,2,3 ,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.
RREFNormalizeType describes, how in each RREF steps the columns are normalized. The options correspond to the ones for NormalizeType and the default is 1 again.
RREFUseEps 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.
The last parameter is RREFType. This must be 1 or 2. If RREFType=1, the rref operates column-wise. Otherwise it works row-wise. The default is 1.
Points := Mat([[2,0,0],[0,3,0],[0,0,1]]); $numerical.BBasisOfPointsInIdeal(Points, 0.001, False,[z,y]); ------------------------------- [z^2 - z, 1/3yz, 1/2xz, 1/9y^2 - 9007199254740991/27021597764222976y, 1/6xy] -------------------------------