Computes a border basis of an almost vanishing ideal for a set of points.
Num.AVI(Points:MAT, Epsilon:RAT):Object Num.AVI(Points:MAT, Epsilon:RAT, Delta:RAT, NormalizeType:INT, RREFNormalizeType:INT, RREFUseEps:BOOL, RREFType:INT):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 it/them.
This function computes an approximate border basis of an almost vanishing ideal for a set of points using the AVI algorithm. AVI is an acronym of "Approximate Vanishing Ideal".
The current ring has to be a ring over the rational numbers 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.
@param Points The points for which a border basis is computed.
@param Epsilon A positive rational number describing which singular values should be treated as 0 (smaller 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.
@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.
The following parameters are optional:
@param Delta A positive 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.
@param NormalizeType 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 absolute value of the corresponding column of the matrix. This ensures that all coordinates of points 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.
@param 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.
@param 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 norm of the matrix.
@param RREFType This must be 1 or 2. If RREFType=1, the RREF operates column-wise, otherwise it works row-wise. The default is 1.
Use P::=QQ[x,y,z]; Points := Mat([[1,0,0],[0,0,1],[0,1,0]]); Num.AVI(Points,0.001); -- CoCoAServer: computing Cpu Time = 0 ------------------------------- [[x + y + 4503599627370495/4503599627370496z - 4503599627370497/4503599627370496, xy, y^2 - y, xz, yz, z^2 - z], [1, z, y]] ------------------------------- -------------------------------