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Numerical.FirstVanishingRelationsInIdeal5

First almost vanishing polynomials for a set of points and ideal
Syntax

$numerical.FirstVanishingRelationsInIdeal5(Points, Epsilon, GetO, GBasis):Object


Description
This function is implemented in ApCoCoALib (requires an active CoCoAServer).

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. Epsilon is a rational >0 describing which singular values should be treated as 0 (smaller values of 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. GetO must be either True or False. If it is true, the command returns a list of two values: the first contains all lowest degree vanishing relations, the second one a vector space basis of P/I comprising those power products lying outside the leading term ideal of I. If GetO is false, the function returns only the relations (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 relations in. Warning: for efficiency the validity of GBasis is not checked.

Example
  Points := Mat([[1,0,0],[0,0,1],[0,2,0]]);
  $numerical.FirstVanishingRelationsInIdeal5(Points,0.001,True,[x]);

-- CoCoAServer: computing Cpu Time = 0
-------------------------------
[[xz, 1/2xy, x^2 - 4503599627370495/4503599627370496x], [x]]


See Also