This method samples a real point on a variety numerically, using a combination of the Nelder-Mead simplex method and line search with gradient descent. This can be much quicker than performing a numerical irreducible decomposition.
The option Tolerance specifies a requested error tolerance for the point, with respect to the generating set of the ideal.
The option Iterations specifies an upper limit on the number of iterations to run the approximation algorithms. If this value is too low, then the method will return a point which may not be within the specified error tolerance.
The following example shows how to sample a point from the 4 x 5 funtf variety.
i1 : (n,r) = (4,5) o1 = (4, 5) o1 : Sequence |
i2 : R = RR[x_(1,1)..x_(n,r)] o2 = R o2 : PolynomialRing |
i3 : A = transpose genericMatrix(R,r,n)
o3 = {-1} | x_(1,1) x_(1,2) x_(1,3) x_(1,4) x_(1,5) |
{-1} | x_(2,1) x_(2,2) x_(2,3) x_(2,4) x_(2,5) |
{-1} | x_(3,1) x_(3,2) x_(3,3) x_(3,4) x_(3,5) |
{-1} | x_(4,1) x_(4,2) x_(4,3) x_(4,4) x_(4,5) |
4 5
o3 : Matrix R <--- R
|
i4 : I1 = ideal(A*transpose A - (r/n)*id_(R^n)); o4 : Ideal of R |
i5 : I2 = ideal apply(entries transpose A, row -> sum(row, v -> v^2) - 1); o5 : Ideal of R |
i6 : I = I1 + I2; o6 : Ideal of R |
i7 : elapsedTime p = realPoint(I, Iterations => 100) -- 0.842436 seconds elapsed o7 = p o7 : Point |
i8 : matrix pack(5, p#Coordinates)
o8 = | .722359 .289465 -.295808 .591752 -.454678 |
| -.110166 .0700102 .908882 .630235 .0984753 |
| .621173 -.330116 -.0305848 .0539234 .866787 |
| .283207 .895726 .292392 -.499725 .179584 |
4 5
o8 : Matrix RR <--- RR
53 53
|
i9 : norm evaluate(gens I, p) o9 = .00000574047029489044 o9 : RR (of precision 53) |
The object realPoint is a method function with options.