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# numericalImageSample -- samples general points on the image of a variety

- Usage:
    `numericalImageSample(F, I, P, s)`
    `numericalImageSample(F, I, s)`
    `numericalImageSample(F, I)`
- Inputs:
  - F, a matrix, or list, or ring map, specifying a map
  - I, an ideal, which is prime, specifying a source variety $V(I)$
  - P, a list, of points on $F(V(I))$
  - s, an integer, the number of points to sample in $F(V(I))$
- Optional inputs:
  - Software => ..., default value M2engine, specify software for homotopy continuation
- Outputs:
  - a list, of sample points on $F(V(I)))$

## Description

This method computes a list of sample points on the image of a variety numerically, by calling numericalSourceSample.

If the number of points $s$ is unspecified, then it is assumed that $s = 1$.

One can optionally provide an initial list of points $P$ on $F(V(I))$, which will then be completed to a list of $s$ points on $F(V(I))$.

The following example samples a point from the twisted cubic. We then independently verify that this point does lie on the twisted cubic.

    i1 : R = CC[s,t];

    i2 : F = {s^3,s^2*t,s*t^2,t^3};

    i3 : p = first numericalImageSample(F, ideal 0_R)

    o3 = p

    o3 : Point

    i4 : A = matrix{p#Coordinates_{0,1,2}, p#Coordinates_{1,2,3}};

                    2         3
    o4 : Matrix CC    <-- CC
                  53        53

    i5 : numericalNullity A == 2

    o5 = true

Here is how to sample a point from the Grassmannian $Gr(2,4)$ of $P^1$'s in $P^3$, under its Pl&uuml;cker embedding in $P^5$. We take maximal minors of a $2 x 4$ matrix, whose row span gives a $P^1$ in $P^3$.

    i6 : R = CC[x_(1,1)..x_(2,4)];

    i7 : F = (minors(2, genericMatrix(R, 2, 4)))_*;

    i8 : numericalImageSample(F, ideal 0_R)

    o8 = {{-.434457-.140153*ii, -.374655-.460763*ii, -.192528-.151059*ii,
         ------------------------------------------------------------------------
         -.113152+.276111*ii, .222593+.353053*ii, .100773+.425004*ii}}

    o8 : List

## See also

- numericalSourceSample -- samples a general point on a variety

## Ways to use numericalImageSample:

- `numericalImageSample(List,Ideal)`
- `numericalImageSample(List,Ideal,List,ZZ)`
- `numericalImageSample(List,Ideal,ZZ)`
- `numericalImageSample(Matrix,Ideal)`
- `numericalImageSample(Matrix,Ideal,List,ZZ)`
- `numericalImageSample(Matrix,Ideal,ZZ)`
- `numericalImageSample(RingMap,Ideal)`
- `numericalImageSample(RingMap,Ideal,List,ZZ)`
- `numericalImageSample(RingMap,Ideal,ZZ)`

## For the programmer

The object numericalImageSample is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NumericalImplicitization/doc.m2:252:0.