Data analysis of solutions

Analysing arrays of vectors

We provide two special functions for analysing data: UniquePoints and Multiplicities. They do the following: suppose that A is an array of real or complex vectors. Then, UniquePoints(A) filters multiple elements of A, such that each entry appears once. On the other hand, multiplicities(A) returns the indices of multiple elements in A.

UniquePoints

Here is the full syntax of UniquePoints

UniquePoints(A, distance; tol=1e-5)

where

  • distance is a function to measure the distance between two vectors. The default option is using the euclidean distance.
  • tol is the value, such that v and w are regarded equal, if distance(v,w) < tol.

Here is an example:

julia> using HomotopyContinuation
julia> A = [[1.0,0.5], [0.99,0.49], [2.0,0.1], [0.5,1.0]]
julia> U = UniquePoints(A)
julia> points(U)
4-element Array{Array{Float64,1},1}:
 [1.0, 0.5]  
 [0.99, 0.49]
 [2.0, 0.1]  
 [0.5, 1.0]

If we relax the tolerance, we get

julia> U = UniquePoints(A, tol = 0.5)
julia> points(U)
3-element Array{Array{Float64,1},1}:
 [1.0, 0.5]
 [2.0, 0.1]
 [0.5, 1.0]

We can also use another distance function:

julia> U = UniquePoints(A, (v,w) -> 0.0)
julia> points(U)
1-element Array{Array{Float64,1},1}:
 [1.0, 0.5]

Multiplicities

Here is the syntax of multiplicities

multiplicities(A; distance=euclidean_distance, tol::Real = 1e-5)

where distance and tol have the same meaning as above.

We use the multiplicities function on the same example as above:

julia> M = multiplicities(A)
0-element Array{Array{Int64,1},1}

For the tol = 0.5 we get one multiplicity:

julia> M = multiplicities(A, tol = 0.5)
1-element Array{Array{Int64,1},1}:
 [1, 2]

This means that the first and the second entry of $A$ are the same up to distance < 0.5. For the all-zero distance function all points are equal:

julia> M = multiplicities(A, distance = (v,w) -> 0.0)
1-element Array{Array{Int64,1},1}:
 [1, 2, 3, 4]

Group Actions

It is also possible to define equality up to group action. For instance, consider the group that interchanges the first and the second entry of a vector in A.

julia> G = GroupActions( x -> ([x[2]; x[1]], ) )

Then, we have

julia> U = UniquePoints(A, group_actions = G)
julia> points(U)
3-element Array{Array{Float64,1},1}:
 [1.0, 0.5]  
 [0.99, 0.49]
 [2.0, 0.1]  

because A[1] = [1.0, 0.5] and A[4] = [0.5, 1.0] are now considered equal:

julia> M = multiplicities(A, group_actions = G)
1-element Array{Array{Int64,1},1}:
 [1, 4]