We provide two special functions for analysing data: unique_points and multiplicities. They do the following: suppose that A is an array of real or complex vectors. Then, unique_points(A) filters multiple elements of A, such that each entry appears once (given a provided tolerance). On the other hand, multiplicities(A) returns the indices of multiple elements in A.
Unique points
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> unique_points(A)
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> unique_points(A, atol = 0.3)
3-element Array{Array{Float64,1},1}:
[1.0, 0.5]
[2.0, 0.1]
[0.5, 1.0]
Note that by default the Euclidean metric is used.
We can also use another metric:
julia> unique_points(A, atol = 0.5, metric = InfNorm())
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, atol = 0.3)
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 < 0.3.
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.
G = GroupActions( x -> ([x[2]; x[1]], ) )
Then, we have
julia> unique_points(A, group_actions = G)
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]