Another Documentation Update

[April-Hannah-Lena]

I've updated the documentation to include some new things:

• detail on how to use BoxSet, ie collect(W), iteration, etc.
• BoxGraph item in general and docstring
• Finite time Lyapunov exponents

The FTLE algorithm has also been updated to make more sense and fit with the GAIO.jl usage pattern.

BoxFun also got two added features:

• precomposition with BoxMaps, i.e. ∘(boxfun::BoxFun, F::BoxMap) just applies the Koopman operator for F to the BoxFun

• more detailed interface for integrating over boxfuns, as shown in the new docstring (the LaTeX is rendered in the actual docs page):

  
  sum(f, boxfun::BoxFun)
  sum(f, μ::BoxFun)
  sum(f, μ::BoxFun, B::BoxSet)
  μ(B) = sum(identity, μ, B)
  Integrate a function `f` using `boxfun` as a density, that is,
  Integrate a function `f` using `μ` as a density, that is,
  if `boxfun` is the discretization of a measure ``\mu`` over the domain 
  ``Q``, then approximate the value of 
  
  \int_Q f \, d\mu .

If a BoxSet B is passed as the third argument, then the integration is restricted to the boxes in B

\int_{Q \cap \bigcup_{b \in B} b} f \, d\mu .

The notation μ(B) is offered to compute \mu (\bigcup_{b \in B} b).

[gaioguy]

Nice, thanks. Great idea to enable the precomposition with the map. Would it be possible to make a BoxFun behave like a vector, i.e. have addition and scalar multiplication?

[April-Hannah-Lena]

Nice, thanks. Great idea to enable the precomposition with the map. Would it be possible to make a BoxFun behave like a vector, i.e. have addition and scalar multiplication?

It definitely would be possible, though the interface for custom broadcasting in julia is a bit complicated. I suppose also we have to ask: fundamentally, should a BoxFun be viewed as a vector (where broadcasting etc makes sense), or as a function (operations should be done through composition with other functions)?

The current functionality to would be to use findnz from SparseArrays to get a vector, perform whatever vector operations on the result, and then create a new BoxFun from that vector. But that requires some knowledge on the internals of the struct. Maybe that workflow can be simplified, e.g. something like

mu = # existing BoxFun

vals = Vector(mu)

A = # some kind of matrix maybe
vals_new = A * vals .+ exp.(vals)    # some operation

mu_new = BoxFun(mu, vals_new)

[gaioguy]

Sorry, I actually meant function, and so more precisely the question is whether we can give the BoxFuns a vector space structure.