Improvements to solve for the association fraction about clapeyron.jl HOT 2 CLOSED

This is written mainly as future reference for anyone interested on how we solve our association
reading 3. we do perform that exact index reduction, but we take a step further: we fuse z[IJ] and ΔIJ (in their [IJ] notation) into one sparse matrix A[IJ]. their proposed iteration is equivalent to ours when dampingfactor = 0.5 but we could use their initial point. we can actually obtain their KIJ matrix by:

julia> model = PCSAFT(["water","methanol"])
PCSAFT{BasicIdeal} with 2 components:
 "water"
 "methanol"
Contains parameters: Mw, segment, sigma, epsilon, epsilon_assoc, bondvol

julia> iszero.(Clapeyron.assoc_site_matrix(model,0.03,373.15,[0.5,0.5]))
4×4 SparseArrays.SparseMatrixCSC{Bool, Int64} with 16 stored entries:
 1  0  1  1
 0  1  1  1
 1  1  1  0
 1  1  0  1
 Note here the `SparseMatrixCSC` type. we only store the non empty values.

The only difference between their algorithm and ourds

All approaches that try to solve association as a fixed iteration problem can be written as:

A*Y⊙Y + Y - 1 = ∅

where ⊙ is the hadamard product between A[I,J]*Y[i] and Y[I]. the fixed point iteration used can be obtained by solving for Y. this is, at the same time, one general case of a quadratic matrix equation:

Mx = a + b(x,x)

where b is a bilinear form, and M in our case is -I. this general form is discussed here (10.1016/j.laa.2011.05.036)(https://doi.org/10.1016/j.laa.2011.05.036) and proposes some alternative iteration methods.

On the optimization approaches, the main roadblock is the allocation of all caches and inverting matrices. it seems that inverting a matrix is a heavy operation, compared to repeated multiplications. on some preliminar tests. if this allocation and solving is done inline (that is, on each eos call) it takes around 5x more time. work in a "inline cached eos" for routines at fixed T,z could help here, but one has to be careful if threads are used.

from clapeyron.jl.

One thing that is complex under our association solver was adding combining rules. a combining rule means a dense matrix of reduced indices, or at least, denser that what gains can be obtained with using an sparse matrix. Because of that, the new combining::Symbol keyword was added to AssocOptions.

• sparse_nocombining: the default. creates an sparse matrix.
• sparse_nocombining: new option. instantiates a dense matrix, but still no combining
• elliott Combining rule.
  we use the general definition of the elliott combining rule:

Δ(i,j,a,b) = √(Δ(i,i,a,b)*Δ(j,j,a,b))

instead on the more EoS-specific values. (CPA and SAFT have different expressions for the mixing rules of the association energy, but they reduce to the equation above).
At the moment, if one term is associating and the other is not, we default to zero. this could be explored further in the future.
this work was made as a byproduct of tackling #75

from clapeyron.jl.