Add Scattering Matrix Calculation about acoustics HOT 6 CLOSED

In testing if S is unitary (i.e. S \dot adjoint(S) = Identity matrix OR det(S)det(adjoint(S))=1), which it isn't, it appears that det(S)det(adjoint(S))=1/(2k)^2 hinting at a problem with the scaling of the incident field, most likely.

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Next steps:

1. Research LU factorization and understand it.

2. Alter computation of scattering matrix. Don't compute Z for each harmonic. Compute once, store LU factors, then generate all RHSs and apply LU factorization to each to get the J vectors. Then stitch together V and J matrices. Alternative: compute V and J matrices straight from Z's LU factors and a matrix of RHSs (e.g. V^T LU V).

3. Compute S for max l = 8. Compute determinant. determinant of S (which is a submatrix of full S) for l = 7 should be nearly the same.

4. Try to find a scale factor on VJ, not S

from acoustics.

to solve Ax=b

1. Call A_factors = factorize(A) or A_lu = lu(A). This may or may not use LU factorization, but it will do a check to see the type of matrix A is and choose a method accordingly (https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#LinearAlgebra.factorize). Directly calling lu(A) might be best since I know A isn't going to have any special properties (unless the mesh has some order to it), but in general A will just be a "general square" matrix. Can probably save (negligible) computation time by using lu(A).

2. A_factors \ b = x will solve the equation for x using the factors of A.

from acoustics.

Determinant is doing weird stuff for max_l = 8. As I truncate S, every two truncations, the determinant flips sign but remains unchanged in magnitude. Not sure about that. Currently I am trying to find a scale factor on VJ

from acoustics.

SCALE FACTOR MIGHT BE THE FACT THAT I AM NOT SCALING INCIDENT FIELD BY K

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All looks good. Awaiting thoughts from Eric before merging

from acoustics.