Update for QuasiArrays.jl

[dlfivefifty]

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[dlfivefifty]

@jagot This is converging on a reasonable revised system, building on QuasiArrays.jl. I'll probably move out the ultraspherical code before merging to try to make this package "bare bones", but thought I'd warn you about what's going to happen.

Probably the nicest change is the addition of an @simplify macro to specify commuting relationships. Here are some examples:

# mass matrix for Legendre
@simplify *(A::QuasiAdjoint{<:Any,<:Legendre}, B::Legendre) = Diagonal(2 ./ (2(0:∞) .+ 1))		     
# Jacobi matrix for Legendre
@simplify \(A::Legendre, *(B::Identity, C::Legendre))  =		
     _BandedMatrix(Vcat(((0:∞)./(1:2:∞))', Zeros(1,∞), ((1:∞)./(1:2:∞))'), ∞, 1,1)		   

The @simplify code adds a bunch of extra functions, for example, if you have @simplify *(::Type1, ::Type2) it automatically also adds the adjoint code *(::QuasiAdjoint{<:Any,<:Type2}, ::QuasiAdjoint{<:Any,<:Type1}) (taking into account Type1/Type2 may also be adjoints).

Other than that its tried to make everything more systematic. One left over unsystematic routine is fullmaterialize, which I think can be redesigned in a more robust way by checking ApplyStyle(*, typeof.(A.args[subinds])...) == SimplifyStyle() to see if a simplifying relationship is known, and this could even happen at compile time now that ApplyStyle depends only on types. I'll probably just leave it for now though.

Another thing left to do is collapse Eye(...) * B to just B. This probably isn't too hard.

[jagot]

That's really cool! 👍