Adapt to LazyArrays.jl

[dlfivefifty]

Sorry for the delay, finally getting into this. Given the new LazyArrays+BandedMatrices+InfiniteArrays framework my view on this has changed, in the sense that we should leverage LazyArrays as much as possible to enable ∞-dimensional usage. For example, the following works and is equivalent to an almost banded matrix:

julia> A = BandedMatrix(1 => Ones(∞), -1 => Fill(2,∞)) # Create Banded Matrix
∞×∞ BandedMatrix{Float64,ApplyArray{Float64,2,typeof(*),Tuple{Array{Float64,2},ApplyArray{Float64,2,typeof(vcat),Tuple{Ones{Float64,2,Tuple{Base.OneTo{Int64},OneToInf{Int64}}},Fill{Int64,2,Tuple{Base.OneTo{Int64},OneToInf{Int64}}}}}}},OneToInf{Int64}} with indices OneToInf()×OneToInf():
 0.0  1.0   ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅   …  
 2.0  0.0  1.0   ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅      
  ⋅   2.0  0.0  1.0   ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅      
  ⋅    ⋅   2.0  0.0  1.0   ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅      
  ⋅    ⋅    ⋅   2.0  0.0  1.0   ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅      
  ⋅    ⋅    ⋅    ⋅   2.0  0.0  1.0   ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅   …  
  ⋅    ⋅    ⋅    ⋅    ⋅   2.0  0.0  1.0   ⋅    ⋅    ⋅    ⋅    ⋅    ⋅      
  ⋅    ⋅    ⋅    ⋅    ⋅    ⋅   2.0  0.0  1.0   ⋅    ⋅    ⋅    ⋅    ⋅      
  ⋅    ⋅    ⋅    ⋅    ⋅    ⋅    ⋅   2.0  0.0  1.0   ⋅    ⋅    ⋅    ⋅      
 ⋮                        ⋮                        ⋮                   ⋱  

julia> B = [[1,2,3]; zeros(∞)] * Ones(1,∞) # Create Low Ranke Matrix
∞×∞ ApplyArray{Float64,2,typeof(*),Tuple{CachedArray{Float64,1,Array{Float64,1},Zeros{Float64,1,Tuple{OneToInf{Int64}}}},Ones{Float64,2,Tuple{Base.OneTo{Int64},OneToInf{Int64}}}}} with indices OneToInf()×OneToInf():
 1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  …  
 2.0  2.0  2.0  2.0  2.0  2.0  2.0  2.0  2.0  2.0  2.0  2.0  2.0  2.0     
 3.0  3.0  3.0  3.0  3.0  3.0  3.0  3.0  3.0  3.0  3.0  3.0  3.0  3.0     
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     
 ⋮                        ⋮                        ⋮                   ⋱  

julia> BroadcastArray(+,A,ApplyArray(triu,B,2)) # almost banded matrix
∞×∞ BroadcastArray{Float64,2,typeof(+),Tuple{BandedMatrix{Float64,ApplyArray{Float64,2,typeof(*),Tuple{Array{Float64,2},ApplyArray{Float64,2,typeof(vcat),Tuple{Ones{Float64,2,Tuple{Base.OneTo{Int64},OneToInf{Int64}}},Fill{Int64,2,Tuple{Base.OneTo{Int64},OneToInf{Int64}}}}}}},OneToInf{Int64}},ApplyArray{Float64,2,typeof(triu),Tuple{ApplyArray{Float64,2,typeof(*),Tuple{CachedArray{Float64,1,Array{Float64,1},Zeros{Float64,1,Tuple{OneToInf{Int64}}}},Ones{Float64,2,Tuple{Base.OneTo{Int64},OneToInf{Int64}}}}},Int64}}}} with indices OneToInf()×OneToInf():
 0.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  …  
 2.0  0.0  1.0  2.0  2.0  2.0  2.0  2.0  2.0  2.0  2.0  2.0  2.0  2.0     
 0.0  2.0  0.0  1.0  3.0  3.0  3.0  3.0  3.0  3.0  3.0  3.0  3.0  3.0     
 0.0  0.0  2.0  0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     
 0.0  0.0  0.0  2.0  0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     
 0.0  0.0  0.0  0.0  2.0  0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  
 0.0  0.0  0.0  0.0  0.0  2.0  0.0  1.0  0.0  0.0  0.0  0.0  0.0  0.0     
 0.0  0.0  0.0  0.0  0.0  0.0  2.0  0.0  1.0  0.0  0.0  0.0  0.0  0.0     
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  2.0  0.0  1.0  0.0  0.0  0.0  0.0     
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  2.0  0.0  1.0  0.0  0.0  0.0     
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  2.0  0.0  1.0  0.0  0.0  …  
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  2.0  0.0  1.0  0.0     
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  2.0  0.0  1.0     
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  2.0  0.0     
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  2.0     
 ⋮                        ⋮                        ⋮                   ⋱  

The one issue with the above construction is it doesn't link the bandwidth of A to the number of diagonals taken in B....