Change definition

[putianyi889]

Currently vr and vc can be modified so that their first entries are different:

julia> T=Toeplitz(rand(3,3))
3×3 Toeplitz{Float64, Vector{Float64}, Vector{Float64}}:
 0.410444  0.893844  0.0538559
 0.477895  0.410444  0.893844
 0.853433  0.477895  0.410444

julia> T.vr[1]=0;

julia> T
3×3 Toeplitz{Float64, Vector{Float64}, Vector{Float64}}:
 0.410444  0.893844  0.0538559
 0.477895  0.410444  0.893844
 0.853433  0.477895  0.410444

julia> Toeplitz(T)
ERROR: First element of the vectors must be the same

It's difficult to keep track of this constraint. How about defining Toeplitz as something like

struct Toeplitz
    first # [1,1]
    vc # [2:end,1]
    vr # [1,2:end]
end

Pros:

• It's easier to convert between types. For example, the following methods could be removed.
  

  ToeplitzMatrices.jl/src/special.jl

  Lines 98 to 99 in c3ac898

  	_circulate(v::AbstractVector) = vcat(v[1],v[end:-1:2])
  	_firstnonzero(v::AbstractVector) = vcat(v[1],zero(view(v,2:lastindex(v))))

• It ensures that the top-left entry is inambiguous.
• It's easier to implement some methods, like isone, istriu, etc.

Cons:

• Getting the full first row/column needs concatenation and is therefore slower. This doesn't affect conversions since they wouldn't need the full row/column.
• A lot of codes need to be rewritten.
• mutable is needed in order to change the first entry.

[dlfivefifty]

@andreasnoack what was the motivation for having the diagonal repeated twice?