A'x falls back on generic mul

[baggepinnen]

Currently, A'x where A::Toeplitz falls back onto a generic and slow method. By defining

function LinearAlgebra.Adjoint(A::Toeplitz{T}) where T
    At = Toeplitz(A.vr, A.vc)
end

the problem is mitigated and the optimized method for mul! is used. Still, if A'x occurs in a loop, the initialization of the Toeplitz object occurs each iteration, which is suboptimal since it performs some allocations and plans an FFT etc. Ideas on how to solve the problem in general is appreciated.

In the meantime, would you be up for a PR that adds the definition above?

[dlfivefifty]

Sounds great! Though you should never overload a type constructor (Adjoint) to return a different type, unless you really have good reasons: it is confusing for users. Instead, overload the function adjoint.

Note if a user is worried about A' being expansive, they can move that out of the loop.

Down the line, this package may move to a trait-based setup using ArrayLayouts.jl, (as BandedMatrices.jl and BlockArrays.jl have). This would allow Adjoint(Toeplitz(A.vr, A.vc)) to conform to a ToeplitzLayout interface. Though that's a ways off from materialising.