Wanted: type indicating efficient M and M⁻¹

[timholy]

In #1043 I noted that the restrictiveness of warm-start for some methods presumably comes down to an API which requires users to specify whether "preconditioners" (see #1046) should be used in the sense of v -> M * v or v -> M \ v; the code tries to avoid making assumptions that the other operation is even available. However, in some cases both operations can be made reasonably efficient (Diagonal being an obvious example, but Tridiagonal and others are also good candidates). One might also wish to follow the example of https://github.com/JuliaStats/PDMats.jl and supply both M and its factorization.

I wonder if it might make sense to provide a way of signaling this. Two potential options:

• allow ldiv to have a third value indicating that M is supplied as a pair of operators (Mmul, Mdiv)
• define a wrapper type that allows one to supply both

[timholy]

A specific proposal:

export PWrapper

struct PWrapper{MMUL, MDIV}
    Mmul::MMUL
    Mdiv::MDIV
end

lsqr(A, b; M=PWrapper(M, nothing))          # equivalent to default ldiv=false
lsqr(A, b; M=PWrapper(nothing, factoredM))  # equivalent to specifying `M` and `ldiv=true`
lsqr(A, b; M=PWrapper(M, factoredM))        # supply both

I'm not wild about the name PWrapper, of course.

[timholy]

I suspect there's a way to add something like this that might be non-breaking; the code could check for a PWrapper. In the longer term, one would want to get rid of ldiv, but we could add a deprecation warning and then remove it in the next breaking release.