MINRES: early termination from poor initialization

[timholy]

This is not sufficiently minimal to use as a test case, but I've seen cases where a poor starting guess can lead to early termination and a result that derails the rest of my computation. The difference between the two solves below is whether we start with the user-supplied guess or at the origin:

julia> sol1, stat1 = minres(Amtrx, collect(b), collect(zθ0); M=Mmtrx, ldiv=false);

julia> sol2, stat2 = minres(Amtrx, collect(b), zero(collect(zθ0)); M=Mmtrx, ldiv=false);

julia> stat1
SimpleStats
 niter: 1
 solved: true
 inconsistent: true
 indefinite: false
 npcCount: 0
 residuals: []
 Aresiduals: []
 κ₂(A): []
 timer: 144.68μs
 status: x is a minimum least-squares solution


julia> stat2
SimpleStats
 niter: 46
 solved: true
 inconsistent: true
 indefinite: false
 npcCount: 0
 residuals: []
 Aresiduals: []
 κ₂(A): []
 timer: 428.50μs
 status: found approximate minimum least-squares solution


julia> dot(sol1, sol2) / (norm(sol1) * norm(sol2))
0.5514977818297163

julia> norm(sol1)
1.2206161359675942e17

julia> norm(sol2)
489.98755083623564

Interestingly, we get a single iteration even if it seems that none of the convergence criteria can be met:

julia> sol1, stat1 = minres(Amtrx, collect(b), collect(zθ0); M=Mmtrx, ldiv=false, atol=0.0, etol=0.0, rtol=0.0); stat1
SimpleStats
 niter: 1
 solved: true
 inconsistent: true
 indefinite: false
 npcCount: 0
 residuals: []
 Aresiduals: []
 κ₂(A): []
 timer: 46.73μs
 status: x is a minimum least-squares solution

I did a little bit of debugging, and the termination arises from

Krylov.jl/src/minres.jl

Line 420 in a29a71a

if iter == 1 && β / β₁ ≤ 10 * ϵM

[dpo]

Most solvers in this library have multiple stopping conditions, including some that try to prevent going beyond reasonable limits given the floating-point system.

[timholy]

I wonder whether solved=true and "found approximate minimum least-squares solution" is really the right status message in that case?

[amontoison]

I don't think it is the correct status message.
I should try to document all stopping conditions.
It is on my todolist since a long time...