GMRES Fails Silently From Stagnation

[packquickly]

GMRES often fails to converge when restart is less than the problem dimension, but will return without indicating failure.

This failure (likely due to stagnation) seems to depend on the absolute difference between the problem dimension and the Krylov subspace dimension (restart.) ie. roughly 90% of the time GMRES will fail silently on a problem of dimension 25 if restart = 24, and will still fail 90% of the time with a problem of dimension 100 if restart = 99. For a difference in dimension of 5, the test below returns an incorrect value greater than 99% of the time.

Reproducing code:

import IterativeSolvers: gmres
import LinearAlgebra: cond, norm

for problem_dim in [25, 100]
    for excess_dim in [0, 1, 5]
        succeeded = 0
        restart = problem_dim - excess_dim
        for i = 1:100 
            condition_number = Inf
            matrix = nothing
            while condition_number > 1000
                matrix = randn(Float64, (problem_dim, problem_dim))
                condition_number = cond(matrix)
            end

            true_vec = randn(Float64, (problem_dim,))
            b = matrix * true_vec

            julia_soln = gmres(
                matrix, b, restart=restart, log=false, abstol=1e-7
            )

            residual_norm = norm(julia_soln - true_vec)

            succeeded += (residual_norm < 1)  # very loose tolerance!
        end
        println("problem dim: $problem_dim. restart dim: $restart. succeeded: $succeeded")
    end
end