GMRES Fails Silently From Stagnation

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 ```