Motivation
LSMR's normal-equation stop is Paige & Saunders' S2, ‖Âᵀr‖ / (‖Â‖ ‖r‖) ≤ tol, and under
preconditioning every quantity in it is measured through M. Scott & Tůma state the consequence
directly for preconditioned LSQR (arXiv:2504.07580, §3.2): the
test ends up based on ‖(A S M_R⁻¹)ᵀ(b − A x)‖, "which depends on S and M_R", and "in some
cases this can lead to early termination".
within's Schwarz metric has dynamic range κ(M⁻¹) ~ 1e8–1e11, so the stop can be met in a metric
that has deflated the direction still carrying the gradient. The true-residual check added in #388
does not see this: it audits the recurrence against ‖b − A x‖, which confirms the Golub-Kahan
relation but says nothing about which directions the metric left reachable. Fong & Saunders derive
‖r_k‖ from orthogonality of U_{k+1} and ‖Aᵀr_k‖ = |ζ̄_{k+1}| from V_{k+1}, and the paper
carries no preconditioner at all.
The audit legs deleted in #388 were reaching for this and were repeatedly wrong because they kept
the metric's scale (#361, #362). The preconditioner-independent form is Gould & Scott's ratio,
ratioGS = (‖Aᵀr⁽ⁱ⁾‖ / ‖r⁽ⁱ⁾‖) / (‖Aᵀr⁽⁰⁾‖ / ‖r⁽⁰⁾‖)
which carries neither A's scale nor M's, and costs one Aᵀ apply when evaluated at the exit
rather than per iteration.
Goal
A tolerance stop's verdict does not depend on the preconditioner: a solve that M has deflated
into meeting S2, while the unpreconditioned normal-equation residual has not moved, is refused.
Cold-start behaviour and the per-iteration cost are unchanged.
Motivation
LSMR's normal-equation stop is Paige & Saunders' S2,
‖Âᵀr‖ / (‖Â‖ ‖r‖) ≤ tol, and underpreconditioning every quantity in it is measured through
M. Scott & Tůma state the consequencedirectly for preconditioned LSQR (arXiv:2504.07580, §3.2): the
test ends up based on
‖(A S M_R⁻¹)ᵀ(b − A x)‖, "which depends onSandM_R", and "in somecases this can lead to early termination".
within's Schwarz metric has dynamic range
κ(M⁻¹) ~ 1e8–1e11, so the stop can be met in a metricthat has deflated the direction still carrying the gradient. The true-residual check added in #388
does not see this: it audits the recurrence against
‖b − A x‖, which confirms the Golub-Kahanrelation but says nothing about which directions the metric left reachable. Fong & Saunders derive
‖r_k‖from orthogonality ofU_{k+1}and‖Aᵀr_k‖ = |ζ̄_{k+1}|fromV_{k+1}, and the papercarries no preconditioner at all.
The audit legs deleted in #388 were reaching for this and were repeatedly wrong because they kept
the metric's scale (#361, #362). The preconditioner-independent form is Gould & Scott's ratio,
which carries neither
A's scale norM's, and costs oneAᵀapply when evaluated at the exitrather than per iteration.
Goal
A tolerance stop's verdict does not depend on the preconditioner: a solve that
Mhas deflatedinto meeting S2, while the unpreconditioned normal-equation residual has not moved, is refused.
Cold-start behaviour and the per-iteration cost are unchanged.