An exact spectral backward-error counterexample for LSMR

An integer $4\times3$ least-squares problem has exact LSMR iterates with increasing optimal matrix-only spectral-norm backward error. A spectral completion gives an upper certificate $1979/2000$ for the squared first error, and a positive-definite rational matrix gives a strict lower certificate $99/100$ for the squared second error. The approximation $\widetilde\mu$ of the backward error increases as well. These certificates use exact rational arithmetic, not a floating-point optimizer or substitution of the Frobenius norm.