Fig. 1.

Left: κ₊(A) versus κ(AN) for different choices of r and s. A ∈ ℝ^{104 × 103} is randomly generated with rank r ∈ {800, 1000} and effective condition number κ₊(A) ∈ {10², 10³, …, 10⁸}. For each (r, s) pair, we take the largest value of κ(AN) in ten independent runs for each κ₊(A) and plot them using circle marks. The estimate $(1+\sqrt{r/s})/(1-\sqrt{r/s})$ is drawn using a solid line for each (r, s) pair. Right: Number of LSQR iterations versus r/s. The number of LSQR iterations is merely a function of r/s, independent of the condition number of the original system.