Table 3 Comparison results of the bounds for largest singular values of Example 1: the bold face numbers confirm our sharp perturbation bounds
From: Rank-one perturbation bounds for singular values of arbitrary matrices
| Â | Â | n | ||
|---|---|---|---|---|
20 | 40 | 60 | ||
Lower bounds | \(\sigma _{2}(A)\) | 18.46 | 45.26 | 37.64 |
\(Lb_{1}\) | 19.68 | 23.89 | 32.70 | |
\(Lb_{2}\) | 21.84 | 46.90 | 40.02 | |
Upper bounds | \(Ub_{1}\) | 50.68 | 72.39 | 81.06 |
\(Ub_{2}\) | 42.04 | 65.79 | 73.80 | |
\(\sigma _{1}(A)+\sigma _{1}(yx^{*})\) | 58.53 | 86.48 | 96.46 | |
Exact values | \(\sigma _{1}(A+yx^{*})\) | 30.58 | 69.57 | 48.19 |