Table 4 Comparison of the bounds in (10) for Example 2 (\(2\leq k\leq n-1\)): the bold face numbers confirm our sharp perturbation bounds
From: Rank-one perturbation bounds for singular values of arbitrary matrices
|  |  | n = 30 | ||
|---|---|---|---|---|
k = 8 | k = 16 | k = 24 | ||
Lower bounds | \(\sigma _{k+1}(A)\) | 16.25 | 12.75 | 26.81 |
\(Lb_{1}\) | 13.96 | 12.91 | 28.54 | |
\(Lb_{2}\) | 17.14 | 10.27 | 19.72 | |
Upper bounds | \(\sigma _{k-1}(A)\) | 53.45 | 36.08 | 51.08 |
\(Ub_{1}\) | 45.26 | 34.77 | 48.76 | |
\(Ub_{2}\) | 40.18 | 34.82 | 50.73 | |
Exact values | \(\sigma _{k}(A+yx^{*})\) | 34.03 | 27.59 | 48.50 |
|  |  | n = 60 | ||
|---|---|---|---|---|
k = 22 | k = 40 | k = 58 | ||
Lower bounds | \(\sigma _{k+1}(A)\) | 30.13 | 23.26 | 10.26 |
\(Lb_{1}\) | 32.53 | 25.11 | 14.96 | |
\(Lb_{2}\) | 36.18 | 10.04 | 15.60 | |
Upper bounds | \(\sigma _{k-1}(A)\) | 68.26 | 60.29 | 49.07 |
\(Ub_{1}\) | 70.24 | 58.65 | 47.43 | |
\(Ub_{2}\) | 68.01 | 54.71 | 45.48 | |
Exact values | \(\sigma _{k}(A+yx^{*})\) | 56.02 | 41.59 | 28.56 |