Table 1 Comparison of the bounds in (10) for Example 1 (\(2\leq k\leq n-1\)): the bold face numbers confirm our sharp perturbation bounds
From: Rank-one perturbation bounds for singular values of arbitrary matrices
|  |  | n = 40 | ||
|---|---|---|---|---|
k = 8 | k = 18 | k = 28 | ||
Lower bounds | \(\sigma _{k+1}(A)\) | 27.46 | 16.34 | 8.41 |
\(Lb_{1}\) | 28.52 | 16.97 | \( \textbf{8}\textbf{.}\textbf{85} \) | |
\(Lb_{2}\) | 25.60 | 17.64 | 7.98 | |
Upper bounds | \(\sigma _{k-1}(A)\) | 66.30 | 45.10 | 21.46 |
\(Ub_{1}\) | 55.24 | 40.65 | \( \textbf{17}\textbf{.}\textbf{46} \) | |
\(Ub_{2}\) | 46.32 | 39.14 | 19.45 | |
Exact values | \(\sigma _{k}(A+yx^{*})\) | 35.23 | 27.35 | 11.46 |
|  |  | n = 80 | ||
|---|---|---|---|---|
k = 38 | k = 52 | k = 66 | ||
Lower bounds | \(\sigma _{k+1}(A)\) | 29.38 | 26.19 | 14.57 |
\(Lb_{1}\) | 36.48 | 24.58 | 13.90 | |
\(Lb_{2}\) | 38.20 | 27.30 | 15.04 | |
Upper bounds | \(\sigma _{k-1}(A)\) | 82.47 | 70.06 | 43.35 |
\(Ub_{1}\) | 81.69 | 71.59 | 42.53 | |
\(Ub_{2}\) | 82.41 | 68.33 | 45.63 | |
Exact values | \(\sigma _{k}(A+yx^{*})\) | 51.24 | 43.45 | 28.24 |
|  |  | n = 100 | ||
|---|---|---|---|---|
k = 16 | k = 44 | k = 72 | ||
Lower bounds | \(\sigma _{k+1}(A)\) | 54.26 | 35.34 | 15.85 |
\(Lb_{1}\) | 56.17 | 31.95 | 10.24 | |
\(Lb_{2}\) | 52.46 | 36.35 | 16.32 | |
Upper bounds | \(\sigma _{k-1}(A)\) | 93.14 | 75.35 | 44.52 |
\(Ub_{1}\) | 86.20 | 70.28 | 43.13 | |
\(Ub_{2}\) | 89.35 | 78.01 | 48.76 | |
Exact values | \(\sigma _{k}(A+yx^{*})\) | 70.56 | 45.15 | 30.42 |