Table 5 Comparison of the bounds in (10) for smallest singular values of Example 2: the bold face numbers confirm our sharp perturbation bounds

Lower bounds

\(Lb_{1}\)

24.14

16.84

11.48

\(Lb_{2}\)

22.03

17.06

10.05

Upper bounds

\(\sigma _{n-1}(A)\)

38.14

39.23

34.29

\(Ub_{1}\)

32.80

38.10

30.40

\(Ub_{2}\)

34.28

37.95

37.22

Exact values

\(\sigma _{n}(A+yx^{*})\)

28.55

27.83

17.74