Inverse of perturbed triangular matrix

Question

For an ill-conditioned triangular matrix $T$, can we give a simple expression for the following

$$(T + \Delta T)^{-1}$$

where $|\Delta T| \leq \epsilon|T|$, $\epsilon \ll 1$.

Answer 1

Using Taylor formula, we obtain $(T+\Delta T)^{-1}= T^{-1}-T^{-1}.\Delta T. T^{-1}+O(||\Delta T||^2||T^{-1}||^3)=$

$T^{-1}-T^{-1}.\Delta T. T^{-1}+O(\epsilon ^2.cond^2(T).||T^{-1}||)$.

Then you must choose $\epsilon \ll \dfrac{1}{cond(T)}$.