For an ill-conditioned triangular matrix $T$, can we give a simple expression for the following
$$(T + \Delta T)^{-1}$$
where $|\Delta T| \leq \epsilon|T|$, $\epsilon \ll 1$.
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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 $\epsilon \ll \dfrac{1}{cond(T)}$.
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What if $\epsilon \approx cond(T)$? – DDaren Apr 23 '16 at 07:16
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@ DDaren. If $T$ is ill-conditioned, then $cond(T)$ can be $10^8$ and more. – Apr 23 '16 at 09:18
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Sorry I mean $\epsilon \approx 1/cond(T)$. Can you still use Taylor series when $\epsilon \geq 1/cond(T)$? In my case $\epsilon$ is fixed but $T$ is not. – DDaren Apr 23 '16 at 09:29
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I see that I spent my time. – Apr 26 '16 at 22:34