I have a matrix $A\in \mathbb{C}^{N\times N}$ and I need to calculate $||A^{-1}||_{2}$ efficiently. Can it be done without having to evaluate the inverse explicitly?
In general, I am looking for methods of calculating $||A^{-1}||_{2}$ that are faster than just doing the following in Matlab:
norm(inv(A))
If no faster method is possible, or actually, just in general, I am alternatively interested in efficient ways to calculate an approximation or upper bound of $||A^{-1}||_{2}$.
Does anyone have any insight on this?
Edit: Note that the form of the matrix $A$ is in fact $A = I-B$ and this $I-B$ is very well conditioned (as $A$ has arisen due to preconditioning). I don't know if this makes any difference to how efficiently $||A^{-1}||_2$ an be computed but I just thought I'd mention it in case.