Sensitivity of linear system

Question

I am trying to find out sensitivity of solution of a linear system, that is given $$(A+\epsilon F)x(\epsilon)=b+\epsilon f$$ how does x changes with $\epsilon$. I am not sure how to go about finding differential of $x(\epsilon)=(A+\epsilon F)^{-1}(b+\epsilon f)$ because of $\epsilon F$ under inverse? Any hint or thoughts on how to go about finding change in $x$ wrt $\epsilon$.

Answer 1

Vague hint: You are basically asking for the differential of the map $A \mapsto A^{-1}$. It is sufficient to find the differential of this map at the identity, which is easier. Once you have it, you can extend it to the general case.

Specific hint:

$$(A+\epsilon F)^{-1}=(A(I+\epsilon A^{-1}F))^{-1}=(I+\epsilon A^{-1} F)^{-1} A^{-1}=\left (\sum_{n=0}^\infty (-1)^n (\epsilon A^{-1} F)^n \right ) A^{-1}$$

where the sum converges for sufficiently small $\epsilon$.